Item parameter estimation using MMLE-EM algorithm

This function fits unidimensional item response theory (IRT) models to mixed-format data comprising both dichotomous and polytomous items, using marginal maximum likelihood estimation via the expectation-maximization (MMLE-EM) algorithm (Bock & Aitkin, 1981). When prior distributions are specified for item parameters, the estimates are the modes of the marginal posterior distribution (Bayes modal estimates). The function also supports fixed item parameter calibration (FIPC; Kim, 2006), a practical method for pretest (or newly developed) item calibration in computerized adaptive testing (CAT). FIPC enables the parameter estimates of pretest items to be placed on the same scale as those of operational items (Ban et al., 2001). For dichotomous items, the function supports the one-, two-, and three-parameter logistic models. For polytomous items, it supports the graded response model (GRM) and the (generalized) partial credit model (GPCM).

Usage

est_irt(
  x = NULL,
  data,
  D = 1,
  model = NULL,
  cats = NULL,
  item.id = NULL,
  fix.a.1pl = FALSE,
  fix.a.gpcm = FALSE,
  fix.g = FALSE,
  a.val.1pl = 1,
  a.val.gpcm = 1,
  g.val = 0.2,
  use.aprior = FALSE,
  use.bprior = FALSE,
  use.gprior = TRUE,
  aprior = list(dist = "lnorm", params = c(0, 0.5)),
  bprior = list(dist = "norm", params = c(0, 1)),
  gprior = list(dist = "beta", params = c(5, 16)),
  missing = NA,
  Quadrature = c(49, 6),
  weights = NULL,
  group.mean = 0,
  group.var = 1,
  EmpHist = FALSE,
  use.startval = FALSE,
  Etol = 1e-04,
  MaxE = 500,
  control = list(eval.max = 500, iter.max = 200, x.tol = 1e-04),
  fipc = FALSE,
  fipc.method = "MEM",
  fix.loc = NULL,
  fix.id = NULL,
  se = TRUE,
  verbose = TRUE
)

Arguments

x

A data frame containing item metadata (item IDs, numbers of score categories, IRT models, and item parameters); see Details for its format. You can create an empty item metadata frame using the function shape_df(). The item parameters in x are used as starting values when use.startval = TRUE and are otherwise ignored, except for the fixed items in FIPC, whose parameters are held at the values in x. If x = NULL, the models and numbers of score categories are taken from model and cats. When fipc = TRUE, x must be supplied. Default is NULL.

data

A matrix of examinees' item responses, with rows for examinees and columns for items in the same order as the items in x. Responses to an item with K score categories must be integers from 0 to K - 1. Missing responses are coded as NA or, in functions that have a missing argument, as the value given in missing.

D

A scaling constant used in IRT models to make the logistic function closely approximate the normal ogive function. A value of 1.702 is commonly used for this purpose. Default is 1.

model

A character vector specifying the IRT model to fit each item. Available values are:

"1PLM", "2PLM", "3PLM", "DRM" for dichotomous items
"GRM", "GPCM" for polytomous items

Here, "GRM" denotes the graded response model and "GPCM" the (generalized) partial credit model. Note that "DRM" is a general label for dichotomous items; in this function, items labeled "DRM" are fitted with the 3PLM, and a warning is issued. If a single model name is provided, it is recycled for all items. This argument is only used when x = NULL and fipc = FALSE. Default is NULL.

cats

Numeric vector specifying the number of score categories per item. For dichotomous items, this should be 2. If a single value is supplied, it will be recycled across all items. When cats = NULL and all models specified in the model argument are dichotomous ("1PLM", "2PLM", "3PLM", or "DRM"), the function defaults to 2 categories per item. This argument is used only when x = NULL and fipc = FALSE. Default is NULL.

item.id

Character vector of item identifiers. If NULL, the IDs in x are used, or IDs are generated automatically when x = NULL. A provided item.id replaces any IDs in x, with or without FIPC, and must have one value per item. With fipc = TRUE, fix.id then refers to the new IDs. Default is NULL.

fix.a.1pl

Logical. If TRUE, the slope parameters of all 1PLM items are fixed to a.val.1pl; otherwise, they are constrained to be equal and estimated. Default is FALSE.

fix.a.gpcm

Logical. If TRUE, GPCM items are calibrated as PCM with slopes fixed to a.val.gpcm; otherwise, each item's slope is estimated. Default is FALSE.

fix.g

Logical. If TRUE, all 3PLM guessing parameters are fixed to g.val; otherwise, each guessing parameter is estimated. The prior in gprior is then not used. Default is FALSE.

a.val.1pl

Numeric. Value to which the slope parameters of 1PLM items are fixed when fix.a.1pl = TRUE. Default is 1.

a.val.gpcm

Numeric. Value to which the slope parameters of GPCM items are fixed when fix.a.gpcm = TRUE. Default is 1.

g.val

Numeric. Value to which the guessing parameters of 3PLM items are fixed when fix.g = TRUE. Default is 0.2.

use.aprior

Logical. If TRUE, applies a prior distribution to all item discrimination (slope) parameters during calibration. Default is FALSE.

use.bprior

Logical. If TRUE, applies a prior distribution to all item difficulty (or threshold) parameters during calibration. Default is FALSE.

use.gprior

Logical. If TRUE, applies the prior distribution in gprior to the guessing parameters of 3PLM items when they are estimated; the prior is not used when fix.g = TRUE. Default is TRUE.

aprior, bprior, gprior

A list specifying the prior distribution for all item discrimination (slope), difficulty (or threshold), and guessing parameters. The slope prior is placed on D times the slope, that is, the density in aprior is evaluated at D * a; with D = 1 (the default), this is the slope itself. The priors also enter the information matrix from which the standard errors are computed. Three distributions are supported: Beta, Log-normal, and Normal. The list must have two elements:

dist: A character string, one of "beta", "lnorm", or "norm".
params: A numeric vector of length two giving the distribution's parameters. For details on each parameterization, see stats::dbeta(), stats::dlnorm(), and stats::dnorm().

Defaults are:

aprior = list(dist = "lnorm", params = c(0.0, 0.5))
bprior = list(dist = "norm", params = c(0.0, 1.0))
gprior = list(dist = "beta", params = c(5, 16))

for discrimination, difficulty, and guessing parameters, respectively.

missing

A value indicating missing responses in the data set, in addition to NA, which is always treated as missing. Missing responses are omitted from the likelihood, which assumes that they are missing at random. Default is NA.

Quadrature

A numeric vector of length two:

first element: number of quadrature points
second element: symmetric bound (absolute value) for those points

For example, c(49, 6) specifies 49 evenly spaced points from -6 to 6. The points are used to approximate the integral over the latent ability distribution in the E-step of the EM algorithm. Quadrature is ignored when weights is supplied. Default is c(49, 6).

weights

A two-column matrix or data frame of quadrature points (first column) and weights (second column) that defines the latent ability distribution; it can be created with gen.weight(). The weights should sum to 1; weights that do not sum to 1 are rescaled to sum to 1 with a warning. The number of points may differ from Quadrature[1]. If supplied, Quadrature, group.mean, and group.var are not used. Without FIPC, the mean and variance of this distribution fix the scale: the distribution is used as it is when EmpHist = FALSE, and its shape is updated but rescaled to its mean and variance when EmpHist = TRUE, in which case the quadrature points must be equally spaced, because the histogram is rescaled by interpolation on an equally spaced grid. With FIPC, it is only the initial distribution of the first EM cycle, which is then updated (see Details).

If NULL, a normal distribution with mean group.mean and variance group.var at the points given by Quadrature is used. Default is NULL.

group.mean

A numeric value specifying the mean of the normal latent ability distribution when weights = NULL. Without FIPC, the mean is fixed at this value to identify the scale of the item parameters, also when EmpHist = TRUE. With FIPC, it is the mean of the initial distribution, and the mean is then estimated. Default is 0.

group.var

A positive numeric value specifying the variance of the normal latent ability distribution when weights = NULL. Without FIPC, the variance is fixed at this value to identify the scale of the item parameters, also when EmpHist = TRUE. With FIPC, it is the variance of the initial distribution, and the variance is then estimated. Default is 1.

EmpHist

Logical. If TRUE, the latent ability distribution is estimated as an empirical histogram on the quadrature points together with the item parameters (Woods, 2007). Item calibration is conducted relative to the estimated empirical histogram. See Details. Default is FALSE.

use.startval

Logical. If TRUE, the item parameters provided in the item metadata (i.e., the x argument) are used as starting values for the items whose parameters are estimated; x must then be supplied. Otherwise, internally generated starting values are used. Default is FALSE.

Etol

A positive numeric value specifying the convergence criterion of the EM algorithm. The EM cycles stop when the largest absolute change in the estimated item parameters between two consecutive cycles is less than or equal to Etol. With FIPC, only the parameters of the items that are not fixed are compared. The mean and variance of the latent ability distribution estimated with FIPC, and the weights of an empirical histogram (EmpHist = TRUE), are not part of the criterion, so they can be less converged than the item parameters; a smaller Etol gives more precise estimates of them. In particular, an estimated histogram can keep changing very slowly over many cycles (toward a spiky distribution) while the log-likelihood barely changes, so group.par and weights then depend on Etol; with small samples, the item parameters can then also need many cycles, or more than MaxE, to meet Etol. When all items are fixed, the criterion is applied to the largest absolute change in the mean and variance of the latent ability distribution. Etol is not used when fipc.method = "OEM" with items to calibrate. The printed output of the fitted object refers to this criterion as the convergence criterion of the E-step. Default is 1e-4.

MaxE

A positive integer specifying the maximum number of EM cycles. It is set to 1 when fipc.method = "OEM". The printed output of the fitted object refers to the cycles as E-step cycles. Default is 500.

control

A named list of options passed directly to stats::nlminb(), which maximizes the M-step objective of each item. The default is list(eval.max = 500, iter.max = 200, x.tol = 1e-4), where

eval.max is the maximum number of objective function evaluations,
iter.max is the maximum number of optimizer iterations, and
x.tol is the X-convergence tolerance of stats::nlminb().

A partial list may be supplied; options that are not supplied keep their default values. Other options of stats::nlminb(), such as rel.tol or trace, can also be supplied.

fipc

Logical. If TRUE, fixed item parameter calibration (FIPC) is applied during item parameter estimation. When fipc = TRUE, the information on which items are fixed must be provided via either fix.loc or fix.id. See Details.

fipc.method

A character string specifying the FIPC method, used only when fipc = TRUE. Available options are:

"OEM": No Prior Weights Updating and One EM Cycle (NWU-OEM; Wainer & Mislevy, 1990).
"MEM": Multiple Prior Weights Updating and Multiple EM Cycles (MWU-MEM; Kim, 2006).

When fipc.method = "OEM", a single EM cycle is run regardless of MaxE. See Details. Default is "MEM".

fix.loc

A vector of positive integers giving the row positions in x of the items to be fixed when fipc = TRUE. For example, suppose that five items located in the 1st, 2nd, 4th, 7th, and 9th rows of x should be fixed. Then use fix.loc = c(1, 2, 4, 7, 9). It must not contain duplicated positions. If fix.id is also supplied, fix.loc is ignored with a warning. See Details.

fix.id

A character vector of the IDs of the items to be fixed when fipc = TRUE. The IDs are matched to the first column (id) of x, or to item.id when it is supplied. For example, use fix.id = c("CMC1", "CMC2", "CMC3", "CMC4", "CMC5") to fix the items with these IDs. If fix.id is not NULL, fix.loc is ignored. See Details.

se

Logical. If FALSE, the standard errors and the variance-covariance matrix of the item parameter estimates are not computed; se.est is then filled with NA and covariance is NULL. Default is TRUE.

verbose

Logical. If FALSE, all progress messages, including information about the EM algorithm process, are suppressed. Default is TRUE.

Value

This function returns an object of class est_irt. The returned object contains the following components:

estimates: A data frame with the item ID, number of score categories, and model of each item, followed by the item parameter estimates and their standard errors in alternating columns (par.1, se.1, par.2, se.2, ...).
par.est: A data frame of item parameter estimates, structured according to the item metadata format.
se.est: A data frame of standard errors for the item parameter estimates, computed from the inverse of the information matrix. The information matrix is the cross-product (empirical) information matrix of the marginal log-likelihood evaluated at the final estimates (Meilijson, 1989), plus the prior information (the negative second derivatives of the log prior densities) for the parameters that have prior distributions. With FIPC, the latent ability distribution is treated as known in this matrix, so the uncertainty of the estimated distribution is not reflected in these standard errors. Standard errors are NA for parameters that are not estimated (for example, fixed slopes or guessing parameters, and the items fixed in FIPC) and for all parameters when se = FALSE. The standard error of a common slope of 1PLM items appears only in the row of the first 1PLM item. A value of 99999 indicates that the standard error could not be obtained.
pos.par: A data frame indicating, for each estimated item parameter, its row (and column) position in covariance. Parameters that are not estimated are NA, and the position of a common slope of 1PLM items is given only in the row of the first 1PLM item. NULL when FIPC is used with all items fixed.
covariance: The variance-covariance matrix of the item parameter estimates (the inverse of the information matrix described in se.est), without row or column names; the order of the parameters is given by pos.par. NULL when se = FALSE or when no item parameter is estimated.
loglikelihood: The marginal log-likelihood of the response data at the final estimates: the sum over examinees of the logarithm of the probability of each examinee's response pattern, integrated over the latent ability distribution with the quadrature points and weights. With FIPC, the responses to all items, including the fixed items, are used. The log prior densities of the item parameters are not included.
aic: Akaike information criterion, -2 * loglikelihood + 2 * npar.est.
bic: Bayesian information criterion, -2 * loglikelihood + npar.est * log(ncase).
group.par: A data frame with columns mu, sigma2, and sigma (mean, variance, and standard deviation of the latent ability distribution) and rows estimates and se. Without FIPC, the values are fixed at group.mean and group.var (or at the mean and variance of weights, if supplied) and se is NA. When EmpHist = TRUE, the empirical histogram is rescaled in each cycle to have this mean and variance. Because the rescaling interpolates on the quadrature points, the mean and variance computed from the returned weights can differ slightly from the reported values. With FIPC, the values are the mean and variance of the final latent distribution (these are the mean and variance of the returned weights), and se gives approximate standard errors, sigma / sqrt(N) for the mean and sigma2 * sqrt(2 / (N - 1)) for the variance, with the delta method for sigma, where N is the number of examinees; these treat the abilities as observed and are therefore usually too small.
weights: A data frame with columns theta and weight giving the quadrature points and the weights of the final latent ability distribution. Without FIPC and with EmpHist = FALSE, it is the distribution given by group.mean and group.var, or weights as supplied (rescaled to sum to 1 if needed). With fipc.method = "OEM", it is the distribution updated once after the item parameters were estimated.
posterior.dist: A matrix of the posterior probabilities of the quadrature points for each examinee, computed at the final estimates, with one row per examinee and one column per quadrature point; each row sums to 1.
data: A matrix of the examinees' response data, with the missing values coded as NA.
scale.D: The scaling factor used in the IRT model.
ncase: The number of examinees (rows of data).
nitem: The total number of items in the response data.
Etol: The convergence criterion of the EM algorithm.
MaxE: The maximum number of EM cycles (1 when fipc.method = "OEM").
aprior: The prior distribution specified for the slope parameters when use.aprior = TRUE; NULL otherwise.
bprior: The prior distribution specified for the difficulty (or threshold) parameters when use.bprior = TRUE; NULL otherwise.
gprior: The prior distribution specified for the guessing parameters when use.gprior = TRUE; NULL otherwise.
npar.est: The number of freely estimated parameters used for aic and bic: the estimated item parameters, the mean and variance of the latent distribution when they are estimated, and the weights of the empirical histogram when EmpHist = TRUE. See Details.
niter: The number of completed EM cycles.
maxpar.diff: The largest absolute change in the estimated item parameters in the last EM cycle (for FIPC with all items fixed, in the mean and variance of the latent ability distribution). With fipc.method = "OEM", it is the change from the starting values.
EMtime: Time (in seconds) spent on EM cycles.
SEtime: Time (in seconds) spent computing standard errors; NULL when they are not computed.
TotalTime: Total computation time (in seconds).
test.1: A message indicating whether the convergence criteria were met: the M-step optimization converged for every item and the largest absolute change in the parameter estimates between two consecutive EM cycles was less than or equal to Etol. When fipc.method = "OEM" with new items to calibrate, the single EM cycle is not judged by Etol and only the M-step check applies. For FIPC with all items fixed, see Etol.
test.2: A message indicating whether the information matrix used for the standard errors (see se.est) is positive definite. Printed results label it "Second-order test", but because the cross-product information matrix is positive semidefinite by construction, this check mainly detects a singular (rank-deficient) information matrix, for example due to unidentified parameters; it is not a second-order test of the log-likelihood surface. The message reports a possible local maximum only when the first-order test is also satisfied. When FIPC is used with all items fixed, no information matrix is computed, and the message says so.
var.note: A note indicating whether the variance-covariance matrix was successfully obtained from the information matrix.
fipc: Logical. Indicates whether FIPC was used.
fipc.method: The FIPC method used; NULL when fipc = FALSE.
fix.loc: An integer vector of the row positions of the fixed items in x, also when the items were specified with fix.id; NULL when fipc = FALSE.
call: The matched call.
npar.detail: An integer vector with the elements item (free item parameters), dist (mean and variance of the latent distribution), and weight (weights of the empirical histogram), whose sum is npar.est. It is the last element of the object; when it is absent, summary() shows the total only.

Note that you can easily extract components from the output using the getirt() function.

Details

A specific format of data frame should be used for the argument x. The first column should contain item IDs, the second column should contain the number of score categories of each item, and the third column should specify the IRT model to be fitted to each item. Available IRT models are:

"1PLM", "2PLM", "3PLM", and "DRM" for dichotomous item data
"GRM" and "GPCM" for polytomous item data

Note that "DRM" serves as a general label covering all dichotomous IRT models (i.e., "1PLM", "2PLM", and "3PLM"), while "GRM" and "GPCM" represent the graded response model and (generalized) partial credit model, respectively. In est_irt(), items labeled "DRM" are calibrated with the 3PLM.

The subsequent columns should contain the item parameters for the specified models. For dichotomous items, the fourth, fifth, and sixth columns represent item discrimination (slope), item difficulty, and item guessing parameters, respectively. When "1PLM" or "2PLM" is specified in the third column, NAs must be entered in the sixth column for the guessing parameters.

For polytomous items, the item discrimination (slope) parameter should appear in the fourth column, and the item difficulty (or threshold) parameters for category boundaries should occupy the fifth through the last columns. When the number of score categories differs across items, unused parameter cells should be filled with NAs.

In the irtQ package, the threshold parameters for GPCM items are expressed as the item location (or overall difficulty) minus the threshold values for each score category. Note that a GPCM item with K score categories has K - 1 threshold parameters, because the term for the lowest score category (score 0) is fixed at 0 and is not included in the metadata. For example, if a GPCM item has five score categories, four threshold parameters must be provided.

An example of a data frame for a single-format test is shown below:

id	cats	model	par.1	par.2	par.3
ITEM1	2	1PLM	1.000	1.461	NA
ITEM2	2	2PLM	1.921	-1.049	NA
ITEM3	2	3PLM	1.736	1.501	0.203
ITEM4	2	3PLM	0.835	-1.049	0.182
ITEM5	2	DRM	0.926	0.394	0.099

An example of a data frame for a mixed-format test is shown below:

id	cats	model	par.1	par.2	par.3	par.4	par.5
ITEM1	2	1PLM	1.000	1.461	NA	NA	NA
ITEM2	2	2PLM	1.921	-1.049	NA	NA	NA
ITEM3	2	3PLM	0.926	0.394	0.099	NA	NA
ITEM4	2	DRM	1.052	-0.407	0.201	NA	NA
ITEM5	4	GRM	1.913	-1.869	-1.238	-0.714	NA
ITEM6	5	GRM	1.278	-0.724	-0.068	0.568	1.072
ITEM7	4	GPCM	1.137	-0.374	0.215	0.848	NA
ITEM8	5	GPCM	1.233	-2.078	-1.347	-0.705	-0.116

See the IRT Models section in the irtQ-package documentation for more details about the IRT models used in the irtQ package. A convenient way to create a data frame for the argument x is by using the function shape_df().

To fit IRT models to data, the item response data must be accompanied by information on the IRT model and the number of score categories for each item. There are two ways to provide this information:

Supply item metadata to the argument x. As explained above, such metadata can be easily created using shape_df().
Specify the IRT models and score category information directly through the arguments model and cats.

If x is supplied, model and cats are ignored. If x = NULL, the function uses model and cats; cats may be omitted when all items are dichotomous.

Item parameters are estimated with the EM algorithm of Bock and Aitkin (1981). In the E-step, the posterior distribution of each examinee over the quadrature points is computed from the current item parameters and the current latent distribution, and the expected numbers of examinees and of responses in each score category are obtained at each quadrature point. In the M-step, the expected complete-data log-likelihood of each item, plus the log prior densities when priors are used, is maximized with stats::nlminb() using analytic gradients and Hessians; the 1PLM items with a common slope are estimated jointly. The cycles are repeated until the convergence criterion given by Etol is met or MaxE cycles are completed. Missing responses are omitted from the likelihood. The slopes of dichotomous items are not bounded during estimation, whereas those of GRM and GPCM items are kept positive.

To implement FIPC, the item metadata must be provided via the x argument. This is because the item parameters of the fixed items in the metadata are used to estimate the characteristics of the underlying latent variable prior distribution when calibrating the remaining (freely estimated) items. More specifically, the latent prior distribution is estimated based on the fixed items, and then used to calibrate the new (pretest) items so that their parameters are placed on the same scale as those of the fixed items (Kim, 2006). The full item metadata, including both fixed and non-fixed items, can be conveniently created using the shape_df_fipc() function.

In terms of approaches for FIPC, Kim (2006) described five different methods. Among them, two methods are available in the est_irt() function. The first method is "NWU-OEM", which uses a single E-step in the EM algorithm (involving only the fixed items) followed by a single M-step (involving only the non-fixed items). This method was proposed by Wainer and Mislevy (1990) in the context of online calibration and can be implemented by setting fipc.method = "OEM". The item parameters are estimated with the initial latent distribution (given by group.mean and group.var, or by weights). The latent distribution returned in group.par and weights is computed once afterward from the posterior distributions based on the fixed items, and EmpHist affects only this returned distribution. When all items are fixed, the function always warns that the convergence criteria are not satisfied.

The second method is "MWU-MEM", which runs multiple EM cycles and updates the weights of the latent ability distribution in every cycle (Kim, 2006; Kim & Kolen, 2016). The first cycle is the same as in NWU-OEM: the posterior distribution of each examinee is computed from the responses to the fixed items only. From the second cycle onward, the posterior distribution is computed from the responses to all items, using the fixed parameters and the current estimates of the other items, and the latent distribution is updated from the expected numbers of examinees at the quadrature points. With EmpHist = TRUE, the updated distribution is this empirical histogram, as in Kim (2006); with EmpHist = FALSE, it is a normal distribution with the mean and variance of the histogram. This method can be implemented by setting fipc.method = "MEM".

When fipc = TRUE, information about which items are to be fixed must be provided via either the fix.loc or fix.id argument. For example, suppose that five items with IDs "CMC1", "CMC2", "CMC3", "CMC4", and "CMC5" should be fixed, and all item IDs are provided via the x or item.id argument. Also, assume these five items are located in the 1st through 5th rows of the item metadata (i.e., x). In this case, the fixed items can be specified using either fix.loc = c(1, 2, 3, 4, 5) or fix.id = c("CMC1", "CMC2", "CMC3", "CMC4", "CMC5"). Note that if both fix.loc and fix.id are not NULL, the information in fix.loc is ignored.

When EmpHist = TRUE, the latent distribution is an empirical histogram: in each EM cycle, the weight at each quadrature point is updated to the average of the posterior probabilities of the examinees at that point (Woods, 2007). Without FIPC, the updated histogram is then rescaled so that its mean and variance equal group.mean and group.var (or the mean and variance of weights, if supplied): the quadrature points are standardized and transformed to the target scale, the weights are interpolated back to the original points (and extrapolated at the ends), and the weights are renormalized. With FIPC, the histogram is not rescaled, so its mean and variance are estimated. When EmpHist = FALSE, a normal distribution is used. With FIPC, its mean and variance are updated in each cycle to those of the histogram; otherwise, they are fixed at group.mean and group.var.

A maximum likelihood estimate does not exist for an item with a score category that has no responses, which includes an item whose responses are all the same. The estimates of such an item can be extreme or remain near their starting values, and their standard errors are unreliable (a standard error of 99999 means that it could not be obtained). Before the estimation, one warning lists the IDs of these items; the items fixed in FIPC are not checked. A parameter with a prior distribution, such as the guessing parameter of a 3PLM item with use.gprior = TRUE, can still be stable.

When prior distributions are used, the item parameters are Bayes modal estimates, and loglikelihood is the log-likelihood at these estimates without the log prior densities.

The information criteria are aic = -2 * loglikelihood + 2 * npar.est and bic = -2 * loglikelihood + npar.est * log(ncase). Here loglikelihood is the marginal log-likelihood without the prior densities of the item parameters, and ncase is the number of examinees. npar.est is the number of freely estimated parameters, which consists of (a) the item parameters that are estimated (a slope shared by 1PLM items is counted once; fixed parameters and the items fixed in FIPC are not counted), (b) the mean and variance of the latent distribution when they are estimated, which is the case with FIPC and EmpHist = FALSE (2 values), and (c) the weights of the empirical histogram when EmpHist = TRUE. Without FIPC, the histogram is rescaled to a fixed mean and variance (given by group.mean and group.var, or by weights), so its Q weights, where Q is the number of quadrature points, have Q - 3 free values (the sum of the weights and the two moments are fixed). With FIPC, the histogram is not rescaled, so the weights have Q - 1 free values. The counts by type are returned in npar.detail.

The print method of this class has an argument digits that is not used. It prints the log-likelihood with the number of significant digits given by getOption("digits") (7 by default).

References

Ban, J. C., Hanson, B. A., Wang, T., Yi, Q., & Harris, D. J. (2001). A comparative study of on-line pretest item calibration/scaling methods in computerized adaptive testing. Journal of Educational Measurement, 38(3), 191-212. doi:10.1111/j.1745-3984.2001.tb01123.x .

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443-459.

Kim, S. (2006). A comparative study of IRT fixed parameter calibration methods. Journal of Educational Measurement, 43(4), 355-381.

Kim, S., & Kolen, M. J. (2016). Multiple group IRT fixed-parameter estimation for maintaining an established ability scale (CASMA Research Report No. 49). Center for Advanced Studies in Measurement and Assessment, University of Iowa.

Meilijson, I. (1989). A fast improvement to the EM algorithm on its own terms. Journal of the Royal Statistical Society: Series B (Methodological), 51, 127-138.

Wainer, H., & Mislevy, R. J. (1990). Item response theory, item calibration, and proficiency estimation. In H. Wainer (Ed.), Computer adaptive testing: A primer (Chap. 4, pp. 65-102). Hillsdale, NJ: Lawrence Erlbaum.

Woods, C. M. (2007). Empirical histograms in item response theory with ordinal data. Educational and Psychological Measurement, 67(1), 73-87.

Author

Hwanggyu Lim hglim83@gmail.com

Examples

# \donttest{

## --------------------------------------------------------------
## 1. Item parameter estimation for dichotomous item data (LSAT6)
## --------------------------------------------------------------
# Fit the 1PL model to LSAT6 data and estimate a common slope parameter
# (i.e., constrain slope parameters to be equal)
(mod.1pl.c <- est_irt(data = LSAT6, D = 1, model = "1PLM", cats = 2,
                      fix.a.1pl = FALSE))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -3182.3860, Max-Change: 2.293929
 EM iteration: 2, Loglike: -2561.3380, Max-Change: 0.58111
 EM iteration: 3, Loglike: -2483.1811, Max-Change: 0.31473
 EM iteration: 4, Loglike: -2469.6884, Max-Change: 0.171175
 EM iteration: 5, Loglike: -2467.5148, Max-Change: 0.096225
 EM iteration: 6, Loglike: -2467.1096, Max-Change: 0.056965
 EM iteration: 7, Loglike: -2467.0029, Max-Change: 0.035311
 EM iteration: 8, Loglike: -2466.9648, Max-Change: 0.022579
 EM iteration: 9, Loglike: -2466.9493, Max-Change: 0.014699
 EM iteration: 10, Loglike: -2466.9427, Max-Change: 0.009659
 EM iteration: 11, Loglike: -2466.9398, Max-Change: 0.006377
 EM iteration: 12, Loglike: -2466.9386, Max-Change: 0.004219
 EM iteration: 13, Loglike: -2466.9380, Max-Change: 0.002795
 EM iteration: 14, Loglike: -2466.9378, Max-Change: 0.001852
 EM iteration: 15, Loglike: -2466.9377, Max-Change: 0.001228
 EM iteration: 16, Loglike: -2466.9376, Max-Change: 0.000814
 EM iteration: 17, Loglike: -2466.9376, Max-Change: 0.000539
 EM iteration: 18, Loglike: -2466.9376, Max-Change: 0.000358
 EM iteration: 19, Loglike: -2466.9376, Max-Change: 0.000237
 EM iteration: 20, Loglike: -2466.9376, Max-Change: 0.000157
 EM iteration: 21, Loglike: -2466.9376, Max-Change: 0.000104
 EM iteration: 22, Loglike: -2466.9376, Max-Change: 6.9e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.13 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "1PLM", cats = 2, fix.a.1pl = FALSE)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 22 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2466.938
#> 

# Display a summary of the estimation results
summary(mod.1pl.c)
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "1PLM", cats = 2, fix.a.1pl = FALSE)
#> 
#> Summary of the Data 
#>  Number of Items: 5
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 6
#>    - Item parameters: 6
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 22
#>  Maximum parameter change: 6.905886e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.11
#>  Standard error computation: 0
#>  Total computation: 0.13
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 4933.875
#>  Akaike Information Criterion (AIC): 4945.875
#>  Bayesian Information Criterion (BIC): 4975.322
#>  Item Parameters: 
#>    id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3
#> 1  V1     2   1PLM   0.76  0.07  -3.62  0.32     NA    NA
#> 2  V2     2   1PLM   0.76    NA  -1.32  0.14     NA    NA
#> 3  V3     2   1PLM   0.76    NA  -0.32  0.10     NA    NA
#> 4  V4     2   1PLM   0.76    NA  -1.73  0.17     NA    NA
#> 5  V5     2   1PLM   0.76    NA  -2.78  0.25     NA    NA
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Extract the item parameter estimates
getirt(mod.1pl.c, what = "par.est")
#>   id cats model     par.1      par.2 par.3
#> 1 V1    2  1PLM 0.7551678 -3.6151311    NA
#> 2 V2    2  1PLM 0.7551678 -1.3223732    NA
#> 3 V3    2  1PLM 0.7551678 -0.3176192    NA
#> 4 V4    2  1PLM 0.7551678 -1.7300276    NA
#> 5 V5    2  1PLM 0.7551678 -2.7800696    NA

# Extract the standard error estimates
getirt(mod.1pl.c, what = "se.est")
#>   id cats model      par.1      par.2 par.3
#> 1 V1    2  1PLM 0.06852372 0.31969228    NA
#> 2 V2    2  1PLM         NA 0.14200939    NA
#> 3 V3    2  1PLM         NA 0.09773711    NA
#> 4 V4    2  1PLM         NA 0.16821833    NA
#> 5 V5    2  1PLM         NA 0.24980846    NA

# Fit the 1PL model to LSAT6 data and fix slope parameters to 1.0
(mod.1pl.f <- est_irt(data = LSAT6, D = 1, model = "1PLM", cats = 2,
                      fix.a.1pl = TRUE, a.val.1pl = 1))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -3182.3860, Max-Change: 2.161943
 EM iteration: 2, Loglike: -2569.0530, Max-Change: 0.399087
 EM iteration: 3, Loglike: -2491.5319, Max-Change: 0.175418
 EM iteration: 4, Loglike: -2476.5061, Max-Change: 0.077114
 EM iteration: 5, Loglike: -2473.6897, Max-Change: 0.033375
 EM iteration: 6, Loglike: -2473.1702, Max-Change: 0.014331
 EM iteration: 7, Loglike: -2473.0751, Max-Change: 0.006132
 EM iteration: 8, Loglike: -2473.0577, Max-Change: 0.00262
 EM iteration: 9, Loglike: -2473.0546, Max-Change: 0.001118
 EM iteration: 10, Loglike: -2473.0540, Max-Change: 0.000477
 EM iteration: 11, Loglike: -2473.0539, Max-Change: 0.000204
 EM iteration: 12, Loglike: -2473.0539, Max-Change: 8.7e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.12 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "1PLM", cats = 2, fix.a.1pl = TRUE, 
#>     a.val.1pl = 1)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 12 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2473.054
#> 

# Display a summary of the estimation results
summary(mod.1pl.f)
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "1PLM", cats = 2, fix.a.1pl = TRUE, 
#>     a.val.1pl = 1)
#> 
#> Summary of the Data 
#>  Number of Items: 5
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 5
#>    - Item parameters: 5
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 12
#>  Maximum parameter change: 8.693637e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.1
#>  Standard error computation: 0
#>  Total computation: 0.12
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 4946.108
#>  Akaike Information Criterion (AIC): 4956.108
#>  Bayesian Information Criterion (BIC): 4980.646
#>  Item Parameters: 
#>    id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3
#> 1  V1     2   1PLM      1    NA  -2.87  0.13     NA    NA
#> 2  V2     2   1PLM      1    NA  -1.06  0.08     NA    NA
#> 3  V3     2   1PLM      1    NA  -0.26  0.08     NA    NA
#> 4  V4     2   1PLM      1    NA  -1.39  0.09     NA    NA
#> 5  V5     2   1PLM      1    NA  -2.22  0.11     NA    NA
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Fit the 2PL model to LSAT6 data
(mod.2pl <- est_irt(data = LSAT6, D = 1, model = "2PLM", cats = 2))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -3182.3860, Max-Change: 2.294233
 EM iteration: 2, Loglike: -2561.0478, Max-Change: 0.58517
 EM iteration: 3, Loglike: -2482.8950, Max-Change: 0.323232
 EM iteration: 4, Loglike: -2469.4379, Max-Change: 0.180623
 EM iteration: 5, Loglike: -2467.2706, Max-Change: 0.106034
 EM iteration: 6, Loglike: -2466.8615, Max-Change: 0.066546
 EM iteration: 7, Loglike: -2466.7487, Max-Change: 0.044361
 EM iteration: 8, Loglike: -2466.7049, Max-Change: 0.030996
 EM iteration: 9, Loglike: -2466.6843, Max-Change: 0.022467
 EM iteration: 10, Loglike: -2466.6736, Max-Change: 0.016791
 EM iteration: 11, Loglike: -2466.6674, Max-Change: 0.012897
 EM iteration: 12, Loglike: -2466.6636, Max-Change: 0.010155
 EM iteration: 13, Loglike: -2466.6611, Max-Change: 0.008179
 EM iteration: 14, Loglike: -2466.6593, Max-Change: 0.00672
 EM iteration: 15, Loglike: -2466.6580, Max-Change: 0.005616
 EM iteration: 16, Loglike: -2466.6571, Max-Change: 0.00476
 EM iteration: 17, Loglike: -2466.6563, Max-Change: 0.004082
 EM iteration: 18, Loglike: -2466.6557, Max-Change: 0.003532
 EM iteration: 19, Loglike: -2466.6553, Max-Change: 0.003079
 EM iteration: 20, Loglike: -2466.6549, Max-Change: 0.00270
 EM iteration: 21, Loglike: -2466.6546, Max-Change: 0.002378
 EM iteration: 22, Loglike: -2466.6544, Max-Change: 0.002102
 EM iteration: 23, Loglike: -2466.6542, Max-Change: 0.001863
 EM iteration: 24, Loglike: -2466.6540, Max-Change: 0.001655
 EM iteration: 25, Loglike: -2466.6539, Max-Change: 0.001473
 EM iteration: 26, Loglike: -2466.6538, Max-Change: 0.001313
 EM iteration: 27, Loglike: -2466.6537, Max-Change: 0.001172
 EM iteration: 28, Loglike: -2466.6537, Max-Change: 0.001047
 EM iteration: 29, Loglike: -2466.6536, Max-Change: 0.000936
 EM iteration: 30, Loglike: -2466.6536, Max-Change: 0.000838
 EM iteration: 31, Loglike: -2466.6535, Max-Change: 0.000751
 EM iteration: 32, Loglike: -2466.6535, Max-Change: 0.000673
 EM iteration: 33, Loglike: -2466.6535, Max-Change: 0.000603
 EM iteration: 34, Loglike: -2466.6535, Max-Change: 0.000541
 EM iteration: 35, Loglike: -2466.6534, Max-Change: 0.000486
 EM iteration: 36, Loglike: -2466.6534, Max-Change: 0.000436
 EM iteration: 37, Loglike: -2466.6534, Max-Change: 0.000392
 EM iteration: 38, Loglike: -2466.6534, Max-Change: 0.000352
 EM iteration: 39, Loglike: -2466.6534, Max-Change: 0.000316
 EM iteration: 40, Loglike: -2466.6534, Max-Change: 0.000284
 EM iteration: 41, Loglike: -2466.6534, Max-Change: 0.000256
 EM iteration: 42, Loglike: -2466.6534, Max-Change: 0.00023
 EM iteration: 43, Loglike: -2466.6534, Max-Change: 0.000207
 EM iteration: 44, Loglike: -2466.6534, Max-Change: 0.000186
 EM iteration: 45, Loglike: -2466.6534, Max-Change: 0.000167
 EM iteration: 46, Loglike: -2466.6534, Max-Change: 0.000151
 EM iteration: 47, Loglike: -2466.6534, Max-Change: 0.000136
 EM iteration: 48, Loglike: -2466.6534, Max-Change: 0.000122
 EM iteration: 49, Loglike: -2466.6534, Max-Change: 0.00011
 EM iteration: 50, Loglike: -2466.6534, Max-Change: 9.9e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.39 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "2PLM", cats = 2)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 50 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2466.653
#> 

# Display a summary of the estimation results
summary(mod.2pl)
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "2PLM", cats = 2)
#> 
#> Summary of the Data 
#>  Number of Items: 5
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 10
#>    - Item parameters: 10
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 50
#>  Maximum parameter change: 9.894598e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.37
#>  Standard error computation: 0
#>  Total computation: 0.39
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 4933.307
#>  Akaike Information Criterion (AIC): 4953.307
#>  Bayesian Information Criterion (BIC): 5002.384
#>  Item Parameters: 
#>    id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3
#> 1  V1     2   2PLM   0.83  0.25  -3.36  0.85     NA    NA
#> 2  V2     2   2PLM   0.72  0.19  -1.37  0.31     NA    NA
#> 3  V3     2   2PLM   0.89  0.23  -0.28  0.10     NA    NA
#> 4  V4     2   2PLM   0.69  0.19  -1.87  0.44     NA    NA
#> 5  V5     2   2PLM   0.66  0.20  -3.13  0.84     NA    NA
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Assess the model fit for the 2PL model using the S-X2 fit statistic
(sx2fit.2pl <- sx2_fit(x = mod.2pl))
#> $fit_stat
#>   id chisq df crit.val     p
#> 1 V1 0.450  2    5.991 0.798
#> 2 V2 1.691  2    5.991 0.429
#> 3 V3 0.666  1    3.841 0.415
#> 4 V4 0.175  2    5.991 0.916
#> 5 V5 0.111  2    5.991 0.946
#> 
#> $item_df
#>   id cats model     par.1      par.2 par.3
#> 1 V1    2  2PLM 0.8255888 -3.3590452     0
#> 2 V2    2  2PLM 0.7229138 -1.3697963     0
#> 3 V3    2  2PLM 0.8904047 -0.2797759     0
#> 4 V4    2  2PLM 0.6884844 -1.8661188     0
#> 5 V5    2  2PLM 0.6570762 -3.1250079     0
#> 
#> $exp_freq
#> $exp_freq$item.1
#>          score.0    score.1
#> score.1 10.74056   9.259444
#> score.2 21.87382  63.126177
#> score.3 26.79306 210.206939
#> score.4 13.70859 343.291413
#> 
#> $exp_freq$item.2
#>           score.0    score.1
#> score.1  18.20255   1.797452
#> score.2  64.54046  20.459542
#> score.3 123.05816 113.941841
#> score.4  82.28224 274.717762
#> 
#> $exp_freq$item.3
#>           score.0    score.1
#> score.1  95.43853   9.561471
#> score.3 177.81316  59.186840
#> score.4 170.73120 186.268805
#> 
#> $exp_freq$item.4
#>          score.0    score.1
#> score.1 17.46393   2.536072
#> score.2 57.69266  27.307338
#> score.3 97.14025 139.859752
#> score.4 61.58031 295.419686
#> 
#> $exp_freq$item.5
#>          score.0    score.1
#> score.1 14.27299   5.727008
#> score.2 34.77451  50.225494
#> score.3 49.19537 187.804628
#> score.4 28.69767 328.302335
#> 
#> 
#> $obs_freq
#> $obs_freq$item.1
#>         score.0 score.1
#> score.1      10      10
#> score.2      23      62
#> score.3      25     212
#> score.4      15     342
#> 
#> $obs_freq$item.2
#>         score.0 score.1
#> score.1      19       1
#> score.2      61      24
#> score.3     128     109
#> score.4      80     277
#> 
#> $obs_freq$item.3
#>         score.0 score.1
#> score.1      97       8
#> score.3     174      63
#> score.4     173     184
#> 
#> $obs_freq$item.4
#>         score.0 score.1
#> score.1      18       2
#> score.2      57      28
#> score.3      98     139
#> score.4      61     296
#> 
#> $obs_freq$item.5
#>         score.0 score.1
#> score.1      14       6
#> score.2      36      49
#> score.3      49     188
#> score.4      28     329
#> 
#> 
#> $exp_prob
#> $exp_prob$item.1
#>           score.0   score.1
#> score.1 0.5370278 0.4629722
#> score.2 0.2573391 0.7426609
#> score.3 0.1130509 0.8869491
#> score.4 0.0383994 0.9616006
#> 
#> $exp_prob$item.2
#>           score.0    score.1
#> score.1 0.9101274 0.08987262
#> score.2 0.7592995 0.24070050
#> score.3 0.5192327 0.48076726
#> score.4 0.2304825 0.76951754
#> 
#> $exp_prob$item.3
#>           score.0    score.1
#> score.1 0.9089384 0.09106163
#> score.3 0.7502665 0.24973350
#> score.4 0.4782386 0.52176136
#> 
#> $exp_prob$item.4
#>           score.0   score.1
#> score.1 0.8731964 0.1268036
#> score.2 0.6787372 0.3212628
#> score.3 0.4098745 0.5901255
#> score.4 0.1724939 0.8275061
#> 
#> $exp_prob$item.5
#>            score.0   score.1
#> score.1 0.71364960 0.2863504
#> score.2 0.40911183 0.5908882
#> score.3 0.20757541 0.7924246
#> score.4 0.08038562 0.9196144
#> 
#> 
#> $obs_prop
#> $obs_prop$item.1
#>            score.0   score.1
#> score.1 0.50000000 0.5000000
#> score.2 0.27058824 0.7294118
#> score.3 0.10548523 0.8945148
#> score.4 0.04201681 0.9579832
#> 
#> $obs_prop$item.2
#>           score.0   score.1
#> score.1 0.9500000 0.0500000
#> score.2 0.7176471 0.2823529
#> score.3 0.5400844 0.4599156
#> score.4 0.2240896 0.7759104
#> 
#> $obs_prop$item.3
#>           score.0    score.1
#> score.1 0.9238095 0.07619048
#> score.3 0.7341772 0.26582278
#> score.4 0.4845938 0.51540616
#> 
#> $obs_prop$item.4
#>           score.0   score.1
#> score.1 0.9000000 0.1000000
#> score.2 0.6705882 0.3294118
#> score.3 0.4135021 0.5864979
#> score.4 0.1708683 0.8291317
#> 
#> $obs_prop$item.5
#>            score.0   score.1
#> score.1 0.70000000 0.3000000
#> score.2 0.42352941 0.5764706
#> score.3 0.20675105 0.7932489
#> score.4 0.07843137 0.9215686
#> 
#> 

# Compute item and test information functions at a range of theta values
theta <- seq(-4, 4, 0.1)
(info.2pl <- info(x = mod.2pl, theta = theta))
#> $iif
#>       theta.1    theta.2    theta.3    theta.4    theta.5    theta.6    theta.7
#> V1 0.15900600 0.16217707 0.16487586 0.16706788 0.16872448 0.16982363 0.17035053
#> V2 0.05908680 0.06229673 0.06559926 0.06898629 0.07244815 0.07597358 0.07954958
#> V3 0.02688417 0.02919633 0.03168883 0.03437232 0.03725741 0.04035457 0.04367392
#> V4 0.07208593 0.07520492 0.07834095 0.08147998 0.08460681 0.08770509 0.09075751
#> V5 0.09948692 0.10123155 0.10279848 0.10417528 0.10535084 0.10631545 0.10706105
#>       theta.8    theta.9   theta.10   theta.11   theta.12   theta.13   theta.14
#> V1 0.17029804 0.16966686 0.16846555 0.16671025 0.16442425 0.16163731 0.15838489
#> V2 0.08316147 0.08679289 0.09042578 0.09404050 0.09761596 0.10112973 0.10455831
#> V3 0.04722501 0.05101661 0.05505640 0.05935068 0.06390397 0.06871866 0.07379458
#> V4 0.09374585 0.09665122 0.09945421 0.10213516 0.10467436 0.10705238 0.10925029
#> V5 0.10758131 0.10787180 0.10793001 0.10775543 0.10734959 0.10671595 0.10585990
#>      theta.15   theta.16   theta.17   theta.18   theta.19  theta.20   theta.21
#> V1 0.15470718 0.15064812 0.14625431 0.14157400 0.13665602 0.1315489 0.12629995
#> V2 0.10787732 0.11106184 0.11408669 0.11692687 0.11955787 0.1219562 0.12409955
#> V3 0.07912854 0.08471383 0.09053979 0.09659131 0.10284840 0.1092858 0.11587245
#> V4 0.11125001 0.11303460 0.11458851 0.11589792 0.11695094 0.1177379 0.11825136
#> V5 0.10478867 0.10351115 0.10203778 0.10038035 0.09855182 0.0965661 0.09443785
#>      theta.22  theta.23   theta.24   theta.25   theta.26  theta.27   theta.28
#> V1 0.12095453 0.1155555 0.11014259 0.10475224 0.09941718 0.0941664 0.08902511
#> V2 0.12596762 0.1275422 0.12880744 0.12975067 0.13036214 0.1306355 0.13056797
#> V3 0.12257173 0.1293409 0.13613120 0.14288849 0.14955322 0.1560614 0.16234528
#> V4 0.11848659 0.1184413 0.11811603 0.11751374 0.11664011 0.1155033 0.11411364
#> V5 0.09218225 0.0898148 0.08735111 0.08480673 0.08219693 0.0795366 0.07684004
#>      theta.29   theta.30   theta.31   theta.32   theta.33   theta.34   theta.35
#> V1 0.08401478 0.07915330 0.07445514 0.06993161 0.06559109 0.06143930 0.05747963
#> V2 0.13016019 0.12941642 0.12834435 0.12695492 0.12526214 0.12328275 0.12103590
#> V3 0.16833484 0.17395884 0.17914650 0.18382918 0.18794219 0.19142653 0.19423064
#> V4 0.11248377 0.11062805 0.10856253 0.10630454 0.10387246 0.10128542 0.09856297
#> V5 0.07412091 0.07139209 0.06866562 0.06595264 0.06326334 0.06060697 0.05799179
#>      theta.36   theta.37   theta.38   theta.39   theta.40   theta.41   theta.42
#> V1 0.05371334 0.05013993 0.04675733 0.04356215 0.04054996 0.03771545 0.03505261
#> V2 0.11854273 0.11582603 0.11290977 0.10981870 0.10657793 0.10321258 0.09974739
#> V3 0.19631198 0.19763838 0.19818906 0.19795531 0.19694085 0.19516161 0.19264528
#> V4 0.09572483 0.09279062 0.08977959 0.08671044 0.08360110 0.08046861 0.07732894
#> V5 0.05542514 0.05291338 0.05046202 0.04807568 0.04575816 0.04351252 0.04134108
#>      theta.43   theta.44   theta.45   theta.46   theta.47   theta.48   theta.49
#> V1 0.03255494 0.03021555 0.02802732 0.02598299 0.02407525 0.02229684 0.02064059
#> V2 0.09620644 0.09261282 0.08898846 0.08535390 0.08172814 0.07812859 0.07457096
#> V3 0.18943035 0.18556490 0.18110513 0.17611364 0.17065770 0.16480741 0.15863401
#> V4 0.07419693 0.07108621 0.06800914 0.06497682 0.06199908 0.05908448 0.05624040
#> V5 0.03924554 0.03722699 0.03528597 0.03342256 0.03163639 0.02992674 0.02829253
#>      theta.50   theta.51   theta.52   theta.53   theta.54   theta.55   theta.56
#> V1 0.01909948 0.01766668 0.01633559 0.01509985 0.01395338 0.01289036 0.01190525
#> V2 0.07106924 0.06763575 0.06428111 0.06101436 0.05784299 0.05477304 0.05180924
#> V3 0.15220825 0.14559892 0.13887163 0.13208781 0.12530392 0.11857090 0.11193391
#> V4 0.05347303 0.05078748 0.04818782 0.04567719 0.04325782 0.04093116 0.03869795
#> V5 0.02673243 0.02524485 0.02382801 0.02247997 0.02119867 0.01998191 0.01882747
#>      theta.57   theta.58    theta.59    theta.60    theta.61   theta.62
#> V1 0.01099280 0.01014806 0.009366322 0.008643188 0.007974505 0.00735638
#> V2 0.04895504 0.04621281 0.043583875 0.041068671 0.038666838 0.03637732
#> V3 0.10543210 0.09909874 0.092961370 0.087042086 0.081357924 0.07592129
#> V4 0.03655826 0.03451164 0.032557110 0.030693273 0.028918377 0.02723036
#> V5 0.01773305 0.01669632 0.015714945 0.014786608 0.013908996 0.01307983
#>       theta.63    theta.64    theta.65    theta.66    theta.67    theta.68
#> V1 0.006785169 0.006257462 0.005770074 0.005320034 0.004904573 0.004521113
#> V2 0.034198476 0.032128146 0.030163756 0.028302390 0.026540854 0.024875744
#> V3 0.070740426 0.065819887 0.061161017 0.056762412 0.052620353 0.048729219
#> V4 0.025626910 0.024105518 0.022663515 0.021298114 0.020006448 0.018785597
#> V5 0.012296878 0.011557949 0.010860910 0.010203692 0.009584286 0.009000755
#>       theta.69    theta.70   theta.71    theta.72    theta.73    theta.74
#> V1 0.004167257 0.003840776 0.00353960 0.003261811 0.003005626 0.002769395
#> V2 0.023303499 0.021820450 0.02042287 0.019106988 0.017869061 0.016705366
#> V3 0.045081847 0.041669876 0.03848404 0.035514409 0.032750652 0.030182186
#> V4 0.017632617 0.016544566 0.01551852 0.014551582 0.013640924 0.012783767
#> V5 0.008451230 0.007933913 0.00744708 0.006989079 0.006558328 0.006153317
#>       theta.75    theta.76    theta.77    theta.78    theta.79    theta.80
#> V1 0.002551590 0.002350795 0.002165699 0.001995091 0.001837849 0.001692938
#> V2 0.015612235 0.014586078 0.013623392 0.012720775 0.011874938 0.011082710
#> V3 0.027798358 0.025588570 0.023542388 0.021649627 0.019900416 0.018285252
#> V4 0.011977407 0.011219221 0.010506671 0.009837308 0.009208777 0.008618815
#> V5 0.005772607 0.005414827 0.005078671 0.004762900 0.004466337 0.004187865
#>       theta.81
#> V1 0.001559401
#> V2 0.010341042
#> V3 0.016795032
#> V4 0.008065259
#> V5 0.003926427
#> 
#> $tif
#>  [1] 0.41654981 0.43010661 0.44330338 0.45608175 0.46838769 0.48017232
#>  [7] 0.49139258 0.50201168 0.51199937 0.52133194 0.52999202 0.53796812
#> [13] 0.54525403 0.55184798 0.55775173 0.56296953 0.56750708 0.57137045
#> [19] 0.57456506 0.57709478 0.57896115 0.58016272 0.58069463 0.58054838
#> [25] 0.57971187 0.57816959 0.57590316 0.57289204 0.56911449 0.56454870
#> [31] 0.55917413 0.55297289 0.54593122 0.53804097 0.52930093 0.51971803
#> [37] 0.50930836 0.49809777 0.48612228 0.47342801 0.46007077 0.44611531
#> [43] 0.43163420 0.41670648 0.40141603 0.38584991 0.37009656 0.35424406
#> [49] 0.33837849 0.32258243 0.30693367 0.29150417 0.27635919 0.26155677
#> [55] 0.24714738 0.23317381 0.21967125 0.20666756 0.19418362 0.18223383
#> [61] 0.17082664 0.15996518 0.14964786 0.13986896 0.13061927 0.12188664
#> [67] 0.11365651 0.10591243 0.09863645 0.09180958 0.08541210 0.07942387
#> [73] 0.07382459 0.06859403 0.06371220 0.05915949 0.05491682 0.05096570
#> [79] 0.04728832 0.04386758 0.04068716
#> 
#> $theta
#>  [1] -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6
#> [16] -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1
#> [31] -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0.0  0.1  0.2  0.3  0.4
#> [46]  0.5  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9
#> [61]  2.0  2.1  2.2  2.3  2.4  2.5  2.6  2.7  2.8  2.9  3.0  3.1  3.2  3.3  3.4
#> [76]  3.5  3.6  3.7  3.8  3.9  4.0
#> 
#> attr(,"class")
#> [1] "info"

# Plot the test characteristic curve (TCC)
(trace.2pl <- traceline(x = mod.2pl, theta = theta))
#> $prob.cats
#> $prob.cats$V1
#>            resp.0    resp.1
#>  [1,] 0.629288374 0.3707116
#>  [2,] 0.609832040 0.3901680
#>  [3,] 0.590019837 0.4099802
#>  [4,] 0.569911111 0.4300889
#>  [5,] 0.549569046 0.4504310
#>  [6,] 0.529059897 0.4709401
#>  [7,] 0.508452158 0.4915478
#>  [8,] 0.487815656 0.5121843
#>  [9,] 0.467220605 0.5327794
#> [10,] 0.446736661 0.5532633
#> [11,] 0.426431978 0.5735680
#> [12,] 0.406372316 0.5936277
#> [13,] 0.386620216 0.6133798
#> [14,] 0.367234260 0.6327657
#> [15,] 0.348268439 0.6517316
#> [16,] 0.329771638 0.6702284
#> [17,] 0.311787248 0.6882128
#> [18,] 0.294352904 0.7056471
#> [19,] 0.277500353 0.7224996
#> [20,] 0.261255434 0.7387446
#> [21,] 0.245638178 0.7543618
#> [22,] 0.230662993 0.7693370
#> [23,] 0.216338941 0.7836611
#> [24,] 0.202670076 0.7973299
#> [25,] 0.189655835 0.8103442
#> [26,] 0.177291467 0.8227085
#> [27,] 0.165568481 0.8344315
#> [28,] 0.154475105 0.8455249
#> [29,] 0.143996737 0.8560033
#> [30,] 0.134116398 0.8658836
#> [31,] 0.124815146 0.8751849
#> [32,] 0.116072485 0.8839275
#> [33,] 0.107866727 0.8921333
#> [34,] 0.100175332 0.8998247
#> [35,] 0.092975214 0.9070248
#> [36,] 0.086243005 0.9137570
#> [37,] 0.079955295 0.9200447
#> [38,] 0.074088833 0.9259112
#> [39,] 0.068620700 0.9313793
#> [40,] 0.063528453 0.9364715
#> [41,] 0.058790242 0.9412098
#> [42,] 0.054384903 0.9456151
#> [43,] 0.050292031 0.9497080
#> [44,] 0.046492035 0.9535080
#> [45,] 0.042966171 0.9570338
#> [46,] 0.039696569 0.9603034
#> [47,] 0.036666243 0.9633338
#> [48,] 0.033859086 0.9661409
#> [49,] 0.031259871 0.9687401
#> [50,] 0.028854228 0.9711458
#> [51,] 0.026628624 0.9733714
#> [52,] 0.024570345 0.9754297
#> [53,] 0.022667456 0.9773325
#> [54,] 0.020908781 0.9790912
#> [55,] 0.019283862 0.9807161
#> [56,] 0.017782929 0.9822171
#> [57,] 0.016396867 0.9836031
#> [58,] 0.015117176 0.9848828
#> [59,] 0.013935943 0.9860641
#> [60,] 0.012845806 0.9871542
#> [61,] 0.011839921 0.9881601
#> [62,] 0.010911930 0.9890881
#> [63,] 0.010055934 0.9899441
#> [64,] 0.009266458 0.9907335
#> [65,] 0.008538428 0.9914616
#> [66,] 0.007867142 0.9921329
#> [67,] 0.007248247 0.9927518
#> [68,] 0.006677711 0.9933223
#> [69,] 0.006151806 0.9938482
#> [70,] 0.005667082 0.9943329
#> [71,] 0.005220351 0.9947796
#> [72,] 0.004808665 0.9951913
#> [73,] 0.004429301 0.9955707
#> [74,] 0.004079743 0.9959203
#> [75,] 0.003757668 0.9962423
#> [76,] 0.003460930 0.9965391
#> [77,] 0.003187551 0.9968124
#> [78,] 0.002935702 0.9970643
#> [79,] 0.002703698 0.9972963
#> [80,] 0.002489982 0.9975100
#> [81,] 0.002293122 0.9977069
#> 
#> $prob.cats$V2
#>           resp.0    resp.1
#>  [1,] 0.87005109 0.1299489
#>  [2,] 0.86165676 0.1383432
#>  [3,] 0.85281192 0.1471881
#>  [4,] 0.84350429 0.1564957
#>  [5,] 0.83372284 0.1662772
#>  [6,] 0.82345799 0.1765420
#>  [7,] 0.81270180 0.1872982
#>  [8,] 0.80144829 0.1985517
#>  [9,] 0.78969360 0.2103064
#> [10,] 0.77743626 0.2225637
#> [11,] 0.76467741 0.2353226
#> [12,] 0.75142100 0.2485790
#> [13,] 0.73767400 0.2623260
#> [14,] 0.72344656 0.2765534
#> [15,] 0.70875214 0.2912479
#> [16,] 0.69360762 0.3063924
#> [17,] 0.67803334 0.3219667
#> [18,] 0.66205312 0.3379469
#> [19,] 0.64569421 0.3543058
#> [20,] 0.62898719 0.3710128
#> [21,] 0.61196580 0.3880342
#> [22,] 0.59466674 0.4053333
#> [23,] 0.57712937 0.4228706
#> [24,] 0.55939542 0.4406046
#> [25,] 0.54150860 0.4584914
#> [26,] 0.52351415 0.4764859
#> [27,] 0.50545844 0.4945416
#> [28,] 0.48738849 0.5126115
#> [29,] 0.46935144 0.5306486
#> [30,] 0.45139411 0.5486059
#> [31,] 0.43356248 0.5664375
#> [32,] 0.41590124 0.5840988
#> [33,] 0.39845335 0.6015466
#> [34,] 0.38125966 0.6187403
#> [35,] 0.36435851 0.6356415
#> [36,] 0.34778545 0.6522145
#> [37,] 0.33157301 0.6684270
#> [38,] 0.31575044 0.6842496
#> [39,] 0.30034366 0.6996563
#> [40,] 0.28537510 0.7146249
#> [41,] 0.27086375 0.7291363
#> [42,] 0.25682511 0.7431749
#> [43,] 0.24327131 0.7567287
#> [44,] 0.23021124 0.7697888
#> [45,] 0.21765063 0.7823494
#> [46,] 0.20559232 0.7944077
#> [47,] 0.19403637 0.8059636
#> [48,] 0.18298035 0.8170196
#> [49,] 0.17241953 0.8275805
#> [50,] 0.16234711 0.8376529
#> [51,] 0.15275449 0.8472455
#> [52,] 0.14363149 0.8563685
#> [53,] 0.13496654 0.8650335
#> [54,] 0.12674697 0.8732530
#> [55,] 0.11895913 0.8810409
#> [56,] 0.11158866 0.8884113
#> [57,] 0.10462063 0.8953794
#> [58,] 0.09803969 0.9019603
#> [59,] 0.09183025 0.9081698
#> [60,] 0.08597661 0.9140234
#> [61,] 0.08046304 0.9195370
#> [62,] 0.07527393 0.9247261
#> [63,] 0.07039385 0.9296061
#> [64,] 0.06580764 0.9341924
#> [65,] 0.06150045 0.9384995
#> [66,] 0.05745784 0.9425422
#> [67,] 0.05366576 0.9463342
#> [68,] 0.05011065 0.9498894
#> [69,] 0.04677940 0.9532206
#> [70,] 0.04365943 0.9563406
#> [71,] 0.04073866 0.9592613
#> [72,] 0.03800551 0.9619945
#> [73,] 0.03544896 0.9645510
#> [74,] 0.03305847 0.9669415
#> [75,] 0.03082403 0.9691760
#> [76,] 0.02873613 0.9712639
#> [77,] 0.02678575 0.9732143
#> [78,] 0.02496434 0.9750357
#> [79,] 0.02326382 0.9767362
#> [80,] 0.02167657 0.9783234
#> [81,] 0.02019537 0.9798046
#> 
#> $prob.cats$V3
#>           resp.0     resp.1
#>  [1,] 0.96485532 0.03514468
#>  [2,] 0.96170780 0.03829220
#>  [3,] 0.95829058 0.04170942
#>  [4,] 0.95458280 0.04541720
#>  [5,] 0.95056243 0.04943757
#>  [6,] 0.94620622 0.05379378
#>  [7,] 0.94148979 0.05851021
#>  [8,] 0.93638764 0.06361236
#>  [9,] 0.93087324 0.06912676
#> [10,] 0.92491915 0.07508085
#> [11,] 0.91849711 0.08150289
#> [12,] 0.91157828 0.08842172
#> [13,] 0.90413341 0.09586659
#> [14,] 0.89613312 0.10386688
#> [15,] 0.88754823 0.11245177
#> [16,] 0.87835009 0.12164991
#> [17,] 0.86851105 0.13148895
#> [18,] 0.85800487 0.14199513
#> [19,] 0.84680730 0.15319270
#> [20,] 0.83489662 0.16510338
#> [21,] 0.82225426 0.17774574
#> [22,] 0.80886548 0.19113452
#> [23,] 0.79471995 0.20528005
#> [24,] 0.77981252 0.22018748
#> [25,] 0.76414380 0.23585620
#> [26,] 0.74772080 0.25227920
#> [27,] 0.73055747 0.26944253
#> [28,] 0.71267516 0.28732484
#> [29,] 0.69410298 0.30589702
#> [30,] 0.67487799 0.32512201
#> [31,] 0.65504521 0.34495479
#> [32,] 0.63465755 0.36534245
#> [33,] 0.61377539 0.38622461
#> [34,] 0.59246609 0.40753391
#> [35,] 0.57080322 0.42919678
#> [36,] 0.54886567 0.45113433
#> [37,] 0.52673650 0.47326350
#> [38,] 0.50450179 0.49549821
#> [39,] 0.48224925 0.51775075
#> [40,] 0.46006691 0.53993309
#> [41,] 0.43804164 0.56195836
#> [42,] 0.41625790 0.58374210
#> [43,] 0.39479641 0.60520359
#> [44,] 0.37373300 0.62626700
#> [45,] 0.35313764 0.64686236
#> [46,] 0.33307362 0.66692638
#> [47,] 0.31359692 0.68640308
#> [48,] 0.29475577 0.70524423
#> [49,] 0.27659047 0.72340953
#> [50,] 0.25913333 0.74086667
#> [51,] 0.24240878 0.75759122
#> [52,] 0.22643375 0.77356625
#> [53,] 0.21121797 0.78878203
#> [54,] 0.19676458 0.80323542
#> [55,] 0.18307067 0.81692933
#> [56,] 0.17012794 0.82987206
#> [57,] 0.15792335 0.84207665
#> [58,] 0.14643979 0.85356021
#> [59,] 0.13565675 0.86434325
#> [60,] 0.12555092 0.87444908
#> [61,] 0.11609681 0.88390319
#> [62,] 0.10726728 0.89273272
#> [63,] 0.09903402 0.90096598
#> [64,] 0.09136803 0.90863197
#> [65,] 0.08423996 0.91576004
#> [66,] 0.07762048 0.92237952
#> [67,] 0.07148055 0.92851945
#> [68,] 0.06579166 0.93420834
#> [69,] 0.06052601 0.93947399
#> [70,] 0.05565670 0.94434330
#> [71,] 0.05115779 0.94884221
#> [72,] 0.04700444 0.95299556
#> [73,] 0.04317294 0.95682706
#> [74,] 0.03964077 0.96035923
#> [75,] 0.03638660 0.96361340
#> [76,] 0.03339027 0.96660973
#> [77,] 0.03063285 0.96936715
#> [78,] 0.02809651 0.97190349
#> [79,] 0.02576460 0.97423540
#> [80,] 0.02362152 0.97637848
#> [81,] 0.02165274 0.97834726
#> 
#> $prob.cats$V4
#>           resp.0    resp.1
#>  [1,] 0.81292724 0.1870728
#>  [2,] 0.80223078 0.1977692
#>  [3,] 0.79107990 0.2089201
#>  [4,] 0.77947313 0.2205269
#>  [5,] 0.76741112 0.2325889
#>  [6,] 0.75489681 0.2451032
#>  [7,] 0.74193561 0.2580644
#>  [8,] 0.72853548 0.2714645
#>  [9,] 0.71470710 0.2852929
#> [10,] 0.70046392 0.2995361
#> [11,] 0.68582225 0.3141778
#> [12,] 0.67080123 0.3291988
#> [13,] 0.65542287 0.3445771
#> [14,] 0.63971197 0.3602880
#> [15,] 0.62369601 0.3763040
#> [16,] 0.60740502 0.3925950
#> [17,] 0.59087141 0.4091286
#> [18,] 0.57412969 0.4258703
#> [19,] 0.55721627 0.4427837
#> [20,] 0.54016913 0.4598309
#> [21,] 0.52302748 0.4769725
#> [22,] 0.50583140 0.4941686
#> [23,] 0.48862152 0.5113785
#> [24,] 0.47143857 0.5285614
#> [25,] 0.45432304 0.5456770
#> [26,] 0.43731478 0.5626852
#> [27,] 0.42045264 0.5795474
#> [28,] 0.40377411 0.5962259
#> [29,] 0.38731504 0.6126850
#> [30,] 0.37110929 0.6288907
#> [31,] 0.35518851 0.6448115
#> [32,] 0.33958194 0.6604181
#> [33,] 0.32431620 0.6756838
#> [34,] 0.30941521 0.6905848
#> [35,] 0.29490004 0.7051000
#> [36,] 0.28078894 0.7192111
#> [37,] 0.26709729 0.7329027
#> [38,] 0.25383762 0.7461624
#> [39,] 0.24101975 0.7589803
#> [40,] 0.22865078 0.7713492
#> [41,] 0.21673532 0.7832647
#> [42,] 0.20527555 0.7947245
#> [43,] 0.19427142 0.8057286
#> [44,] 0.18372082 0.8162792
#> [45,] 0.17361973 0.8263803
#> [46,] 0.16396246 0.8360375
#> [47,] 0.15474177 0.8452582
#> [48,] 0.14594910 0.8540509
#> [49,] 0.13757472 0.8624253
#> [50,] 0.12960793 0.8703921
#> [51,] 0.12203721 0.8779628
#> [52,] 0.11485036 0.8851496
#> [53,] 0.10803467 0.8919653
#> [54,] 0.10157703 0.8984230
#> [55,] 0.09546408 0.9045359
#> [56,] 0.08968228 0.9103177
#> [57,] 0.08421806 0.9157819
#> [58,] 0.07905785 0.9209421
#> [59,] 0.07418821 0.9258118
#> [60,] 0.06959584 0.9304042
#> [61,] 0.06526771 0.9347323
#> [62,] 0.06119105 0.9388090
#> [63,] 0.05735339 0.9426466
#> [64,] 0.05374263 0.9462574
#> [65,] 0.05034706 0.9496529
#> [66,] 0.04715533 0.9528447
#> [67,] 0.04415653 0.9558435
#> [68,] 0.04134016 0.9586598
#> [69,] 0.03869616 0.9613038
#> [70,] 0.03621487 0.9637851
#> [71,] 0.03388707 0.9661129
#> [72,] 0.03170398 0.9682960
#> [73,] 0.02965721 0.9703428
#> [74,] 0.02773880 0.9722612
#> [75,] 0.02594116 0.9740588
#> [76,] 0.02425711 0.9757429
#> [77,] 0.02267984 0.9773202
#> [78,] 0.02120290 0.9787971
#> [79,] 0.01982019 0.9801798
#> [80,] 0.01852595 0.9814740
#> [81,] 0.01731473 0.9826853
#> 
#> $prob.cats$V5
#>            resp.0    resp.1
#>  [1,] 0.639901461 0.3600985
#>  [2,] 0.624625684 0.3753743
#>  [3,] 0.609097751 0.3909022
#>  [4,] 0.593345666 0.4066543
#>  [5,] 0.577399227 0.4226008
#>  [6,] 0.561289820 0.4387102
#>  [7,] 0.545050183 0.4549498
#>  [8,] 0.528714153 0.4712858
#>  [9,] 0.512316384 0.4876836
#> [10,] 0.495892063 0.5041079
#> [11,] 0.479476603 0.5205234
#> [12,] 0.463105341 0.5368947
#> [13,] 0.446813235 0.5531868
#> [14,] 0.430634567 0.5693654
#> [15,] 0.414602656 0.5853973
#> [16,] 0.398749596 0.6012504
#> [17,] 0.383106005 0.6168940
#> [18,] 0.367700809 0.6322992
#> [19,] 0.352561046 0.6474390
#> [20,] 0.337711710 0.6622883
#> [21,] 0.323175614 0.6768244
#> [22,] 0.308973302 0.6910267
#> [23,] 0.295122977 0.7048770
#> [24,] 0.281640474 0.7183595
#> [25,] 0.268539254 0.7314607
#> [26,] 0.255830431 0.7441696
#> [27,] 0.243522822 0.7564772
#> [28,] 0.231623022 0.7683770
#> [29,] 0.220135496 0.7798645
#> [30,] 0.209062687 0.7909373
#> [31,] 0.198405144 0.8015949
#> [32,] 0.188161649 0.8118384
#> [33,] 0.178329363 0.8216706
#> [34,] 0.168903964 0.8310960
#> [35,] 0.159879802 0.8401202
#> [36,] 0.151250036 0.8487500
#> [37,] 0.143006785 0.8569932
#> [38,] 0.135141264 0.8648587
#> [39,] 0.127643921 0.8723561
#> [40,] 0.120504559 0.8794954
#> [41,] 0.113712464 0.8862875
#> [42,] 0.107256510 0.8927435
#> [43,] 0.101125268 0.8988747
#> [44,] 0.095307096 0.9046929
#> [45,] 0.089790231 0.9102098
#> [46,] 0.084562861 0.9154371
#> [47,] 0.079613197 0.9203868
#> [48,] 0.074929535 0.9250705
#> [49,] 0.070500308 0.9294997
#> [50,] 0.066314132 0.9336859
#> [51,] 0.062359848 0.9376402
#> [52,] 0.058626550 0.9413735
#> [53,] 0.055103619 0.9448964
#> [54,] 0.051780739 0.9482193
#> [55,] 0.048647921 0.9513521
#> [56,] 0.045695510 0.9543045
#> [57,] 0.042914196 0.9570858
#> [58,] 0.040295021 0.9597050
#> [59,] 0.037829385 0.9621706
#> [60,] 0.035509037 0.9644910
#> [61,] 0.033326083 0.9666739
#> [62,] 0.031272977 0.9687270
#> [63,] 0.029342517 0.9706575
#> [64,] 0.027527836 0.9724722
#> [65,] 0.025822398 0.9741776
#> [66,] 0.024219987 0.9757800
#> [67,] 0.022714694 0.9772853
#> [68,] 0.021300914 0.9786991
#> [69,] 0.019973331 0.9800267
#> [70,] 0.018726906 0.9812731
#> [71,] 0.017556871 0.9824431
#> [72,] 0.016458711 0.9835413
#> [73,] 0.015428162 0.9845718
#> [74,] 0.014461191 0.9855388
#> [75,] 0.013553991 0.9864460
#> [76,] 0.012702969 0.9872970
#> [77,] 0.011904736 0.9880953
#> [78,] 0.011156096 0.9888439
#> [79,] 0.010454036 0.9895460
#> [80,] 0.009795720 0.9902043
#> [81,] 0.009178475 0.9908215
#> 
#> 
#> $icc
#>              V1        V2         V3        V4        V5
#>  [1,] 0.3707116 0.1299489 0.03514468 0.1870728 0.3600985
#>  [2,] 0.3901680 0.1383432 0.03829220 0.1977692 0.3753743
#>  [3,] 0.4099802 0.1471881 0.04170942 0.2089201 0.3909022
#>  [4,] 0.4300889 0.1564957 0.04541720 0.2205269 0.4066543
#>  [5,] 0.4504310 0.1662772 0.04943757 0.2325889 0.4226008
#>  [6,] 0.4709401 0.1765420 0.05379378 0.2451032 0.4387102
#>  [7,] 0.4915478 0.1872982 0.05851021 0.2580644 0.4549498
#>  [8,] 0.5121843 0.1985517 0.06361236 0.2714645 0.4712858
#>  [9,] 0.5327794 0.2103064 0.06912676 0.2852929 0.4876836
#> [10,] 0.5532633 0.2225637 0.07508085 0.2995361 0.5041079
#> [11,] 0.5735680 0.2353226 0.08150289 0.3141778 0.5205234
#> [12,] 0.5936277 0.2485790 0.08842172 0.3291988 0.5368947
#> [13,] 0.6133798 0.2623260 0.09586659 0.3445771 0.5531868
#> [14,] 0.6327657 0.2765534 0.10386688 0.3602880 0.5693654
#> [15,] 0.6517316 0.2912479 0.11245177 0.3763040 0.5853973
#> [16,] 0.6702284 0.3063924 0.12164991 0.3925950 0.6012504
#> [17,] 0.6882128 0.3219667 0.13148895 0.4091286 0.6168940
#> [18,] 0.7056471 0.3379469 0.14199513 0.4258703 0.6322992
#> [19,] 0.7224996 0.3543058 0.15319270 0.4427837 0.6474390
#> [20,] 0.7387446 0.3710128 0.16510338 0.4598309 0.6622883
#> [21,] 0.7543618 0.3880342 0.17774574 0.4769725 0.6768244
#> [22,] 0.7693370 0.4053333 0.19113452 0.4941686 0.6910267
#> [23,] 0.7836611 0.4228706 0.20528005 0.5113785 0.7048770
#> [24,] 0.7973299 0.4406046 0.22018748 0.5285614 0.7183595
#> [25,] 0.8103442 0.4584914 0.23585620 0.5456770 0.7314607
#> [26,] 0.8227085 0.4764859 0.25227920 0.5626852 0.7441696
#> [27,] 0.8344315 0.4945416 0.26944253 0.5795474 0.7564772
#> [28,] 0.8455249 0.5126115 0.28732484 0.5962259 0.7683770
#> [29,] 0.8560033 0.5306486 0.30589702 0.6126850 0.7798645
#> [30,] 0.8658836 0.5486059 0.32512201 0.6288907 0.7909373
#> [31,] 0.8751849 0.5664375 0.34495479 0.6448115 0.8015949
#> [32,] 0.8839275 0.5840988 0.36534245 0.6604181 0.8118384
#> [33,] 0.8921333 0.6015466 0.38622461 0.6756838 0.8216706
#> [34,] 0.8998247 0.6187403 0.40753391 0.6905848 0.8310960
#> [35,] 0.9070248 0.6356415 0.42919678 0.7051000 0.8401202
#> [36,] 0.9137570 0.6522145 0.45113433 0.7192111 0.8487500
#> [37,] 0.9200447 0.6684270 0.47326350 0.7329027 0.8569932
#> [38,] 0.9259112 0.6842496 0.49549821 0.7461624 0.8648587
#> [39,] 0.9313793 0.6996563 0.51775075 0.7589803 0.8723561
#> [40,] 0.9364715 0.7146249 0.53993309 0.7713492 0.8794954
#> [41,] 0.9412098 0.7291363 0.56195836 0.7832647 0.8862875
#> [42,] 0.9456151 0.7431749 0.58374210 0.7947245 0.8927435
#> [43,] 0.9497080 0.7567287 0.60520359 0.8057286 0.8988747
#> [44,] 0.9535080 0.7697888 0.62626700 0.8162792 0.9046929
#> [45,] 0.9570338 0.7823494 0.64686236 0.8263803 0.9102098
#> [46,] 0.9603034 0.7944077 0.66692638 0.8360375 0.9154371
#> [47,] 0.9633338 0.8059636 0.68640308 0.8452582 0.9203868
#> [48,] 0.9661409 0.8170196 0.70524423 0.8540509 0.9250705
#> [49,] 0.9687401 0.8275805 0.72340953 0.8624253 0.9294997
#> [50,] 0.9711458 0.8376529 0.74086667 0.8703921 0.9336859
#> [51,] 0.9733714 0.8472455 0.75759122 0.8779628 0.9376402
#> [52,] 0.9754297 0.8563685 0.77356625 0.8851496 0.9413735
#> [53,] 0.9773325 0.8650335 0.78878203 0.8919653 0.9448964
#> [54,] 0.9790912 0.8732530 0.80323542 0.8984230 0.9482193
#> [55,] 0.9807161 0.8810409 0.81692933 0.9045359 0.9513521
#> [56,] 0.9822171 0.8884113 0.82987206 0.9103177 0.9543045
#> [57,] 0.9836031 0.8953794 0.84207665 0.9157819 0.9570858
#> [58,] 0.9848828 0.9019603 0.85356021 0.9209421 0.9597050
#> [59,] 0.9860641 0.9081698 0.86434325 0.9258118 0.9621706
#> [60,] 0.9871542 0.9140234 0.87444908 0.9304042 0.9644910
#> [61,] 0.9881601 0.9195370 0.88390319 0.9347323 0.9666739
#> [62,] 0.9890881 0.9247261 0.89273272 0.9388090 0.9687270
#> [63,] 0.9899441 0.9296061 0.90096598 0.9426466 0.9706575
#> [64,] 0.9907335 0.9341924 0.90863197 0.9462574 0.9724722
#> [65,] 0.9914616 0.9384995 0.91576004 0.9496529 0.9741776
#> [66,] 0.9921329 0.9425422 0.92237952 0.9528447 0.9757800
#> [67,] 0.9927518 0.9463342 0.92851945 0.9558435 0.9772853
#> [68,] 0.9933223 0.9498894 0.93420834 0.9586598 0.9786991
#> [69,] 0.9938482 0.9532206 0.93947399 0.9613038 0.9800267
#> [70,] 0.9943329 0.9563406 0.94434330 0.9637851 0.9812731
#> [71,] 0.9947796 0.9592613 0.94884221 0.9661129 0.9824431
#> [72,] 0.9951913 0.9619945 0.95299556 0.9682960 0.9835413
#> [73,] 0.9955707 0.9645510 0.95682706 0.9703428 0.9845718
#> [74,] 0.9959203 0.9669415 0.96035923 0.9722612 0.9855388
#> [75,] 0.9962423 0.9691760 0.96361340 0.9740588 0.9864460
#> [76,] 0.9965391 0.9712639 0.96660973 0.9757429 0.9872970
#> [77,] 0.9968124 0.9732143 0.96936715 0.9773202 0.9880953
#> [78,] 0.9970643 0.9750357 0.97190349 0.9787971 0.9888439
#> [79,] 0.9972963 0.9767362 0.97423540 0.9801798 0.9895460
#> [80,] 0.9975100 0.9783234 0.97637848 0.9814740 0.9902043
#> [81,] 0.9977069 0.9798046 0.97834726 0.9826853 0.9908215
#> 
#> $tcc
#>  [1] 1.082977 1.139947 1.198700 1.259183 1.321335 1.385089 1.450370 1.517099
#>  [9] 1.585189 1.654552 1.725095 1.796722 1.869336 1.942840 2.017133 2.092116
#> [17] 2.167691 2.243759 2.320221 2.396980 2.473939 2.551000 2.628067 2.705043
#> [25] 2.781829 2.858328 2.934440 3.010064 3.085098 3.159440 3.232984 3.305625
#> [33] 3.377259 3.447780 3.517083 3.585067 3.651631 3.716680 3.780123 3.841874
#> [41] 3.901857 3.960000 4.016244 4.070536 4.122836 4.173112 4.221345 4.267526
#> [49] 4.311655 4.353743 4.393811 4.431888 4.468010 4.502222 4.534574 4.565123
#> [57] 4.593927 4.621050 4.646559 4.670522 4.693006 4.714083 4.733820 4.752287
#> [65] 4.769552 4.785679 4.800734 4.814779 4.827873 4.840075 4.851439 4.862019
#> [73] 4.871863 4.881021 4.889537 4.897453 4.904809 4.911644 4.917994 4.923890
#> [81] 4.929366
#> 
#> $theta
#>  [1] -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6
#> [16] -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1
#> [31] -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0.0  0.1  0.2  0.3  0.4
#> [46]  0.5  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9
#> [61]  2.0  2.1  2.2  2.3  2.4  2.5  2.6  2.7  2.8  2.9  3.0  3.1  3.2  3.3  3.4
#> [76]  3.5  3.6  3.7  3.8  3.9  4.0
#> 
#> attr(,"class")
#> [1] "traceline"
plot(trace.2pl)


# Plot the item characteristic curve (ICC) for the first item
plot(trace.2pl, item.loc = 1)


# Fit the 2PL model and simultaneously estimate an empirical histogram
# of the latent variable prior distribution
# Also apply a looser convergence threshold for the EM algorithm
(mod.2pl.hist <- est_irt(data = LSAT6, D = 1, model = "2PLM", cats = 2,
                         EmpHist = TRUE, Etol = 0.001))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -3182.0286, Max-Change: 2.294233
 EM iteration: 2, Loglike: -2561.3283, Max-Change: 0.586799
 EM iteration: 3, Loglike: -2483.1569, Max-Change: 0.32487
 EM iteration: 4, Loglike: -2469.4065, Max-Change: 0.181068
 EM iteration: 5, Loglike: -2467.1665, Max-Change: 0.101828
 EM iteration: 6, Loglike: -2466.7828, Max-Change: 0.059822
 EM iteration: 7, Loglike: -2466.6906, Max-Change: 0.038796
 EM iteration: 8, Loglike: -2466.6550, Max-Change: 0.026996
 EM iteration: 9, Loglike: -2466.6368, Max-Change: 0.019699
 EM iteration: 10, Loglike: -2466.6258, Max-Change: 0.015007
 EM iteration: 11, Loglike: -2466.6182, Max-Change: 0.011822
 EM iteration: 12, Loglike: -2466.6122, Max-Change: 0.00960
 EM iteration: 13, Loglike: -2466.6073, Max-Change: 0.00804
 EM iteration: 14, Loglike: -2466.6029, Max-Change: 0.006884
 EM iteration: 15, Loglike: -2466.5988, Max-Change: 0.005994
 EM iteration: 16, Loglike: -2466.5950, Max-Change: 0.005292
 EM iteration: 17, Loglike: -2466.5914, Max-Change: 0.004726
 EM iteration: 18, Loglike: -2466.5879, Max-Change: 0.004262
 EM iteration: 19, Loglike: -2466.5844, Max-Change: 0.003877
 EM iteration: 20, Loglike: -2466.5810, Max-Change: 0.003554
 EM iteration: 21, Loglike: -2466.5777, Max-Change: 0.00328
 EM iteration: 22, Loglike: -2466.5744, Max-Change: 0.003046
 EM iteration: 23, Loglike: -2466.5712, Max-Change: 0.002846
 EM iteration: 24, Loglike: -2466.5680, Max-Change: 0.002674
 EM iteration: 25, Loglike: -2466.5647, Max-Change: 0.002525
 EM iteration: 26, Loglike: -2466.5616, Max-Change: 0.002396
 EM iteration: 27, Loglike: -2466.5584, Max-Change: 0.002284
 EM iteration: 28, Loglike: -2466.5553, Max-Change: 0.002186
 EM iteration: 29, Loglike: -2466.5522, Max-Change: 0.002101
 EM iteration: 30, Loglike: -2466.5491, Max-Change: 0.002026
 EM iteration: 31, Loglike: -2466.5461, Max-Change: 0.00196
 EM iteration: 32, Loglike: -2466.5432, Max-Change: 0.001956
 EM iteration: 33, Loglike: -2466.5402, Max-Change: 0.002012
 EM iteration: 34, Loglike: -2466.5374, Max-Change: 0.002058
 EM iteration: 35, Loglike: -2466.5346, Max-Change: 0.002097
 EM iteration: 36, Loglike: -2466.5318, Max-Change: 0.002127
 EM iteration: 37, Loglike: -2466.5291, Max-Change: 0.00215
 EM iteration: 38, Loglike: -2466.5265, Max-Change: 0.002166
 EM iteration: 39, Loglike: -2466.5240, Max-Change: 0.002174
 EM iteration: 40, Loglike: -2466.5216, Max-Change: 0.002175
 EM iteration: 41, Loglike: -2466.5192, Max-Change: 0.002169
 EM iteration: 42, Loglike: -2466.5169, Max-Change: 0.002157
 EM iteration: 43, Loglike: -2466.5147, Max-Change: 0.002138
 EM iteration: 44, Loglike: -2466.5126, Max-Change: 0.002112
 EM iteration: 45, Loglike: -2466.5105, Max-Change: 0.002081
 EM iteration: 46, Loglike: -2466.5086, Max-Change: 0.002044
 EM iteration: 47, Loglike: -2466.5067, Max-Change: 0.002002
 EM iteration: 48, Loglike: -2466.5049, Max-Change: 0.001954
 EM iteration: 49, Loglike: -2466.5032, Max-Change: 0.001902
 EM iteration: 50, Loglike: -2466.5015, Max-Change: 0.001846
 EM iteration: 51, Loglike: -2466.4999, Max-Change: 0.001786
 EM iteration: 52, Loglike: -2466.4984, Max-Change: 0.001723
 EM iteration: 53, Loglike: -2466.4970, Max-Change: 0.001657
 EM iteration: 54, Loglike: -2466.4956, Max-Change: 0.001588
 EM iteration: 55, Loglike: -2466.4943, Max-Change: 0.001517
 EM iteration: 56, Loglike: -2466.4930, Max-Change: 0.001445
 EM iteration: 57, Loglike: -2466.4918, Max-Change: 0.001371
 EM iteration: 58, Loglike: -2466.4906, Max-Change: 0.001296
 EM iteration: 59, Loglike: -2466.4895, Max-Change: 0.001221
 EM iteration: 60, Loglike: -2466.4884, Max-Change: 0.001146
 EM iteration: 61, Loglike: -2466.4874, Max-Change: 0.001071
 EM iteration: 62, Loglike: -2466.4864, Max-Change: 0.000997 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.52 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "2PLM", cats = 2, EmpHist = TRUE, 
#>     Etol = 0.001)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 62 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2466.486
#> 
(emphist <- getirt(mod.2pl.hist, what = "weights"))
#>    theta       weight
#> 1  -6.00 4.611236e-05
#> 2  -5.75 7.025257e-05
#> 3  -5.50 9.886358e-05
#> 4  -5.25 1.328055e-04
#> 5  -5.00 1.705644e-04
#> 6  -4.75 2.103039e-04
#> 7  -4.50 2.512197e-04
#> 8  -4.25 2.957909e-04
#> 9  -4.00 3.510166e-04
#> 10 -3.75 4.314610e-04
#> 11 -3.50 5.651813e-04
#> 12 -3.25 7.997833e-04
#> 13 -3.00 1.215539e-03
#> 14 -2.75 1.963256e-03
#> 15 -2.50 3.324013e-03
#> 16 -2.25 5.788969e-03
#> 17 -2.00 1.013301e-02
#> 18 -1.75 1.738845e-02
#> 19 -1.50 2.854741e-02
#> 20 -1.25 4.387088e-02
#> 21 -1.00 6.210588e-02
#> 22 -0.75 8.040515e-02
#> 23 -0.50 9.523228e-02
#> 24 -0.25 1.036868e-01
#> 25  0.00 1.044838e-01
#> 26  0.25 9.809202e-02
#> 27  0.50 8.621777e-02
#> 28  0.75 7.140521e-02
#> 29  1.00 5.622430e-02
#> 30  1.25 4.233697e-02
#> 31  1.50 3.047311e-02
#> 32  1.75 2.092058e-02
#> 33  2.00 1.366054e-02
#> 34  2.25 8.460080e-03
#> 35  2.50 4.959317e-03
#> 36  2.75 2.754582e-03
#> 37  3.00 1.467955e-03
#> 38  3.25 7.618452e-04
#> 39  3.50 3.816747e-04
#> 40  3.75 1.805755e-04
#> 41  4.00 8.038312e-05
#> 42  4.25 3.360738e-05
#> 43  4.50 1.318299e-05
#> 44  4.75 4.855794e-06
#> 45  5.00 1.704295e-06
#> 46  5.25 6.092706e-07
#> 47  5.50 2.180233e-07
#> 48  5.75 7.322877e-08
#> 49  6.00 1.927910e-08
plot(emphist$weight ~ emphist$theta, type = "h")


# Fit the 3PL model and apply a Beta prior to the guessing parameters
(mod.3pl <- est_irt(
  data = LSAT6, D = 1, model = "3PLM", cats = 2, use.gprior = TRUE,
  gprior = list(dist = "beta", params = c(5, 16))
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -2938.6296, Max-Change: 2.625751
 EM iteration: 2, Loglike: -2493.4333, Max-Change: 0.362646
 EM iteration: 3, Loglike: -2469.7317, Max-Change: 0.125925
 EM iteration: 4, Loglike: -2467.2541, Max-Change: 0.052351
 EM iteration: 5, Loglike: -2466.9797, Max-Change: 0.02802
 EM iteration: 6, Loglike: -2466.9281, Max-Change: 0.019165
 EM iteration: 7, Loglike: -2466.9042, Max-Change: 0.015209
 EM iteration: 8, Loglike: -2466.8875, Max-Change: 0.012905
 EM iteration: 9, Loglike: -2466.8747, Max-Change: 0.01125
 EM iteration: 10, Loglike: -2466.8647, Max-Change: 0.00992
 EM iteration: 11, Loglike: -2466.8566, Max-Change: 0.00880
 EM iteration: 12, Loglike: -2466.8502, Max-Change: 0.007834
 EM iteration: 13, Loglike: -2466.8450, Max-Change: 0.007092
 EM iteration: 14, Loglike: -2466.8407, Max-Change: 0.006541
 EM iteration: 15, Loglike: -2466.8373, Max-Change: 0.006028
 EM iteration: 16, Loglike: -2466.8344, Max-Change: 0.005552
 EM iteration: 17, Loglike: -2466.8321, Max-Change: 0.005113
 EM iteration: 18, Loglike: -2466.8302, Max-Change: 0.004709
 EM iteration: 19, Loglike: -2466.8287, Max-Change: 0.004336
 EM iteration: 20, Loglike: -2466.8274, Max-Change: 0.003993
 EM iteration: 21, Loglike: -2466.8263, Max-Change: 0.003678
 EM iteration: 22, Loglike: -2466.8254, Max-Change: 0.003388
 EM iteration: 23, Loglike: -2466.8247, Max-Change: 0.003121
 EM iteration: 24, Loglike: -2466.8241, Max-Change: 0.002876
 EM iteration: 25, Loglike: -2466.8237, Max-Change: 0.00265
 EM iteration: 26, Loglike: -2466.8233, Max-Change: 0.002442
 EM iteration: 27, Loglike: -2466.8229, Max-Change: 0.002251
 EM iteration: 28, Loglike: -2466.8227, Max-Change: 0.002075
 EM iteration: 29, Loglike: -2466.8224, Max-Change: 0.001913
 EM iteration: 30, Loglike: -2466.8223, Max-Change: 0.001764
 EM iteration: 31, Loglike: -2466.8221, Max-Change: 0.001626
 EM iteration: 32, Loglike: -2466.8220, Max-Change: 0.00150
 EM iteration: 33, Loglike: -2466.8219, Max-Change: 0.001383
 EM iteration: 34, Loglike: -2466.8218, Max-Change: 0.001275
 EM iteration: 35, Loglike: -2466.8218, Max-Change: 0.001176
 EM iteration: 36, Loglike: -2466.8217, Max-Change: 0.001085
 EM iteration: 37, Loglike: -2466.8217, Max-Change: 0.001001
 EM iteration: 38, Loglike: -2466.8217, Max-Change: 0.000923
 EM iteration: 39, Loglike: -2466.8216, Max-Change: 0.000852
 EM iteration: 40, Loglike: -2466.8216, Max-Change: 0.000786
 EM iteration: 41, Loglike: -2466.8216, Max-Change: 0.000725
 EM iteration: 42, Loglike: -2466.8216, Max-Change: 0.000669
 EM iteration: 43, Loglike: -2466.8216, Max-Change: 0.000617
 EM iteration: 44, Loglike: -2466.8216, Max-Change: 0.000569
 EM iteration: 45, Loglike: -2466.8216, Max-Change: 0.000525
 EM iteration: 46, Loglike: -2466.8216, Max-Change: 0.000485
 EM iteration: 47, Loglike: -2466.8216, Max-Change: 0.000447
 EM iteration: 48, Loglike: -2466.8216, Max-Change: 0.000413
 EM iteration: 49, Loglike: -2466.8216, Max-Change: 0.000381
 EM iteration: 50, Loglike: -2466.8216, Max-Change: 0.000351
 EM iteration: 51, Loglike: -2466.8216, Max-Change: 0.000324
 EM iteration: 52, Loglike: -2466.8217, Max-Change: 0.000299
 EM iteration: 53, Loglike: -2466.8217, Max-Change: 0.000276
 EM iteration: 54, Loglike: -2466.8217, Max-Change: 0.000255
 EM iteration: 55, Loglike: -2466.8217, Max-Change: 0.000235
 EM iteration: 56, Loglike: -2466.8217, Max-Change: 0.000217
 EM iteration: 57, Loglike: -2466.8217, Max-Change: 2e-04
 EM iteration: 58, Loglike: -2466.8217, Max-Change: 0.000185
 EM iteration: 59, Loglike: -2466.8217, Max-Change: 0.000171
 EM iteration: 60, Loglike: -2466.8217, Max-Change: 0.000158
 EM iteration: 61, Loglike: -2466.8217, Max-Change: 0.000145
 EM iteration: 62, Loglike: -2466.8217, Max-Change: 0.000134
 EM iteration: 63, Loglike: -2466.8217, Max-Change: 0.000124
 EM iteration: 64, Loglike: -2466.8217, Max-Change: 0.000114
 EM iteration: 65, Loglike: -2466.8217, Max-Change: 0.000105
 EM iteration: 66, Loglike: -2466.8217, Max-Change: 9.7e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.73 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "3PLM", cats = 2, use.gprior = TRUE, 
#>     gprior = list(dist = "beta", params = c(5, 16)))
#> 
#> Item parameter estimation using MMLE-EM. 
#> 66 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2466.822
#> 

# Display a summary of the estimation results
summary(mod.3pl)
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "3PLM", cats = 2, use.gprior = TRUE, 
#>     gprior = list(dist = "beta", params = c(5, 16)))
#> 
#> Summary of the Data 
#>  Number of Items: 5
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 15
#>    - Item parameters: 15
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 66
#>  Maximum parameter change: 9.73637e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.7
#>  Standard error computation: 0.01
#>  Total computation: 0.73
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 4933.643
#>  Akaike Information Criterion (AIC): 4963.643
#>  Bayesian Information Criterion (BIC): 5037.26
#>  Item Parameters: 
#>    id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3
#> 1  V1     2   3PLM   0.85  0.28  -2.96  0.83   0.21  0.09
#> 2  V2     2   3PLM   0.86  0.26  -0.73  0.36   0.21  0.09
#> 3  V3     2   3PLM   1.24  0.51   0.25  0.24   0.20  0.09
#> 4  V4     2   3PLM   0.77  0.23  -1.24  0.44   0.21  0.09
#> 5  V5     2   3PLM   0.71  0.23  -2.51  0.74   0.21  0.09
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Fit the 3PL model and fix the guessing parameters at 0.2
(mod.3pl.f <- est_irt(data = LSAT6, D = 1, model = "3PLM", cats = 2,
                      fix.g = TRUE, g.val = 0.2))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -2938.6296, Max-Change: 2.65963
 EM iteration: 2, Loglike: -2493.2423, Max-Change: 0.336936
 EM iteration: 3, Loglike: -2469.7540, Max-Change: 0.123232
 EM iteration: 4, Loglike: -2467.2643, Max-Change: 0.052978
 EM iteration: 5, Loglike: -2466.9883, Max-Change: 0.028639
 EM iteration: 6, Loglike: -2466.9373, Max-Change: 0.019494
 EM iteration: 7, Loglike: -2466.9131, Max-Change: 0.015392
 EM iteration: 8, Loglike: -2466.8954, Max-Change: 0.013058
 EM iteration: 9, Loglike: -2466.8813, Max-Change: 0.011428
 EM iteration: 10, Loglike: -2466.8700, Max-Change: 0.01014
 EM iteration: 11, Loglike: -2466.8607, Max-Change: 0.009058
 EM iteration: 12, Loglike: -2466.8532, Max-Change: 0.008122
 EM iteration: 13, Loglike: -2466.8471, Max-Change: 0.00732
 EM iteration: 14, Loglike: -2466.8420, Max-Change: 0.00670
 EM iteration: 15, Loglike: -2466.8379, Max-Change: 0.006135
 EM iteration: 16, Loglike: -2466.8345, Max-Change: 0.005619
 EM iteration: 17, Loglike: -2466.8316, Max-Change: 0.005148
 EM iteration: 18, Loglike: -2466.8293, Max-Change: 0.004718
 EM iteration: 19, Loglike: -2466.8273, Max-Change: 0.004326
 EM iteration: 20, Loglike: -2466.8257, Max-Change: 0.003967
 EM iteration: 21, Loglike: -2466.8244, Max-Change: 0.00364
 EM iteration: 22, Loglike: -2466.8233, Max-Change: 0.00334
 EM iteration: 23, Loglike: -2466.8223, Max-Change: 0.003066
 EM iteration: 24, Loglike: -2466.8215, Max-Change: 0.002815
 EM iteration: 25, Loglike: -2466.8209, Max-Change: 0.002586
 EM iteration: 26, Loglike: -2466.8203, Max-Change: 0.002376
 EM iteration: 27, Loglike: -2466.8199, Max-Change: 0.002183
 EM iteration: 28, Loglike: -2466.8195, Max-Change: 0.002006
 EM iteration: 29, Loglike: -2466.8191, Max-Change: 0.001844
 EM iteration: 30, Loglike: -2466.8189, Max-Change: 0.001695
 EM iteration: 31, Loglike: -2466.8186, Max-Change: 0.001559
 EM iteration: 32, Loglike: -2466.8184, Max-Change: 0.001433
 EM iteration: 33, Loglike: -2466.8183, Max-Change: 0.001318
 EM iteration: 34, Loglike: -2466.8181, Max-Change: 0.001213
 EM iteration: 35, Loglike: -2466.8180, Max-Change: 0.001116
 EM iteration: 36, Loglike: -2466.8179, Max-Change: 0.001026
 EM iteration: 37, Loglike: -2466.8178, Max-Change: 0.000944
 EM iteration: 38, Loglike: -2466.8178, Max-Change: 0.000869
 EM iteration: 39, Loglike: -2466.8177, Max-Change: 8e-04
 EM iteration: 40, Loglike: -2466.8176, Max-Change: 0.000736
 EM iteration: 41, Loglike: -2466.8176, Max-Change: 0.000677
 EM iteration: 42, Loglike: -2466.8176, Max-Change: 0.000624
 EM iteration: 43, Loglike: -2466.8175, Max-Change: 0.000574
 EM iteration: 44, Loglike: -2466.8175, Max-Change: 0.000528
 EM iteration: 45, Loglike: -2466.8175, Max-Change: 0.000486
 EM iteration: 46, Loglike: -2466.8175, Max-Change: 0.000448
 EM iteration: 47, Loglike: -2466.8175, Max-Change: 0.000412
 EM iteration: 48, Loglike: -2466.8174, Max-Change: 0.00038
 EM iteration: 49, Loglike: -2466.8174, Max-Change: 0.000349
 EM iteration: 50, Loglike: -2466.8174, Max-Change: 0.000322
 EM iteration: 51, Loglike: -2466.8174, Max-Change: 0.000296
 EM iteration: 52, Loglike: -2466.8174, Max-Change: 0.000273
 EM iteration: 53, Loglike: -2466.8174, Max-Change: 0.000251
 EM iteration: 54, Loglike: -2466.8174, Max-Change: 0.000231
 EM iteration: 55, Loglike: -2466.8174, Max-Change: 0.000213
 EM iteration: 56, Loglike: -2466.8174, Max-Change: 0.000196
 EM iteration: 57, Loglike: -2466.8174, Max-Change: 0.000181
 EM iteration: 58, Loglike: -2466.8174, Max-Change: 0.000166
 EM iteration: 59, Loglike: -2466.8174, Max-Change: 0.000153
 EM iteration: 60, Loglike: -2466.8174, Max-Change: 0.000141
 EM iteration: 61, Loglike: -2466.8174, Max-Change: 0.00013
 EM iteration: 62, Loglike: -2466.8174, Max-Change: 0.00012
 EM iteration: 63, Loglike: -2466.8174, Max-Change: 0.00011
 EM iteration: 64, Loglike: -2466.8174, Max-Change: 0.000101
 EM iteration: 65, Loglike: -2466.8174, Max-Change: 9.3e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.52 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "3PLM", cats = 2, fix.g = TRUE, 
#>     g.val = 0.2)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 65 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2466.817
#> 

# Display a summary of the estimation results
summary(mod.3pl.f)
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = "3PLM", cats = 2, fix.g = TRUE, 
#>     g.val = 0.2)
#> 
#> Summary of the Data 
#>  Number of Items: 5
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 10
#>    - Item parameters: 10
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 65
#>  Maximum parameter change: 9.346778e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.49
#>  Standard error computation: 0.01
#>  Total computation: 0.52
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 4933.635
#>  Akaike Information Criterion (AIC): 4953.635
#>  Bayesian Information Criterion (BIC): 5002.712
#>  Item Parameters: 
#>    id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3
#> 1  V1     2   3PLM   0.85  0.28  -2.99  0.80    0.2    NA
#> 2  V2     2   3PLM   0.85  0.24  -0.76  0.20    0.2    NA
#> 3  V3     2   3PLM   1.25  0.41   0.25  0.09    0.2    NA
#> 4  V4     2   3PLM   0.76  0.22  -1.27  0.32    0.2    NA
#> 5  V5     2   3PLM   0.71  0.23  -2.54  0.70    0.2    NA
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Fit different dichotomous models to each item in the LSAT6 data:
# Fit the constrained 1PL model to items 1-3, the 2PL model to item 4,
# and the 3PL model with a Beta prior on guessing to item 5
(mod.drm.mix <- est_irt(
  data = LSAT6, D = 1, model = c("1PLM", "1PLM", "1PLM", "2PLM", "3PLM"),
  cats = 2, fix.a.1pl = FALSE, use.gprior = TRUE,
  gprior = list(dist = "beta", params = c(5, 16))
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -3100.6555, Max-Change: 2.331776
 EM iteration: 2, Loglike: -2544.0863, Max-Change: 0.520835
 EM iteration: 3, Loglike: -2479.3669, Max-Change: 0.26437
 EM iteration: 4, Loglike: -2468.8666, Max-Change: 0.13618
 EM iteration: 5, Loglike: -2467.2368, Max-Change: 0.073111
 EM iteration: 6, Loglike: -2466.9318, Max-Change: 0.041681
 EM iteration: 7, Loglike: -2466.8475, Max-Change: 0.026892
 EM iteration: 8, Loglike: -2466.8155, Max-Change: 0.018499
 EM iteration: 9, Loglike: -2466.8015, Max-Change: 0.013086
 EM iteration: 10, Loglike: -2466.7950, Max-Change: 0.00941
 EM iteration: 11, Loglike: -2466.7919, Max-Change: 0.00684
 EM iteration: 12, Loglike: -2466.7904, Max-Change: 0.005015
 EM iteration: 13, Loglike: -2466.7897, Max-Change: 0.003705
 EM iteration: 14, Loglike: -2466.7893, Max-Change: 0.002759
 EM iteration: 15, Loglike: -2466.7891, Max-Change: 0.00207
 EM iteration: 16, Loglike: -2466.7890, Max-Change: 0.001566
 EM iteration: 17, Loglike: -2466.7889, Max-Change: 0.001194
 EM iteration: 18, Loglike: -2466.7889, Max-Change: 0.000917
 EM iteration: 19, Loglike: -2466.7888, Max-Change: 0.00071
 EM iteration: 20, Loglike: -2466.7888, Max-Change: 0.000554
 EM iteration: 21, Loglike: -2466.7888, Max-Change: 0.000436
 EM iteration: 22, Loglike: -2466.7888, Max-Change: 0.000345
 EM iteration: 23, Loglike: -2466.7888, Max-Change: 0.000275
 EM iteration: 24, Loglike: -2466.7888, Max-Change: 0.000221
 EM iteration: 25, Loglike: -2466.7888, Max-Change: 0.000178
 EM iteration: 26, Loglike: -2466.7888, Max-Change: 0.000145
 EM iteration: 27, Loglike: -2466.7888, Max-Change: 0.000118
 EM iteration: 28, Loglike: -2466.7888, Max-Change: 9.7e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.22 seconds. 
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = c("1PLM", "1PLM", "1PLM", 
#>     "2PLM", "3PLM"), cats = 2, fix.a.1pl = FALSE, use.gprior = TRUE, 
#>     gprior = list(dist = "beta", params = c(5, 16)))
#> 
#> Item parameter estimation using MMLE-EM. 
#> 28 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -2466.789
#> 

# Display a summary of the estimation results
summary(mod.drm.mix)
#> 
#> Call:
#> est_irt(data = LSAT6, D = 1, model = c("1PLM", "1PLM", "1PLM", 
#>     "2PLM", "3PLM"), cats = 2, fix.a.1pl = FALSE, use.gprior = TRUE, 
#>     gprior = list(dist = "beta", params = c(5, 16)))
#> 
#> Summary of the Data 
#>  Number of Items: 5
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 9
#>    - Item parameters: 9
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 28
#>  Maximum parameter change: 9.690417e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.2
#>  Standard error computation: 0
#>  Total computation: 0.22
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 4933.578
#>  Akaike Information Criterion (AIC): 4951.578
#>  Bayesian Information Criterion (BIC): 4995.747
#>  Item Parameters: 
#>    id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3
#> 1  V1     2   1PLM   0.80  0.10  -3.43  0.39     NA    NA
#> 2  V2     2   1PLM   0.80    NA  -1.26  0.16     NA    NA
#> 3  V3     2   1PLM   0.80    NA  -0.30  0.10     NA    NA
#> 4  V4     2   2PLM   0.68  0.18  -1.87  0.44     NA    NA
#> 5  V5     2   3PLM   0.72  0.23  -2.48  0.72   0.21  0.09
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

## -------------------------------------------------------------------
## 2. Item parameter estimation for mixed-format data (simulated data)
## -------------------------------------------------------------------
## Import the "-prm.txt" output file from flexMIRT
flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt",
                        package = "irtQ")

# Extract item metadata
x <- bring.flexmirt(file = flex_sam, "par")$Group1$full_df

# Modify the item metadata so that the 39th and 40th items use the GPCM
x[39:40, 3] <- "GPCM"

# Generate 1,000 examinees' latent abilities from N(0, 1)
set.seed(37)
score1 <- rnorm(1000, mean = 0, sd = 1)

# Simulate item response data
sim.dat1 <- simdat(x = x, theta = score1, D = 1)

# Fit the 3PL model to all dichotomous items, the GPCM to items 39 and 40,
# and the GRM to items 53, 54, and 55.
# Use a Beta prior for guessing parameters, a log-normal prior for slope
# parameters, and a normal prior for difficulty (threshold) parameters.
# Also, specify the argument `x` to provide IRT model and score category
# information.
item.meta <- shape_df(item.id = x$id, cats = x$cats, model = x$model,
  default.par = TRUE)
(mod.mix1 <- est_irt(
  x = item.meta, data = sim.dat1, D = 1, use.aprior = TRUE,
  use.bprior = TRUE,
  use.gprior = TRUE,
  aprior = list(dist = "lnorm", params = c(0.0, 0.5)),
  bprior = list(dist = "norm", params = c(0.0, 2.0)),
  gprior = list(dist = "beta", params = c(5, 16))
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -39032.8987, Max-Change: 2.339566
 EM iteration: 2, Loglike: -33980.7119, Max-Change: 0.316376
 EM iteration: 3, Loglike: -33966.2963, Max-Change: 0.085202
 EM iteration: 4, Loglike: -33965.3783, Max-Change: 0.021805
 EM iteration: 5, Loglike: -33965.3380, Max-Change: 0.005862
 EM iteration: 6, Loglike: -33965.3884, Max-Change: 0.001885
 EM iteration: 7, Loglike: -33965.4397, Max-Change: 0.001309
 EM iteration: 8, Loglike: -33965.4817, Max-Change: 0.00093
 EM iteration: 9, Loglike: -33965.5149, Max-Change: 0.000717
 EM iteration: 10, Loglike: -33965.5412, Max-Change: 0.000561
 EM iteration: 11, Loglike: -33965.5620, Max-Change: 0.000448
 EM iteration: 12, Loglike: -33965.5784, Max-Change: 0.000357
 EM iteration: 13, Loglike: -33965.5915, Max-Change: 0.000285
 EM iteration: 14, Loglike: -33965.6019, Max-Change: 0.000226
 EM iteration: 15, Loglike: -33965.6102, Max-Change: 0.000183
 EM iteration: 16, Loglike: -33965.6168, Max-Change: 0.000149
 EM iteration: 17, Loglike: -33965.6220, Max-Change: 0.000121
 EM iteration: 18, Loglike: -33965.6262, Max-Change: 9.9e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 2.85 seconds. 
#> 
#> Call:
#> est_irt(x = item.meta, data = sim.dat1, D = 1, use.aprior = TRUE, 
#>     use.bprior = TRUE, use.gprior = TRUE, aprior = list(dist = "lnorm", 
#>         params = c(0, 0.5)), bprior = list(dist = "norm", params = c(0, 
#>         2)), gprior = list(dist = "beta", params = c(5, 16)))
#> 
#> Item parameter estimation using MMLE-EM. 
#> 18 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -33965.63
#> 

# Display a summary of the estimation results
summary(mod.mix1)
#> 
#> Call:
#> est_irt(x = item.meta, data = sim.dat1, D = 1, use.aprior = TRUE, 
#>     use.bprior = TRUE, use.gprior = TRUE, aprior = list(dist = "lnorm", 
#>         params = c(0, 0.5)), bprior = list(dist = "norm", params = c(0, 
#>         2)), gprior = list(dist = "beta", params = c(5, 16)))
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 175
#>    - Item parameters: 175
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 18
#>  Maximum parameter change: 9.925478e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 2.75
#>  Standard error computation: 0.05
#>  Total computation: 2.85
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 67931.26
#>  Akaike Information Criterion (AIC): 68281.26
#>  Bayesian Information Criterion (BIC): 69140.12
#>  Item Parameters: 
#>        id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1    CMC1     2   3PLM   0.66  0.15   1.17  0.34   0.19  0.07     NA    NA
#> 2    CMC2     2   3PLM   1.64  0.19  -1.04  0.15   0.15  0.06     NA    NA
#> 3    CMC3     2   3PLM   0.95  0.16   0.49  0.19   0.15  0.06     NA    NA
#> 4    CMC4     2   3PLM   1.21  0.18  -0.31  0.21   0.22  0.07     NA    NA
#> 5    CMC5     2   3PLM   0.88  0.17   0.25  0.31   0.25  0.08     NA    NA
#> 6    CMC6     2   3PLM   1.50  0.22   0.73  0.10   0.10  0.03     NA    NA
#> 7    CMC7     2   3PLM   0.77  0.14   1.13  0.23   0.12  0.05     NA    NA
#> 8    CMC8     2   3PLM   0.84  0.17   1.10  0.22   0.14  0.05     NA    NA
#> 9    CMC9     2   3PLM   0.84  0.16   0.57  0.26   0.18  0.07     NA    NA
#> 10  CMC10     2   3PLM   1.76  0.27   0.32  0.12   0.22  0.05     NA    NA
#> 11  CMC11     2   3PLM   0.96  0.12  -0.55  0.21   0.15  0.07     NA    NA
#> 12  CMC12     2   3PLM   0.97  0.19   1.32  0.19   0.13  0.04     NA    NA
#> 13  CMC13     2   3PLM   1.21  0.26   1.37  0.16   0.16  0.04     NA    NA
#> 14  CMC14     2   3PLM   1.58  0.23   0.04  0.14   0.23  0.06     NA    NA
#> 15  CMC15     2   3PLM   1.16  0.16  -0.18  0.18   0.15  0.06     NA    NA
#> 16  CMC16     2   3PLM   2.08  0.22   0.08  0.07   0.09  0.03     NA    NA
#> 17  CMC17     2   3PLM   1.29  0.18  -0.18  0.17   0.16  0.06     NA    NA
#> 18  CMC18     2   3PLM   1.34  0.32   1.36  0.16   0.25  0.04     NA    NA
#> 19  CMC19     2   3PLM   2.06  0.30  -0.99  0.15   0.20  0.07     NA    NA
#> 20  CMC20     2   3PLM   1.49  0.21  -1.58  0.22   0.21  0.09     NA    NA
#> 21  CMC21     2   3PLM   2.07  0.29  -0.78  0.15   0.26  0.07     NA    NA
#> 22  CMC22     2   3PLM   0.97  0.12  -0.67  0.22   0.15  0.07     NA    NA
#> 23  CMC23     2   3PLM   0.87  0.14  -0.10  0.26   0.18  0.07     NA    NA
#> 24  CMC24     2   3PLM   1.12  0.32   1.70  0.22   0.24  0.04     NA    NA
#> 25  CMC25     2   3PLM   0.72  0.11  -1.51  0.40   0.23  0.10     NA    NA
#> 26  CMC26     2   3PLM   1.12  0.14  -1.73  0.24   0.18  0.08     NA    NA
#> 27  CMC27     2   3PLM   1.25  0.15   0.05  0.14   0.12  0.05     NA    NA
#> 28  CMC28     2   3PLM   1.97  0.24  -0.19  0.10   0.15  0.05     NA    NA
#> 29  CMC29     2   3PLM   1.11  0.16  -1.46  0.29   0.24  0.10     NA    NA
#> 30  CMC30     2   3PLM   1.14  0.22   0.46  0.20   0.24  0.06     NA    NA
#> 31  CMC31     2   3PLM   0.84  0.14   0.74  0.20   0.12  0.05     NA    NA
#> 32  CMC32     2   3PLM   1.60  0.25  -0.78  0.21   0.29  0.08     NA    NA
#> 33  CMC33     2   3PLM   1.05  0.13  -1.44  0.25   0.18  0.08     NA    NA
#> 34  CMC34     2   3PLM   1.30  0.21   0.31  0.16   0.20  0.06     NA    NA
#> 35  CMC35     2   3PLM   1.40  0.24  -0.06  0.19   0.25  0.07     NA    NA
#> 36  CMC36     2   3PLM   1.09  0.23   1.20  0.17   0.16  0.05     NA    NA
#> 37  CMC37     2   3PLM   1.93  0.24  -0.30  0.11   0.15  0.05     NA    NA
#> 38  CMC38     2   3PLM   0.63  0.11  -0.48  0.42   0.22  0.09     NA    NA
#> 39   CFR1     5   GPCM   1.84  0.15  -1.79  0.13  -1.22  0.10  -0.63  0.08
#> 40   CFR2     5   GPCM   1.25  0.09  -0.60  0.09   0.00  0.09   0.46  0.10
#> 41   AMC1     2   3PLM   1.46  0.29   0.72  0.14   0.24  0.05     NA    NA
#> 42   AMC2     2   3PLM   1.65  0.23  -1.50  0.21   0.24  0.09     NA    NA
#> 43   AMC3     2   3PLM   1.37  0.20   0.64  0.12   0.14  0.04     NA    NA
#> 44   AMC4     2   3PLM   1.11  0.19   0.02  0.23   0.23  0.07     NA    NA
#> 45   AMC5     2   3PLM   1.13  0.41   2.52  0.40   0.21  0.03     NA    NA
#> 46   AMC6     2   3PLM   2.25  0.61   1.67  0.13   0.19  0.02     NA    NA
#> 47   AMC7     2   3PLM   1.27  0.17  -0.02  0.15   0.14  0.05     NA    NA
#> 48   AMC8     2   3PLM   1.94  0.30   0.39  0.11   0.25  0.04     NA    NA
#> 49   AMC9     2   3PLM   1.31  0.21   0.48  0.15   0.17  0.05     NA    NA
#> 50  AMC10     2   3PLM   1.34  0.23   1.35  0.13   0.08  0.03     NA    NA
#> 51  AMC11     2   3PLM   1.55  0.17  -1.20  0.16   0.15  0.07     NA    NA
#> 52  AMC12     2   3PLM   0.91  0.13  -0.73  0.26   0.18  0.08     NA    NA
#> 53   AFR1     5    GRM   1.15  0.09  -0.20  0.07   0.34  0.07   0.96  0.09
#> 54   AFR2     5    GRM   1.18  0.09  -2.09  0.16  -1.39  0.11  -0.73  0.08
#> 55   AFR3     5    GRM   0.91  0.08  -0.62  0.10   0.04  0.08   0.75  0.10
#>     par.5  se.5
#> 1      NA    NA
#> 2      NA    NA
#> 3      NA    NA
#> 4      NA    NA
#> 5      NA    NA
#> 6      NA    NA
#> 7      NA    NA
#> 8      NA    NA
#> 9      NA    NA
#> 10     NA    NA
#> 11     NA    NA
#> 12     NA    NA
#> 13     NA    NA
#> 14     NA    NA
#> 15     NA    NA
#> 16     NA    NA
#> 17     NA    NA
#> 18     NA    NA
#> 19     NA    NA
#> 20     NA    NA
#> 21     NA    NA
#> 22     NA    NA
#> 23     NA    NA
#> 24     NA    NA
#> 25     NA    NA
#> 26     NA    NA
#> 27     NA    NA
#> 28     NA    NA
#> 29     NA    NA
#> 30     NA    NA
#> 31     NA    NA
#> 32     NA    NA
#> 33     NA    NA
#> 34     NA    NA
#> 35     NA    NA
#> 36     NA    NA
#> 37     NA    NA
#> 38     NA    NA
#> 39  -0.31  0.07
#> 40   1.19  0.11
#> 41     NA    NA
#> 42     NA    NA
#> 43     NA    NA
#> 44     NA    NA
#> 45     NA    NA
#> 46     NA    NA
#> 47     NA    NA
#> 48     NA    NA
#> 49     NA    NA
#> 50     NA    NA
#> 51     NA    NA
#> 52     NA    NA
#> 53   1.56  0.12
#> 54  -0.09  0.07
#> 55   1.30  0.13
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Estimate examinees' latent scores using MLE and the estimated item
# parameters
(score.mle <- est_score(x = mod.mix1, method = "ML", range = c(-4, 4)))
#>          est.theta   se.theta
#> 1     0.3926614617  0.2628543
#> 2     0.4443005364  0.2647684
#> 3     0.9146240121  0.2891789
#> 4    -0.2689015108  0.2535997
#> 5    -1.2823351205  0.3120158
#> 6    -0.4824293381  0.2571234
#> 7    -0.1337304061  0.2531543
#> 8     1.3066135528  0.3163575
#> 9     0.5642779904  0.2698642
#> 10    0.2118210724  0.2575001
#> 11   -0.5222620236  0.2582119
#> 12   -0.0138651672  0.2537883
#> 13    1.4181322324  0.3250707
#> 14    1.0817463397  0.3001784
#> 15    0.3297683667  0.2607503
#> 16   -1.8112605445  0.3965064
#> 17   -4.0000000000 99.9999000
#> 18   -1.5109466231  0.3426391
#> 19    0.4400644006  0.2646030
#> 20   -1.1349969012  0.2962157
#> 21   -0.0927053493  0.2532653
#> 22    1.1911097549  0.3078493
#> 23    1.1917876135  0.3078916
#> 24   -2.4034989063  0.5635697
#> 25    1.0712352159  0.2994607
#> 26   -0.2476724872  0.2534446
#> 27    0.0636515397  0.2546944
#> 28   -0.0960126240  0.2532522
#> 29   -2.0077792782  0.4419792
#> 30    0.3524368912  0.2614823
#> 31   -0.0322240041  0.2536307
#> 32   -0.7042280911  0.2651656
#> 33    0.9146236525  0.2891802
#> 34   -0.5111733237  0.2578938
#> 35    0.2292634214  0.2579234
#> 36    0.1330862299  0.2558322
#> 37   -0.5670456935  0.2596169
#> 38   -0.0224202094  0.2537125
#> 39   -0.0417403135  0.2535574
#> 40    0.8563442382  0.2855741
#> 41   -0.0668801468  0.2533923
#> 42   -0.8509147525  0.2733404
#> 43   -2.3465575664  0.5433223
#> 44    1.7896381422  0.3601105
#> 45   -2.0226323331  0.4457870
#> 46    0.6546339220  0.2742578
#> 47   -0.2553007350  0.2534965
#> 48    1.3184090045  0.3172516
#> 49   -0.7723936316  0.2686704
#> 50   -0.7827766638  0.2692450
#> 51    0.4057191678  0.2633235
#> 52    0.1461847300  0.2560819
#> 53    1.7054768871  0.3510912
#> 54    1.8679713032  0.3692045
#> 55    0.4217902136  0.2639125
#> 56    1.0074112031  0.2951693
#> 57   -1.5835244745  0.3541084
#> 58   -0.7816307263  0.2691810
#> 59    1.6701288778  0.3475345
#> 60   -0.8066618169  0.2706187
#> 61    0.2026314999  0.2572843
#> 62    0.0868330044  0.2550397
#> 63   -1.5608641850  0.3504282
#> 64    1.6048624531  0.3412416
#> 65   -1.7602364053  0.3860946
#> 66    1.3739267494  0.3215527
#> 67    0.3175578676  0.2603722
#> 68    0.7436710796  0.2790121
#> 69    0.5203588823  0.2678954
#> 70   -1.1657900675  0.2992900
#> 71   -0.5225600883  0.2582211
#> 72   -0.9345620943  0.2790776
#> 73    1.3388662605  0.3188267
#> 74   -2.9962795006  0.8371436
#> 75    0.0677370535  0.2547518
#> 76    0.9840140906  0.2936396
#> 77    0.1399441952  0.2559620
#> 78   -1.0248941889  0.2862061
#> 79    0.9095591240  0.2888621
#> 80    1.3128351470  0.3168284
#> 81   -0.8154635932  0.2711430
#> 82    0.0179493470  0.2541126
#> 83   -1.1959281470  0.3024115
#> 84   -1.1447615703  0.2971774
#> 85    0.8784704214  0.2869292
#> 86   -1.8149213216  0.3972440
#> 87   -0.7320859546  0.2665366
#> 88    0.5751775752  0.2703651
#> 89    0.1247754914  0.2556793
#> 90   -4.0000000000 99.9999000
#> 91   -1.7696465422  0.3879873
#> 92   -0.5967873250  0.2606546
#> 93   -1.2845273328  0.3122720
#> 94    1.0066166962  0.2951203
#> 95   -0.2337532917  0.2533600
#> 96    0.0479131207  0.2544772
#> 97   -0.4613900569  0.2566060
#> 98   -0.0792958319  0.2533256
#> 99   -0.3857790724  0.2550653
#> 100   2.9131251092  0.5702092
#> 101   1.5386520697  0.3352406
#> 102   1.5283327125  0.3343362
#> 103  -0.5472149063  0.2589687
#> 104   1.3109931639  0.3166866
#> 105   2.9671257370  0.5846962
#> 106  -0.7142350884  0.2656491
#> 107   0.8707177145  0.2864539
#> 108   1.0608471300  0.2987539
#> 109  -0.2232294669  0.2533058
#> 110   2.1521816370  0.4089149
#> 111  -0.1029210277  0.2532272
#> 112   0.9095116348  0.2888584
#> 113  -0.3927258378  0.2551873
#> 114  -0.0569056441  0.2534523
#> 115   0.6451144513  0.2737697
#> 116  -0.6277646052  0.2618376
#> 117   1.6302594911  0.3436410
#> 118  -1.1492419400  0.2976236
#> 119  -0.1590032235  0.2531416
#> 120  -0.8693307930  0.2745383
#> 121   0.8483249454  0.2850911
#> 122   1.2124227650  0.3093800
#> 123  -0.9701682633  0.2817761
#> 124   1.4593482360  0.3284565
#> 125  -0.0787164199  0.2533285
#> 126   0.6908251826  0.2761403
#> 127   2.1281674082  0.4051343
#> 128   0.7907171710  0.2816865
#> 129   0.1670112219  0.2565021
#> 130  -0.3218531916  0.2541323
#> 131  -0.4018269110  0.2553502
#> 132  -0.0849546258  0.2532985
#> 133   1.3427777146  0.3191246
#> 134  -0.0792944873  0.2533257
#> 135   0.1535439260  0.2562283
#> 136  -0.1018630921  0.2532308
#> 137  -0.8177675039  0.2712817
#> 138  -1.0101869312  0.2849802
#> 139  -1.8180664633  0.3979139
#> 140   1.0975102090  0.3012570
#> 141   0.7138103426  0.2773696
#> 142  -0.7279171042  0.2663266
#> 143   0.3186841611  0.2604074
#> 144  -0.6571051636  0.2630427
#> 145   1.3121598607  0.3167782
#> 146   1.4708224663  0.3294118
#> 147  -1.4236166027  0.3299965
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#> 155   2.1533968710  0.4091240
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#> 159  -0.5583351468  0.2593286
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#> 176  -0.9022646621  0.2767687
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#> 192  -1.1794903315  0.3006947
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#> 201  -1.2394835241  0.3071259
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#> 215  -1.1228352035  0.2950368
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#> 231  -0.3915808879  0.2551663
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#> 235  -1.3183286922  0.3163241
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#> 239  -1.1484091744  0.2975407
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#> 241   0.2008364385  0.2572430
#> 242  -0.7626790447  0.2681337
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#> 245  -1.0183047701  0.2856491
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#> 249   3.0430803461  0.6057768
#> 250  -0.4756974826  0.2569527
#> 251  -4.0000000000 99.9999000
#> 252   0.8763000464  0.2867946
#> 253  -0.3692877187  0.2547934
#> 254  -0.7978234807  0.2701074
#> 255   1.5635213054  0.3374668
#> 256   0.7341221726  0.2784832
#> 257  -1.3914176219  0.3256508
#> 258  -1.0487042823  0.2882514
#> 259  -0.5963159189  0.2606412
#> 260  -0.6338544679  0.2620767
#> 261  -2.5397447844  0.6159126
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#> 265  -0.3079324520  0.2539715
#> 266  -1.4064552513  0.3276568
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#> 268   0.3353185138  0.2609266
#> 269  -1.5734520961  0.3524604
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#> 271   0.4772469345  0.2660797
#> 272  -1.9438589172  0.4261782
#> 273   1.0336803698  0.2969150
#> 274   0.9095407595  0.2888609
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#> 277  -1.4375858644  0.3319396
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#> 279  -2.0764891208  0.4600687
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#> 282  -1.0734797271  0.2904501
#> 283  -0.2625591444  0.2535501
#> 284  -1.0592504472  0.2891744
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#> 288   1.3921296468  0.3229926
#> 289  -0.4936751553  0.2574164
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#> 293  -1.0379188454  0.2873208
#> 294  -0.2714127388  0.2536201
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#> 296   0.5220415468  0.2679679
#> 297  -1.2713047850  0.3107336
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#> 300   0.5355299879  0.2685604
#> 301  -0.2482649189  0.2534482
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#> 304  -0.3708564273  0.2548183
#> 305  -0.4630012830  0.2566435
#> 306  -0.6508846548  0.2627806
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#> 309   0.4018466833  0.2631816
#> 310  -1.9195200978  0.4204544
#> 311   0.2597898074  0.2587148
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#> 313   0.6817904902  0.2756626
#> 314   1.5950837610  0.3403426
#> 315   1.8785835350  0.3705086
#> 316  -0.5423380688  0.2588171
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#> 318   0.7893077722  0.2816035
#> 319   1.7267364126  0.3533013
#> 320   0.6715419493  0.2751255
#> 321   2.0339456228  0.3910707
#> 322   1.6508221232  0.3456368
#> 323  -1.2411905749  0.3073204
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#> 325   0.5019193240  0.2671027
#> 326   1.4580177678  0.3283418
#> 327   1.1640300348  0.3059092
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#> 329  -1.5087267768  0.3422928
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#> 331   0.8939025910  0.2878853
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#> 333   0.2736656212  0.2590938
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#> 335  -0.4709011609  0.2568345
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#> 341  -0.5580328322  0.2593192
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#> 345  -1.5198055316  0.3439835
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#> 351  -0.5306916905  0.2584614
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#> 353  -1.4751054768  0.3373148
#> 354  -1.6932895505  0.3732645
#> 355  -1.1555234592  0.2982588
#> 356   1.3073570251  0.3164109
#> 357   0.7508703795  0.2794175
#> 358  -0.4358041166  0.2560288
#> 359   1.6647495870  0.3469972
#> 360  -1.9819059266  0.4354781
#> 361  -0.6105104929  0.2611670
#> 362  -0.7040298528  0.2651591
#> 363   0.4324653932  0.2643119
#> 364   0.7281159869  0.2781521
#> 365  -0.3295419282  0.2542274
#> 366  -0.5855975184  0.2602567
#> 367   3.5885332662  0.7840008
#> 368   0.9517293492  0.2915438
#> 369   0.4678349171  0.2656985
#> 370   1.2533042805  0.3123666
#> 371   1.4823825999  0.3303866
#> 372   0.3110013328  0.2601737
#> 373  -0.9894190619  0.2832935
#> 374  -0.1742739850  0.2531550
#> 375  -0.4750432434  0.2569383
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#> 377   1.3516927066  0.3198173
#> 378  -0.3221692645  0.2541354
#> 379   0.4299641890  0.2642187
#> 380   2.3869882363  0.4501739
#> 381   0.1656211554  0.2564729
#> 382  -0.4180690184  0.2556627
#> 383   0.7292944010  0.2782180
#> 384  -1.3857688351  0.3248934
#> 385   1.2482089753  0.3119944
#> 386  -0.9175732385  0.2778528
#> 387   0.0359724033  0.2543256
#> 388   1.0185908600  0.2959129
#> 389  -1.0964907960  0.2925533
#> 390   0.4764874595  0.2660490
#> 391  -1.6940285675  0.3734036
#> 392  -0.9033898230  0.2768470
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#> 396  -0.3757120436  0.2548969
#> 397  -0.3393278660  0.2543544
#> 398   0.8322567099  0.2841269
#> 399  -0.8115329392  0.2709072
#> 400  -0.5325094577  0.2585164
#> 401  -0.1100064247  0.2532049
#> 402  -4.0000000000 99.9999000
#> 403  -0.0331461744  0.2536234
#> 404  -0.7275999458  0.2663112
#> 405  -0.8174873736  0.2712625
#> 406  -0.6316659446  0.2619886
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#> 409   1.2196144867  0.3099016
#> 410  -2.0659680166  0.4572212
#> 411  -0.7561336349  0.2677857
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#> 413   1.7234063444  0.3529453
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#> 415  -0.1002406554  0.2532366
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#> 419  -0.3061070683  0.2539516
#> 420  -1.6969914074  0.3739483
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#> 422   2.4832454555  0.4693045
#> 423   0.2414440543  0.2582302
#> 424   0.8042775657  0.2824732
#> 425  -1.0735431977  0.2904396
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#> 429  -2.0738981108  0.4593724
#> 430  -0.6138555163  0.2612931
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#> 446  -0.6798981925  0.2640418
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#> 450  -0.0782292929  0.2533311
#> 451  -0.1038297286  0.2532239
#> 452   2.0719603049  0.3965976
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#> 454  -0.9379736653  0.2793353
#> 455   0.3048760353  0.2599894
#> 456  -0.9705027190  0.2818091
#> 457   0.5801719847  0.2706022
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#> 459  -0.1797217053  0.2531637
#> 460  -2.6375214631  0.6571548
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#> 462   0.8422280275  0.2847257
#> 463  -1.8300343187  0.4004537
#> 464   0.7367466192  0.2786257
#> 465   1.7855243819  0.3596495
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#> 467  -0.2562912291  0.2535035
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#> 469   0.2387552622  0.2581619
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#> 471  -1.8226818005  0.3988893
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#> 475   0.2123534698  0.2575122
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#> 477  -0.6074870008  0.2610521
#> 478  -1.0133069255  0.2852348
#> 479  -1.5000456072  0.3409839
#> 480  -0.3316943643  0.2542550
#> 481   0.5438689934  0.2689317
#> 482  -0.7212102816  0.2659929
#> 483  -0.8517921158  0.2733926
#> 484  -0.9007684188  0.2766654
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#> 486   2.3607297042  0.4451724
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#> 488   0.6958422124  0.2764084
#> 489  -0.0624767107  0.2534179
#> 490  -0.0542895267  0.2534695
#> 491  -0.0057816627  0.2538647
#> 492  -4.0000000000 99.9999000
#> 493   1.3174829539  0.3171835
#> 494  -0.6805674364  0.2640743
#> 495   0.1176043320  0.2555510
#> 496  -0.4878263336  0.2572618
#> 497  -0.8584450709  0.2738228
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#> 500   0.5115506790  0.2675146
#> 501  -0.2035187347  0.2532252
#> 502  -1.4802008432  0.3380473
#> 503   0.6031902315  0.2716992
#> 504   1.4693329813  0.3292863
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#> 506  -1.1173607916  0.2945151
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#> 508  -0.6109449152  0.2611833
#> 509  -0.3549388488  0.2545751
#> 510   0.6812110699  0.2756316
#> 511  -0.2177716422  0.2532804
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#> 513  -0.2739791853  0.2536415
#> 514   1.2468965347  0.3118982
#> 515  -0.1816887079  0.2531671
#> 516  -0.2123635939  0.2532578
#> 517   0.8050220919  0.2825165
#> 518  -1.3582389891  0.3213217
#> 519   0.2582557111  0.2586722
#> 520   2.5655435897  0.4866817
#> 521  -1.0223792350  0.2859966
#> 522   1.4440136033  0.3271834
#> 523  -1.2730074472  0.3109467
#> 524   1.1965600593  0.3082362
#> 525   0.1292239899  0.2557607
#> 526  -2.5239968455  0.6095662
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#> 528   0.7051188982  0.2769008
#> 529   0.4165442644  0.2637177
#> 530   1.1538236638  0.3051870
#> 531  -1.5853743983  0.3544042
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#> 533   1.0440732686  0.2976192
#> 534  -0.1927027235  0.2531924
#> 535   1.3150334680  0.3169959
#> 536  -0.7440590724  0.2671503
#> 537  -0.6431668662  0.2624565
#> 538  -0.6442913351  0.2625042
#> 539   0.1255200944  0.2556946
#> 540   0.3811144566  0.2624493
#> 541   1.1738585150  0.3066111
#> 542   0.2695016043  0.2589769
#> 543   2.0440253252  0.3925162
#> 544   0.9475760747  0.2912766
#> 545  -0.6332212167  0.2620539
#> 546   0.9289077953  0.2900805
#> 547   0.0903733712  0.2550956
#> 548  -0.8638542713  0.2741757
#> 549  -0.9529321346  0.2804564
#> 550   1.2470972960  0.3119063
#> 551   0.9809300821  0.2934375
#> 552   1.0146236535  0.2956498
#> 553  -0.8606149559  0.2739631
#> 554   0.1438260406  0.2560366
#> 555   0.0059059686  0.2539823
#> 556   1.4099657691  0.3244147
#> 557  -1.7220113475  0.3786415
#> 558  -1.0967196564  0.2925717
#> 559  -1.2493646680  0.3082351
#> 560   0.9033406034  0.2884725
#> 561  -1.0638000428  0.2895818
#> 562  -0.4844181226  0.2571738
#> 563   0.1299254890  0.2557733
#> 564  -0.5889898393  0.2603766
#> 565  -0.8069303884  0.2706363
#> 566   1.4105670296  0.3244621
#> 567  -0.5636408651  0.2595048
#> 568  -0.1712344945  0.2531510
#> 569   0.8375238450  0.2844421
#> 570  -1.3292863006  0.3176722
#> 571   1.1499429629  0.3049155
#> 572   0.2196726186  0.2576884
#> 573   0.5838397978  0.2707720
#> 574  -0.1771396427  0.2531594
#> 575   0.2994354883  0.2598292
#> 576   0.9372428542  0.2906179
#> 577  -0.4796477755  0.2570518
#> 578   1.9391204643  0.3781433
#> 579   3.9294276840  0.9233797
#> 580  -1.4180200018  0.3292202
#> 581  -0.2669823883  0.2535846
#> 582   0.4089738549  0.2634382
#> 583   0.0733274281  0.2548335
#> 584  -0.7714621872  0.2686177
#> 585  -2.5413187041  0.6165463
#> 586  -0.0009107443  0.2539131
#> 587   0.0692596220  0.2547749
#> 588  -0.0183563046  0.2537476
#> 589   0.1637927966  0.2564354
#> 590  -0.9805928427  0.2825899
#> 591   0.6287586678  0.2729487
#> 592  -2.8307010636  0.7482457
#> 593   1.2803856817  0.3143847
#> 594   2.2747067256  0.4294799
#> 595  -0.3382535437  0.2543409
#> 596  -0.8419489525  0.2727670
#> 597  -1.0467126765  0.2880789
#> 598   1.1508851804  0.3049860
#> 599  -1.5934433878  0.3557609
#> 600  -0.3364741710  0.2543163
#> 601   0.3364691567  0.2609632
#> 602  -1.4113091326  0.3283133
#> 603   0.7502000623  0.2793760
#> 604   1.5584061876  0.3369974
#> 605  -0.0578200911  0.2534468
#> 606   0.2922835418  0.2596208
#> 607  -0.8860083765  0.2756512
#> 608   0.0763084634  0.2548780
#> 609  -0.2387302103  0.2533884
#> 610  -0.6070942185  0.2610383
#> 611   0.5168101551  0.2677408
#> 612   2.5848255532  0.4908677
#> 613  -0.0936588630  0.2532615
#> 614   0.5846842518  0.2708128
#> 615   0.7203585117  0.2777278
#> 616   0.4297119479  0.2642116
#> 617   0.6401212928  0.2735168
#> 618  -0.3291033203  0.2542216
#> 619   1.0965984405  0.3011944
#> 620   2.6201745894  0.4987238
#> 621  -3.2525334538  0.9961975
#> 622   0.6823590870  0.2756951
#> 623  -2.2592179346  0.5141823
#> 624   1.5195955697  0.3335683
#> 625   0.2431513031  0.2582747
#> 626  -0.2671711543  0.2535857
#> 627  -3.5449855727  1.2135794
#> 628  -1.4082352785  0.3278964
#> 629   0.0927345351  0.2551320
#> 630  -0.9516380575  0.2803574
#> 631   0.2865982420  0.2594570
#> 632  -0.7098815707  0.2654368
#> 633   0.9761364906  0.2931245
#> 634   1.5871083679  0.3396118
#> 635   2.3594622072  0.4449355
#> 636  -0.0338055074  0.2536184
#> 637   0.3351128340  0.2609184
#> 638  -0.3264353742  0.2541879
#> 639   0.7136647669  0.2773630
#> 640   1.2986569300  0.3157570
#> 641   0.0876792022  0.2550531
#> 642  -0.2848172934  0.2537375
#> 643  -0.8438124427  0.2728855
#> 644  -0.5582401675  0.2593265
#> 645  -0.6841086277  0.2642325
#> 646   0.0189222922  0.2541235
#> 647   0.5921771530  0.2711676
#> 648  -0.6494140257  0.2627157
#> 649  -0.0392581588  0.2535759
#> 650   1.7107726227  0.3516399
#> 651  -0.5691554795  0.2596899
#> 652   0.3757144016  0.2622621
#> 653  -2.1889688055  0.4923254
#> 654   0.5393067760  0.2687242
#> 655  -0.5513557357  0.2591029
#> 656  -0.1377090287  0.2531494
#> 657   1.1691910421  0.3062778
#> 658   1.6265434824  0.3432959
#> 659   0.6210346281  0.2725650
#> 660   0.4572062326  0.2652746
#> 661   0.7612278567  0.2800011
#> 662   1.0372571611  0.2971599
#> 663   0.1027756932  0.2552976
#> 664   0.1261774916  0.2557033
#> 665  -0.2639646455  0.2535606
#> 666  -0.4080779764  0.2554688
#> 667  -2.5314295813  0.6125484
#> 668   0.5215403765  0.2679428
#> 669   1.5598173229  0.3371259
#> 670  -1.9254956016  0.4218276
#> 671   0.1700807849  0.2565655
#> 672   1.4983950855  0.3317369
#> 673  -1.3894025766  0.3253805
#> 674  -1.5314365262  0.3457672
#> 675   1.2526699705  0.3123290
#> 676   1.3836303267  0.3223181
#> 677   0.1312218920  0.2557973
#> 678  -0.6442052350  0.2625012
#> 679  -0.9751904155  0.2821750
#> 680   0.6987448638  0.2765613
#> 681   1.7919114478  0.3603624
#> 682  -0.2065153414  0.2532355
#> 683   0.6525913724  0.2741514
#> 684   0.9260460098  0.2899009
#> 685  -0.4667539140  0.2567333
#> 686   0.6032468570  0.2716981
#> 687  -0.7435361306  0.2671241
#> 688  -1.2939086323  0.3133891
#> 689  -0.1307752942  0.2531585
#> 690  -1.3467615977  0.3198576
#> 691  -1.9130884691  0.4189359
#> 692   0.0443633033  0.2544315
#> 693   0.1220387924  0.2556302
#> 694   0.7496683829  0.2793478
#> 695  -0.5897902493  0.2604028
#> 696   1.0867012959  0.3005197
#> 697  -2.2795795986  0.5207750
#> 698  -1.3396288416  0.3189624
#> 699   1.0410717547  0.2974172
#> 700   0.0322065574  0.2542792
#> 701   0.3199659014  0.2604469
#> 702   0.0566438939  0.2545955
#> 703   0.7705846401  0.2805296
#> 704  -1.0989088317  0.2927780
#> 705   0.3252021989  0.2606061
#> 706  -0.5730385899  0.2598209
#> 707  -0.3430006907  0.2544048
#> 708   0.1578718198  0.2563143
#> 709   1.6768230557  0.3481971
#> 710   0.0661811666  0.2547296
#> 711   0.3867932247  0.2626472
#> 712  -1.1404426745  0.2967538
#> 713   0.0779197802  0.2549021
#> 714  -0.7792097848  0.2690470
#> 715  -0.6062355387  0.2610070
#> 716   1.1673362700  0.3061447
#> 717   0.3580907662  0.2616670
#> 718   1.1311186598  0.3035928
#> 719  -1.1715925623  0.2998813
#> 720  -2.0679813357  0.4577346
#> 721  -1.3567406898  0.3211371
#> 722   0.2130600077  0.2575296
#> 723   2.2669633998  0.4281334
#> 724   1.2267859076  0.3104182
#> 725   1.2054637295  0.3088751
#> 726  -0.2867676930  0.2537558
#> 727  -2.7418176049  0.7047235
#> 728  -0.9402658902  0.2795070
#> 729  -0.3925550045  0.2551819
#> 730  -0.3622666380  0.2546846
#> 731   1.4817809266  0.3303353
#> 732   1.0114713660  0.2954417
#> 733  -0.0627809847  0.2534162
#> 734  -1.2074868745  0.3036358
#> 735  -0.5829373962  0.2601630
#> 736  -0.6746520018  0.2638072
#> 737  -1.0759694383  0.2906667
#> 738   1.7792154959  0.3589462
#> 739   1.6055347444  0.3413111
#> 740   0.5389009795  0.2687100
#> 741   1.7455979154  0.3552979
#> 742  -0.1561050445  0.2531409
#> 743  -0.4988667226  0.2575546
#> 744   0.0305177135  0.2542591
#> 745   1.5162112389  0.3332740
#> 746  -1.2921420881  0.3131856
#> 747   0.1134083570  0.2554789
#> 748  -0.3360168718  0.2543117
#> 749   1.3085594517  0.3165044
#> 750  -0.4271836773  0.2558475
#> 751  -0.9701458782  0.2817722
#> 752   1.4713360168  0.3294550
#> 753  -2.5518458364  0.6208921
#> 754  -1.7272607137  0.3796486
#> 755  -0.5318379968  0.2584936
#> 756   0.6044664977  0.2717554
#> 757  -1.0156664158  0.2854296
#> 758  -0.0545084667  0.2534682
#> 759  -0.1684326806  0.2531480
#> 760  -0.6560701933  0.2629991
#> 761  -0.5886557914  0.2603658
#> 762   1.4671589385  0.3291068
#> 763  -0.0867180949  0.2532910
#> 764  -0.9639479519  0.2813014
#> 765   0.1865842507  0.2569222
#> 766   0.5126118037  0.2675592
#> 767   1.5407279025  0.3354226
#> 768  -1.2113236514  0.3040480
#> 769  -0.9286767358  0.2786495
#> 770   0.8241041499  0.2836395
#> 771  -0.4499906161  0.2563412
#> 772   0.9077124652  0.2887464
#> 773  -0.8246886352  0.2716999
#> 774  -0.0764438974  0.2533400
#> 775  -0.4481470698  0.2562997
#> 776  -0.8991176433  0.2765550
#> 777   0.4147616062  0.2636534
#> 778   0.3020794638  0.2599048
#> 779  -4.0000000000 99.9999000
#> 780  -0.4260986882  0.2558256
#> 781   1.3318535866  0.3182806
#> 782  -0.1543200151  0.2531407
#> 783  -1.0228849286  0.2860357
#> 784  -0.2529838714  0.2534801
#> 785  -0.0532517144  0.2534765
#> 786   0.5161077686  0.2677072
#> 787  -0.6682378847  0.2635229
#> 788  -0.1748177762  0.2531557
#> 789   1.2171870955  0.3097202
#> 790  -0.4150607721  0.2556029
#> 791   1.4408276474  0.3269227
#> 792  -0.5544916016  0.2592040
#> 793   1.8149277881  0.3629731
#> 794  -1.0727583775  0.2903781
#> 795   2.4749337767  0.4676084
#> 796  -1.0958293212  0.2924878
#> 797   0.6831849382  0.2757364
#> 798   0.0636392460  0.2546933
#> 799   1.5468313232  0.3359652
#> 800   0.5946543594  0.2712847
#> 801   0.5232399131  0.2680187
#> 802   0.0900437245  0.2550913
#> 803   1.2436005764  0.3116518
#> 804  -0.0456935284  0.2535287
#> 805   0.4682822191  0.2657166
#> 806   1.5170794243  0.3333514
#> 807  -1.0986852731  0.2927551
#> 808  -0.6521326507  0.2628330
#> 809   0.1530084141  0.2562164
#> 810  -0.3138401500  0.2540391
#> 811   0.4314014618  0.2642744
#> 812   0.4560225400  0.2652278
#> 813   0.1352877146  0.2558725
#> 814   0.0655501270  0.2547213
#> 815  -1.4445891685  0.3329354
#> 816   1.7730903746  0.3582782
#> 817  -1.3369252389  0.3186258
#> 818  -0.1150000512  0.2531912
#> 819  -0.6583428939  0.2630952
#> 820  -0.1057109269  0.2532178
#> 821  -1.2276828113  0.3058247
#> 822   0.9271641007  0.2899737
#> 823   0.6433975092  0.2736784
#> 824   0.2765363159  0.2591717
#> 825  -1.1690971721  0.2996258
#> 826   0.4242722030  0.2640062
#> 827   1.7540570714  0.3562050
#> 828  -0.5784964079  0.2600074
#> 829  -0.7290843174  0.2663858
#> 830   2.0139053633  0.3882322
#> 831   0.1935187774  0.2570757
#> 832   0.8054342556  0.2825433
#> 833  -0.2896960322  0.2537845
#> 834   1.9215876048  0.3758810
#> 835   0.1235180486  0.2556571
#> 836  -1.2385607266  0.3070185
#> 837   0.6318556368  0.2730985
#> 838   0.7053375529  0.2769164
#> 839   0.1504372279  0.2561660
#> 840   2.2415810319  0.4236998
#> 841   2.1077490842  0.4019817
#> 842  -0.2016101493  0.2532184
#> 843  -0.5245285391  0.2582786
#> 844   1.6536965893  0.3459146
#> 845  -0.0119929707  0.2538055
#> 846   1.0648688952  0.2990267
#> 847   0.0284955424  0.2542349
#> 848  -0.2127543590  0.2532595
#> 849   1.2409265426  0.3114598
#> 850   1.4569385619  0.3282569
#> 851  -0.7328908527  0.2665750
#> 852  -0.7061026959  0.2652572
#> 853  -0.2942474843  0.2538291
#> 854   0.5533561562  0.2693605
#> 855  -0.7627458535  0.2681424
#> 856   0.6252569673  0.2727738
#> 857   0.5894930155  0.2710398
#> 858   2.6143972210  0.4974081
#> 859  -0.2272759770  0.2533257
#> 860   0.2841313442  0.2593864
#> 861  -0.1792199373  0.2531628
#> 862   0.7209363783  0.2777581
#> 863  -0.9276306254  0.2785779
#> 864  -0.4826538928  0.2571291
#> 865   0.9456839617  0.2911519
#> 866  -0.3163244629  0.2540667
#> 867  -0.0789336051  0.2533272
#> 868   0.0178756859  0.2541120
#> 869   0.4940334228  0.2667763
#> 870  -0.5952585350  0.2606019
#> 871  -0.9555417562  0.2806581
#> 872  -0.1148860794  0.2531915
#> 873  -0.8283744985  0.2719254
#> 874  -0.6518438874  0.2628233
#> 875  -0.2328649987  0.2533552
#> 876  -1.0164400285  0.2855019
#> 877   0.1252758286  0.2556886
#> 878   1.5387311317  0.3352430
#> 879  -1.7237545995  0.3789739
#> 880  -0.8863372417  0.2756760
#> 881  -0.4551943799  0.2564607
#> 882  -1.4745677991  0.3372212
#> 883  -0.8239591649  0.2716513
#> 884  -2.5611067901  0.6246952
#> 885   0.3697381355  0.2620577
#> 886   0.3824283423  0.2624932
#> 887  -0.0430117250  0.2535486
#> 888  -0.9125675215  0.2774975
#> 889  -0.3571900798  0.2546080
#> 890   1.8190027805  0.3634349
#> 891  -1.3227333475  0.3168681
#> 892  -0.9545377576  0.2805776
#> 893  -1.0908342475  0.2920252
#> 894   0.8275472135  0.2838452
#> 895  -0.7650989066  0.2682702
#> 896   0.2675173100  0.2589226
#> 897   0.7410658337  0.2788665
#> 898   1.0671930907  0.2991859
#> 899   1.2298331027  0.3106511
#> 900  -0.2527517797  0.2534784
#> 901  -0.2412334438  0.2534036
#> 902  -0.1286118101  0.2531620
#> 903  -0.6322309587  0.2620111
#> 904  -1.7657996993  0.3872081
#> 905  -1.6777264901  0.3704204
#> 906   0.8181563639  0.2832871
#> 907  -0.2833496628  0.2537248
#> 908   2.0546250099  0.3940519
#> 909   0.0408825649  0.2543874
#> 910  -2.8934060545  0.7807520
#> 911  -0.6497372141  0.2627296
#> 912   1.1263172108  0.3032574
#> 913  -0.7443549365  0.2671710
#> 914   0.8217554783  0.2835027
#> 915  -1.0185257802  0.2856725
#> 916  -0.6538130545  0.2629037
#> 917   0.3733515273  0.2621827
#> 918  -0.1933005889  0.2531937
#> 919  -2.4142105387  0.5674713
#> 920  -0.4050197137  0.2554109
#> 921  -0.3670652030  0.2547584
#> 922   1.5111833315  0.3328376
#> 923  -0.2344838788  0.2533642
#> 924   0.5071622029  0.2673261
#> 925  -0.1867677762  0.2531778
#> 926  -0.4646535680  0.2566825
#> 927  -2.1262731616  0.4739157
#> 928  -0.2604409718  0.2535338
#> 929  -1.3501562777  0.3202901
#> 930  -0.1671392916  0.2531468
#> 931   0.0956407408  0.2551797
#> 932  -1.1359693374  0.2963148
#> 933   0.2660972185  0.2588865
#> 934   2.2710150748  0.4288368
#> 935   0.4761449935  0.2660339
#> 936  -0.3046093802  0.2539355
#> 937   3.0391028119  0.6046393
#> 938  -0.4058073839  0.2554252
#> 939   0.3949534575  0.2629330
#> 940   0.2816547443  0.2593165
#> 941   0.7155962705  0.2774688
#> 942  -0.3574909436  0.2546115
#> 943   1.7603861834  0.3568900
#> 944   0.1933766711  0.2570735
#> 945  -0.7278558202  0.2663198
#> 946  -0.8612260632  0.2740037
#> 947  -0.0080486924  0.2538427
#> 948   1.9481029004  0.3793141
#> 949   1.0504891125  0.2980517
#> 950  -1.7531097596  0.3846738
#> 951  -0.6981544086  0.2648784
#> 952  -0.0638428425  0.2534100
#> 953   0.4187820954  0.2638016
#> 954  -1.9774215292  0.4343752
#> 955   0.5557374354  0.2694695
#> 956  -4.0000000000 99.9999000
#> 957   0.7509108812  0.2794177
#> 958  -1.2799329715  0.3117325
#> 959  -0.3700972730  0.2548062
#> 960  -1.3976686934  0.3264774
#> 961  -1.0171616179  0.2855548
#> 962   0.1213112102  0.2556170
#> 963   0.6044841953  0.2717538
#> 964   0.3030554181  0.2599332
#> 965   0.2462081521  0.2583554
#> 966   1.4688078702  0.3292365
#> 967   1.3655797045  0.3208994
#> 968   1.0578005867  0.2985477
#> 969   0.1587176427  0.2563309
#> 970  -0.7769319195  0.2689243
#> 971  -2.9773218577  0.8264513
#> 972  -0.0897241557  0.2532777
#> 973   0.2355674401  0.2580817
#> 974   0.0689668381  0.2547706
#> 975   1.4257003984  0.3256870
#> 976   0.8303376123  0.2840122
#> 977   1.2014428204  0.3085871
#> 978  -1.7113805426  0.3766375
#> 979  -3.1877067115  0.9533483
#> 980   0.7253437662  0.2780003
#> 981  -0.4671781586  0.2567455
#> 982  -0.1159301501  0.2531888
#> 983   0.5587253613  0.2696057
#> 984  -1.5503138974  0.3487533
#> 985  -1.0773078630  0.2907819
#> 986  -0.4641504635  0.2566726
#> 987  -0.6684342201  0.2635329
#> 988   0.3454868770  0.2612533
#> 989   0.7976198047  0.2820858
#> 990  -2.9347024907  0.8028866
#> 991   2.9293488392  0.5745182
#> 992   1.6610346707  0.3466379
#> 993  -0.2582238219  0.2535173
#> 994   0.8417380521  0.2846952
#> 995  -0.1529120814  0.2531407
#> 996  -0.3679423408  0.2547722
#> 997   0.0687481824  0.2547667
#> 998  -1.6396224333  0.3636097
#> 999   1.3880922727  0.3226753
#> 1000 -0.5666130989  0.2596022

# Compute traditional model-fit statistics
(fit.mix1 <- irtfit(
  x = mod.mix1, score = score.mle$est.theta, group.method = "equal.width",
  n.width = 10, loc.theta = "middle"
))
#> 
#> Call:
#> irtfit.est_irt(x = mod.mix1, score = score.mle$est.theta, group.method = "equal.width", 
#>     n.width = 10, loc.theta = "middle")
#> 
#> Significance level for chi-square fit statistic: 0.05 
#> 
#> Item fit statistics: 
#>        id      X2      G2  df.X2  df.G2  crit.val.X2  crit.val.G2   p.X2   p.G2
#> 1    CMC1   5.698   5.180      6      9        12.59        16.92  0.458  0.818
#> 2    CMC2   1.411   1.376      4      7         9.49        14.07  0.842  0.986
#> 3    CMC3  11.013  12.851      6      9        12.59        16.92  0.088  0.169
#> 4    CMC4   5.336   5.101      5      8        11.07        15.51  0.376  0.747
#> 5    CMC5   4.969   4.833      6      9        12.59        16.92  0.548  0.849
#> 6    CMC6   5.392   6.127      4      7         9.49        14.07  0.249  0.525
#> 7    CMC7   5.776   8.270      6      9        12.59        16.92  0.449  0.507
#> 8    CMC8  12.872  13.616      6      9        12.59        16.92  0.045  0.137
#> 9    CMC9  10.895  10.358      6      9        12.59        16.92  0.092  0.322
#> 10  CMC10  11.541  10.602      5      8        11.07        15.51  0.042  0.225
#> 11  CMC11   8.065  10.521      5      8        11.07        15.51  0.153  0.230
#> 12  CMC12   6.389   6.396      6      9        12.59        16.92  0.381  0.700
#> 13  CMC13   3.740   3.718      6      9        12.59        16.92  0.712  0.929
#> 14  CMC14   5.574   4.819      5      8        11.07        15.51  0.350  0.777
#> 15  CMC15   5.943   6.899      5      8        11.07        15.51  0.312  0.548
#> 16  CMC16   3.330   3.289      4      7         9.49        14.07  0.504  0.857
#> 17  CMC17   2.495   2.525      5      8        11.07        15.51  0.777  0.961
#> 18  CMC18   7.061   6.331      6      9        12.59        16.92  0.315  0.706
#> 19  CMC19   6.011   8.038      4      7         9.49        14.07  0.198  0.329
#> 20  CMC20   5.991   8.467      4      7         9.49        14.07  0.200  0.293
#> 21  CMC21   3.832   3.705      4      7         9.49        14.07  0.429  0.813
#> 22  CMC22   9.238  10.649      5      8        11.07        15.51  0.100  0.222
#> 23  CMC23   4.001   3.846      6      9        12.59        16.92  0.677  0.921
#> 24  CMC24   4.775   4.623      6      9        12.59        16.92  0.573  0.866
#> 25  CMC25   8.384   8.557      5      8        11.07        15.51  0.136  0.381
#> 26  CMC26   4.523   4.736      4      7         9.49        14.07  0.340  0.692
#> 27  CMC27   9.621   9.579      5      8        11.07        15.51  0.087  0.296
#> 28  CMC28   3.657   3.171      4      7         9.49        14.07  0.454  0.869
#> 29  CMC29  13.714  14.806      5      8        11.07        15.51  0.018  0.063
#> 30  CMC30   5.333   5.093      5      8        11.07        15.51  0.377  0.748
#> 31  CMC31  10.647  10.609      6      9        12.59        16.92  0.100  0.303
#> 32  CMC32   5.816   6.477      4      7         9.49        14.07  0.213  0.485
#> 33  CMC33   1.834   1.835      5      8        11.07        15.51  0.872  0.986
#> 34  CMC34   7.554   6.919      5      8        11.07        15.51  0.183  0.545
#> 35  CMC35   2.907   3.228      5      8        11.07        15.51  0.714  0.919
#> 36  CMC36   4.197   3.906      6      9        12.59        16.92  0.650  0.918
#> 37  CMC37   3.160   2.882      4      7         9.49        14.07  0.531  0.896
#> 38  CMC38   3.252   3.316      6      9        12.59        16.92  0.777  0.950
#> 39   CFR1   8.414  11.733      3      8         7.81        15.51  0.038  0.164
#> 40   CFR2   2.710   3.071      3      8         7.81        15.51  0.439  0.930
#> 41   AMC1   1.981   1.960      5      8        11.07        15.51  0.852  0.982
#> 42   AMC2   4.920   4.627      4      7         9.49        14.07  0.296  0.705
#> 43   AMC3   2.770   2.477      5      8        11.07        15.51  0.735  0.963
#> 44   AMC4   2.490   2.490      5      8        11.07        15.51  0.778  0.962
#> 45   AMC5   6.316   6.460      6      9        12.59        16.92  0.389  0.693
#> 46   AMC6   9.627  10.044      6      9        12.59        16.92  0.141  0.347
#> 47   AMC7   6.731   6.639      5      8        11.07        15.51  0.241  0.576
#> 48   AMC8   4.819   4.618      5      8        11.07        15.51  0.438  0.797
#> 49   AMC9   3.667   3.624      5      8        11.07        15.51  0.598  0.889
#> 50  AMC10   3.960   5.918      5      8        11.07        15.51  0.555  0.656
#> 51  AMC11   5.215   5.355      4      7         9.49        14.07  0.266  0.617
#> 52  AMC12  10.566  11.863      5      8        11.07        15.51  0.061  0.157
#> 53   AFR1  30.614  29.665     15     20        25.00        31.41  0.010  0.075
#> 54   AFR2  24.576  24.423     15     20        25.00        31.41  0.056  0.224
#> 55   AFR3  20.662  18.878     19     24        30.14        36.42  0.356  0.758
#>     outfit  infit     N  overSR.prop
#> 1    1.004  1.004  1000         0.00
#> 2    0.912  0.964  1000         0.00
#> 3    0.984  1.000  1000         0.10
#> 4    0.977  1.000  1000         0.00
#> 5    1.005  1.006  1000         0.00
#> 6    0.954  0.981  1000         0.00
#> 7    0.991  0.998  1000         0.00
#> 8    0.989  0.998  1000         0.00
#> 9    1.008  1.003  1000         0.10
#> 10   0.982  0.992  1000         0.10
#> 11   1.013  0.995  1000         0.10
#> 12   0.990  0.994  1000         0.00
#> 13   0.987  0.991  1000         0.00
#> 14   1.021  0.992  1000         0.00
#> 15   0.993  0.994  1000         0.10
#> 16   0.943  0.960  1000         0.00
#> 17   0.968  0.996  1000         0.00
#> 18   1.004  0.998  1000         0.00
#> 19   0.717  0.974  1000         0.00
#> 20   0.806  0.979  1000         0.00
#> 21   0.862  0.974  1000         0.00
#> 22   1.004  0.993  1000         0.00
#> 23   1.014  1.001  1000         0.00
#> 24   0.996  0.998  1000         0.00
#> 25   0.992  1.001  1000         0.00
#> 26   1.042  0.960  1000         0.00
#> 27   1.010  0.988  1000         0.10
#> 28   0.997  0.971  1000         0.00
#> 29   0.929  0.998  1000         0.20
#> 30   1.001  1.003  1000         0.00
#> 31   1.000  0.999  1000         0.20
#> 32   0.859  0.994  1000         0.00
#> 33   0.992  0.984  1000         0.00
#> 34   1.007  0.996  1000         0.00
#> 35   0.948  1.001  1000         0.00
#> 36   0.994  0.995  1000         0.00
#> 37   0.940  0.971  1000         0.00
#> 38   0.998  1.005  1000         0.00
#> 39   0.840  0.805  1000         0.08
#> 40   0.917  0.903  1000         0.04
#> 41   0.981  0.998  1000         0.00
#> 42   0.819  0.978  1000         0.00
#> 43   1.016  0.988  1000         0.10
#> 44   0.974  1.004  1000         0.00
#> 45   0.997  0.996  1000         0.00
#> 46   0.973  0.974  1000         0.00
#> 47   0.971  0.993  1000         0.00
#> 48   1.016  0.988  1000         0.00
#> 49   0.996  0.992  1000         0.00
#> 50   0.951  0.970  1000         0.00
#> 51   1.021  0.956  1000         0.00
#> 52   0.978  1.000  1000         0.00
#> 53   1.051  0.991  1000         0.12
#> 54   0.979  0.988  1000         0.04
#> 55   0.991  0.996  1000         0.08
#> 
#> Caution is needed in interpreting infit and outfit statistics for non-Rasch models. 

# Residual plot for the first item (dichotomous)
plot(
  x = fit.mix1, item.loc = 1, type = "both", ci.method = "wald",
  show.table = TRUE, ylim.sr.adjust = TRUE
)

#>                    interval      point total obs.freq.0 obs.freq.1 obs.prop.0
#> 1            [-4,-3.207057) -3.6035286     9          5          4  0.5555556
#> 2     [-3.207057,-2.414114) -2.8105858    16         13          3  0.8125000
#> 3     [-2.414114,-1.621172) -2.0176431    47         35         12  0.7446809
#> 4    [-1.621172,-0.8282289) -1.2247003   139         95         44  0.6834532
#> 5  [-0.8282289,-0.03528616) -0.4317575   277        159        118  0.5740072
#> 6   [-0.03528616,0.7576566)  0.3611852   267        141        126  0.5280899
#> 7      [0.7576566,1.550599)  1.1541280   166         69         97  0.4156627
#> 8       [1.550599,2.343542)  1.9470708    60         19         41  0.3166667
#> 9       [2.343542,3.136485)  2.7400135    17          6         11  0.3529412
#> 10      [3.136485,3.929428]  3.5329563     2          0          2  0.0000000
#>    obs.prop.1 exp.prob.0 exp.prob.1    raw.rsd.0    raw.rsd.1       se.0
#> 1   0.4444444  0.7773236  0.2226764 -0.221768093  0.221768093 0.13868093
#> 2   0.1875000  0.7557618  0.2442382  0.056738171 -0.056738171 0.10740865
#> 3   0.2553191  0.7220652  0.2779348  0.022615607 -0.022615607 0.06534475
#> 4   0.3165468  0.6716815  0.3283185  0.011771718 -0.011771718 0.03983107
#> 5   0.4259928  0.6011152  0.3988848 -0.027107980  0.027107980 0.02942136
#> 6   0.4719101  0.5108265  0.4891735  0.017263390 -0.017263390 0.03059233
#> 7   0.5843373  0.4077811  0.5922189  0.007881574 -0.007881574 0.03814175
#> 8   0.6833333  0.3044560  0.6955440  0.012210669 -0.012210669 0.05940855
#> 9   0.6470588  0.2134602  0.7865398  0.139481004 -0.139481004 0.09937893
#> 10  1.0000000  0.1420386  0.8579614 -0.142038618  0.142038618 0.24684373
#>          se.1  std.rsd.0  std.rsd.1
#> 1  0.13868093 -1.5991247  1.5991247
#> 2  0.10740865  0.5282458 -0.5282458
#> 3  0.06534475  0.3460968 -0.3460968
#> 4  0.03983107  0.2955411 -0.2955411
#> 5  0.02942136 -0.9213708  0.9213708
#> 6  0.03059233  0.5643045 -0.5643045
#> 7  0.03814175  0.2066390 -0.2066390
#> 8  0.05940855  0.2055372 -0.2055372
#> 9  0.09937893  1.4035269 -1.4035269
#> 10 0.24684373 -0.5754192  0.5754192

# Residual plot for the last item (polytomous)
plot(
  x = fit.mix1, item.loc = 55, type = "both", ci.method = "wald",
  show.table = FALSE, ylim.sr.adjust = TRUE
)


# Fit the 2PL model to all dichotomous items, the GPCM to items 39 and 40,
# and the GRM to items 53, 54, and 55.
# Provide IRT model and score category information via `model` and `cats`
# arguments.
(mod.mix2 <- est_irt(
  data = sim.dat1, D = 1,
  model = c(rep("2PLM", 38), rep("GPCM", 2), rep("2PLM", 12), rep("GRM", 3)),
  cats = c(rep(2, 38), rep(5, 2), rep(2, 12), rep(5, 3))
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -39015.7262, Max-Change: 2.725763
 EM iteration: 2, Loglike: -34112.9883, Max-Change: 0.495202
 EM iteration: 3, Loglike: -34092.3716, Max-Change: 0.072635
 EM iteration: 4, Loglike: -34085.1295, Max-Change: 0.031067
 EM iteration: 5, Loglike: -34079.2149, Max-Change: 0.028824
 EM iteration: 6, Loglike: -34073.9510, Max-Change: 0.026982
 EM iteration: 7, Loglike: -34069.2334, Max-Change: 0.025244
 EM iteration: 8, Loglike: -34065.0073, Max-Change: 0.023615
 EM iteration: 9, Loglike: -34061.2267, Max-Change: 0.022095
 EM iteration: 10, Loglike: -34057.8496, Max-Change: 0.02068
 EM iteration: 11, Loglike: -34054.8370, Max-Change: 0.019361
 EM iteration: 12, Loglike: -34052.1530, Max-Change: 0.018132
 EM iteration: 13, Loglike: -34049.7647, Max-Change: 0.016987
 EM iteration: 14, Loglike: -34047.6419, Max-Change: 0.015918
 EM iteration: 15, Loglike: -34045.7571, Max-Change: 0.01492
 EM iteration: 16, Loglike: -34044.0851, Max-Change: 0.013987
 EM iteration: 17, Loglike: -34042.6034, Max-Change: 0.013115
 EM iteration: 18, Loglike: -34041.2912, Max-Change: 0.012299
 EM iteration: 19, Loglike: -34040.1302, Max-Change: 0.011536
 EM iteration: 20, Loglike: -34039.1035, Max-Change: 0.010821
 EM iteration: 21, Loglike: -34038.1964, Max-Change: 0.010151
 EM iteration: 22, Loglike: -34037.3952, Max-Change: 0.009523
 EM iteration: 23, Loglike: -34036.6882, Max-Change: 0.008934
 EM iteration: 24, Loglike: -34036.0644, Max-Change: 0.008382
 EM iteration: 25, Loglike: -34035.5145, Max-Change: 0.007863
 EM iteration: 26, Loglike: -34035.0298, Max-Change: 0.007377
 EM iteration: 27, Loglike: -34034.6029, Max-Change: 0.00692
 EM iteration: 28, Loglike: -34034.2269, Max-Change: 0.006492
 EM iteration: 29, Loglike: -34033.8959, Max-Change: 0.006089
 EM iteration: 30, Loglike: -34033.6047, Max-Change: 0.005711
 EM iteration: 31, Loglike: -34033.3485, Max-Change: 0.005357
 EM iteration: 32, Loglike: -34033.1232, Max-Change: 0.005024
 EM iteration: 33, Loglike: -34032.9252, Max-Change: 0.004711
 EM iteration: 34, Loglike: -34032.7511, Max-Change: 0.004417
 EM iteration: 35, Loglike: -34032.5981, Max-Change: 0.004142
 EM iteration: 36, Loglike: -34032.4637, Max-Change: 0.003883
 EM iteration: 37, Loglike: -34032.3457, Max-Change: 0.00364
 EM iteration: 38, Loglike: -34032.2421, Max-Change: 0.003412
 EM iteration: 39, Loglike: -34032.1511, Max-Change: 0.003198
 EM iteration: 40, Loglike: -34032.0712, Max-Change: 0.002998
 EM iteration: 41, Loglike: -34032.0011, Max-Change: 0.002809
 EM iteration: 42, Loglike: -34031.9395, Max-Change: 0.002632
 EM iteration: 43, Loglike: -34031.8856, Max-Change: 0.002467
 EM iteration: 44, Loglike: -34031.8382, Max-Change: 0.002311
 EM iteration: 45, Loglike: -34031.7966, Max-Change: 0.002165
 EM iteration: 46, Loglike: -34031.7602, Max-Change: 0.002029
 EM iteration: 47, Loglike: -34031.7283, Max-Change: 0.00190
 EM iteration: 48, Loglike: -34031.7002, Max-Change: 0.00178
 EM iteration: 49, Loglike: -34031.6757, Max-Change: 0.001667
 EM iteration: 50, Loglike: -34031.6541, Max-Change: 0.001562
 EM iteration: 51, Loglike: -34031.6352, Max-Change: 0.001463
 EM iteration: 52, Loglike: -34031.6187, Max-Change: 0.00137
 EM iteration: 53, Loglike: -34031.6042, Max-Change: 0.001283
 EM iteration: 54, Loglike: -34031.5915, Max-Change: 0.001201
 EM iteration: 55, Loglike: -34031.5803, Max-Change: 0.001125
 EM iteration: 56, Loglike: -34031.5706, Max-Change: 0.001053
 EM iteration: 57, Loglike: -34031.5620, Max-Change: 0.000986
 EM iteration: 58, Loglike: -34031.5545, Max-Change: 0.000923
 EM iteration: 59, Loglike: -34031.5479, Max-Change: 0.000864
 EM iteration: 60, Loglike: -34031.5422, Max-Change: 0.000809
 EM iteration: 61, Loglike: -34031.5371, Max-Change: 0.000757
 EM iteration: 62, Loglike: -34031.5327, Max-Change: 0.000709
 EM iteration: 63, Loglike: -34031.5288, Max-Change: 0.000664
 EM iteration: 64, Loglike: -34031.5255, Max-Change: 0.000621
 EM iteration: 65, Loglike: -34031.5225, Max-Change: 0.000582
 EM iteration: 66, Loglike: -34031.5199, Max-Change: 0.000544
 EM iteration: 67, Loglike: -34031.5176, Max-Change: 0.00051
 EM iteration: 68, Loglike: -34031.5156, Max-Change: 0.000477
 EM iteration: 69, Loglike: -34031.5139, Max-Change: 0.000446
 EM iteration: 70, Loglike: -34031.5123, Max-Change: 0.000417
 EM iteration: 71, Loglike: -34031.5110, Max-Change: 0.000391
 EM iteration: 72, Loglike: -34031.5098, Max-Change: 0.000366
 EM iteration: 73, Loglike: -34031.5088, Max-Change: 0.000343
 EM iteration: 74, Loglike: -34031.5079, Max-Change: 0.000321
 EM iteration: 75, Loglike: -34031.5071, Max-Change: 3e-04
 EM iteration: 76, Loglike: -34031.5064, Max-Change: 0.000281
 EM iteration: 77, Loglike: -34031.5058, Max-Change: 0.000263
 EM iteration: 78, Loglike: -34031.5053, Max-Change: 0.000246
 EM iteration: 79, Loglike: -34031.5048, Max-Change: 0.00023
 EM iteration: 80, Loglike: -34031.5044, Max-Change: 0.000215
 EM iteration: 81, Loglike: -34031.5040, Max-Change: 0.000202
 EM iteration: 82, Loglike: -34031.5037, Max-Change: 0.000189
 EM iteration: 83, Loglike: -34031.5035, Max-Change: 0.000177
 EM iteration: 84, Loglike: -34031.5032, Max-Change: 0.000165
 EM iteration: 85, Loglike: -34031.5030, Max-Change: 0.000155
 EM iteration: 86, Loglike: -34031.5028, Max-Change: 0.000145
 EM iteration: 87, Loglike: -34031.5027, Max-Change: 0.000135
 EM iteration: 88, Loglike: -34031.5025, Max-Change: 0.000127
 EM iteration: 89, Loglike: -34031.5024, Max-Change: 0.000119
 EM iteration: 90, Loglike: -34031.5023, Max-Change: 0.000111
 EM iteration: 91, Loglike: -34031.5022, Max-Change: 0.000104
 EM iteration: 92, Loglike: -34031.5021, Max-Change: 9.7e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 5.65 seconds. 
#> 
#> Call:
#> est_irt(data = sim.dat1, D = 1, model = c(rep("2PLM", 38), rep("GPCM", 
#>     2), rep("2PLM", 12), rep("GRM", 3)), cats = c(rep(2, 38), 
#>     rep(5, 2), rep(2, 12), rep(5, 3)))
#> 
#> Item parameter estimation using MMLE-EM. 
#> 92 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -34031.5
#> 

# Display a summary of the estimation results
summary(mod.mix2)
#> 
#> Call:
#> est_irt(data = sim.dat1, D = 1, model = c(rep("2PLM", 38), rep("GPCM", 
#>     2), rep("2PLM", 12), rep("GRM", 3)), cats = c(rep(2, 38), 
#>     rep(5, 2), rep(2, 12), rep(5, 3)))
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 125
#>    - Item parameters: 125
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 92
#>  Maximum parameter change: 9.717858e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 5.58
#>  Standard error computation: 0.04
#>  Total computation: 5.65
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 68063
#>  Akaike Information Criterion (AIC): 68313
#>  Bayesian Information Criterion (BIC): 68926.47
#>  Item Parameters: 
#>      id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4  par.5
#> 1    V1     2   2PLM   0.47  0.08   0.37  0.16     NA    NA     NA    NA     NA
#> 2    V2     2   2PLM   1.69  0.16  -1.18  0.10     NA    NA     NA    NA     NA
#> 3    V3     2   2PLM   0.79  0.09   0.04  0.10     NA    NA     NA    NA     NA
#> 4    V4     2   2PLM   1.03  0.11  -0.82  0.11     NA    NA     NA    NA     NA
#> 5    V5     2   2PLM   0.66  0.09  -0.62  0.13     NA    NA     NA    NA     NA
#> 6    V6     2   2PLM   1.17  0.11   0.48  0.08     NA    NA     NA    NA     NA
#> 7    V7     2   2PLM   0.62  0.08   0.72  0.15     NA    NA     NA    NA     NA
#> 8    V8     2   2PLM   0.65  0.09   0.64  0.13     NA    NA     NA    NA     NA
#> 9    V9     2   2PLM   0.66  0.09  -0.07  0.11     NA    NA     NA    NA     NA
#> 10  V10     2   2PLM   1.15  0.11  -0.21  0.08     NA    NA     NA    NA     NA
#> 11  V11     2   2PLM   0.92  0.10  -0.89  0.12     NA    NA     NA    NA     NA
#> 12  V12     2   2PLM   0.69  0.09   0.98  0.15     NA    NA     NA    NA     NA
#> 13  V13     2   2PLM   0.72  0.09   1.04  0.15     NA    NA     NA    NA     NA
#> 14  V14     2   2PLM   1.15  0.11  -0.51  0.08     NA    NA     NA    NA     NA
#> 15  V15     2   2PLM   1.04  0.10  -0.54  0.09     NA    NA     NA    NA     NA
#> 16  V16     2   2PLM   1.79  0.14  -0.10  0.06     NA    NA     NA    NA     NA
#> 17  V17     2   2PLM   1.14  0.11  -0.55  0.09     NA    NA     NA    NA     NA
#> 18  V18     2   2PLM   0.57  0.08   0.73  0.16     NA    NA     NA    NA     NA
#> 19  V19     2   2PLM   2.05  0.21  -1.18  0.09     NA    NA     NA    NA     NA
#> 20  V20     2   2PLM   1.58  0.19  -1.72  0.15     NA    NA     NA    NA     NA
#> 21  V21     2   2PLM   1.76  0.17  -1.17  0.09     NA    NA     NA    NA     NA
#> 22  V22     2   2PLM   0.93  0.10  -1.01  0.12     NA    NA     NA    NA     NA
#> 23  V23     2   2PLM   0.75  0.09  -0.66  0.12     NA    NA     NA    NA     NA
#> 24  V24     2   2PLM   0.47  0.08   1.14  0.23     NA    NA     NA    NA     NA
#> 25  V25     2   2PLM   0.69  0.11  -2.11  0.31     NA    NA     NA    NA     NA
#> 26  V26     2   2PLM   1.16  0.13  -1.89  0.18     NA    NA     NA    NA     NA
#> 27  V27     2   2PLM   1.10  0.10  -0.25  0.08     NA    NA     NA    NA     NA
#> 28  V28     2   2PLM   1.65  0.14  -0.48  0.07     NA    NA     NA    NA     NA
#> 29  V29     2   2PLM   1.11  0.14  -1.80  0.18     NA    NA     NA    NA     NA
#> 30  V30     2   2PLM   0.77  0.09  -0.28  0.10     NA    NA     NA    NA     NA
#> 31  V31     2   2PLM   0.71  0.09   0.35  0.11     NA    NA     NA    NA     NA
#> 32  V32     2   2PLM   1.36  0.15  -1.28  0.12     NA    NA     NA    NA     NA
#> 33  V33     2   2PLM   1.07  0.12  -1.68  0.17     NA    NA     NA    NA     NA
#> 34  V34     2   2PLM   0.97  0.10  -0.21  0.09     NA    NA     NA    NA     NA
#> 35  V35     2   2PLM   1.07  0.11  -0.67  0.09     NA    NA     NA    NA     NA
#> 36  V36     2   2PLM   0.70  0.09   0.78  0.14     NA    NA     NA    NA     NA
#> 37  V37     2   2PLM   1.68  0.14  -0.58  0.07     NA    NA     NA    NA     NA
#> 38  V38     2   2PLM   0.54  0.09  -1.34  0.24     NA    NA     NA    NA     NA
#> 39  V39     5   GPCM   1.98  0.16  -1.59  0.12  -1.16  0.10  -0.66  0.08  -0.34
#> 40  V40     5   GPCM   1.31  0.10  -0.59  0.09  -0.07  0.09   0.41  0.09   1.19
#> 41  V41     2   2PLM   0.84  0.09   0.05  0.09     NA    NA     NA    NA     NA
#> 42  V42     2   2PLM   1.72  0.20  -1.67  0.14     NA    NA     NA    NA     NA
#> 43  V43     2   2PLM   1.02  0.10   0.29  0.08     NA    NA     NA    NA     NA
#> 44  V44     2   2PLM   0.87  0.10  -0.63  0.11     NA    NA     NA    NA     NA
#> 45  V45     2   2PLM   0.30  0.08   3.26  0.85     NA    NA     NA    NA     NA
#> 46  V46     2   2PLM   0.58  0.08   1.84  0.27     NA    NA     NA    NA     NA
#> 47  V47     2   2PLM   1.12  0.11  -0.35  0.08     NA    NA     NA    NA     NA
#> 48  V48     2   2PLM   1.14  0.11  -0.20  0.08     NA    NA     NA    NA     NA
#> 49  V49     2   2PLM   0.99  0.10   0.04  0.08     NA    NA     NA    NA     NA
#> 50  V50     2   2PLM   1.01  0.10   1.23  0.13     NA    NA     NA    NA     NA
#> 51  V51     2   2PLM   1.60  0.15  -1.33  0.11     NA    NA     NA    NA     NA
#> 52  V52     2   2PLM   0.88  0.10  -1.14  0.14     NA    NA     NA    NA     NA
#> 53  V53     5    GRM   1.18  0.09  -0.21  0.07   0.31  0.07   0.92  0.09   1.51
#> 54  V54     5    GRM   1.25  0.10  -1.96  0.15  -1.32  0.11  -0.71  0.08  -0.10
#> 55  V55     5    GRM   0.94  0.08  -0.61  0.10   0.03  0.08   0.72  0.10   1.26
#>     se.5
#> 1     NA
#> 2     NA
#> 3     NA
#> 4     NA
#> 5     NA
#> 6     NA
#> 7     NA
#> 8     NA
#> 9     NA
#> 10    NA
#> 11    NA
#> 12    NA
#> 13    NA
#> 14    NA
#> 15    NA
#> 16    NA
#> 17    NA
#> 18    NA
#> 19    NA
#> 20    NA
#> 21    NA
#> 22    NA
#> 23    NA
#> 24    NA
#> 25    NA
#> 26    NA
#> 27    NA
#> 28    NA
#> 29    NA
#> 30    NA
#> 31    NA
#> 32    NA
#> 33    NA
#> 34    NA
#> 35    NA
#> 36    NA
#> 37    NA
#> 38    NA
#> 39  0.07
#> 40  0.11
#> 41    NA
#> 42    NA
#> 43    NA
#> 44    NA
#> 45    NA
#> 46    NA
#> 47    NA
#> 48    NA
#> 49    NA
#> 50    NA
#> 51    NA
#> 52    NA
#> 53  0.12
#> 54  0.07
#> 55  0.13
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

# Fit the 2PL model to all dichotomous items, the GPCM to items 39 and 40,
# and the GRM to items 53, 54, and 55.
# Also estimate the empirical histogram of the latent prior distribution.
# Provide IRT model and score category information via `model` and `cats`
# arguments.
(mod.mix3 <- est_irt(
  data = sim.dat1, D = 1,
  model = c(rep("2PLM", 38), rep("GPCM", 2), rep("2PLM", 12), rep("GRM", 3)),
  cats = c(rep(2, 38), rep(5, 2), rep(2, 12), rep(5, 3)), EmpHist = TRUE
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -39020.4583, Max-Change: 2.725763
 EM iteration: 2, Loglike: -34115.0550, Max-Change: 0.511114
 EM iteration: 3, Loglike: -34091.1908, Max-Change: 0.101242
 EM iteration: 4, Loglike: -34081.4188, Max-Change: 0.043922
 EM iteration: 5, Loglike: -34072.9699, Max-Change: 0.040887
 EM iteration: 6, Loglike: -34065.3505, Max-Change: 0.037099
 EM iteration: 7, Loglike: -34058.5666, Max-Change: 0.033332
 EM iteration: 8, Loglike: -34052.6086, Max-Change: 0.029914
 EM iteration: 9, Loglike: -34047.6502, Max-Change: 0.026719
 EM iteration: 10, Loglike: -34043.8647, Max-Change: 0.024096
 EM iteration: 11, Loglike: -34040.5190, Max-Change: 0.022003
 EM iteration: 12, Loglike: -34037.5390, Max-Change: 0.019994
 EM iteration: 13, Loglike: -34034.8823, Max-Change: 0.01815
 EM iteration: 14, Loglike: -34032.5100, Max-Change: 0.016449
 EM iteration: 15, Loglike: -34030.3905, Max-Change: 0.014881
 EM iteration: 16, Loglike: -34028.4975, Max-Change: 0.01344
 EM iteration: 17, Loglike: -34026.8081, Max-Change: 0.012115
 EM iteration: 18, Loglike: -34025.3021, Max-Change: 0.010897
 EM iteration: 19, Loglike: -34023.9617, Max-Change: 0.009777
 EM iteration: 20, Loglike: -34022.7708, Max-Change: 0.008748
 EM iteration: 21, Loglike: -34021.7144, Max-Change: 0.008054
 EM iteration: 22, Loglike: -34020.7788, Max-Change: 0.007545
 EM iteration: 23, Loglike: -34019.9512, Max-Change: 0.007126
 EM iteration: 24, Loglike: -34019.2196, Max-Change: 0.006717
 EM iteration: 25, Loglike: -34018.5732, Max-Change: 0.006322
 EM iteration: 26, Loglike: -34018.0020, Max-Change: 0.006099
 EM iteration: 27, Loglike: -34017.4971, Max-Change: 0.005931
 EM iteration: 28, Loglike: -34017.0505, Max-Change: 0.005758
 EM iteration: 29, Loglike: -34016.6550, Max-Change: 0.005581
 EM iteration: 30, Loglike: -34016.3045, Max-Change: 0.005401
 EM iteration: 31, Loglike: -34015.9935, Max-Change: 0.005219
 EM iteration: 32, Loglike: -34015.7170, Max-Change: 0.005036
 EM iteration: 33, Loglike: -34015.4709, Max-Change: 0.004852
 EM iteration: 34, Loglike: -34015.2514, Max-Change: 0.004669
 EM iteration: 35, Loglike: -34015.0555, Max-Change: 0.004487
 EM iteration: 36, Loglike: -34014.8801, Max-Change: 0.004307
 EM iteration: 37, Loglike: -34014.7230, Max-Change: 0.004129
 EM iteration: 38, Loglike: -34014.5818, Max-Change: 0.003954
 EM iteration: 39, Loglike: -34014.4547, Max-Change: 0.003782
 EM iteration: 40, Loglike: -34014.3401, Max-Change: 0.003613
 EM iteration: 41, Loglike: -34014.2364, Max-Change: 0.003447
 EM iteration: 42, Loglike: -34014.1423, Max-Change: 0.003285
 EM iteration: 43, Loglike: -34014.0568, Max-Change: 0.003128
 EM iteration: 44, Loglike: -34013.9789, Max-Change: 0.002974
 EM iteration: 45, Loglike: -34013.9075, Max-Change: 0.002825
 EM iteration: 46, Loglike: -34013.8420, Max-Change: 0.002679
 EM iteration: 47, Loglike: -34013.7816, Max-Change: 0.002539
 EM iteration: 48, Loglike: -34013.7256, Max-Change: 0.002402
 EM iteration: 49, Loglike: -34013.6735, Max-Change: 0.002271
 EM iteration: 50, Loglike: -34013.6250, Max-Change: 0.002144
 EM iteration: 51, Loglike: -34013.5796, Max-Change: 0.002021
 EM iteration: 52, Loglike: -34013.5370, Max-Change: 0.001902
 EM iteration: 53, Loglike: -34013.4968, Max-Change: 0.001788
 EM iteration: 54, Loglike: -34013.4586, Max-Change: 0.001678
 EM iteration: 55, Loglike: -34013.4223, Max-Change: 0.001573
 EM iteration: 56, Loglike: -34013.3876, Max-Change: 0.001473
 EM iteration: 57, Loglike: -34013.3542, Max-Change: 0.001377
 EM iteration: 58, Loglike: -34013.3221, Max-Change: 0.001287
 EM iteration: 59, Loglike: -34013.2912, Max-Change: 0.00120
 EM iteration: 60, Loglike: -34013.2612, Max-Change: 0.001118
 EM iteration: 61, Loglike: -34013.2322, Max-Change: 0.001039
 EM iteration: 62, Loglike: -34013.2040, Max-Change: 0.000961
 EM iteration: 63, Loglike: -34013.1766, Max-Change: 0.000886
 EM iteration: 64, Loglike: -34013.1498, Max-Change: 0.000814
 EM iteration: 65, Loglike: -34013.1237, Max-Change: 0.000745
 EM iteration: 66, Loglike: -34013.0982, Max-Change: 0.00068
 EM iteration: 67, Loglike: -34013.0733, Max-Change: 0.000622
 EM iteration: 68, Loglike: -34013.0488, Max-Change: 0.000568
 EM iteration: 69, Loglike: -34013.0248, Max-Change: 0.000542
 EM iteration: 70, Loglike: -34013.0013, Max-Change: 0.000518
 EM iteration: 71, Loglike: -34012.9782, Max-Change: 0.000495
 EM iteration: 72, Loglike: -34012.9555, Max-Change: 0.000475
 EM iteration: 73, Loglike: -34012.9334, Max-Change: 0.000449
 EM iteration: 74, Loglike: -34012.9118, Max-Change: 0.000425
 EM iteration: 75, Loglike: -34012.8910, Max-Change: 0.000403
 EM iteration: 76, Loglike: -34012.8711, Max-Change: 0.000381
 EM iteration: 77, Loglike: -34012.8521, Max-Change: 0.000361
 EM iteration: 78, Loglike: -34012.8338, Max-Change: 0.000341
 EM iteration: 79, Loglike: -34012.8162, Max-Change: 0.000322
 EM iteration: 80, Loglike: -34012.7993, Max-Change: 0.000304
 EM iteration: 81, Loglike: -34012.7831, Max-Change: 0.000288
 EM iteration: 82, Loglike: -34012.7674, Max-Change: 0.000272
 EM iteration: 83, Loglike: -34012.7522, Max-Change: 0.000257
 EM iteration: 84, Loglike: -34012.7375, Max-Change: 0.000243
 EM iteration: 85, Loglike: -34012.7232, Max-Change: 0.00023
 EM iteration: 86, Loglike: -34012.7094, Max-Change: 0.000218
 EM iteration: 87, Loglike: -34012.6959, Max-Change: 0.000206
 EM iteration: 88, Loglike: -34012.6827, Max-Change: 0.000195
 EM iteration: 89, Loglike: -34012.6698, Max-Change: 0.000184
 EM iteration: 90, Loglike: -34012.6572, Max-Change: 0.000174
 EM iteration: 91, Loglike: -34012.6448, Max-Change: 0.000165
 EM iteration: 92, Loglike: -34012.6327, Max-Change: 0.000156
 EM iteration: 93, Loglike: -34012.6207, Max-Change: 0.000148
 EM iteration: 94, Loglike: -34012.6089, Max-Change: 0.000143
 EM iteration: 95, Loglike: -34012.5973, Max-Change: 0.000138
 EM iteration: 96, Loglike: -34012.5858, Max-Change: 0.000134
 EM iteration: 97, Loglike: -34012.5745, Max-Change: 0.000129
 EM iteration: 98, Loglike: -34012.5633, Max-Change: 0.000125
 EM iteration: 99, Loglike: -34012.5522, Max-Change: 0.00012
 EM iteration: 100, Loglike: -34012.5412, Max-Change: 0.000116
 EM iteration: 101, Loglike: -34012.5303, Max-Change: 0.000112
 EM iteration: 102, Loglike: -34012.5195, Max-Change: 0.000108
 EM iteration: 103, Loglike: -34012.5087, Max-Change: 0.000104
 EM iteration: 104, Loglike: -34012.4981, Max-Change: 1e-04
 EM iteration: 105, Loglike: -34012.4876, Max-Change: 9.6e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 5.88 seconds. 
#> 
#> Call:
#> est_irt(data = sim.dat1, D = 1, model = c(rep("2PLM", 38), rep("GPCM", 
#>     2), rep("2PLM", 12), rep("GRM", 3)), cats = c(rep(2, 38), 
#>     rep(5, 2), rep(2, 12), rep(5, 3)), EmpHist = TRUE)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 105 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -34012.49
#> 
(emphist <- getirt(mod.mix3, what = "weights"))
#>    theta       weight
#> 1  -6.00 1.000156e-23
#> 2  -5.75 1.000156e-23
#> 3  -5.50 1.000156e-23
#> 4  -5.25 1.000156e-23
#> 5  -5.00 1.000156e-23
#> 6  -4.75 1.000156e-23
#> 7  -4.50 1.000156e-23
#> 8  -4.25 1.000156e-23
#> 9  -4.00 1.000156e-23
#> 10 -3.75 1.000156e-23
#> 11 -3.50 1.000156e-23
#> 12 -3.25 1.000156e-23
#> 13 -3.00 1.000156e-23
#> 14 -2.75 1.198185e-15
#> 15 -2.50 4.824176e-08
#> 16 -2.25 5.397742e-05
#> 17 -2.00 3.373625e-03
#> 18 -1.75 3.474347e-02
#> 19 -1.50 3.877324e-02
#> 20 -1.25 4.400393e-02
#> 21 -1.00 6.037273e-02
#> 22 -0.75 1.175590e-01
#> 23 -0.50 8.686911e-02
#> 24 -0.25 1.059898e-01
#> 25  0.00 1.038976e-01
#> 26  0.25 5.964898e-02
#> 27  0.50 9.217908e-02
#> 28  0.75 7.221697e-02
#> 29  1.00 3.495477e-02
#> 30  1.25 3.988313e-02
#> 31  1.50 4.847406e-02
#> 32  1.75 2.391442e-02
#> 33  2.00 7.502915e-03
#> 34  2.25 3.890058e-03
#> 35  2.50 4.867261e-03
#> 36  2.75 8.228783e-03
#> 37  3.00 7.155109e-03
#> 38  3.25 1.399605e-03
#> 39  3.50 4.777224e-05
#> 40  3.75 4.670538e-07
#> 41  4.00 1.773812e-09
#> 42  4.25 3.020610e-12
#> 43  4.50 2.581355e-15
#> 44  4.75 1.127167e-18
#> 45  5.00 2.831752e-22
#> 46  5.25 1.000156e-23
#> 47  5.50 1.000156e-23
#> 48  5.75 1.000156e-23
#> 49  6.00 1.000156e-23
plot(emphist$weight ~ emphist$theta, type = "h")


# Fit the 2PL model to all dichotomous items, the PCM to items 39 and 40 by
# fixing slope parameters to 1, and the GRM to items 53, 54, and 55.
# Provide IRT model and score category information via `model` and `cats`
# arguments.
(mod.mix4 <- est_irt(
  data = sim.dat1, D = 1,
  model = c(rep("2PLM", 38), rep("GPCM", 2), rep("2PLM", 12), rep("GRM", 3)),
  cats = c(rep(2, 38), rep(5, 2), rep(2, 12), rep(5, 3)),
  fix.a.gpcm = TRUE, a.val.gpcm = 1
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -39015.7262, Max-Change: 2.725763
 EM iteration: 2, Loglike: -34173.8698, Max-Change: 0.522828
 EM iteration: 3, Loglike: -34149.9420, Max-Change: 0.117157
 EM iteration: 4, Loglike: -34137.5850, Max-Change: 0.059167
 EM iteration: 5, Loglike: -34127.7660, Max-Change: 0.052432
 EM iteration: 6, Loglike: -34119.6424, Max-Change: 0.046519
 EM iteration: 7, Loglike: -34112.8425, Max-Change: 0.041219
 EM iteration: 8, Loglike: -34107.1103, Max-Change: 0.036545
 EM iteration: 9, Loglike: -34102.2556, Max-Change: 0.032463
 EM iteration: 10, Loglike: -34098.1318, Max-Change: 0.028912
 EM iteration: 11, Loglike: -34094.6229, Max-Change: 0.025825
 EM iteration: 12, Loglike: -34091.6349, Max-Change: 0.023139
 EM iteration: 13, Loglike: -34089.0901, Max-Change: 0.020795
 EM iteration: 14, Loglike: -34086.9233, Max-Change: 0.018742
 EM iteration: 15, Loglike: -34085.0791, Max-Change: 0.016937
 EM iteration: 16, Loglike: -34083.5106, Max-Change: 0.015343
 EM iteration: 17, Loglike: -34082.1773, Max-Change: 0.01393
 EM iteration: 18, Loglike: -34081.0447, Max-Change: 0.012671
 EM iteration: 19, Loglike: -34080.0833, Max-Change: 0.011546
 EM iteration: 20, Loglike: -34079.2677, Max-Change: 0.010535
 EM iteration: 21, Loglike: -34078.5761, Max-Change: 0.009625
 EM iteration: 22, Loglike: -34077.9901, Max-Change: 0.008803
 EM iteration: 23, Loglike: -34077.4936, Max-Change: 0.008058
 EM iteration: 24, Loglike: -34077.0733, Max-Change: 0.007381
 EM iteration: 25, Loglike: -34076.7176, Max-Change: 0.006765
 EM iteration: 26, Loglike: -34076.4167, Max-Change: 0.006204
 EM iteration: 27, Loglike: -34076.1622, Max-Change: 0.005692
 EM iteration: 28, Loglike: -34075.9470, Max-Change: 0.005223
 EM iteration: 29, Loglike: -34075.7651, Max-Change: 0.004795
 EM iteration: 30, Loglike: -34075.6114, Max-Change: 0.004402
 EM iteration: 31, Loglike: -34075.4816, Max-Change: 0.004042
 EM iteration: 32, Loglike: -34075.3719, Max-Change: 0.003712
 EM iteration: 33, Loglike: -34075.2792, Max-Change: 0.003409
 EM iteration: 34, Loglike: -34075.2010, Max-Change: 0.003132
 EM iteration: 35, Loglike: -34075.1350, Max-Change: 0.002876
 EM iteration: 36, Loglike: -34075.0792, Max-Change: 0.002642
 EM iteration: 37, Loglike: -34075.0322, Max-Change: 0.002427
 EM iteration: 38, Loglike: -34074.9925, Max-Change: 0.002229
 EM iteration: 39, Loglike: -34074.9590, Max-Change: 0.002048
 EM iteration: 40, Loglike: -34074.9307, Max-Change: 0.001881
 EM iteration: 41, Loglike: -34074.9068, Max-Change: 0.001728
 EM iteration: 42, Loglike: -34074.8867, Max-Change: 0.001587
 EM iteration: 43, Loglike: -34074.8697, Max-Change: 0.001458
 EM iteration: 44, Loglike: -34074.8554, Max-Change: 0.001339
 EM iteration: 45, Loglike: -34074.8434, Max-Change: 0.00123
 EM iteration: 46, Loglike: -34074.8332, Max-Change: 0.001129
 EM iteration: 47, Loglike: -34074.8246, Max-Change: 0.001037
 EM iteration: 48, Loglike: -34074.8173, Max-Change: 0.000953
 EM iteration: 49, Loglike: -34074.8112, Max-Change: 0.000875
 EM iteration: 50, Loglike: -34074.8061, Max-Change: 0.000803
 EM iteration: 51, Loglike: -34074.8017, Max-Change: 0.000738
 EM iteration: 52, Loglike: -34074.7981, Max-Change: 0.000678
 EM iteration: 53, Loglike: -34074.7950, Max-Change: 0.000622
 EM iteration: 54, Loglike: -34074.7924, Max-Change: 0.000571
 EM iteration: 55, Loglike: -34074.7902, Max-Change: 0.000525
 EM iteration: 56, Loglike: -34074.7884, Max-Change: 0.000482
 EM iteration: 57, Loglike: -34074.7868, Max-Change: 0.000442
 EM iteration: 58, Loglike: -34074.7855, Max-Change: 0.000406
 EM iteration: 59, Loglike: -34074.7844, Max-Change: 0.000373
 EM iteration: 60, Loglike: -34074.7834, Max-Change: 0.000342
 EM iteration: 61, Loglike: -34074.7827, Max-Change: 0.000314
 EM iteration: 62, Loglike: -34074.7820, Max-Change: 0.000289
 EM iteration: 63, Loglike: -34074.7814, Max-Change: 0.000265
 EM iteration: 64, Loglike: -34074.7810, Max-Change: 0.000243
 EM iteration: 65, Loglike: -34074.7806, Max-Change: 0.000223
 EM iteration: 66, Loglike: -34074.7802, Max-Change: 0.000205
 EM iteration: 67, Loglike: -34074.7799, Max-Change: 0.000188
 EM iteration: 68, Loglike: -34074.7797, Max-Change: 0.000173
 EM iteration: 69, Loglike: -34074.7795, Max-Change: 0.000159
 EM iteration: 70, Loglike: -34074.7793, Max-Change: 0.000146
 EM iteration: 71, Loglike: -34074.7792, Max-Change: 0.000134
 EM iteration: 72, Loglike: -34074.7791, Max-Change: 0.000123
 EM iteration: 73, Loglike: -34074.7790, Max-Change: 0.000113
 EM iteration: 74, Loglike: -34074.7789, Max-Change: 0.000104
 EM iteration: 75, Loglike: -34074.7788, Max-Change: 9.5e-05 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 4.41 seconds. 
#> 
#> Call:
#> est_irt(data = sim.dat1, D = 1, model = c(rep("2PLM", 38), rep("GPCM", 
#>     2), rep("2PLM", 12), rep("GRM", 3)), cats = c(rep(2, 38), 
#>     rep(5, 2), rep(2, 12), rep(5, 3)), fix.a.gpcm = TRUE, a.val.gpcm = 1)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 75 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -34074.78
#> 

# Display a summary of the estimation results
summary(mod.mix4)
#> 
#> Call:
#> est_irt(data = sim.dat1, D = 1, model = c(rep("2PLM", 38), rep("GPCM", 
#>     2), rep("2PLM", 12), rep("GRM", 3)), cats = c(rep(2, 38), 
#>     rep(5, 2), rep(2, 12), rep(5, 3)), fix.a.gpcm = TRUE, a.val.gpcm = 1)
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 1e-04
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 123
#>    - Item parameters: 123
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 0
#>  Number of fixed items: 0
#>  Number of E-step cycles completed: 75
#>  Maximum parameter change: 9.509097e-05
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 4.32
#>  Standard error computation: 0.04
#>  Total computation: 4.41
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 68149.56
#>  Akaike Information Criterion (AIC): 68395.56
#>  Bayesian Information Criterion (BIC): 68999.21
#>  Item Parameters: 
#>      id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4  par.5
#> 1    V1     2   2PLM   0.44  0.07   0.40  0.17     NA    NA     NA    NA     NA
#> 2    V2     2   2PLM   1.57  0.15  -1.27  0.10     NA    NA     NA    NA     NA
#> 3    V3     2   2PLM   0.74  0.08   0.05  0.10     NA    NA     NA    NA     NA
#> 4    V4     2   2PLM   0.96  0.10  -0.88  0.11     NA    NA     NA    NA     NA
#> 5    V5     2   2PLM   0.61  0.08  -0.66  0.14     NA    NA     NA    NA     NA
#> 6    V6     2   2PLM   1.09  0.10   0.52  0.08     NA    NA     NA    NA     NA
#> 7    V7     2   2PLM   0.57  0.08   0.78  0.16     NA    NA     NA    NA     NA
#> 8    V8     2   2PLM   0.60  0.08   0.69  0.14     NA    NA     NA    NA     NA
#> 9    V9     2   2PLM   0.62  0.08  -0.07  0.12     NA    NA     NA    NA     NA
#> 10  V10     2   2PLM   1.07  0.10  -0.23  0.08     NA    NA     NA    NA     NA
#> 11  V11     2   2PLM   0.86  0.09  -0.95  0.12     NA    NA     NA    NA     NA
#> 12  V12     2   2PLM   0.65  0.08   1.05  0.16     NA    NA     NA    NA     NA
#> 13  V13     2   2PLM   0.68  0.08   1.11  0.16     NA    NA     NA    NA     NA
#> 14  V14     2   2PLM   1.08  0.10  -0.54  0.09     NA    NA     NA    NA     NA
#> 15  V15     2   2PLM   0.97  0.09  -0.57  0.09     NA    NA     NA    NA     NA
#> 16  V16     2   2PLM   1.66  0.13  -0.11  0.06     NA    NA     NA    NA     NA
#> 17  V17     2   2PLM   1.06  0.10  -0.59  0.09     NA    NA     NA    NA     NA
#> 18  V18     2   2PLM   0.53  0.08   0.78  0.17     NA    NA     NA    NA     NA
#> 19  V19     2   2PLM   1.93  0.20  -1.26  0.09     NA    NA     NA    NA     NA
#> 20  V20     2   2PLM   1.47  0.18  -1.85  0.16     NA    NA     NA    NA     NA
#> 21  V21     2   2PLM   1.63  0.16  -1.26  0.09     NA    NA     NA    NA     NA
#> 22  V22     2   2PLM   0.88  0.09  -1.08  0.13     NA    NA     NA    NA     NA
#> 23  V23     2   2PLM   0.70  0.08  -0.71  0.13     NA    NA     NA    NA     NA
#> 24  V24     2   2PLM   0.44  0.07   1.22  0.24     NA    NA     NA    NA     NA
#> 25  V25     2   2PLM   0.64  0.10  -2.27  0.33     NA    NA     NA    NA     NA
#> 26  V26     2   2PLM   1.09  0.12  -2.03  0.19     NA    NA     NA    NA     NA
#> 27  V27     2   2PLM   1.02  0.09  -0.27  0.08     NA    NA     NA    NA     NA
#> 28  V28     2   2PLM   1.53  0.13  -0.51  0.07     NA    NA     NA    NA     NA
#> 29  V29     2   2PLM   1.04  0.13  -1.93  0.19     NA    NA     NA    NA     NA
#> 30  V30     2   2PLM   0.72  0.08  -0.30  0.11     NA    NA     NA    NA     NA
#> 31  V31     2   2PLM   0.66  0.08   0.37  0.12     NA    NA     NA    NA     NA
#> 32  V32     2   2PLM   1.27  0.14  -1.37  0.12     NA    NA     NA    NA     NA
#> 33  V33     2   2PLM   0.99  0.11  -1.81  0.18     NA    NA     NA    NA     NA
#> 34  V34     2   2PLM   0.90  0.09  -0.23  0.09     NA    NA     NA    NA     NA
#> 35  V35     2   2PLM   0.99  0.10  -0.72  0.10     NA    NA     NA    NA     NA
#> 36  V36     2   2PLM   0.66  0.08   0.82  0.14     NA    NA     NA    NA     NA
#> 37  V37     2   2PLM   1.56  0.13  -0.62  0.07     NA    NA     NA    NA     NA
#> 38  V38     2   2PLM   0.50  0.08  -1.45  0.26     NA    NA     NA    NA     NA
#> 39  V39     5   GPCM   1.00    NA  -1.78  0.21  -1.34  0.16  -0.74  0.13  -0.78
#> 40  V40     5   GPCM   1.00    NA  -0.58  0.11  -0.05  0.11   0.44  0.12   1.28
#> 41  V41     2   2PLM   0.79  0.08   0.06  0.10     NA    NA     NA    NA     NA
#> 42  V42     2   2PLM   1.62  0.19  -1.79  0.14     NA    NA     NA    NA     NA
#> 43  V43     2   2PLM   0.96  0.09   0.31  0.09     NA    NA     NA    NA     NA
#> 44  V44     2   2PLM   0.81  0.09  -0.68  0.12     NA    NA     NA    NA     NA
#> 45  V45     2   2PLM   0.28  0.07   3.48  0.91     NA    NA     NA    NA     NA
#> 46  V46     2   2PLM   0.54  0.08   1.98  0.30     NA    NA     NA    NA     NA
#> 47  V47     2   2PLM   1.04  0.10  -0.38  0.08     NA    NA     NA    NA     NA
#> 48  V48     2   2PLM   1.07  0.10  -0.21  0.08     NA    NA     NA    NA     NA
#> 49  V49     2   2PLM   0.92  0.09   0.04  0.09     NA    NA     NA    NA     NA
#> 50  V50     2   2PLM   0.94  0.10   1.31  0.14     NA    NA     NA    NA     NA
#> 51  V51     2   2PLM   1.48  0.14  -1.44  0.11     NA    NA     NA    NA     NA
#> 52  V52     2   2PLM   0.82  0.09  -1.22  0.15     NA    NA     NA    NA     NA
#> 53  V53     5    GRM   1.10  0.08  -0.23  0.08   0.34  0.08   0.99  0.09   1.63
#> 54  V54     5    GRM   1.17  0.09  -2.11  0.16  -1.42  0.12  -0.76  0.09  -0.11
#> 55  V55     5    GRM   0.87  0.07  -0.66  0.10   0.03  0.09   0.78  0.11   1.36
#>     se.5
#> 1     NA
#> 2     NA
#> 3     NA
#> 4     NA
#> 5     NA
#> 6     NA
#> 7     NA
#> 8     NA
#> 9     NA
#> 10    NA
#> 11    NA
#> 12    NA
#> 13    NA
#> 14    NA
#> 15    NA
#> 16    NA
#> 17    NA
#> 18    NA
#> 19    NA
#> 20    NA
#> 21    NA
#> 22    NA
#> 23    NA
#> 24    NA
#> 25    NA
#> 26    NA
#> 27    NA
#> 28    NA
#> 29    NA
#> 30    NA
#> 31    NA
#> 32    NA
#> 33    NA
#> 34    NA
#> 35    NA
#> 36    NA
#> 37    NA
#> 38    NA
#> 39  0.10
#> 40  0.13
#> 41    NA
#> 42    NA
#> 43    NA
#> 44    NA
#> 45    NA
#> 46    NA
#> 47    NA
#> 48    NA
#> 49    NA
#> 50    NA
#> 51    NA
#> 52    NA
#> 53  0.13
#> 54  0.07
#> 55  0.14
#>  Group Parameters: 
#>            mu  sigma2  sigma
#> estimates   0       1      1
#> se         NA      NA     NA
#> 

## ----------------------------------------------------------------
## 3. Fixed item parameter calibration (FIPC) for mixed-format data
##    (simulated)
## ----------------------------------------------------------------
## Import the "-prm.txt" output file from flexMIRT
flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt",
                        package = "irtQ")

# Select item metadata
x <- bring.flexmirt(file = flex_sam, "par")$Group1$full_df

# Generate 1,000 examinees' latent abilities from N(0.4, 1.3)
set.seed(20)
score2 <- rnorm(1000, mean = 0.4, sd = 1.3)

# Simulate response data
sim.dat2 <- simdat(x = x, theta = score2, D = 1)

# Fit the 3PL model to all dichotomous items and the GRM to all polytomous
# items
# Fix five 3PL items (1st - 5th) and three GRM items (53rd - 55th)
# Also estimate the empirical histogram of the latent variable distribution
# Use the MEM method
fix.loc <- c(1:5, 53:55)
(mod.fix1 <- est_irt(
  x = x, data = sim.dat2, D = 1, use.gprior = TRUE,
  gprior = list(dist = "beta", params = c(5, 16)), EmpHist = TRUE,
  Etol = 1e-3, fipc = TRUE, fipc.method = "MEM", fix.loc = fix.loc
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -6799.7353, Max-Change: 3.222649
 EM iteration: 2, Loglike: -31337.6995, Max-Change: 0.867069
 EM iteration: 3, Loglike: -31073.9672, Max-Change: 0.237885
 EM iteration: 4, Loglike: -31063.5310, Max-Change: 0.099662
 EM iteration: 5, Loglike: -31059.2570, Max-Change: 0.07400
 EM iteration: 6, Loglike: -31056.5795, Max-Change: 0.054547
 EM iteration: 7, Loglike: -31054.7969, Max-Change: 0.040231
 EM iteration: 8, Loglike: -31053.5770, Max-Change: 0.02986
 EM iteration: 9, Loglike: -31052.7248, Max-Change: 0.022385
 EM iteration: 10, Loglike: -31052.1188, Max-Change: 0.016975
 EM iteration: 11, Loglike: -31051.6809, Max-Change: 0.013019
 EM iteration: 12, Loglike: -31051.3592, Max-Change: 0.010476
 EM iteration: 13, Loglike: -31051.1191, Max-Change: 0.008984
 EM iteration: 14, Loglike: -31050.9366, Max-Change: 0.008296
 EM iteration: 15, Loglike: -31050.7953, Max-Change: 0.007586
 EM iteration: 16, Loglike: -31050.6836, Max-Change: 0.00689
 EM iteration: 17, Loglike: -31050.5935, Max-Change: 0.006231
 EM iteration: 18, Loglike: -31050.5191, Max-Change: 0.005619
 EM iteration: 19, Loglike: -31050.4563, Max-Change: 0.005059
 EM iteration: 20, Loglike: -31050.4021, Max-Change: 0.004551
 EM iteration: 21, Loglike: -31050.3545, Max-Change: 0.004095
 EM iteration: 22, Loglike: -31050.3118, Max-Change: 0.003686
 EM iteration: 23, Loglike: -31050.2729, Max-Change: 0.00332
 EM iteration: 24, Loglike: -31050.2371, Max-Change: 0.002993
 EM iteration: 25, Loglike: -31050.2037, Max-Change: 0.002701
 EM iteration: 26, Loglike: -31050.1723, Max-Change: 0.00244
 EM iteration: 27, Loglike: -31050.1425, Max-Change: 0.002207
 EM iteration: 28, Loglike: -31050.1142, Max-Change: 0.001997
 EM iteration: 29, Loglike: -31050.0870, Max-Change: 0.00181
 EM iteration: 30, Loglike: -31050.0609, Max-Change: 0.001641
 EM iteration: 31, Loglike: -31050.0358, Max-Change: 0.001488
 EM iteration: 32, Loglike: -31050.0115, Max-Change: 0.001351
 EM iteration: 33, Loglike: -31049.9881, Max-Change: 0.001227
 EM iteration: 34, Loglike: -31049.9654, Max-Change: 0.001115
 EM iteration: 35, Loglike: -31049.9433, Max-Change: 0.001013
 EM iteration: 36, Loglike: -31049.9220, Max-Change: 0.00092 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 2.94 seconds. 
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = TRUE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 36 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -31049.92
#> 

# Extract group-level parameter estimates
(prior.par <- mod.fix1$group.par)
#>                   mu     sigma2      sigma
#> estimates 0.39695509 1.87889646 1.37072844
#> se        0.04334624 0.08406885 0.03066576

# Visualize the empirical prior distribution
(emphist <- getirt(mod.fix1, what = "weights"))
#>    theta       weight
#> 1  -6.00 2.301249e-10
#> 2  -5.75 1.434594e-09
#> 3  -5.50 8.649272e-09
#> 4  -5.25 5.019863e-08
#> 5  -5.00 2.782909e-07
#> 6  -4.75 1.456750e-06
#> 7  -4.50 7.085624e-06
#> 8  -4.25 3.134805e-05
#> 9  -4.00 1.227011e-04
#> 10 -3.75 4.100601e-04
#> 11 -3.50 1.119741e-03
#> 12 -3.25 2.386353e-03
#> 13 -3.00 3.889321e-03
#> 14 -2.75 5.167719e-03
#> 15 -2.50 6.767224e-03
#> 16 -2.25 1.053445e-02
#> 17 -2.00 1.742567e-02
#> 18 -1.75 2.236173e-02
#> 19 -1.50 2.498734e-02
#> 20 -1.25 3.374293e-02
#> 21 -1.00 4.224443e-02
#> 22 -0.75 4.948940e-02
#> 23 -0.50 7.710828e-02
#> 24 -0.25 8.024288e-02
#> 25  0.00 4.752956e-02
#> 26  0.25 5.636581e-02
#> 27  0.50 8.674316e-02
#> 28  0.75 6.936383e-02
#> 29  1.00 6.035964e-02
#> 30  1.25 6.234864e-02
#> 31  1.50 5.768393e-02
#> 32  1.75 4.254008e-02
#> 33  2.00 3.148380e-02
#> 34  2.25 3.111071e-02
#> 35  2.50 2.935626e-02
#> 36  2.75 1.950494e-02
#> 37  3.00 1.019680e-02
#> 38  3.25 5.154557e-03
#> 39  3.50 2.816466e-03
#> 40  3.75 1.753617e-03
#> 41  4.00 1.282170e-03
#> 42  4.25 1.097172e-03
#> 43  4.50 1.050444e-03
#> 44  4.75 1.046478e-03
#> 45  5.00 1.004637e-03
#> 46  5.25 8.721994e-04
#> 47  5.50 6.557355e-04
#> 48  5.75 4.168380e-04
#> 49  6.00 2.220842e-04
plot(emphist$weight ~ emphist$theta, type = "h")


# Display a summary of the estimation results
summary(mod.fix1)
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = TRUE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 0.001
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 193
#>    - Item parameters: 145
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 48
#>  Number of fixed items: 8
#>  Number of E-step cycles completed: 36
#>  Maximum parameter change: 0.000920385
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 2.84
#>  Standard error computation: 0.04
#>  Total computation: 2.94
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 62099.84
#>  Akaike Information Criterion (AIC): 62485.84
#>  Bayesian Information Criterion (BIC): 63433.04
#>  Item Parameters: 
#>        id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1    CMC1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2    CMC2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3    CMC3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4    CMC4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5    CMC5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6    CMC6     2   3PLM   1.47  0.15   0.61  0.09   0.07  0.03     NA    NA
#> 7    CMC7     2   3PLM   1.45  0.25   1.23  0.14   0.24  0.04     NA    NA
#> 8    CMC8     2   3PLM   0.80  0.11   0.82  0.21   0.12  0.05     NA    NA
#> 9    CMC9     2   3PLM   0.81  0.13   0.63  0.28   0.20  0.07     NA    NA
#> 10  CMC10     2   3PLM   1.55  0.21   0.16  0.14   0.18  0.05     NA    NA
#> 11  CMC11     2   3PLM   0.99  0.17  -0.01  0.33   0.32  0.08     NA    NA
#> 12  CMC12     2   3PLM   0.86  0.13   1.30  0.18   0.11  0.04     NA    NA
#> 13  CMC13     2   3PLM   1.48  0.26   1.61  0.12   0.18  0.03     NA    NA
#> 14  CMC14     2   3PLM   1.53  0.21   0.25  0.16   0.27  0.05     NA    NA
#> 15  CMC15     2   3PLM   1.53  0.18  -0.11  0.13   0.14  0.05     NA    NA
#> 16  CMC16     2   3PLM   2.16  0.22   0.02  0.07   0.08  0.03     NA    NA
#> 17  CMC17     2   3PLM   1.39  0.19   0.03  0.17   0.20  0.06     NA    NA
#> 18  CMC18     2   3PLM   1.36  0.27   1.34  0.16   0.27  0.04     NA    NA
#> 19  CMC19     2   3PLM   2.48  0.37  -0.94  0.12   0.19  0.06     NA    NA
#> 20  CMC20     2   3PLM   1.80  0.37  -1.21  0.26   0.40  0.10     NA    NA
#> 21  CMC21     2   3PLM   1.76  0.22  -0.98  0.17   0.21  0.07     NA    NA
#> 22  CMC22     2   3PLM   0.94  0.13  -0.51  0.27   0.19  0.08     NA    NA
#> 23  CMC23     2   3PLM   0.83  0.10  -0.37  0.23   0.13  0.06     NA    NA
#> 24  CMC24     2   3PLM   0.98  0.21   1.86  0.20   0.22  0.04     NA    NA
#> 25  CMC25     2   3PLM   0.63  0.09  -2.01  0.47   0.21  0.09     NA    NA
#> 26  CMC26     2   3PLM   1.13  0.14  -1.68  0.28   0.22  0.09     NA    NA
#> 27  CMC27     2   3PLM   1.19  0.14   0.01  0.16   0.14  0.05     NA    NA
#> 28  CMC28     2   3PLM   2.23  0.26  -0.13  0.09   0.15  0.04     NA    NA
#> 29  CMC29     2   3PLM   1.31  0.16  -1.32  0.22   0.19  0.08     NA    NA
#> 30  CMC30     2   3PLM   1.63  0.30   1.03  0.15   0.37  0.04     NA    NA
#> 31  CMC31     2   3PLM   1.03  0.15   0.93  0.17   0.15  0.05     NA    NA
#> 32  CMC32     2   3PLM   1.55  0.21  -0.75  0.20   0.26  0.08     NA    NA
#> 33  CMC33     2   3PLM   1.24  0.19  -1.09  0.30   0.31  0.10     NA    NA
#> 34  CMC34     2   3PLM   1.34  0.16   0.31  0.15   0.17  0.05     NA    NA
#> 35  CMC35     2   3PLM   1.24  0.15  -0.36  0.20   0.19  0.07     NA    NA
#> 36  CMC36     2   3PLM   1.06  0.17   1.05  0.17   0.15  0.05     NA    NA
#> 37  CMC37     2   3PLM   2.11  0.26  -0.29  0.11   0.16  0.05     NA    NA
#> 38  CMC38     2   3PLM   0.57  0.11  -0.30  0.55   0.26  0.10     NA    NA
#> 39   CFR1     5    GRM   2.09  0.13  -1.81  0.10  -1.14  0.07  -0.68  0.06
#> 40   CFR2     5    GRM   1.38  0.08  -0.70  0.08  -0.08  0.07   0.48  0.06
#> 41   AMC1     2   3PLM   1.25  0.18   0.62  0.16   0.18  0.05     NA    NA
#> 42   AMC2     2   3PLM   1.79  0.22  -1.61  0.18   0.17  0.07     NA    NA
#> 43   AMC3     2   3PLM   1.37  0.17   0.64  0.12   0.12  0.04     NA    NA
#> 44   AMC4     2   3PLM   0.94  0.11  -0.22  0.23   0.16  0.06     NA    NA
#> 45   AMC5     2   3PLM   1.11  0.33   2.83  0.26   0.21  0.03     NA    NA
#> 46   AMC6     2   3PLM   2.22  0.37   1.70  0.09   0.19  0.02     NA    NA
#> 47   AMC7     2   3PLM   1.16  0.13   0.02  0.14   0.10  0.04     NA    NA
#> 48   AMC8     2   3PLM   1.31  0.16   0.33  0.15   0.18  0.05     NA    NA
#> 49   AMC9     2   3PLM   1.22  0.13   0.30  0.12   0.09  0.04     NA    NA
#> 50  AMC10     2   3PLM   1.83  0.28   1.48  0.09   0.15  0.03     NA    NA
#> 51  AMC11     2   3PLM   1.68  0.22  -1.08  0.17   0.19  0.07     NA    NA
#> 52  AMC12     2   3PLM   0.91  0.13  -0.82  0.35   0.26  0.09     NA    NA
#> 53   AFR1     5    GRM   1.14    NA  -0.37    NA   0.22    NA   0.85    NA
#> 54   AFR2     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 55   AFR3     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#>     par.5  se.5
#> 1      NA    NA
#> 2      NA    NA
#> 3      NA    NA
#> 4      NA    NA
#> 5      NA    NA
#> 6      NA    NA
#> 7      NA    NA
#> 8      NA    NA
#> 9      NA    NA
#> 10     NA    NA
#> 11     NA    NA
#> 12     NA    NA
#> 13     NA    NA
#> 14     NA    NA
#> 15     NA    NA
#> 16     NA    NA
#> 17     NA    NA
#> 18     NA    NA
#> 19     NA    NA
#> 20     NA    NA
#> 21     NA    NA
#> 22     NA    NA
#> 23     NA    NA
#> 24     NA    NA
#> 25     NA    NA
#> 26     NA    NA
#> 27     NA    NA
#> 28     NA    NA
#> 29     NA    NA
#> 30     NA    NA
#> 31     NA    NA
#> 32     NA    NA
#> 33     NA    NA
#> 34     NA    NA
#> 35     NA    NA
#> 36     NA    NA
#> 37     NA    NA
#> 38     NA    NA
#> 39  -0.24  0.05
#> 40   1.05  0.07
#> 41     NA    NA
#> 42     NA    NA
#> 43     NA    NA
#> 44     NA    NA
#> 45     NA    NA
#> 46     NA    NA
#> 47     NA    NA
#> 48     NA    NA
#> 49     NA    NA
#> 50     NA    NA
#> 51     NA    NA
#> 52     NA    NA
#> 53   1.38    NA
#> 54  -0.12    NA
#> 55   1.25    NA
#>  Group Parameters: 
#>              mu  sigma2  sigma
#> estimates  0.40    1.88   1.37
#> se         0.04    0.08   0.03
#> 

# Alternatively, fix the same items by providing their item IDs
# using the `fix.id` argument. In this case, set `fix.loc = NULL`
fix.id <- c(x$id[1:5], x$id[53:55])
(mod.fix1 <- est_irt(
  x = x, data = sim.dat2, D = 1, use.gprior = TRUE,
  gprior = list(dist = "beta", params = c(5, 16)), EmpHist = TRUE,
  Etol = 1e-3, fipc = TRUE, fipc.method = "MEM", fix.loc = NULL,
  fix.id = fix.id
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -6799.7353, Max-Change: 3.222649
 EM iteration: 2, Loglike: -31337.6995, Max-Change: 0.867069
 EM iteration: 3, Loglike: -31073.9672, Max-Change: 0.237885
 EM iteration: 4, Loglike: -31063.5310, Max-Change: 0.099662
 EM iteration: 5, Loglike: -31059.2570, Max-Change: 0.07400
 EM iteration: 6, Loglike: -31056.5795, Max-Change: 0.054547
 EM iteration: 7, Loglike: -31054.7969, Max-Change: 0.040231
 EM iteration: 8, Loglike: -31053.5770, Max-Change: 0.02986
 EM iteration: 9, Loglike: -31052.7248, Max-Change: 0.022385
 EM iteration: 10, Loglike: -31052.1188, Max-Change: 0.016975
 EM iteration: 11, Loglike: -31051.6809, Max-Change: 0.013019
 EM iteration: 12, Loglike: -31051.3592, Max-Change: 0.010476
 EM iteration: 13, Loglike: -31051.1191, Max-Change: 0.008984
 EM iteration: 14, Loglike: -31050.9366, Max-Change: 0.008296
 EM iteration: 15, Loglike: -31050.7953, Max-Change: 0.007586
 EM iteration: 16, Loglike: -31050.6836, Max-Change: 0.00689
 EM iteration: 17, Loglike: -31050.5935, Max-Change: 0.006231
 EM iteration: 18, Loglike: -31050.5191, Max-Change: 0.005619
 EM iteration: 19, Loglike: -31050.4563, Max-Change: 0.005059
 EM iteration: 20, Loglike: -31050.4021, Max-Change: 0.004551
 EM iteration: 21, Loglike: -31050.3545, Max-Change: 0.004095
 EM iteration: 22, Loglike: -31050.3118, Max-Change: 0.003686
 EM iteration: 23, Loglike: -31050.2729, Max-Change: 0.00332
 EM iteration: 24, Loglike: -31050.2371, Max-Change: 0.002993
 EM iteration: 25, Loglike: -31050.2037, Max-Change: 0.002701
 EM iteration: 26, Loglike: -31050.1723, Max-Change: 0.00244
 EM iteration: 27, Loglike: -31050.1425, Max-Change: 0.002207
 EM iteration: 28, Loglike: -31050.1142, Max-Change: 0.001997
 EM iteration: 29, Loglike: -31050.0870, Max-Change: 0.00181
 EM iteration: 30, Loglike: -31050.0609, Max-Change: 0.001641
 EM iteration: 31, Loglike: -31050.0358, Max-Change: 0.001488
 EM iteration: 32, Loglike: -31050.0115, Max-Change: 0.001351
 EM iteration: 33, Loglike: -31049.9881, Max-Change: 0.001227
 EM iteration: 34, Loglike: -31049.9654, Max-Change: 0.001115
 EM iteration: 35, Loglike: -31049.9433, Max-Change: 0.001013
 EM iteration: 36, Loglike: -31049.9220, Max-Change: 0.00092 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 2.96 seconds. 
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = TRUE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = NULL, fix.id = fix.id)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 36 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -31049.92
#> 

# Display a summary of the estimation results
summary(mod.fix1)
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = TRUE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = NULL, fix.id = fix.id)
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 0.001
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 193
#>    - Item parameters: 145
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 48
#>  Number of fixed items: 8
#>  Number of E-step cycles completed: 36
#>  Maximum parameter change: 0.000920385
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 2.85
#>  Standard error computation: 0.04
#>  Total computation: 2.96
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 62099.84
#>  Akaike Information Criterion (AIC): 62485.84
#>  Bayesian Information Criterion (BIC): 63433.04
#>  Item Parameters: 
#>        id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1    CMC1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2    CMC2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3    CMC3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4    CMC4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5    CMC5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6    CMC6     2   3PLM   1.47  0.15   0.61  0.09   0.07  0.03     NA    NA
#> 7    CMC7     2   3PLM   1.45  0.25   1.23  0.14   0.24  0.04     NA    NA
#> 8    CMC8     2   3PLM   0.80  0.11   0.82  0.21   0.12  0.05     NA    NA
#> 9    CMC9     2   3PLM   0.81  0.13   0.63  0.28   0.20  0.07     NA    NA
#> 10  CMC10     2   3PLM   1.55  0.21   0.16  0.14   0.18  0.05     NA    NA
#> 11  CMC11     2   3PLM   0.99  0.17  -0.01  0.33   0.32  0.08     NA    NA
#> 12  CMC12     2   3PLM   0.86  0.13   1.30  0.18   0.11  0.04     NA    NA
#> 13  CMC13     2   3PLM   1.48  0.26   1.61  0.12   0.18  0.03     NA    NA
#> 14  CMC14     2   3PLM   1.53  0.21   0.25  0.16   0.27  0.05     NA    NA
#> 15  CMC15     2   3PLM   1.53  0.18  -0.11  0.13   0.14  0.05     NA    NA
#> 16  CMC16     2   3PLM   2.16  0.22   0.02  0.07   0.08  0.03     NA    NA
#> 17  CMC17     2   3PLM   1.39  0.19   0.03  0.17   0.20  0.06     NA    NA
#> 18  CMC18     2   3PLM   1.36  0.27   1.34  0.16   0.27  0.04     NA    NA
#> 19  CMC19     2   3PLM   2.48  0.37  -0.94  0.12   0.19  0.06     NA    NA
#> 20  CMC20     2   3PLM   1.80  0.37  -1.21  0.26   0.40  0.10     NA    NA
#> 21  CMC21     2   3PLM   1.76  0.22  -0.98  0.17   0.21  0.07     NA    NA
#> 22  CMC22     2   3PLM   0.94  0.13  -0.51  0.27   0.19  0.08     NA    NA
#> 23  CMC23     2   3PLM   0.83  0.10  -0.37  0.23   0.13  0.06     NA    NA
#> 24  CMC24     2   3PLM   0.98  0.21   1.86  0.20   0.22  0.04     NA    NA
#> 25  CMC25     2   3PLM   0.63  0.09  -2.01  0.47   0.21  0.09     NA    NA
#> 26  CMC26     2   3PLM   1.13  0.14  -1.68  0.28   0.22  0.09     NA    NA
#> 27  CMC27     2   3PLM   1.19  0.14   0.01  0.16   0.14  0.05     NA    NA
#> 28  CMC28     2   3PLM   2.23  0.26  -0.13  0.09   0.15  0.04     NA    NA
#> 29  CMC29     2   3PLM   1.31  0.16  -1.32  0.22   0.19  0.08     NA    NA
#> 30  CMC30     2   3PLM   1.63  0.30   1.03  0.15   0.37  0.04     NA    NA
#> 31  CMC31     2   3PLM   1.03  0.15   0.93  0.17   0.15  0.05     NA    NA
#> 32  CMC32     2   3PLM   1.55  0.21  -0.75  0.20   0.26  0.08     NA    NA
#> 33  CMC33     2   3PLM   1.24  0.19  -1.09  0.30   0.31  0.10     NA    NA
#> 34  CMC34     2   3PLM   1.34  0.16   0.31  0.15   0.17  0.05     NA    NA
#> 35  CMC35     2   3PLM   1.24  0.15  -0.36  0.20   0.19  0.07     NA    NA
#> 36  CMC36     2   3PLM   1.06  0.17   1.05  0.17   0.15  0.05     NA    NA
#> 37  CMC37     2   3PLM   2.11  0.26  -0.29  0.11   0.16  0.05     NA    NA
#> 38  CMC38     2   3PLM   0.57  0.11  -0.30  0.55   0.26  0.10     NA    NA
#> 39   CFR1     5    GRM   2.09  0.13  -1.81  0.10  -1.14  0.07  -0.68  0.06
#> 40   CFR2     5    GRM   1.38  0.08  -0.70  0.08  -0.08  0.07   0.48  0.06
#> 41   AMC1     2   3PLM   1.25  0.18   0.62  0.16   0.18  0.05     NA    NA
#> 42   AMC2     2   3PLM   1.79  0.22  -1.61  0.18   0.17  0.07     NA    NA
#> 43   AMC3     2   3PLM   1.37  0.17   0.64  0.12   0.12  0.04     NA    NA
#> 44   AMC4     2   3PLM   0.94  0.11  -0.22  0.23   0.16  0.06     NA    NA
#> 45   AMC5     2   3PLM   1.11  0.33   2.83  0.26   0.21  0.03     NA    NA
#> 46   AMC6     2   3PLM   2.22  0.37   1.70  0.09   0.19  0.02     NA    NA
#> 47   AMC7     2   3PLM   1.16  0.13   0.02  0.14   0.10  0.04     NA    NA
#> 48   AMC8     2   3PLM   1.31  0.16   0.33  0.15   0.18  0.05     NA    NA
#> 49   AMC9     2   3PLM   1.22  0.13   0.30  0.12   0.09  0.04     NA    NA
#> 50  AMC10     2   3PLM   1.83  0.28   1.48  0.09   0.15  0.03     NA    NA
#> 51  AMC11     2   3PLM   1.68  0.22  -1.08  0.17   0.19  0.07     NA    NA
#> 52  AMC12     2   3PLM   0.91  0.13  -0.82  0.35   0.26  0.09     NA    NA
#> 53   AFR1     5    GRM   1.14    NA  -0.37    NA   0.22    NA   0.85    NA
#> 54   AFR2     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 55   AFR3     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#>     par.5  se.5
#> 1      NA    NA
#> 2      NA    NA
#> 3      NA    NA
#> 4      NA    NA
#> 5      NA    NA
#> 6      NA    NA
#> 7      NA    NA
#> 8      NA    NA
#> 9      NA    NA
#> 10     NA    NA
#> 11     NA    NA
#> 12     NA    NA
#> 13     NA    NA
#> 14     NA    NA
#> 15     NA    NA
#> 16     NA    NA
#> 17     NA    NA
#> 18     NA    NA
#> 19     NA    NA
#> 20     NA    NA
#> 21     NA    NA
#> 22     NA    NA
#> 23     NA    NA
#> 24     NA    NA
#> 25     NA    NA
#> 26     NA    NA
#> 27     NA    NA
#> 28     NA    NA
#> 29     NA    NA
#> 30     NA    NA
#> 31     NA    NA
#> 32     NA    NA
#> 33     NA    NA
#> 34     NA    NA
#> 35     NA    NA
#> 36     NA    NA
#> 37     NA    NA
#> 38     NA    NA
#> 39  -0.24  0.05
#> 40   1.05  0.07
#> 41     NA    NA
#> 42     NA    NA
#> 43     NA    NA
#> 44     NA    NA
#> 45     NA    NA
#> 46     NA    NA
#> 47     NA    NA
#> 48     NA    NA
#> 49     NA    NA
#> 50     NA    NA
#> 51     NA    NA
#> 52     NA    NA
#> 53   1.38    NA
#> 54  -0.12    NA
#> 55   1.25    NA
#>  Group Parameters: 
#>              mu  sigma2  sigma
#> estimates  0.40    1.88   1.37
#> se         0.04    0.08   0.03
#> 

# Fit the 3PL model to all dichotomous items and the GRM to all polytomous
# items
# Fix the same items as before (1st - 5th and 53rd - 55th)
# This time, do not estimate the empirical histogram of the latent prior
# Instead, estimate the scale of the normal prior distribution
# Use the MEM method
fix.loc <- c(1:5, 53:55)
(mod.fix2 <- est_irt(
  x = x, data = sim.dat2, D = 1, use.gprior = TRUE,
  gprior = list(dist = "beta", params = c(5, 16)), EmpHist = FALSE,
  Etol = 1e-3, fipc = TRUE, fipc.method = "MEM", fix.loc = fix.loc
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -6800.6017, Max-Change: 3.222649
 EM iteration: 2, Loglike: -31330.2705, Max-Change: 0.799829
 EM iteration: 3, Loglike: -31073.8931, Max-Change: 0.197432
 EM iteration: 4, Loglike: -31063.4621, Max-Change: 0.083067
 EM iteration: 5, Loglike: -31059.7618, Max-Change: 0.048836
 EM iteration: 6, Loglike: -31057.5133, Max-Change: 0.033394
 EM iteration: 7, Loglike: -31056.0049, Max-Change: 0.026974
 EM iteration: 8, Loglike: -31054.9624, Max-Change: 0.022481
 EM iteration: 9, Loglike: -31054.2321, Max-Change: 0.018797
 EM iteration: 10, Loglike: -31053.7165, Max-Change: 0.015766
 EM iteration: 11, Loglike: -31053.3505, Max-Change: 0.013259
 EM iteration: 12, Loglike: -31053.0897, Max-Change: 0.011177
 EM iteration: 13, Loglike: -31052.9032, Max-Change: 0.009441
 EM iteration: 14, Loglike: -31052.7695, Max-Change: 0.007988
 EM iteration: 15, Loglike: -31052.6735, Max-Change: 0.006769
 EM iteration: 16, Loglike: -31052.6045, Max-Change: 0.005743
 EM iteration: 17, Loglike: -31052.5547, Max-Change: 0.004878
 EM iteration: 18, Loglike: -31052.5187, Max-Change: 0.004148
 EM iteration: 19, Loglike: -31052.4928, Max-Change: 0.003556
 EM iteration: 20, Loglike: -31052.4740, Max-Change: 0.003059
 EM iteration: 21, Loglike: -31052.4604, Max-Change: 0.002633
 EM iteration: 22, Loglike: -31052.4506, Max-Change: 0.002267
 EM iteration: 23, Loglike: -31052.4435, Max-Change: 0.001953
 EM iteration: 24, Loglike: -31052.4383, Max-Change: 0.001683
 EM iteration: 25, Loglike: -31052.4346, Max-Change: 0.001451
 EM iteration: 26, Loglike: -31052.4319, Max-Change: 0.001252
 EM iteration: 27, Loglike: -31052.4300, Max-Change: 0.00108
 EM iteration: 28, Loglike: -31052.4286, Max-Change: 0.000933 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 2.49 seconds. 
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = FALSE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 28 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -31052.43
#> 

# Extract group-level parameter estimates
(prior.par <- mod.fix2$group.par)
#>                   mu     sigma2      sigma
#> estimates 0.39326312 1.86244126 1.36471289
#> se        0.04315601 0.08333258 0.03053118

# Visualize the prior distribution
(emphist <- getirt(mod.fix2, what = "weights"))
#>    theta       weight
#> 1  -6.00 1.256938e-06
#> 2  -5.75 2.915002e-06
#> 3  -5.50 6.537229e-06
#> 4  -5.25 1.417680e-05
#> 5  -5.00 2.972983e-05
#> 6  -4.75 6.028877e-05
#> 7  -4.50 1.182252e-04
#> 8  -4.25 2.241886e-04
#> 9  -4.00 4.110994e-04
#> 10 -3.75 7.289703e-04
#> 11 -3.50 1.249978e-03
#> 12 -3.25 2.072645e-03
#> 13 -3.00 3.323359e-03
#> 14 -2.75 5.152988e-03
#> 15 -2.50 7.726287e-03
#> 16 -2.25 1.120243e-02
#> 17 -2.00 1.570665e-02
#> 18 -1.75 2.129533e-02
#> 19 -1.50 2.791998e-02
#> 20 -1.25 3.539774e-02
#> 21 -1.00 4.339760e-02
#> 22 -0.75 5.145003e-02
#> 23 -0.50 5.898414e-02
#> 24 -0.25 6.539050e-02
#> 25  0.00 7.010094e-02
#> 26  0.25 7.267127e-02
#> 27  0.50 7.285030e-02
#> 28  0.75 7.062033e-02
#> 29  1.00 6.619999e-02
#> 30  1.25 6.000891e-02
#> 31  1.50 5.260214e-02
#> 32  1.75 4.458829e-02
#> 33  2.00 3.654836e-02
#> 34  2.25 2.896975e-02
#> 35  2.50 2.220503e-02
#> 36  2.75 1.645840e-02
#> 37  3.00 1.179652e-02
#> 38  3.25 8.176165e-03
#> 39  3.50 5.479933e-03
#> 40  3.75 3.551653e-03
#> 41  4.00 2.225951e-03
#> 42  4.25 1.349058e-03
#> 43  4.50 7.906332e-04
#> 44  4.75 4.480736e-04
#> 45  5.00 2.455576e-04
#> 46  5.25 1.301329e-04
#> 47  5.50 6.668847e-05
#> 48  5.75 3.304792e-05
#> 49  6.00 1.583679e-05
plot(emphist$weight ~ emphist$theta, type = "h")


# Fit the 3PL model to all dichotomous items and the GRM to all polytomous
# items
# Fix only the five 3PL items (1st to 5th) and estimate the empirical
# histogram
# Use the OEM method (i.e., only one EM cycle is used)
fix.loc <- c(1:5)
(mod.fix3 <- est_irt(
  x = x, data = sim.dat2, D = 1, use.gprior = TRUE,
  gprior = list(dist = "beta", params = c(5, 16)), EmpHist = TRUE,
  Etol = 1e-3, fipc = TRUE, fipc.method = "OEM", fix.loc = fix.loc
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -2966.5364, Max-Change: 4.112978 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 0.5 seconds. 
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = TRUE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "OEM", fix.loc = fix.loc)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 1 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: Solution is a possible local maximum.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Log-likelihood: -32191.75
#> 

# Extract group-level parameter estimates
(prior.par <- mod.fix3$group.par)
#>                   mu    sigma2      sigma
#> estimates 0.11933891 1.0777865 1.03816496
#> se        0.03282966 0.0482242 0.02322569

# Visualize the prior distribution
(emphist <- getirt(mod.fix3, what = "weights"))
#>    theta       weight
#> 1  -6.00 1.504352e-09
#> 2  -5.75 6.532989e-09
#> 3  -5.50 2.664992e-08
#> 4  -5.25 1.021152e-07
#> 5  -5.00 3.675193e-07
#> 6  -4.75 1.242357e-06
#> 7  -4.50 3.944238e-06
#> 8  -4.25 1.175972e-05
#> 9  -4.00 3.292330e-05
#> 10 -3.75 8.654148e-05
#> 11 -3.50 2.135422e-04
#> 12 -3.25 4.945193e-04
#> 13 -3.00 1.074478e-03
#> 14 -2.75 2.189644e-03
#> 15 -2.50 4.183470e-03
#> 16 -2.25 7.490661e-03
#> 17 -2.00 1.256715e-02
#> 18 -1.75 1.976148e-02
#> 19 -1.50 2.916253e-02
#> 20 -1.25 4.049533e-02
#> 21 -1.00 5.311614e-02
#> 22 -0.75 6.606955e-02
#> 23 -0.50 7.813346e-02
#> 24 -0.25 8.787908e-02
#> 25  0.00 9.385941e-02
#> 26  0.25 9.495165e-02
#> 27  0.50 9.072679e-02
#> 28  0.75 8.166412e-02
#> 29  1.00 6.908936e-02
#> 30  1.25 5.483993e-02
#> 31  1.50 4.078625e-02
#> 32  1.75 2.839721e-02
#> 33  2.00 1.849942e-02
#> 34  2.25 1.127370e-02
#> 35  2.50 6.426944e-03
#> 36  2.75 3.428038e-03
#> 37  3.00 1.711230e-03
#> 38  3.25 7.997202e-04
#> 39  3.50 3.500189e-04
#> 40  3.75 1.435247e-04
#> 41  4.00 5.515645e-05
#> 42  4.25 1.987202e-05
#> 43  4.50 6.714177e-06
#> 44  4.75 2.127959e-06
#> 45  5.00 6.327824e-07
#> 46  5.25 1.765846e-07
#> 47  5.50 4.625232e-08
#> 48  5.75 1.137259e-08
#> 49  6.00 2.625333e-09
plot(emphist$weight ~ emphist$theta, type = "h")


# Display a summary of the estimation results
summary(mod.fix3)
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, use.gprior = TRUE, gprior = list(dist = "beta", 
#>     params = c(5, 16)), EmpHist = TRUE, Etol = 0.001, fipc = TRUE, 
#>     fipc.method = "OEM", fix.loc = fix.loc)
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 1
#>  Convergence criterion of E-step: 0.001
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 208
#>    - Item parameters: 160
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 48
#>  Number of fixed items: 5
#>  Number of E-step cycles completed: 1
#>  Maximum parameter change: 4.112978
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.37
#>  Standard error computation: 0.05
#>  Total computation: 0.5
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: Solution is a possible local maximum.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 64383.51
#>  Akaike Information Criterion (AIC): 64799.51
#>  Bayesian Information Criterion (BIC): 65820.32
#>  Item Parameters: 
#>        id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1    CMC1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2    CMC2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3    CMC3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4    CMC4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5    CMC5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6    CMC6     2   3PLM   0.70  0.12   0.68  0.27   0.13  0.06     NA    NA
#> 7    CMC7     2   3PLM   0.53  0.13   1.25  0.43   0.19  0.07     NA    NA
#> 8    CMC8     2   3PLM   0.48  0.10   1.11  0.47   0.18  0.08     NA    NA
#> 9    CMC9     2   3PLM   0.50  0.09   0.37  0.47   0.18  0.08     NA    NA
#> 10  CMC10     2   3PLM   0.73  0.12  -0.22  0.33   0.18  0.07     NA    NA
#> 11  CMC11     2   3PLM   0.44  0.08  -1.20  0.60   0.20  0.09     NA    NA
#> 12  CMC12     2   3PLM   0.58  0.11   1.42  0.31   0.14  0.06     NA    NA
#> 13  CMC13     2   3PLM   0.52  0.14   2.12  0.37   0.16  0.06     NA    NA
#> 14  CMC14     2   3PLM   0.60  0.11  -0.43  0.44   0.20  0.08     NA    NA
#> 15  CMC15     2   3PLM   0.73  0.12  -0.46  0.33   0.17  0.07     NA    NA
#> 16  CMC16     2   3PLM   0.89  0.15  -0.19  0.24   0.14  0.06     NA    NA
#> 17  CMC17     2   3PLM   0.64  0.12  -0.38  0.42   0.20  0.08     NA    NA
#> 18  CMC18     2   3PLM   0.54  0.11   0.99  0.40   0.18  0.07     NA    NA
#> 19  CMC19     2   3PLM   0.95  0.21  -1.57  0.35   0.23  0.09     NA    NA
#> 20  CMC20     2   3PLM   0.71  0.16  -2.88  0.51   0.22  0.10     NA    NA
#> 21  CMC21     2   3PLM   0.85  0.14  -1.67  0.38   0.23  0.09     NA    NA
#> 22  CMC22     2   3PLM   0.57  0.09  -1.08  0.45   0.19  0.08     NA    NA
#> 23  CMC23     2   3PLM   0.54  0.08  -0.64  0.47   0.19  0.08     NA    NA
#> 24  CMC24     2   3PLM   0.41  0.14   2.77  0.58   0.21  0.08     NA    NA
#> 25  CMC25     2   3PLM   0.41  0.07  -3.25  0.74   0.21  0.09     NA    NA
#> 26  CMC26     2   3PLM   0.66  0.10  -2.69  0.47   0.22  0.09     NA    NA
#> 27  CMC27     2   3PLM   0.58  0.10  -0.25  0.42   0.18  0.08     NA    NA
#> 28  CMC28     2   3PLM   0.87  0.16  -0.50  0.29   0.18  0.08     NA    NA
#> 29  CMC29     2   3PLM   0.68  0.11  -2.26  0.44   0.21  0.09     NA    NA
#> 30  CMC30     2   3PLM   0.41  0.09   0.18  0.62   0.20  0.09     NA    NA
#> 31  CMC31     2   3PLM   0.50  0.11   1.21  0.45   0.18  0.08     NA    NA
#> 32  CMC32     2   3PLM   0.80  0.12  -1.51  0.36   0.21  0.09     NA    NA
#> 33  CMC33     2   3PLM   0.52  0.10  -2.77  0.60   0.21  0.09     NA    NA
#> 34  CMC34     2   3PLM   0.67  0.11   0.01  0.34   0.17  0.07     NA    NA
#> 35  CMC35     2   3PLM   0.63  0.10  -0.91  0.43   0.20  0.09     NA    NA
#> 36  CMC36     2   3PLM   0.50  0.12   1.34  0.43   0.17  0.07     NA    NA
#> 37  CMC37     2   3PLM   0.88  0.16  -0.70  0.31   0.19  0.08     NA    NA
#> 38  CMC38     2   3PLM   0.27  0.06  -1.65  1.03   0.21  0.09     NA    NA
#> 39   CFR1     5    GRM   0.88  0.10  -2.98  0.25  -1.97  0.16  -1.30  0.13
#> 40   CFR2     5    GRM   0.67  0.07  -1.47  0.18  -0.55  0.14   0.27  0.12
#> 41   AMC1     2   3PLM   0.61  0.11   0.41  0.37   0.17  0.07     NA    NA
#> 42   AMC2     2   3PLM   0.85  0.15  -2.49  0.39   0.22  0.09     NA    NA
#> 43   AMC3     2   3PLM   0.64  0.12   0.63  0.32   0.15  0.06     NA    NA
#> 44   AMC4     2   3PLM   0.53  0.09  -0.59  0.49   0.20  0.08     NA    NA
#> 45   AMC5     2   3PLM   0.45  0.21   4.11  0.73   0.18  0.06     NA    NA
#> 46   AMC6     2   3PLM   0.58  0.17   2.37  0.34   0.17  0.06     NA    NA
#> 47   AMC7     2   3PLM   0.68  0.11  -0.11  0.35   0.17  0.07     NA    NA
#> 48   AMC8     2   3PLM   0.61  0.11   0.04  0.39   0.18  0.07     NA    NA
#> 49   AMC9     2   3PLM   0.67  0.11   0.24  0.31   0.15  0.07     NA    NA
#> 50  AMC10     2   3PLM   0.63  0.13   1.72  0.26   0.12  0.05     NA    NA
#> 51  AMC11     2   3PLM   0.81  0.15  -1.77  0.39   0.22  0.09     NA    NA
#> 52  AMC12     2   3PLM   0.53  0.08  -1.79  0.53   0.21  0.09     NA    NA
#> 53   AFR1     5    GRM   0.53  0.06  -1.18  0.20  -0.20  0.15   0.78  0.15
#> 54   AFR2     5    GRM   0.60  0.06  -3.88  0.36  -2.60  0.25  -1.61  0.19
#> 55   AFR3     5    GRM   0.47  0.05  -1.61  0.25  -0.50  0.18   0.59  0.16
#>     par.5  se.5
#> 1      NA    NA
#> 2      NA    NA
#> 3      NA    NA
#> 4      NA    NA
#> 5      NA    NA
#> 6      NA    NA
#> 7      NA    NA
#> 8      NA    NA
#> 9      NA    NA
#> 10     NA    NA
#> 11     NA    NA
#> 12     NA    NA
#> 13     NA    NA
#> 14     NA    NA
#> 15     NA    NA
#> 16     NA    NA
#> 17     NA    NA
#> 18     NA    NA
#> 19     NA    NA
#> 20     NA    NA
#> 21     NA    NA
#> 22     NA    NA
#> 23     NA    NA
#> 24     NA    NA
#> 25     NA    NA
#> 26     NA    NA
#> 27     NA    NA
#> 28     NA    NA
#> 29     NA    NA
#> 30     NA    NA
#> 31     NA    NA
#> 32     NA    NA
#> 33     NA    NA
#> 34     NA    NA
#> 35     NA    NA
#> 36     NA    NA
#> 37     NA    NA
#> 38     NA    NA
#> 39  -0.71  0.11
#> 40   1.08  0.14
#> 41     NA    NA
#> 42     NA    NA
#> 43     NA    NA
#> 44     NA    NA
#> 45     NA    NA
#> 46     NA    NA
#> 47     NA    NA
#> 48     NA    NA
#> 49     NA    NA
#> 50     NA    NA
#> 51     NA    NA
#> 52     NA    NA
#> 53   1.72  0.19
#> 54  -0.56  0.14
#> 55   1.43  0.20
#>  Group Parameters: 
#>              mu  sigma2  sigma
#> estimates  0.12    1.08   1.04
#> se         0.03    0.05   0.02
#> 

# Fit the 3PL model to all dichotomous items and the GRM to all polytomous
# items
# Fix all 55 items and estimate only the latent ability distribution
# Use the MEM method
fix.loc <- c(1:55)
(mod.fix4 <- est_irt(
  x = x, data = sim.dat2, D = 1, EmpHist = TRUE,
  Etol = 1e-3, fipc = TRUE, fipc.method = "MEM", fix.loc = fix.loc
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -31130.3586, Max-Change: 0.528058
 EM iteration: 2, Loglike: -31125.5269, Max-Change: 0.14546
 EM iteration: 3, Loglike: -31124.6717, Max-Change: 0.040942
 EM iteration: 4, Loglike: -31124.3171, Max-Change: 0.015804
 EM iteration: 5, Loglike: -31124.1190, Max-Change: 0.008253
 EM iteration: 6, Loglike: -31123.9854, Max-Change: 0.00518
 EM iteration: 7, Loglike: -31123.8815, Max-Change: 0.00365
 EM iteration: 8, Loglike: -31123.7927, Max-Change: 0.002811
 EM iteration: 9, Loglike: -31123.7123, Max-Change: 0.002331
 EM iteration: 10, Loglike: -31123.6373, Max-Change: 0.002051
 EM iteration: 11, Loglike: -31123.5659, Max-Change: 0.001887
 EM iteration: 12, Loglike: -31123.4976, Max-Change: 0.001791
 EM iteration: 13, Loglike: -31123.4317, Max-Change: 0.001732
 EM iteration: 14, Loglike: -31123.3682, Max-Change: 0.001695
 EM iteration: 15, Loglike: -31123.3067, Max-Change: 0.001669
 EM iteration: 16, Loglike: -31123.2472, Max-Change: 0.001648
 EM iteration: 17, Loglike: -31123.1897, Max-Change: 0.001628
 EM iteration: 18, Loglike: -31123.1340, Max-Change: 0.001606
 EM iteration: 19, Loglike: -31123.0800, Max-Change: 0.001583
 EM iteration: 20, Loglike: -31123.0278, Max-Change: 0.001558
 EM iteration: 21, Loglike: -31122.9773, Max-Change: 0.001531
 EM iteration: 22, Loglike: -31122.9284, Max-Change: 0.001503
 EM iteration: 23, Loglike: -31122.8810, Max-Change: 0.001474
 EM iteration: 24, Loglike: -31122.8352, Max-Change: 0.001444
 EM iteration: 25, Loglike: -31122.7908, Max-Change: 0.001414
 EM iteration: 26, Loglike: -31122.7478, Max-Change: 0.001383
 EM iteration: 27, Loglike: -31122.7061, Max-Change: 0.001353
 EM iteration: 28, Loglike: -31122.6658, Max-Change: 0.001323
 EM iteration: 29, Loglike: -31122.6267, Max-Change: 0.001294
 EM iteration: 30, Loglike: -31122.5889, Max-Change: 0.001265
 EM iteration: 31, Loglike: -31122.5522, Max-Change: 0.001237
 EM iteration: 32, Loglike: -31122.5167, Max-Change: 0.001209
 EM iteration: 33, Loglike: -31122.4822, Max-Change: 0.001183
 EM iteration: 34, Loglike: -31122.4489, Max-Change: 0.001157
 EM iteration: 35, Loglike: -31122.4165, Max-Change: 0.001131
 EM iteration: 36, Loglike: -31122.3851, Max-Change: 0.001107
 EM iteration: 37, Loglike: -31122.3547, Max-Change: 0.001083
 EM iteration: 38, Loglike: -31122.3252, Max-Change: 0.00106
 EM iteration: 39, Loglike: -31122.2966, Max-Change: 0.001037
 EM iteration: 40, Loglike: -31122.2688, Max-Change: 0.001015
 EM iteration: 41, Loglike: -31122.2419, Max-Change: 0.000993 
#> Estimation is finished in 0.65 seconds. 
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, EmpHist = TRUE, Etol = 0.001, 
#>     fipc = TRUE, fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 41 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: No item parameters are estimated, so the information matrix is not computed.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates was not estimated.
#> 
#> Log-likelihood: -31122.24
#> 

# Extract group-level parameter estimates
(prior.par <- mod.fix4$group.par)
#>                   mu     sigma2      sigma
#> estimates 0.37036369 1.79753259 1.34072092
#> se        0.04239732 0.08042833 0.02999443

# Visualize the prior distribution
(emphist <- getirt(mod.fix4, what = "weights"))
#>    theta       weight
#> 1  -6.00 3.677170e-12
#> 2  -5.75 2.188873e-11
#> 3  -5.50 1.295005e-10
#> 4  -5.25 7.667966e-10
#> 5  -5.00 4.573080e-09
#> 6  -4.75 2.760422e-08
#> 7  -4.50 1.689564e-07
#> 8  -4.25 1.044329e-06
#> 9  -4.00 6.419815e-06
#> 10 -3.75 3.787026e-05
#> 11 -3.50 1.999956e-04
#> 12 -3.25 8.447505e-04
#> 13 -3.00 2.488156e-03
#> 14 -2.75 4.758835e-03
#> 15 -2.50 6.860545e-03
#> 16 -2.25 1.071855e-02
#> 17 -2.00 1.994145e-02
#> 18 -1.75 2.694815e-02
#> 19 -1.50 2.452203e-02
#> 20 -1.25 3.155507e-02
#> 21 -1.00 4.641204e-02
#> 22 -0.75 5.181014e-02
#> 23 -0.50 7.316168e-02
#> 24 -0.25 8.304573e-02
#> 25  0.00 4.728912e-02
#> 26  0.25 5.720852e-02
#> 27  0.50 9.156180e-02
#> 28  0.75 6.741923e-02
#> 29  1.00 6.068827e-02
#> 30  1.25 7.113787e-02
#> 31  1.50 5.970141e-02
#> 32  1.75 3.169634e-02
#> 33  2.00 2.427735e-02
#> 34  2.25 3.288801e-02
#> 35  2.50 3.409639e-02
#> 36  2.75 1.861733e-02
#> 37  3.00 7.217021e-03
#> 38  3.25 2.868751e-03
#> 39  3.50 1.407924e-03
#> 40  3.75 9.019656e-04
#> 41  4.00 7.485840e-04
#> 42  4.25 7.645929e-04
#> 43  4.50 8.834549e-04
#> 44  4.75 1.048111e-03
#> 45  5.00 1.168700e-03
#> 46  5.25 1.145483e-03
#> 47  5.50 9.459614e-04
#> 48  5.75 6.445798e-04
#> 49  6.00 3.605678e-04
plot(emphist$weight ~ emphist$theta, type = "h")


# Display a summary of the estimation results
summary(mod.fix4)
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, EmpHist = TRUE, Etol = 0.001, 
#>     fipc = TRUE, fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 0.001
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 48
#>    - Item parameters: 0
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 48
#>  Number of fixed items: 55
#>  Number of E-step cycles completed: 41
#>  Maximum parameter change: 0.000993239
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.32
#>  Standard error computation: 
#>  Total computation: 0.65
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: No item parameters are estimated, so the information matrix is not computed.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates was not estimated.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 62244.48
#>  Akaike Information Criterion (AIC): 62340.48
#>  Bayesian Information Criterion (BIC): 62576.06
#>  Item Parameters: 
#>        id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1    CMC1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2    CMC2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3    CMC3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4    CMC4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5    CMC5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6    CMC6     2   3PLM   1.70    NA   0.63    NA   0.07    NA     NA    NA
#> 7    CMC7     2   3PLM   0.91    NA   1.02    NA   0.12    NA     NA    NA
#> 8    CMC8     2   3PLM   0.84    NA   0.80    NA   0.11    NA     NA    NA
#> 9    CMC9     2   3PLM   0.85    NA   0.85    NA   0.26    NA     NA    NA
#> 10  CMC10     2   3PLM   1.53    NA   0.09    NA   0.14    NA     NA    NA
#> 11  CMC11     2   3PLM   1.00    NA  -0.46    NA   0.13    NA     NA    NA
#> 12  CMC12     2   3PLM   0.88    NA   1.18    NA   0.09    NA     NA    NA
#> 13  CMC13     2   3PLM   1.46    NA   1.41    NA   0.18    NA     NA    NA
#> 14  CMC14     2   3PLM   1.51    NA   0.18    NA   0.25    NA     NA    NA
#> 15  CMC15     2   3PLM   1.30    NA  -0.23    NA   0.11    NA     NA    NA
#> 16  CMC16     2   3PLM   2.05    NA  -0.09    NA   0.05    NA     NA    NA
#> 17  CMC17     2   3PLM   1.40    NA  -0.13    NA   0.18    NA     NA    NA
#> 18  CMC18     2   3PLM   1.70    NA   1.25    NA   0.27    NA     NA    NA
#> 19  CMC19     2   3PLM   2.31    NA  -1.01    NA   0.18    NA     NA    NA
#> 20  CMC20     2   3PLM   1.45    NA  -1.65    NA   0.19    NA     NA    NA
#> 21  CMC21     2   3PLM   1.63    NA  -1.19    NA   0.12    NA     NA    NA
#> 22  CMC22     2   3PLM   0.83    NA  -0.68    NA   0.20    NA     NA    NA
#> 23  CMC23     2   3PLM   0.98    NA  -0.26    NA   0.13    NA     NA    NA
#> 24  CMC24     2   3PLM   1.14    NA   1.68    NA   0.25    NA     NA    NA
#> 25  CMC25     2   3PLM   0.79    NA  -1.39    NA   0.26    NA     NA    NA
#> 26  CMC26     2   3PLM   1.09    NA  -1.85    NA   0.17    NA     NA    NA
#> 27  CMC27     2   3PLM   1.17    NA   0.07    NA   0.13    NA     NA    NA
#> 28  CMC28     2   3PLM   2.15    NA  -0.09    NA   0.21    NA     NA    NA
#> 29  CMC29     2   3PLM   1.28    NA  -1.38    NA   0.20    NA     NA    NA
#> 30  CMC30     2   3PLM   1.35    NA   0.82    NA   0.32    NA     NA    NA
#> 31  CMC31     2   3PLM   0.82    NA   0.71    NA   0.08    NA     NA    NA
#> 32  CMC32     2   3PLM   1.52    NA  -0.89    NA   0.26    NA     NA    NA
#> 33  CMC33     2   3PLM   1.27    NA  -1.31    NA   0.19    NA     NA    NA
#> 34  CMC34     2   3PLM   1.31    NA   0.19    NA   0.16    NA     NA    NA
#> 35  CMC35     2   3PLM   1.47    NA  -0.14    NA   0.23    NA     NA    NA
#> 36  CMC36     2   3PLM   0.89    NA   1.10    NA   0.13    NA     NA    NA
#> 37  CMC37     2   3PLM   1.73    NA  -0.41    NA   0.10    NA     NA    NA
#> 38  CMC38     2   3PLM   0.73    NA  -0.37    NA   0.22    NA     NA    NA
#> 39   CFR1     5    GRM   1.91    NA  -1.87    NA  -1.24    NA  -0.71    NA
#> 40   CFR2     5    GRM   1.28    NA  -0.72    NA  -0.07    NA   0.57    NA
#> 41   AMC1     2   3PLM   1.47    NA   0.64    NA   0.23    NA     NA    NA
#> 42   AMC2     2   3PLM   1.76    NA  -1.53    NA   0.16    NA     NA    NA
#> 43   AMC3     2   3PLM   1.44    NA   0.54    NA   0.14    NA     NA    NA
#> 44   AMC4     2   3PLM   0.98    NA  -0.37    NA   0.13    NA     NA    NA
#> 45   AMC5     2   3PLM   0.99    NA   2.37    NA   0.16    NA     NA    NA
#> 46   AMC6     2   3PLM   2.27    NA   1.62    NA   0.18    NA     NA    NA
#> 47   AMC7     2   3PLM   1.23    NA  -0.07    NA   0.13    NA     NA    NA
#> 48   AMC8     2   3PLM   1.64    NA   0.17    NA   0.18    NA     NA    NA
#> 49   AMC9     2   3PLM   1.21    NA   0.24    NA   0.08    NA     NA    NA
#> 50  AMC10     2   3PLM   1.32    NA   1.34    NA   0.08    NA     NA    NA
#> 51  AMC11     2   3PLM   1.74    NA  -1.00    NA   0.25    NA     NA    NA
#> 52  AMC12     2   3PLM   0.97    NA  -0.73    NA   0.22    NA     NA    NA
#> 53   AFR1     5    GRM   1.14    NA  -0.37    NA   0.22    NA   0.85    NA
#> 54   AFR2     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 55   AFR3     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#>     par.5  se.5
#> 1      NA    NA
#> 2      NA    NA
#> 3      NA    NA
#> 4      NA    NA
#> 5      NA    NA
#> 6      NA    NA
#> 7      NA    NA
#> 8      NA    NA
#> 9      NA    NA
#> 10     NA    NA
#> 11     NA    NA
#> 12     NA    NA
#> 13     NA    NA
#> 14     NA    NA
#> 15     NA    NA
#> 16     NA    NA
#> 17     NA    NA
#> 18     NA    NA
#> 19     NA    NA
#> 20     NA    NA
#> 21     NA    NA
#> 22     NA    NA
#> 23     NA    NA
#> 24     NA    NA
#> 25     NA    NA
#> 26     NA    NA
#> 27     NA    NA
#> 28     NA    NA
#> 29     NA    NA
#> 30     NA    NA
#> 31     NA    NA
#> 32     NA    NA
#> 33     NA    NA
#> 34     NA    NA
#> 35     NA    NA
#> 36     NA    NA
#> 37     NA    NA
#> 38     NA    NA
#> 39  -0.23    NA
#> 40   1.07    NA
#> 41     NA    NA
#> 42     NA    NA
#> 43     NA    NA
#> 44     NA    NA
#> 45     NA    NA
#> 46     NA    NA
#> 47     NA    NA
#> 48     NA    NA
#> 49     NA    NA
#> 50     NA    NA
#> 51     NA    NA
#> 52     NA    NA
#> 53   1.38    NA
#> 54  -0.12    NA
#> 55   1.25    NA
#>  Group Parameters: 
#>              mu  sigma2  sigma
#> estimates  0.37    1.80   1.34
#> se         0.04    0.08   0.03
#> 

# Alternatively, fix all 55 items by providing their item IDs
# using the `fix.id` argument. In this case, set `fix.loc = NULL`
fix.id <- x$id
(mod.fix4 <- est_irt(
  x = x, data = sim.dat2, D = 1, EmpHist = TRUE,
  Etol = 1e-3, fipc = TRUE, fipc.method = "MEM", fix.loc = NULL,
  fix.id = fix.id
))
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -31130.3586, Max-Change: 0.528058
 EM iteration: 2, Loglike: -31125.5269, Max-Change: 0.14546
 EM iteration: 3, Loglike: -31124.6717, Max-Change: 0.040942
 EM iteration: 4, Loglike: -31124.3171, Max-Change: 0.015804
 EM iteration: 5, Loglike: -31124.1190, Max-Change: 0.008253
 EM iteration: 6, Loglike: -31123.9854, Max-Change: 0.00518
 EM iteration: 7, Loglike: -31123.8815, Max-Change: 0.00365
 EM iteration: 8, Loglike: -31123.7927, Max-Change: 0.002811
 EM iteration: 9, Loglike: -31123.7123, Max-Change: 0.002331
 EM iteration: 10, Loglike: -31123.6373, Max-Change: 0.002051
 EM iteration: 11, Loglike: -31123.5659, Max-Change: 0.001887
 EM iteration: 12, Loglike: -31123.4976, Max-Change: 0.001791
 EM iteration: 13, Loglike: -31123.4317, Max-Change: 0.001732
 EM iteration: 14, Loglike: -31123.3682, Max-Change: 0.001695
 EM iteration: 15, Loglike: -31123.3067, Max-Change: 0.001669
 EM iteration: 16, Loglike: -31123.2472, Max-Change: 0.001648
 EM iteration: 17, Loglike: -31123.1897, Max-Change: 0.001628
 EM iteration: 18, Loglike: -31123.1340, Max-Change: 0.001606
 EM iteration: 19, Loglike: -31123.0800, Max-Change: 0.001583
 EM iteration: 20, Loglike: -31123.0278, Max-Change: 0.001558
 EM iteration: 21, Loglike: -31122.9773, Max-Change: 0.001531
 EM iteration: 22, Loglike: -31122.9284, Max-Change: 0.001503
 EM iteration: 23, Loglike: -31122.8810, Max-Change: 0.001474
 EM iteration: 24, Loglike: -31122.8352, Max-Change: 0.001444
 EM iteration: 25, Loglike: -31122.7908, Max-Change: 0.001414
 EM iteration: 26, Loglike: -31122.7478, Max-Change: 0.001383
 EM iteration: 27, Loglike: -31122.7061, Max-Change: 0.001353
 EM iteration: 28, Loglike: -31122.6658, Max-Change: 0.001323
 EM iteration: 29, Loglike: -31122.6267, Max-Change: 0.001294
 EM iteration: 30, Loglike: -31122.5889, Max-Change: 0.001265
 EM iteration: 31, Loglike: -31122.5522, Max-Change: 0.001237
 EM iteration: 32, Loglike: -31122.5167, Max-Change: 0.001209
 EM iteration: 33, Loglike: -31122.4822, Max-Change: 0.001183
 EM iteration: 34, Loglike: -31122.4489, Max-Change: 0.001157
 EM iteration: 35, Loglike: -31122.4165, Max-Change: 0.001131
 EM iteration: 36, Loglike: -31122.3851, Max-Change: 0.001107
 EM iteration: 37, Loglike: -31122.3547, Max-Change: 0.001083
 EM iteration: 38, Loglike: -31122.3252, Max-Change: 0.00106
 EM iteration: 39, Loglike: -31122.2966, Max-Change: 0.001037
 EM iteration: 40, Loglike: -31122.2688, Max-Change: 0.001015
 EM iteration: 41, Loglike: -31122.2419, Max-Change: 0.000993 
#> Estimation is finished in 0.37 seconds. 
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, EmpHist = TRUE, Etol = 0.001, 
#>     fipc = TRUE, fipc.method = "MEM", fix.loc = NULL, fix.id = fix.id)
#> 
#> Item parameter estimation using MMLE-EM. 
#> 41 E-step cycles were completed using 49 quadrature points.
#> First-order test: Convergence criteria are satisfied.
#> Second-order test: No item parameters are estimated, so the information matrix is not computed.
#> Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates was not estimated.
#> 
#> Log-likelihood: -31122.24
#> 

# Display a summary of the estimation results
summary(mod.fix4)
#> 
#> Call:
#> est_irt(x = x, data = sim.dat2, D = 1, EmpHist = TRUE, Etol = 0.001, 
#>     fipc = TRUE, fipc.method = "MEM", fix.loc = NULL, fix.id = fix.id)
#> 
#> Summary of the Data 
#>  Number of Items: 55
#>  Number of Cases: 1000
#> 
#> Summary of Estimation Process 
#>  Maximum number of EM cycles: 500
#>  Convergence criterion of E-step: 0.001
#>  Number of rectangular quadrature points: 49
#>  Minimum & Maximum quadrature points: -6, 6
#>  Number of free parameters: 48
#>    - Item parameters: 0
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 48
#>  Number of fixed items: 55
#>  Number of E-step cycles completed: 41
#>  Maximum parameter change: 0.000993239
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.33
#>  Standard error computation: 
#>  Total computation: 0.37
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are satisfied.
#>  Second-order test: No item parameters are estimated, so the information matrix is not computed.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates was not estimated.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 62244.48
#>  Akaike Information Criterion (AIC): 62340.48
#>  Bayesian Information Criterion (BIC): 62576.06
#>  Item Parameters: 
#>        id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1    CMC1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2    CMC2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3    CMC3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4    CMC4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5    CMC5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6    CMC6     2   3PLM   1.70    NA   0.63    NA   0.07    NA     NA    NA
#> 7    CMC7     2   3PLM   0.91    NA   1.02    NA   0.12    NA     NA    NA
#> 8    CMC8     2   3PLM   0.84    NA   0.80    NA   0.11    NA     NA    NA
#> 9    CMC9     2   3PLM   0.85    NA   0.85    NA   0.26    NA     NA    NA
#> 10  CMC10     2   3PLM   1.53    NA   0.09    NA   0.14    NA     NA    NA
#> 11  CMC11     2   3PLM   1.00    NA  -0.46    NA   0.13    NA     NA    NA
#> 12  CMC12     2   3PLM   0.88    NA   1.18    NA   0.09    NA     NA    NA
#> 13  CMC13     2   3PLM   1.46    NA   1.41    NA   0.18    NA     NA    NA
#> 14  CMC14     2   3PLM   1.51    NA   0.18    NA   0.25    NA     NA    NA
#> 15  CMC15     2   3PLM   1.30    NA  -0.23    NA   0.11    NA     NA    NA
#> 16  CMC16     2   3PLM   2.05    NA  -0.09    NA   0.05    NA     NA    NA
#> 17  CMC17     2   3PLM   1.40    NA  -0.13    NA   0.18    NA     NA    NA
#> 18  CMC18     2   3PLM   1.70    NA   1.25    NA   0.27    NA     NA    NA
#> 19  CMC19     2   3PLM   2.31    NA  -1.01    NA   0.18    NA     NA    NA
#> 20  CMC20     2   3PLM   1.45    NA  -1.65    NA   0.19    NA     NA    NA
#> 21  CMC21     2   3PLM   1.63    NA  -1.19    NA   0.12    NA     NA    NA
#> 22  CMC22     2   3PLM   0.83    NA  -0.68    NA   0.20    NA     NA    NA
#> 23  CMC23     2   3PLM   0.98    NA  -0.26    NA   0.13    NA     NA    NA
#> 24  CMC24     2   3PLM   1.14    NA   1.68    NA   0.25    NA     NA    NA
#> 25  CMC25     2   3PLM   0.79    NA  -1.39    NA   0.26    NA     NA    NA
#> 26  CMC26     2   3PLM   1.09    NA  -1.85    NA   0.17    NA     NA    NA
#> 27  CMC27     2   3PLM   1.17    NA   0.07    NA   0.13    NA     NA    NA
#> 28  CMC28     2   3PLM   2.15    NA  -0.09    NA   0.21    NA     NA    NA
#> 29  CMC29     2   3PLM   1.28    NA  -1.38    NA   0.20    NA     NA    NA
#> 30  CMC30     2   3PLM   1.35    NA   0.82    NA   0.32    NA     NA    NA
#> 31  CMC31     2   3PLM   0.82    NA   0.71    NA   0.08    NA     NA    NA
#> 32  CMC32     2   3PLM   1.52    NA  -0.89    NA   0.26    NA     NA    NA
#> 33  CMC33     2   3PLM   1.27    NA  -1.31    NA   0.19    NA     NA    NA
#> 34  CMC34     2   3PLM   1.31    NA   0.19    NA   0.16    NA     NA    NA
#> 35  CMC35     2   3PLM   1.47    NA  -0.14    NA   0.23    NA     NA    NA
#> 36  CMC36     2   3PLM   0.89    NA   1.10    NA   0.13    NA     NA    NA
#> 37  CMC37     2   3PLM   1.73    NA  -0.41    NA   0.10    NA     NA    NA
#> 38  CMC38     2   3PLM   0.73    NA  -0.37    NA   0.22    NA     NA    NA
#> 39   CFR1     5    GRM   1.91    NA  -1.87    NA  -1.24    NA  -0.71    NA
#> 40   CFR2     5    GRM   1.28    NA  -0.72    NA  -0.07    NA   0.57    NA
#> 41   AMC1     2   3PLM   1.47    NA   0.64    NA   0.23    NA     NA    NA
#> 42   AMC2     2   3PLM   1.76    NA  -1.53    NA   0.16    NA     NA    NA
#> 43   AMC3     2   3PLM   1.44    NA   0.54    NA   0.14    NA     NA    NA
#> 44   AMC4     2   3PLM   0.98    NA  -0.37    NA   0.13    NA     NA    NA
#> 45   AMC5     2   3PLM   0.99    NA   2.37    NA   0.16    NA     NA    NA
#> 46   AMC6     2   3PLM   2.27    NA   1.62    NA   0.18    NA     NA    NA
#> 47   AMC7     2   3PLM   1.23    NA  -0.07    NA   0.13    NA     NA    NA
#> 48   AMC8     2   3PLM   1.64    NA   0.17    NA   0.18    NA     NA    NA
#> 49   AMC9     2   3PLM   1.21    NA   0.24    NA   0.08    NA     NA    NA
#> 50  AMC10     2   3PLM   1.32    NA   1.34    NA   0.08    NA     NA    NA
#> 51  AMC11     2   3PLM   1.74    NA  -1.00    NA   0.25    NA     NA    NA
#> 52  AMC12     2   3PLM   0.97    NA  -0.73    NA   0.22    NA     NA    NA
#> 53   AFR1     5    GRM   1.14    NA  -0.37    NA   0.22    NA   0.85    NA
#> 54   AFR2     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 55   AFR3     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#>     par.5  se.5
#> 1      NA    NA
#> 2      NA    NA
#> 3      NA    NA
#> 4      NA    NA
#> 5      NA    NA
#> 6      NA    NA
#> 7      NA    NA
#> 8      NA    NA
#> 9      NA    NA
#> 10     NA    NA
#> 11     NA    NA
#> 12     NA    NA
#> 13     NA    NA
#> 14     NA    NA
#> 15     NA    NA
#> 16     NA    NA
#> 17     NA    NA
#> 18     NA    NA
#> 19     NA    NA
#> 20     NA    NA
#> 21     NA    NA
#> 22     NA    NA
#> 23     NA    NA
#> 24     NA    NA
#> 25     NA    NA
#> 26     NA    NA
#> 27     NA    NA
#> 28     NA    NA
#> 29     NA    NA
#> 30     NA    NA
#> 31     NA    NA
#> 32     NA    NA
#> 33     NA    NA
#> 34     NA    NA
#> 35     NA    NA
#> 36     NA    NA
#> 37     NA    NA
#> 38     NA    NA
#> 39  -0.23    NA
#> 40   1.07    NA
#> 41     NA    NA
#> 42     NA    NA
#> 43     NA    NA
#> 44     NA    NA
#> 45     NA    NA
#> 46     NA    NA
#> 47     NA    NA
#> 48     NA    NA
#> 49     NA    NA
#> 50     NA    NA
#> 51     NA    NA
#> 52     NA    NA
#> 53   1.38    NA
#> 54  -0.12    NA
#> 55   1.25    NA
#>  Group Parameters: 
#>              mu  sigma2  sigma
#> estimates  0.37    1.80   1.34
#> se         0.04    0.08   0.03
#> 

# }