Multiple-group item calibration using MMLE-EM algorithm

This function performs multiple-group item calibration (Bock & Zimowski, 1997) 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). It also supports multiple-group fixed item parameter calibration (MG-FIPC; e.g., Kim & Kolen, 2016), which extends the single-group FIPC method (Kim, 2006) to multiple-group settings. For dichotomous items, the function supports one-, two-, and three-parameter logistic IRT models. For polytomous items, the graded response model (GRM) and the (generalized) partial credit model (GPCM) are available.

Usage

est_mg(
  x = NULL,
  data,
  group.name = NULL,
  D = 1,
  model = NULL,
  cats = NULL,
  item.id = NULL,
  free.group = 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 = 0.001,
  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 list containing item metadata for all groups to be analyzed. For example, if five groups are analyzed, the list should contain five elements, each representing the item metadata for one group. The order of the elements in the list must match the order of group names specified in the group.name argument.

Each group's item metadata includes essential information for each item (e.g., number of score categories, IRT model type, etc.) required for calibration. See est_irt() or simdat() for more details about the item metadata.

When use.startval = TRUE, the item parameters specified in the metadata will be used as starting values for parameter estimation; otherwise they are ignored, except for the fixed items in MG-FIPC, whose parameters are held at the values in x. A common item must have the same model and number of score categories in every group that includes it. If x = NULL, the model, cats, and item.id arguments are used. Note that when fipc = TRUE to implement MG-FIPC, the x argument must be specified and cannot be NULL. Default is NULL.

data

A list containing the item response data of all groups to be analyzed. For example, if five groups are analyzed, the list should include five elements, each representing the response data of one group. The elements in the list must be ordered to match the group names specified in the group.name argument. Each element is a matrix or data frame whose rows represent the examinees of the group and whose columns represent the items of the group's form, in the same order as in the group's item metadata. Responses to an item with K score categories must be integers from 0 to K - 1, and missing responses are coded as NA or as the value given in missing. Items are matched across groups by their IDs, and the items that are not in a group's form are treated as missing for the examinees of that group.

group.name

A character vector indicating the names of the groups, in the order of the elements of data. For example, if five groups are analyzed, use group.name = c("G1", "G2", "G3", "G4", "G5"). Group names can be any valid character strings and label the group-level elements of the returned object. If NULL, the names "g1", "g2", and so on are used. Default is NULL.

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 list containing character vectors specifying the IRT models used to calibrate items across all groups. For example, if five groups are analyzed, the list should contain five elements, each being a character vector of IRT model names for one group. The elements in the list must be ordered to match the group names specified in the group.name argument.

Available IRT models include:

"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; items labeled "DRM" are fitted with the 3PLM, and a warning is issued. If a single model name is provided in any element of the list, it will be recycled across all items within that group.

This argument is used only when x = NULL and fipc = FALSE. Default is NULL.

cats

A list containing numeric vectors specifying the number of score categories for items in each group. For example, if five groups are analyzed, the list should contain five numeric vectors corresponding to the five groups. The elements in the list must be ordered consistently with the group names specified in the group.name argument.

If a single numeric value is specified in any element of the list, it will be recycled across all items in the corresponding group. If cats = NULL and all models specified in the model argument are dichotomous (i.e., "1PLM", "2PLM", "3PLM", or "DRM"), the function assumes that all items have two score categories across all groups.

This argument is used only when x = NULL and fipc = FALSE. Default is NULL.

item.id

A list containing character vectors of item IDs for each group to be analyzed. For example, if five groups are analyzed, the list should contain five character vectors corresponding to the five groups. The elements in the list must be ordered consistently with the group names specified in the group.name argument.

Items with the same ID in different groups are treated as common items. A provided item.id replaces any IDs in x, with or without FIPC, and each element must have one value per item of its group. With fipc = TRUE, fix.id then refers to the new IDs. It is required when x = NULL. Default is NULL.

free.group

A numeric or character vector of the positions or names of the groups whose latent ability distributions are freely estimated: their means and variances, or their empirical histograms when EmpHist = TRUE. The distributions of the remaining groups (reference groups, those not specified in this argument) are fixed using the values provided in the group.mean and group.var arguments, or from the weights argument. Each group in free.group must share common items with another group or, with FIPC, include fixed items.

For example, suppose that five groups are analyzed with group names "G1", "G2", "G3", "G4", and "G5". To freely estimate the distributions of groups 2 through 5, set free.group = c(2, 3, 4, 5) or free.group = c("G2", "G3", "G4", "G5"). In this case, the first group ("G1") will have a fixed distribution (e.g., a mean of 0 and variance of 1 when group.mean = 0, group.var = 1, and weights = NULL).

Without FIPC, at least one group must be a reference group to identify the scale; if free.group = NULL, all groups are reference groups. With FIPC, free.group must be supplied, and it usually includes all groups so that every distribution is estimated on the scale of the fixed items (see Details). 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 containing the quadrature points (in the first column) and the corresponding weights (in the second column) that define the initial latent ability distribution of every group. The weights and points can be conveniently generated using the function 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. For the groups not specified in free.group, 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 this 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. For the groups in free.group, it is only the initial distribution.

If NULL, a normal prior density is used instead, based on the information provided in the Quadrature, group.mean, and group.var arguments. Default is NULL.

group.mean

A numeric value specifying the mean of the normal latent ability distribution when weights = NULL. Default is 0. The means of the groups not specified in the free.group argument are fixed at this value to resolve the indeterminacy of the item parameter scale, also when EmpHist = TRUE. For the groups in free.group, it is the mean of the initial distribution.

group.var

A positive numeric value specifying the variance of the normal latent ability distribution when weights = NULL. Default is 1. The variances of the groups not specified in the free.group argument are fixed at this value to resolve the indeterminacy of the item parameter scale, also when EmpHist = TRUE. For the groups in free.group, it is the variance of the initial distribution.

EmpHist

Logical. If TRUE, the latent ability distribution of each group is estimated as an empirical histogram on the quadrature points together with the item parameters, using the approach proposed by Woods (2007). The histograms of the groups not in free.group are rescaled to the fixed mean and variance in every EM cycle, and those of the groups in free.group are not rescaled. If FALSE, the distribution of each group in free.group is a normal distribution whose mean and variance are updated in every EM cycle. 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 MG-FIPC, only the parameters of the items that are not fixed are compared. The means and variances of the latent ability distributions of the groups in free.group, and the weights of the empirical histograms (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 means and variances of the latent ability distributions of all groups; those of the groups not in free.group stay at their fixed values, apart from small changes due to the rescaling of their histograms when EmpHist = TRUE. 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-3, which is larger than the default of est_irt().

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, multiple-group fixed item parameter calibration (MG-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 list of positive integer vectors. Each internal vector specifies the positions of the items to be fixed in the item metadata (i.e., x) for each group when MG-FIPC is implemented (i.e., fipc = TRUE). The internal objects in the list must follow the same order as the group names provided in the group.name argument. Use NULL for a group without fixed items. An item fixed in one group is fixed in every group that includes it.

For example, suppose three groups are analyzed. In the first group, the 1st, 3rd, and 5th items are fixed; in the second group, the 2nd, 3rd, 4th, and 7th items are fixed; and in the third group, the 1st, 2nd, and 6th items are fixed. Then fix.loc = list(c(1, 3, 5), c(2, 3, 4, 7), c(1, 2, 6)). Note that if the fix.id argument is not NULL, the information in fix.loc will be ignored. See Details.

fix.id

A vector of character strings specifying the IDs of items to be fixed when MG-FIPC is implemented (i.e., fipc = TRUE). It is a single character vector for all groups, not a list.

For example, suppose that three groups are analyzed. In the first group, three items with IDs G1I1, C1I1, and C1I2 are fixed. In the second group, four items with IDs C1I1, C1I2, C2I1, and C2I2 are fixed. In the third group, three items with IDs C2I1, C2I2, and G3I1 are fixed.

In this case, there are six unique items fixed across the groups (G1I1, C1I1, C1I2, C2I1, C2I2, and G3I1), because C1I1 and C1I2 appear in both the first and second groups, while C2I1 and C2I2 appear in both the second and third groups. Thus, you should specify fix.id = c("G1I1", "C1I1", "C1I2", "C2I1", "C2I2", "G3I1"). Note that if the fix.id argument is not NULL, the information provided in 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_mg. The returned object contains the following components:

estimates: A list containing two internal elements: overall and group. The overall element is a data frame with item parameter estimates and their standard errors, computed from the combined data across all groups. This data frame includes only the unique items across all groups. The group element is a list of group-specific data frames, each containing item parameter estimates and standard errors for that particular group.
par.est: A list with the same structure as estimates, containing only the item parameter estimates (excluding standard errors), formatted according to the item metadata structure.
se.est: A list with the same structure as estimates, but containing only the standard errors of the item parameter estimates. Note that the standard errors are computed from the inverse of the information matrix, which 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. The latent ability distributions of the groups are treated as known in this matrix, so the uncertainty of the estimated distributions is not reflected in these standard errors. Parameters that are not estimated, including those of the items fixed in FIPC, have NA standard errors, and all are NA when se = FALSE.
pos.par: A data frame of the positions of the estimated item parameters in covariance, for the unique items across all groups (the overall element of estimates). Parameters that are not estimated, including those of the items fixed in FIPC, are NA. It is NULL when all items are fixed in FIPC.
covariance: The variance-covariance matrix of the estimated item parameters of the unique items across all groups, in the order given by pos.par. It is NULL when se = FALSE or when all items are fixed in FIPC.
loglikelihood: A list with two elements. overall is the marginal log-likelihood of the response data at the final estimates: the sum over all examinees of the logarithm of the probability of each examinee's response pattern, integrated over the latent ability distribution of the examinee's group. group is a named list of the same sums within the groups, which add up to overall. The log prior densities of the item parameters are not included.
aic: Akaike information criterion, -2 * loglikelihood$overall + 2 * npar.est, calculated from the overall log-likelihood.
bic: Bayesian information criterion, -2 * loglikelihood$overall + npar.est * log(ncase$overall), calculated from the overall log-likelihood and the overall number of examinees.
group.par: A named list with one data frame per group, with rows estimates and se and columns mu, sigma2, and sigma (the mean, variance, and standard deviation of the latent ability distribution). For a group not in free.group, the estimates are the fixed values given by group.mean and group.var (or by weights), and the standard errors are NA; with EmpHist = TRUE, the histogram of such a group is rescaled in each cycle to have this mean and variance, but because the rescaling interpolates on the quadrature points, the mean and variance computed from the returned weights can differ slightly from the reported values. For a group in free.group, the estimates are the mean and variance of the final distribution (the same as those computed from the returned weights), and the standard errors are approximations that treat the abilities as observed and are therefore usually too small: sigma / sqrt(N), sigma2 * sqrt(2 / (N - 1)), and se(sigma2) / (2 * sigma), where N is the number of examinees in the group.
weights: A list of two-column data frames, where the first column contains quadrature points and the second column contains the corresponding weights of the (updated) latent variable prior distributions for each group.
posterior.dist: A matrix of the posterior probabilities of the quadrature points for each examinee, computed with the final estimates and the latent distribution of the examinee's group. Rows represent the examinees of all groups, stacked in the order of group.name, and columns represent quadrature points; each row sums to 1.
data: A list with two elements. overall is a matrix of the responses of all examinees to all unique items, with NA for the items not in a group's form and for missing responses. group is the list of response data sets named by group.name, in which the missing responses are also recoded as NA.
scale.D: The scaling factor used in the IRT model.
ncase: A list with two elements: overall, the total number of examinees, and group, a named vector of the numbers of examinees in the groups.
nitem: A list with two elements: overall, the number of unique items across all groups, and group, a named vector of the numbers of items in the groups.
Etol: The convergence criterion of the EM algorithm; see Etol in Arguments.
MaxE: The maximum number of EM cycles (1 when fipc.method = "OEM").
aprior: A list describing the prior distribution used for discrimination parameters. It is NULL when use.aprior = FALSE.
bprior: A list describing the prior distribution used for difficulty parameters. It is NULL when use.bprior = FALSE.
gprior: A list describing the prior distribution used for guessing parameters. It is NULL when use.gprior = FALSE.
npar.est: The number of freely estimated parameters used for aic and bic: the estimated item parameters of all unique items, the means and variances of the free groups when they have normal distributions, and the weights of the empirical histograms 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 MG-FIPC with all items fixed, in the means and variances of the latent ability distributions of all groups; see Etol). 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 method used for FIPC, or NULL when fipc = FALSE.
fix.loc: A list with two elements: overall, the positions of the fixed items among the unique items across all groups (the rows of par.est$overall), and group, a named list of the positions of the fixed items in each group's form. Both are NULL when fipc = FALSE.
call: The matched call.
npar.detail: An integer vector with the elements item (free item parameters), dist (means and variances of the latent distributions of all groups), and weight (weights of the empirical histograms of all groups), 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

Multiple-group (MG) item calibration (Bock & Zimowski, 1997) provides a unified framework for handling testing scenarios involving multiple groups, such as nonequivalent groups equating, vertical scaling, and the identification of differential item functioning (DIF). In such applications, examinees from different groups typically respond to either the same test form or to different forms that share common (anchor) items.

The goal of MG item calibration is to estimate both item parameters and latent ability distributions for all groups simultaneously (Bock & Zimowski, 1997). The irtQ package implements MG calibration via the est_mg() function, which uses marginal maximum likelihood estimation through the expectation-maximization (MMLE-EM) algorithm (Bock & Aitkin, 1981). In addition, the function supports multiple-group fixed item parameter calibration (MG-FIPC; e.g., Kim & Kolen, 2016), which allows the parameters of specific items to be fixed across groups.

In MG IRT analyses, it is common for multiple groups' test forms to share some common (anchor) items. The est_mg() function treats items with the same item ID in different groups as common items and constrains them to have the same parameter estimates.

Most of the features of the est_mg() function are similar to those of the est_irt() function. The main difference is that the arguments x, data, model, cats, item.id, and fix.loc accept list objects containing elements for each group to be analyzed, in the order of the group names in group.name. In contrast, fix.id is a single character vector of the IDs of the fixed items across all groups.

Additionally, est_mg() introduces two new arguments: group.name and free.group. The group.name argument assigns a unique identifier to each group.

The free.group argument indicates the groups whose latent ability distributions are freely estimated; the distributions of the other groups are fixed using the values provided in the group.mean and group.var arguments or from the weights argument. When no item parameters are fixed (i.e., fipc = FALSE), at least one group must have a fixed latent ability distribution (e.g., mean = 0 and variance = 1) among the multiple groups sharing common items, in order to resolve the scale indeterminacy inherent in IRT estimation.

Situations requiring the implementation of MG-FIPC typically arise when new latent ability scales from multiple-group (MG) test data need to be linked to an established scale (e.g., that of an existing item bank). In a single run of the MG-FIPC procedure, the parameters of non-fixed (freed) items across multiple test forms, as well as the latent ability distributions for multiple groups, can be estimated on the same scale as the fixed items (Kim & Kolen, 2016).

For example, suppose that three different test forms (Form 1, Form 2, and Form 3) are administered to three nonequivalent groups: Group1, Group2, and Group3. Form 1 and Form 2 share 12 common items (C1I1 to C1I12), while Form 2 and Form 3 share 10 common items (C2I1 to C2I10). There are no common items between Form 1 and Form 3. Also, assume that all unique items in Form 1 are from an existing item bank and have already been calibrated on the item bank's scale.

In this case, the goal of MG-FIPC is to estimate the parameters of all items across the three test forms, except the unique items in Form 1, and the latent ability distributions of the three groups, all on the same scale as the item bank. To achieve this, the unique items in Form 1 must be fixed during MG-FIPC to link the current MG test data to the item bank scale.

The est_mg() function can implement MG-FIPC by setting fipc = TRUE. In this case, the information on which items to fix must be provided through either the fix.loc or fix.id argument. When using fix.loc, you must supply a list of item positions (locations) to be fixed in each group's test form. For example, suppose that the test data from the three groups above are analyzed. In the first group, the 1st, 3rd, and 5th items are fixed; in the second group, the 2nd, 3rd, 4th, and 7th items are fixed; and in the third group, the 1st, 2nd, and 6th items are fixed. In this case, you would specify: fix.loc = list(c(1, 3, 5), c(2, 3, 4, 7), c(1, 2, 6)).

Alternatively, you can use fix.id to specify a character vector of item IDs to be fixed across groups. In the first group, the items with IDs G1I1, C1I1, and C1I2 are fixed; in the second group, the items with IDs C1I1, C1I2, C2I1, and C2I2 are fixed; and in the third group, the items with IDs C2I1, C2I2, and G3I1 are fixed. In this case, there are six unique items to be fixed across all groups: G1I1, C1I1, C1I2, C2I1, C2I2, and G3I1. You would then specify: fix.id = c("G1I1", "C1I1", "C1I2", "C2I1", "C2I2", "G3I1").

Note that when both fix.loc and fix.id are provided, the information in fix.id takes precedence and overrides fix.loc.

Two methods are available for MG-FIPC through the fipc.method argument, as in est_irt(). The method "MEM" runs multiple EM cycles. The posterior distributions of the first EM cycle are computed from the responses to the fixed items only, and those of the later cycles from the responses to all items, as in Kim and Kolen (2016, Equations 13 to 20). The latent ability distribution of each group in free.group is updated in every EM cycle without rescaling, so that the item parameters and the distributions are placed on the scale of the fixed items. With EmpHist = TRUE, the distribution is the histogram of the expected numbers of examinees at the quadrature points, as in Kim and Kolen (2016); with EmpHist = FALSE, it is a normal distribution with the mean and variance of that histogram. The distributions of the groups not in free.group remain fixed also in MG-FIPC, whereas Kim and Kolen (2016) estimate the distributions of all groups; to follow their method, include all groups in free.group. The method "OEM" (Wainer & Mislevy, 1990) runs a single EM cycle: the posterior distributions are computed from the fixed items and the initial distributions, the parameters of the other items are estimated once, and the returned distributions are updated once afterward.

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. Because the parameters of an item are common to all the groups that include it, the responses of an item are pooled over these groups; an item that is constant in one group but varies in another is not flagged. The items fixed in MG-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 computed as aic = -2 * loglikelihood$overall + 2 * npar.est and bic = -2 * loglikelihood$overall + npar.est * log(ncase$overall), where loglikelihood$overall is the overall marginal log-likelihood without the prior densities of the item parameters and ncase$overall is the overall number of examinees. npar.est is the number of freely estimated parameters, which consists of (a) the item parameters that are estimated across all unique items (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 each group in free.group when EmpHist = FALSE (2 values per group), and (c) the weights of the empirical histograms when EmpHist = TRUE. The histogram of a group in free.group is not rescaled, so its Q weights, where Q is the number of quadrature points, have Q - 1 free values (the sum of the weights is fixed). The histogram of any other group is rescaled to a fixed mean and variance, so its weights have Q - 3 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

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

Bock, R. D., & Zimowski, M. F. (1997). Multiple group IRT. In W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 433-448). Springer. doi:10.1007/978-1-4757-2691-6_25 .

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. MG calibration using the simMG data
#  - Details:
#    (a) Constrain common items between groups to have
#        identical item parameters (i.e., items C1I1 - C1I12 between
#        Groups 1 and 2, and items C2I1 - C2I10 between Groups 2 and 3).
#    (b) Freely estimate the means and variances of the ability
#        distributions for all groups except the reference group,
#        where the mean and variance are fixed to 0 and 1, respectively.
## ------------------------------------------------------------------------------
# 1-(1). Freely estimate the means and variances of Groups 2 and 3
# Import the true item metadata for the three groups
x <- simMG$item.prm

# Extract model, score category, and item ID information
# from the item metadata for the three groups
model <- list(x$Group1$model, x$Group2$model, x$Group3$model)
cats <- list(x$Group1$cats, x$Group2$cats, x$Group3$cats)
item.id <- list(x$Group1$id, x$Group2$id, x$Group3$id)

# Import the simulated response data sets for the three groups
data <- simMG$res.dat

# Import the group names for the three groups
group.name <- simMG$group.name

# Specify Groups 2 and 3 as the free groups where the scale
# of the ability distributions will be freely estimated.
# Group 1 will serve as the reference group, where the scale
# of the ability distribution is fixed to the values specified
# via the 'group.mean' and 'group.var' arguments
free.group <- c(2, 3) # or use 'free.group <- group.name[2:3]'

# Estimate IRT parameters:
# As long as common items across groups share the same item IDs,
# their item parameters will be constrained to be equal across groups
# unless FIPC is implemented.
fit.1 <-
  est_mg(
    data = data, group.name = group.name, model = model,
    cats = cats, item.id = item.id, D = 1, free.group = free.group,
    use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 16)),
    group.mean = 0, group.var = 1, EmpHist = TRUE, Etol = 0.001, MaxE = 500
  )
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -184619.3757, Max-Change: 2.188782
 EM iteration: 2, Loglike: -159724.2438, Max-Change: 0.42374
 EM iteration: 3, Loglike: -159631.0877, Max-Change: 0.166346
 EM iteration: 4, Loglike: -159617.3743, Max-Change: 0.083614
 EM iteration: 5, Loglike: -159611.4702, Max-Change: 0.052115
 EM iteration: 6, Loglike: -159608.0480, Max-Change: 0.038966
 EM iteration: 7, Loglike: -159605.7836, Max-Change: 0.032285
 EM iteration: 8, Loglike: -159604.1351, Max-Change: 0.027248
 EM iteration: 9, Loglike: -159602.8431, Max-Change: 0.023166
 EM iteration: 10, Loglike: -159601.7733, Max-Change: 0.01977
 EM iteration: 11, Loglike: -159600.8519, Max-Change: 0.016922
 EM iteration: 12, Loglike: -159600.0358, Max-Change: 0.014527
 EM iteration: 13, Loglike: -159599.2989, Max-Change: 0.012509
 EM iteration: 14, Loglike: -159598.6241, Max-Change: 0.010869
 EM iteration: 15, Loglike: -159597.9998, Max-Change: 0.009797
 EM iteration: 16, Loglike: -159597.4177, Max-Change: 0.00890
 EM iteration: 17, Loglike: -159596.8716, Max-Change: 0.008145
 EM iteration: 18, Loglike: -159596.3569, Max-Change: 0.007856
 EM iteration: 19, Loglike: -159595.8700, Max-Change: 0.007658
 EM iteration: 20, Loglike: -159595.4080, Max-Change: 0.00744
 EM iteration: 21, Loglike: -159594.9687, Max-Change: 0.007208
 EM iteration: 22, Loglike: -159594.5501, Max-Change: 0.006968
 EM iteration: 23, Loglike: -159594.1507, Max-Change: 0.006723
 EM iteration: 24, Loglike: -159593.7692, Max-Change: 0.006477
 EM iteration: 25, Loglike: -159593.4043, Max-Change: 0.006232
 EM iteration: 26, Loglike: -159593.0551, Max-Change: 0.00599
 EM iteration: 27, Loglike: -159592.7206, Max-Change: 0.005753
 EM iteration: 28, Loglike: -159592.4000, Max-Change: 0.005522
 EM iteration: 29, Loglike: -159592.0925, Max-Change: 0.005298
 EM iteration: 30, Loglike: -159591.7975, Max-Change: 0.005081
 EM iteration: 31, Loglike: -159591.5143, Max-Change: 0.004871
 EM iteration: 32, Loglike: -159591.2423, Max-Change: 0.00467
 EM iteration: 33, Loglike: -159590.9808, Max-Change: 0.004476
 EM iteration: 34, Loglike: -159590.7295, Max-Change: 0.00429
 EM iteration: 35, Loglike: -159590.4879, Max-Change: 0.004112
 EM iteration: 36, Loglike: -159590.2553, Max-Change: 0.003942
 EM iteration: 37, Loglike: -159590.0316, Max-Change: 0.003779
 EM iteration: 38, Loglike: -159589.8161, Max-Change: 0.003624
 EM iteration: 39, Loglike: -159589.6086, Max-Change: 0.003475
 EM iteration: 40, Loglike: -159589.4087, Max-Change: 0.003333
 EM iteration: 41, Loglike: -159589.2160, Max-Change: 0.003198
 EM iteration: 42, Loglike: -159589.0303, Max-Change: 0.003068
 EM iteration: 43, Loglike: -159588.8512, Max-Change: 0.002945
 EM iteration: 44, Loglike: -159588.6785, Max-Change: 0.002827
 EM iteration: 45, Loglike: -159588.5119, Max-Change: 0.002715
 EM iteration: 46, Loglike: -159588.3511, Max-Change: 0.002607
 EM iteration: 47, Loglike: -159588.1959, Max-Change: 0.002505
 EM iteration: 48, Loglike: -159588.0460, Max-Change: 0.002407
 EM iteration: 49, Loglike: -159587.9013, Max-Change: 0.002313
 EM iteration: 50, Loglike: -159587.7616, Max-Change: 0.002228
 EM iteration: 51, Loglike: -159587.6266, Max-Change: 0.002156
 EM iteration: 52, Loglike: -159587.4961, Max-Change: 0.002086
 EM iteration: 53, Loglike: -159587.3700, Max-Change: 0.002019
 EM iteration: 54, Loglike: -159587.2481, Max-Change: 0.001955
 EM iteration: 55, Loglike: -159587.1302, Max-Change: 0.001893
 EM iteration: 56, Loglike: -159587.0163, Max-Change: 0.001833
 EM iteration: 57, Loglike: -159586.9060, Max-Change: 0.001776
 EM iteration: 58, Loglike: -159586.7993, Max-Change: 0.001721
 EM iteration: 59, Loglike: -159586.6961, Max-Change: 0.001667
 EM iteration: 60, Loglike: -159586.5962, Max-Change: 0.001616
 EM iteration: 61, Loglike: -159586.4995, Max-Change: 0.001566
 EM iteration: 62, Loglike: -159586.4058, Max-Change: 0.001519
 EM iteration: 63, Loglike: -159586.3151, Max-Change: 0.001473
 EM iteration: 64, Loglike: -159586.2272, Max-Change: 0.001429
 EM iteration: 65, Loglike: -159586.1421, Max-Change: 0.001386
 EM iteration: 66, Loglike: -159586.0596, Max-Change: 0.001358
 EM iteration: 67, Loglike: -159585.9796, Max-Change: 0.001334
 EM iteration: 68, Loglike: -159585.9021, Max-Change: 0.00131
 EM iteration: 69, Loglike: -159585.8269, Max-Change: 0.001287
 EM iteration: 70, Loglike: -159585.7540, Max-Change: 0.001264
 EM iteration: 71, Loglike: -159585.6832, Max-Change: 0.001241
 EM iteration: 72, Loglike: -159585.6146, Max-Change: 0.001219
 EM iteration: 73, Loglike: -159585.5479, Max-Change: 0.001197
 EM iteration: 74, Loglike: -159585.4832, Max-Change: 0.001175
 EM iteration: 75, Loglike: -159585.4204, Max-Change: 0.001153
 EM iteration: 76, Loglike: -159585.3593, Max-Change: 0.001132
 EM iteration: 77, Loglike: -159585.3000, Max-Change: 0.001111
 EM iteration: 78, Loglike: -159585.2424, Max-Change: 0.001091
 EM iteration: 79, Loglike: -159585.1864, Max-Change: 0.00107
 EM iteration: 80, Loglike: -159585.1319, Max-Change: 0.00105
 EM iteration: 81, Loglike: -159585.0790, Max-Change: 0.001031
 EM iteration: 82, Loglike: -159585.0274, Max-Change: 0.001011
 EM iteration: 83, Loglike: -159584.9773, Max-Change: 0.000992 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 16.24 seconds. 

# Summary of the estimation
summary(fit.1)
#> 
#> Call:
#> est_mg(data = data, group.name = group.name, D = 1, model = model, 
#>     cats = cats, item.id = item.id, free.group = free.group, 
#>     use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 
#>         16)), group.mean = 0, group.var = 1, EmpHist = TRUE, 
#>     Etol = 0.001, MaxE = 500)
#> 
#> Summary of the Data 
#>  Number of Items: 
#>   Overall: 116 unique items 
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of Cases: 
#>   Overall: 6000
#>   By group: 2000(Group1), 2000(Group2), 2000(Group3)
#> 
#> 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: 500
#>    - Item parameters: 358
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 142
#>  Number of fixed items: 
#>   Overall: 0
#>   By group: 0(Group1), 0(Group2), 0(Group3)
#>  Number of E-step cycles completed: 83
#>  Maximum parameter change: 0.0009920582
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 15.23
#>  Standard error computation: 0.2
#>  Total computation: 16.24
#> 
#> 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: 
#>   Overall: 319169.9
#>   By group: 120347.958(Group1), 113944.899(Group2), 84877.056(Group3)
#> 
#>  Akaike Information Criterion (AIC): 320169.9
#>  Bayesian Information Criterion (BIC): 323519.7
#>  Item Parameters (Overall): 
#>         id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1     C1I1     2   3PLM   0.90  0.18   1.36  0.16   0.28  0.05     NA    NA
#> 2     C1I2     2   3PLM   2.12  0.15  -0.99  0.10   0.18  0.06     NA    NA
#> 3     C1I3     2   3PLM   1.05  0.12   0.61  0.13   0.18  0.05     NA    NA
#> 4     C1I4     2   3PLM   1.08  0.13  -0.17  0.22   0.28  0.07     NA    NA
#> 5     C1I5     2   3PLM   0.87  0.09  -0.15  0.22   0.18  0.07     NA    NA
#> 6     C1I6     2   3PLM   1.91  0.13   0.60  0.04   0.09  0.02     NA    NA
#> 7     C1I7     2   3PLM   1.10  0.13   1.10  0.09   0.15  0.03     NA    NA
#> 8     C1I8     2   3PLM   0.94  0.12   0.89  0.14   0.16  0.05     NA    NA
#> 9     C1I9     2   3PLM   0.89  0.12   0.63  0.18   0.21  0.06     NA    NA
#> 10   C1I10     2   3PLM   1.49  0.12   0.13  0.08   0.15  0.04     NA    NA
#> 11    G1I1     2   3PLM   0.96  0.11  -0.46  0.20   0.16  0.07     NA    NA
#> 12    G1I2     2   3PLM   0.90  0.14   1.20  0.14   0.11  0.04     NA    NA
#> 13    G1I3     2   3PLM   1.54  0.26   1.30  0.09   0.19  0.03     NA    NA
#> 14    G1I4     2   3PLM   1.57  0.22   0.28  0.12   0.30  0.04     NA    NA
#> 15    G1I5     2   3PLM   1.35  0.14  -0.17  0.13   0.16  0.05     NA    NA
#> 16    G1I6     2   3PLM   2.17  0.18   0.02  0.06   0.09  0.03     NA    NA
#> 17    G1I7     2   3PLM   1.47  0.16  -0.06  0.12   0.20  0.05     NA    NA
#> 18    G1I8     2   3PLM   2.53  0.47   1.17  0.07   0.32  0.02     NA    NA
#> 19    G1I9     2   3PLM   2.40  0.27  -0.93  0.11   0.24  0.06     NA    NA
#> 20   G1I10     2   3PLM   1.26  0.13  -1.76  0.23   0.23  0.10     NA    NA
#> 21   G1I11     2   3PLM   1.55  0.15  -1.11  0.16   0.21  0.08     NA    NA
#> 22   G1I12     2   3PLM   0.76  0.09  -0.80  0.31   0.20  0.08     NA    NA
#> 23   G1I13     2   3PLM   1.05  0.14  -0.12  0.21   0.21  0.07     NA    NA
#> 24   G1I14     2   3PLM   1.51  0.38   1.74  0.15   0.30  0.02     NA    NA
#> 25   G1I15     2   3PLM   0.87  0.10  -1.43  0.30   0.21  0.09     NA    NA
#> 26   G1I16     2   3PLM   1.05  0.11  -1.94  0.27   0.23  0.10     NA    NA
#> 27   G1I17     2   3PLM   1.06  0.13   0.26  0.15   0.16  0.05     NA    NA
#> 28   G1I18     2   3PLM   2.11  0.21  -0.08  0.08   0.24  0.04     NA    NA
#> 29   G1I19     2   3PLM   1.32  0.13  -1.39  0.19   0.20  0.08     NA    NA
#> 30   G1I20     2   3PLM   1.03  0.18   0.52  0.19   0.23  0.06     NA    NA
#> 31   G1I21     2   3PLM   0.92  0.13   0.76  0.15   0.13  0.05     NA    NA
#> 32   G1I22     2   3PLM   1.79  0.22  -0.63  0.16   0.36  0.06     NA    NA
#> 33   G1I23     2   3PLM   1.31  0.15  -1.16  0.23   0.26  0.09     NA    NA
#> 34   G1I24     2   3PLM   1.66  0.20   0.32  0.09   0.23  0.04     NA    NA
#> 35   G1I25     2   3PLM   1.60  0.18  -0.12  0.12   0.25  0.05     NA    NA
#> 36   G1I26     2   3PLM   1.91  0.26   0.66  0.08   0.25  0.03     NA    NA
#> 37   G1I27     2   3PLM   1.62  0.18  -1.55  0.20   0.27  0.10     NA    NA
#> 38   G1I28     2   3PLM   1.35  0.17   0.57  0.10   0.15  0.04     NA    NA
#> 39   G1I29     2   3PLM   0.92  0.09  -0.37  0.17   0.12  0.06     NA    NA
#> 40   G1I30     2   3PLM   1.06  0.29   2.24  0.25   0.17  0.03     NA    NA
#> 41   G1I31     2   3PLM   2.45  0.49   1.62  0.09   0.18  0.01     NA    NA
#> 42   G1I32     2   3PLM   1.11  0.12  -0.07  0.15   0.15  0.05     NA    NA
#> 43   G1I33     2   3PLM   1.62  0.18   0.18  0.09   0.16  0.04     NA    NA
#> 44   G1I34     2   3PLM   1.34  0.14   0.24  0.09   0.11  0.04     NA    NA
#> 45   G1I35     2   3PLM   1.35  0.17   1.28  0.08   0.07  0.02     NA    NA
#> 46   G1I36     2   3PLM   1.44  0.15  -1.23  0.19   0.22  0.09     NA    NA
#> 47   G1I37     2   3PLM   1.06  0.15  -0.58  0.27   0.27  0.09     NA    NA
#> 48   G1I38     5    GRM   1.06  0.06  -0.37  0.05   0.21  0.05   0.86  0.06
#> 49   C1I11     5    GRM   1.19  0.05  -2.21  0.09  -1.45  0.07  -0.75  0.05
#> 50   C1I12     5    GRM   0.91  0.04  -0.69  0.06   0.03  0.04   0.68  0.04
#> 51    G2I1     2   3PLM   1.76  0.19  -0.86  0.18   0.24  0.10     NA    NA
#> 52    G2I2     2   3PLM   0.84  0.11  -0.44  0.31   0.21  0.09     NA    NA
#> 53    G2I3     2   3PLM   1.09  0.14   0.09  0.20   0.18  0.08     NA    NA
#> 54    G2I4     2   3PLM   1.51  0.29   1.45  0.09   0.18  0.04     NA    NA
#> 55    G2I5     2   3PLM   0.71  0.10  -1.66  0.40   0.20  0.09     NA    NA
#> 56    G2I6     2   3PLM   1.09  0.15  -1.60  0.30   0.22  0.10     NA    NA
#> 57    G2I7     2   3PLM   1.33  0.14   0.26  0.12   0.13  0.05     NA    NA
#> 58    G2I8     2   3PLM   2.22  0.19  -0.09  0.08   0.14  0.05     NA    NA
#> 59    G2I9     2   3PLM   1.08  0.13  -1.56  0.28   0.21  0.09     NA    NA
#> 60   G2I10     2   3PLM   1.65  0.28   0.88  0.11   0.29  0.05     NA    NA
#> 61   G2I11     2   3PLM   0.96  0.13   0.86  0.14   0.11  0.05     NA    NA
#> 62   G2I12     2   3PLM   1.68  0.19  -0.68  0.19   0.26  0.10     NA    NA
#> 63   G2I13     2   3PLM   1.17  0.13  -1.30  0.24   0.21  0.09     NA    NA
#> 64   G2I14     2   3PLM   1.43  0.14   0.22  0.11   0.13  0.05     NA    NA
#> 65   G2I15     2   3PLM   1.58  0.21   0.05  0.17   0.27  0.08     NA    NA
#> 66   G2I16     2   3PLM   1.77  0.24   0.74  0.09   0.22  0.05     NA    NA
#> 67   G2I17     2   3PLM   2.11  0.22  -1.21  0.15   0.20  0.09     NA    NA
#> 68   G2I18     2   3PLM   1.79  0.21   0.63  0.08   0.14  0.04     NA    NA
#> 69   G2I19     2   3PLM   1.12  0.15   0.05  0.22   0.21  0.08     NA    NA
#> 70   G2I20     2   3PLM   1.70  0.47   2.16  0.18   0.16  0.02     NA    NA
#> 71   G2I21     2   3PLM   2.49  0.38   1.53  0.06   0.12  0.02     NA    NA
#> 72   G2I22     2   3PLM   1.50  0.19   0.24  0.15   0.22  0.07     NA    NA
#> 73   G2I23     2   3PLM   2.19  0.28   0.45  0.09   0.28  0.05     NA    NA
#> 74   G2I24     2   3PLM   1.43  0.12   0.36  0.07   0.07  0.03     NA    NA
#> 75   G2I25     2   3PLM   1.80  0.23   1.37  0.06   0.07  0.02     NA    NA
#> 76   G2I26     2   3PLM   1.90  0.20  -0.88  0.17   0.25  0.10     NA    NA
#> 77   G2I27     2   3PLM   0.99  0.12  -0.58  0.25   0.20  0.09     NA    NA
#> 78   G2I28     5    GRM   1.14  0.07  -0.37  0.07   0.14  0.05   0.78  0.05
#> 79    C2I1     2   3PLM   1.07  0.08  -0.29  0.11   0.13  0.04     NA    NA
#> 80    C2I2     2   3PLM   1.00  0.09   1.23  0.07   0.06  0.02     NA    NA
#> 81    C2I3     2   3PLM   1.63  0.16   1.37  0.05   0.13  0.01     NA    NA
#> 82    C2I4     2   3PLM   1.55  0.11   0.24  0.06   0.18  0.02     NA    NA
#> 83    C2I5     2   3PLM   1.43  0.10  -0.07  0.07   0.12  0.03     NA    NA
#> 84    C2I6     2   3PLM   2.04  0.10  -0.06  0.04   0.05  0.01     NA    NA
#> 85    C2I7     2   3PLM   1.56  0.11   0.00  0.07   0.18  0.02     NA    NA
#> 86    C2I8     2   3PLM   1.53  0.15   1.27  0.06   0.19  0.02     NA    NA
#> 87    C2I9     2   3PLM   2.17  0.13  -1.01  0.07   0.14  0.03     NA    NA
#> 88   C2I10     2   3PLM   1.58  0.12  -1.35  0.14   0.26  0.05     NA    NA
#> 89    G3I1     2   3PLM   1.52  0.14  -1.10  0.13   0.16  0.05     NA    NA
#> 90    G3I2     2   3PLM   0.77  0.10  -0.60  0.27   0.19  0.07     NA    NA
#> 91    G3I3     2   3PLM   1.15  0.12   0.08  0.12   0.17  0.03     NA    NA
#> 92    G3I4     2   3PLM   1.31  0.24   1.71  0.12   0.22  0.02     NA    NA
#> 93    G3I5     2   3PLM   0.79  0.11  -1.00  0.36   0.29  0.08     NA    NA
#> 94    G3I6     2   3PLM   1.18  0.15  -1.30  0.25   0.33  0.07     NA    NA
#> 95    G3I7     2   3PLM   1.37  0.13   0.26  0.09   0.14  0.03     NA    NA
#> 96    G3I8     2   3PLM   1.92  0.18  -0.08  0.07   0.15  0.02     NA    NA
#> 97    G3I9     2   3PLM   1.13  0.12  -1.20  0.21   0.21  0.06     NA    NA
#> 98   G3I10     2   3PLM   1.51  0.21   0.97  0.09   0.29  0.02     NA    NA
#> 99   G3I11     2   3PLM   0.90  0.10   0.80  0.11   0.08  0.03     NA    NA
#> 100  G3I12     2   3PLM   1.42  0.17  -0.78  0.15   0.26  0.05     NA    NA
#> 101  G3I13     2   3PLM   1.11  0.13  -1.10  0.22   0.24  0.06     NA    NA
#> 102  G3I14     2   3PLM   1.38  0.14   0.30  0.09   0.17  0.03     NA    NA
#> 103  G3I15     2   3PLM   1.29  0.13  -0.06  0.11   0.19  0.03     NA    NA
#> 104  G3I16     2   3PLM   1.58  0.18   0.74  0.08   0.20  0.02     NA    NA
#> 105  G3I17     2   3PLM   1.61  0.16  -1.55  0.15   0.19  0.06     NA    NA
#> 106  G3I18     2   3PLM   1.38  0.14   0.71  0.08   0.12  0.02     NA    NA
#> 107  G3I19     2   3PLM   0.99  0.11   0.09  0.14   0.17  0.04     NA    NA
#> 108  G3I20     2   3PLM   1.16  0.24   2.27  0.20   0.12  0.02     NA    NA
#> 109  G3I21     2   3PLM   3.06  0.57   1.63  0.06   0.14  0.01     NA    NA
#> 110  G3I22     2   3PLM   1.17  0.11   0.09  0.10   0.10  0.03     NA    NA
#> 111  G3I23     2   3PLM   1.79  0.17   0.32  0.06   0.15  0.02     NA    NA
#> 112  G3I24     2   3PLM   1.13  0.09   0.32  0.08   0.05  0.02     NA    NA
#> 113  G3I25     2   3PLM   1.38  0.15   1.35  0.08   0.05  0.01     NA    NA
#> 114  G3I26     2   3PLM   1.66  0.17  -1.00  0.12   0.22  0.04     NA    NA
#> 115  G3I27     2   3PLM   0.92  0.10  -0.62  0.21   0.18  0.06     NA    NA
#> 116  G3I28     5    GRM   0.94  0.05  -0.38  0.06   0.15  0.06   0.81  0.07
#>      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      NA    NA
#> 40      NA    NA
#> 41      NA    NA
#> 42      NA    NA
#> 43      NA    NA
#> 44      NA    NA
#> 45      NA    NA
#> 46      NA    NA
#> 47      NA    NA
#> 48    1.42  0.08
#> 49   -0.12  0.03
#> 50    1.25  0.06
#> 51      NA    NA
#> 52      NA    NA
#> 53      NA    NA
#> 54      NA    NA
#> 55      NA    NA
#> 56      NA    NA
#> 57      NA    NA
#> 58      NA    NA
#> 59      NA    NA
#> 60      NA    NA
#> 61      NA    NA
#> 62      NA    NA
#> 63      NA    NA
#> 64      NA    NA
#> 65      NA    NA
#> 66      NA    NA
#> 67      NA    NA
#> 68      NA    NA
#> 69      NA    NA
#> 70      NA    NA
#> 71      NA    NA
#> 72      NA    NA
#> 73      NA    NA
#> 74      NA    NA
#> 75      NA    NA
#> 76      NA    NA
#> 77      NA    NA
#> 78    1.44  0.07
#> 79      NA    NA
#> 80      NA    NA
#> 81      NA    NA
#> 82      NA    NA
#> 83      NA    NA
#> 84      NA    NA
#> 85      NA    NA
#> 86      NA    NA
#> 87      NA    NA
#> 88      NA    NA
#> 89      NA    NA
#> 90      NA    NA
#> 91      NA    NA
#> 92      NA    NA
#> 93      NA    NA
#> 94      NA    NA
#> 95      NA    NA
#> 96      NA    NA
#> 97      NA    NA
#> 98      NA    NA
#> 99      NA    NA
#> 100     NA    NA
#> 101     NA    NA
#> 102     NA    NA
#> 103     NA    NA
#> 104     NA    NA
#> 105     NA    NA
#> 106     NA    NA
#> 107     NA    NA
#> 108     NA    NA
#> 109     NA    NA
#> 110     NA    NA
#> 111     NA    NA
#> 112     NA    NA
#> 113     NA    NA
#> 114     NA    NA
#> 115     NA    NA
#> 116   1.55  0.09
#>  Group Parameters: 
#>                      mu  sigma2  sigma
#> estimate(Group1)   0.00    1.00   1.00
#> se(Group1)           NA      NA     NA
#> estimate(Group2)   0.49    0.56   0.75
#> se(Group2)         0.02    0.02   0.01
#> estimate(Group3)  -0.37    1.95   1.40
#> se(Group3)         0.03    0.06   0.02
#> 

# Extract the item parameter estimates
getirt(fit.1, what = "par.est")
#> $overall
#>        id cats model     par.1         par.2       par.3      par.4      par.5
#> 1    C1I1    2  3PLM 0.9002993  1.3630996575  0.27971247         NA         NA
#> 2    C1I2    2  3PLM 2.1246417 -0.9940070182  0.18249617         NA         NA
#> 3    C1I3    2  3PLM 1.0541278  0.6086911594  0.18187156         NA         NA
#> 4    C1I4    2  3PLM 1.0789640 -0.1731865971  0.28172552         NA         NA
#> 5    C1I5    2  3PLM 0.8731636 -0.1539491072  0.18023951         NA         NA
#> 6    C1I6    2  3PLM 1.9100015  0.5956052555  0.08680994         NA         NA
#> 7    C1I7    2  3PLM 1.1042207  1.1038327947  0.15045515         NA         NA
#> 8    C1I8    2  3PLM 0.9407054  0.8862776985  0.15890193         NA         NA
#> 9    C1I9    2  3PLM 0.8926878  0.6255808132  0.20799520         NA         NA
#> 10  C1I10    2  3PLM 1.4872042  0.1323954156  0.15372172         NA         NA
#> 11   G1I1    2  3PLM 0.9648995 -0.4558711253  0.15992372         NA         NA
#> 12   G1I2    2  3PLM 0.8982317  1.2036385733  0.10668307         NA         NA
#> 13   G1I3    2  3PLM 1.5352805  1.2976450244  0.18624929         NA         NA
#> 14   G1I4    2  3PLM 1.5653523  0.2810572503  0.30326526         NA         NA
#> 15   G1I5    2  3PLM 1.3511034 -0.1735122321  0.16083579         NA         NA
#> 16   G1I6    2  3PLM 2.1741319  0.0170068934  0.08620387         NA         NA
#> 17   G1I7    2  3PLM 1.4655950 -0.0639529911  0.20160863         NA         NA
#> 18   G1I8    2  3PLM 2.5274624  1.1707732481  0.32453925         NA         NA
#> 19   G1I9    2  3PLM 2.3986360 -0.9296897935  0.24418284         NA         NA
#> 20  G1I10    2  3PLM 1.2585300 -1.7572134100  0.23391662         NA         NA
#> 21  G1I11    2  3PLM 1.5470937 -1.1118172625  0.20751568         NA         NA
#> 22  G1I12    2  3PLM 0.7556135 -0.8038513476  0.19656034         NA         NA
#> 23  G1I13    2  3PLM 1.0527888 -0.1185472910  0.20738601         NA         NA
#> 24  G1I14    2  3PLM 1.5102932  1.7406214138  0.29709350         NA         NA
#> 25  G1I15    2  3PLM 0.8664971 -1.4323893982  0.21465862         NA         NA
#> 26  G1I16    2  3PLM 1.0542506 -1.9446624559  0.22812622         NA         NA
#> 27  G1I17    2  3PLM 1.0571687  0.2574260236  0.15670798         NA         NA
#> 28  G1I18    2  3PLM 2.1077524 -0.0802270383  0.24053595         NA         NA
#> 29  G1I19    2  3PLM 1.3243225 -1.3873897664  0.20073972         NA         NA
#> 30  G1I20    2  3PLM 1.0303144  0.5198099414  0.22883325         NA         NA
#> 31  G1I21    2  3PLM 0.9230319  0.7599364216  0.13215608         NA         NA
#> 32  G1I22    2  3PLM 1.7870120 -0.6306797452  0.36010326         NA         NA
#> 33  G1I23    2  3PLM 1.3094006 -1.1637343298  0.25562388         NA         NA
#> 34  G1I24    2  3PLM 1.6645514  0.3233691792  0.23193826         NA         NA
#> 35  G1I25    2  3PLM 1.5989408 -0.1175965320  0.25385326         NA         NA
#> 36  G1I26    2  3PLM 1.9085326  0.6574101910  0.25475783         NA         NA
#> 37  G1I27    2  3PLM 1.6232889 -1.5482047540  0.26787427         NA         NA
#> 38  G1I28    2  3PLM 1.3497743  0.5653525107  0.15089732         NA         NA
#> 39  G1I29    2  3PLM 0.9229732 -0.3681170370  0.12451485         NA         NA
#> 40  G1I30    2  3PLM 1.0636199  2.2379302346  0.17259660         NA         NA
#> 41  G1I31    2  3PLM 2.4508248  1.6191425517  0.18202915         NA         NA
#> 42  G1I32    2  3PLM 1.1137113 -0.0722062405  0.14770787         NA         NA
#> 43  G1I33    2  3PLM 1.6242308  0.1841468641  0.16019712         NA         NA
#> 44  G1I34    2  3PLM 1.3396394  0.2438713586  0.10954369         NA         NA
#> 45  G1I35    2  3PLM 1.3498016  1.2826823305  0.06760955         NA         NA
#> 46  G1I36    2  3PLM 1.4377572 -1.2263600162  0.22297804         NA         NA
#> 47  G1I37    2  3PLM 1.0646367 -0.5799937848  0.27097889         NA         NA
#> 48  G1I38    5   GRM 1.0649768 -0.3653605937  0.21365349  0.8571501  1.4220840
#> 49  C1I11    5   GRM 1.1892683 -2.2080814013 -1.44963215 -0.7471002 -0.1188008
#> 50  C1I12    5   GRM 0.9141155 -0.6885203546  0.02631147  0.6789189  1.2487151
#> 51   G2I1    2  3PLM 1.7609541 -0.8578505763  0.24035894         NA         NA
#> 52   G2I2    2  3PLM 0.8436960 -0.4382789590  0.21438678         NA         NA
#> 53   G2I3    2  3PLM 1.0886052  0.0911696953  0.18458253         NA         NA
#> 54   G2I4    2  3PLM 1.5129142  1.4502930938  0.18027120         NA         NA
#> 55   G2I5    2  3PLM 0.7139700 -1.6581797011  0.19781812         NA         NA
#> 56   G2I6    2  3PLM 1.0919249 -1.5999033264  0.21671254         NA         NA
#> 57   G2I7    2  3PLM 1.3325688  0.2633001084  0.13036039         NA         NA
#> 58   G2I8    2  3PLM 2.2199875 -0.0891740323  0.13780823         NA         NA
#> 59   G2I9    2  3PLM 1.0832071 -1.5619697258  0.20682223         NA         NA
#> 60  G2I10    2  3PLM 1.6525972  0.8838581723  0.29075920         NA         NA
#> 61  G2I11    2  3PLM 0.9630202  0.8571579302  0.11294805         NA         NA
#> 62  G2I12    2  3PLM 1.6845080 -0.6793797269  0.25993819         NA         NA
#> 63  G2I13    2  3PLM 1.1744864 -1.3046157573  0.20979661         NA         NA
#> 64  G2I14    2  3PLM 1.4329862  0.2158522639  0.12986846         NA         NA
#> 65  G2I15    2  3PLM 1.5763766  0.0515443558  0.27114790         NA         NA
#> 66  G2I16    2  3PLM 1.7692947  0.7418363904  0.21979684         NA         NA
#> 67  G2I17    2  3PLM 2.1059175 -1.2085830250  0.20089939         NA         NA
#> 68  G2I18    2  3PLM 1.7861465  0.6306602727  0.14248762         NA         NA
#> 69  G2I19    2  3PLM 1.1236963  0.0462753636  0.21071941         NA         NA
#> 70  G2I20    2  3PLM 1.6957930  2.1576947259  0.16005055         NA         NA
#> 71  G2I21    2  3PLM 2.4882719  1.5329597736  0.11756098         NA         NA
#> 72  G2I22    2  3PLM 1.4964966  0.2397808198  0.21725633         NA         NA
#> 73  G2I23    2  3PLM 2.1906673  0.4452744277  0.28388427         NA         NA
#> 74  G2I24    2  3PLM 1.4315517  0.3567207429  0.07277537         NA         NA
#> 75  G2I25    2  3PLM 1.8049099  1.3706377057  0.07249339         NA         NA
#> 76  G2I26    2  3PLM 1.8993430 -0.8763222582  0.24753546         NA         NA
#> 77  G2I27    2  3PLM 0.9922602 -0.5809395811  0.20058057         NA         NA
#> 78  G2I28    5   GRM 1.1428941 -0.3726286557  0.13986303  0.7760909  1.4399545
#> 79   C2I1    2  3PLM 1.0724788 -0.2925185560  0.12706645         NA         NA
#> 80   C2I2    2  3PLM 1.0026458  1.2329410896  0.06038371         NA         NA
#> 81   C2I3    2  3PLM 1.6298126  1.3717691195  0.12593191         NA         NA
#> 82   C2I4    2  3PLM 1.5539481  0.2386833733  0.18401402         NA         NA
#> 83   C2I5    2  3PLM 1.4284843 -0.0690446128  0.11886473         NA         NA
#> 84   C2I6    2  3PLM 2.0356386 -0.0609918387  0.04627366         NA         NA
#> 85   C2I7    2  3PLM 1.5601571  0.0009650238  0.17678863         NA         NA
#> 86   C2I8    2  3PLM 1.5324423  1.2675649741  0.18822170         NA         NA
#> 87   C2I9    2  3PLM 2.1652174 -1.0147828245  0.14241930         NA         NA
#> 88  C2I10    2  3PLM 1.5756487 -1.3546338562  0.26239714         NA         NA
#> 89   G3I1    2  3PLM 1.5177028 -1.1012930860  0.16029848         NA         NA
#> 90   G3I2    2  3PLM 0.7684745 -0.5958229695  0.19239503         NA         NA
#> 91   G3I3    2  3PLM 1.1511024  0.0760256826  0.16953254         NA         NA
#> 92   G3I4    2  3PLM 1.3078975  1.7075770987  0.22441977         NA         NA
#> 93   G3I5    2  3PLM 0.7934450 -1.0028686365  0.28801113         NA         NA
#> 94   G3I6    2  3PLM 1.1835586 -1.2995355416  0.32753494         NA         NA
#> 95   G3I7    2  3PLM 1.3696476  0.2557793128  0.14270858         NA         NA
#> 96   G3I8    2  3PLM 1.9236418 -0.0843458748  0.14793716         NA         NA
#> 97   G3I9    2  3PLM 1.1300723 -1.1978738749  0.21206463         NA         NA
#> 98  G3I10    2  3PLM 1.5126694  0.9703312346  0.29033962         NA         NA
#> 99  G3I11    2  3PLM 0.9021794  0.7968831683  0.08388012         NA         NA
#> 100 G3I12    2  3PLM 1.4214420 -0.7766765557  0.25975672         NA         NA
#> 101 G3I13    2  3PLM 1.1082496 -1.0983147672  0.23504140         NA         NA
#> 102 G3I14    2  3PLM 1.3797649  0.3022772316  0.16690037         NA         NA
#> 103 G3I15    2  3PLM 1.2876135 -0.0573943519  0.19245588         NA         NA
#> 104 G3I16    2  3PLM 1.5824294  0.7372872341  0.20217816         NA         NA
#> 105 G3I17    2  3PLM 1.6058953 -1.5522762053  0.18645088         NA         NA
#> 106 G3I18    2  3PLM 1.3800008  0.7080448816  0.11881653         NA         NA
#> 107 G3I19    2  3PLM 0.9933705  0.0911167346  0.16557109         NA         NA
#> 108 G3I20    2  3PLM 1.1645170  2.2678742079  0.11821851         NA         NA
#> 109 G3I21    2  3PLM 3.0616274  1.6301700614  0.14040136         NA         NA
#> 110 G3I22    2  3PLM 1.1712500  0.0856353364  0.10276551         NA         NA
#> 111 G3I23    2  3PLM 1.7915979  0.3172563768  0.15045425         NA         NA
#> 112 G3I24    2  3PLM 1.1259988  0.3187456987  0.05486984         NA         NA
#> 113 G3I25    2  3PLM 1.3848088  1.3476897612  0.05145709         NA         NA
#> 114 G3I26    2  3PLM 1.6627860 -0.9983631667  0.21786458         NA         NA
#> 115 G3I27    2  3PLM 0.9242740 -0.6199576206  0.18433956         NA         NA
#> 116 G3I28    5   GRM 0.9426521 -0.3795334501  0.14801722  0.8117698  1.5515925
#> 
#> $group
#> $group$Group1
#>       id cats model     par.1       par.2       par.3      par.4      par.5
#> 1   C1I1    2  3PLM 0.9002993  1.36309966  0.27971247         NA         NA
#> 2   C1I2    2  3PLM 2.1246417 -0.99400702  0.18249617         NA         NA
#> 3   C1I3    2  3PLM 1.0541278  0.60869116  0.18187156         NA         NA
#> 4   C1I4    2  3PLM 1.0789640 -0.17318660  0.28172552         NA         NA
#> 5   C1I5    2  3PLM 0.8731636 -0.15394911  0.18023951         NA         NA
#> 6   C1I6    2  3PLM 1.9100015  0.59560526  0.08680994         NA         NA
#> 7   C1I7    2  3PLM 1.1042207  1.10383279  0.15045515         NA         NA
#> 8   C1I8    2  3PLM 0.9407054  0.88627770  0.15890193         NA         NA
#> 9   C1I9    2  3PLM 0.8926878  0.62558081  0.20799520         NA         NA
#> 10 C1I10    2  3PLM 1.4872042  0.13239542  0.15372172         NA         NA
#> 11  G1I1    2  3PLM 0.9648995 -0.45587113  0.15992372         NA         NA
#> 12  G1I2    2  3PLM 0.8982317  1.20363857  0.10668307         NA         NA
#> 13  G1I3    2  3PLM 1.5352805  1.29764502  0.18624929         NA         NA
#> 14  G1I4    2  3PLM 1.5653523  0.28105725  0.30326526         NA         NA
#> 15  G1I5    2  3PLM 1.3511034 -0.17351223  0.16083579         NA         NA
#> 16  G1I6    2  3PLM 2.1741319  0.01700689  0.08620387         NA         NA
#> 17  G1I7    2  3PLM 1.4655950 -0.06395299  0.20160863         NA         NA
#> 18  G1I8    2  3PLM 2.5274624  1.17077325  0.32453925         NA         NA
#> 19  G1I9    2  3PLM 2.3986360 -0.92968979  0.24418284         NA         NA
#> 20 G1I10    2  3PLM 1.2585300 -1.75721341  0.23391662         NA         NA
#> 21 G1I11    2  3PLM 1.5470937 -1.11181726  0.20751568         NA         NA
#> 22 G1I12    2  3PLM 0.7556135 -0.80385135  0.19656034         NA         NA
#> 23 G1I13    2  3PLM 1.0527888 -0.11854729  0.20738601         NA         NA
#> 24 G1I14    2  3PLM 1.5102932  1.74062141  0.29709350         NA         NA
#> 25 G1I15    2  3PLM 0.8664971 -1.43238940  0.21465862         NA         NA
#> 26 G1I16    2  3PLM 1.0542506 -1.94466246  0.22812622         NA         NA
#> 27 G1I17    2  3PLM 1.0571687  0.25742602  0.15670798         NA         NA
#> 28 G1I18    2  3PLM 2.1077524 -0.08022704  0.24053595         NA         NA
#> 29 G1I19    2  3PLM 1.3243225 -1.38738977  0.20073972         NA         NA
#> 30 G1I20    2  3PLM 1.0303144  0.51980994  0.22883325         NA         NA
#> 31 G1I21    2  3PLM 0.9230319  0.75993642  0.13215608         NA         NA
#> 32 G1I22    2  3PLM 1.7870120 -0.63067975  0.36010326         NA         NA
#> 33 G1I23    2  3PLM 1.3094006 -1.16373433  0.25562388         NA         NA
#> 34 G1I24    2  3PLM 1.6645514  0.32336918  0.23193826         NA         NA
#> 35 G1I25    2  3PLM 1.5989408 -0.11759653  0.25385326         NA         NA
#> 36 G1I26    2  3PLM 1.9085326  0.65741019  0.25475783         NA         NA
#> 37 G1I27    2  3PLM 1.6232889 -1.54820475  0.26787427         NA         NA
#> 38 G1I28    2  3PLM 1.3497743  0.56535251  0.15089732         NA         NA
#> 39 G1I29    2  3PLM 0.9229732 -0.36811704  0.12451485         NA         NA
#> 40 G1I30    2  3PLM 1.0636199  2.23793023  0.17259660         NA         NA
#> 41 G1I31    2  3PLM 2.4508248  1.61914255  0.18202915         NA         NA
#> 42 G1I32    2  3PLM 1.1137113 -0.07220624  0.14770787         NA         NA
#> 43 G1I33    2  3PLM 1.6242308  0.18414686  0.16019712         NA         NA
#> 44 G1I34    2  3PLM 1.3396394  0.24387136  0.10954369         NA         NA
#> 45 G1I35    2  3PLM 1.3498016  1.28268233  0.06760955         NA         NA
#> 46 G1I36    2  3PLM 1.4377572 -1.22636002  0.22297804         NA         NA
#> 47 G1I37    2  3PLM 1.0646367 -0.57999378  0.27097889         NA         NA
#> 48 G1I38    5   GRM 1.0649768 -0.36536059  0.21365349  0.8571501  1.4220840
#> 49 C1I11    5   GRM 1.1892683 -2.20808140 -1.44963215 -0.7471002 -0.1188008
#> 50 C1I12    5   GRM 0.9141155 -0.68852035  0.02631147  0.6789189  1.2487151
#> 
#> $group$Group2
#>       id cats model     par.1         par.2       par.3      par.4      par.5
#> 1   C1I1    2  3PLM 0.9002993  1.3630996575  0.27971247         NA         NA
#> 2   C1I2    2  3PLM 2.1246417 -0.9940070182  0.18249617         NA         NA
#> 3   C1I3    2  3PLM 1.0541278  0.6086911594  0.18187156         NA         NA
#> 4   C1I4    2  3PLM 1.0789640 -0.1731865971  0.28172552         NA         NA
#> 5   C1I5    2  3PLM 0.8731636 -0.1539491072  0.18023951         NA         NA
#> 6   C1I6    2  3PLM 1.9100015  0.5956052555  0.08680994         NA         NA
#> 7   C1I7    2  3PLM 1.1042207  1.1038327947  0.15045515         NA         NA
#> 8   C1I8    2  3PLM 0.9407054  0.8862776985  0.15890193         NA         NA
#> 9   C1I9    2  3PLM 0.8926878  0.6255808132  0.20799520         NA         NA
#> 10 C1I10    2  3PLM 1.4872042  0.1323954156  0.15372172         NA         NA
#> 11 C1I11    5   GRM 1.1892683 -2.2080814013 -1.44963215 -0.7471002 -0.1188008
#> 12 C1I12    5   GRM 0.9141155 -0.6885203546  0.02631147  0.6789189  1.2487151
#> 13  G2I1    2  3PLM 1.7609541 -0.8578505763  0.24035894         NA         NA
#> 14  G2I2    2  3PLM 0.8436960 -0.4382789590  0.21438678         NA         NA
#> 15  G2I3    2  3PLM 1.0886052  0.0911696953  0.18458253         NA         NA
#> 16  G2I4    2  3PLM 1.5129142  1.4502930938  0.18027120         NA         NA
#> 17  G2I5    2  3PLM 0.7139700 -1.6581797011  0.19781812         NA         NA
#> 18  G2I6    2  3PLM 1.0919249 -1.5999033264  0.21671254         NA         NA
#> 19  G2I7    2  3PLM 1.3325688  0.2633001084  0.13036039         NA         NA
#> 20  G2I8    2  3PLM 2.2199875 -0.0891740323  0.13780823         NA         NA
#> 21  G2I9    2  3PLM 1.0832071 -1.5619697258  0.20682223         NA         NA
#> 22 G2I10    2  3PLM 1.6525972  0.8838581723  0.29075920         NA         NA
#> 23 G2I11    2  3PLM 0.9630202  0.8571579302  0.11294805         NA         NA
#> 24 G2I12    2  3PLM 1.6845080 -0.6793797269  0.25993819         NA         NA
#> 25 G2I13    2  3PLM 1.1744864 -1.3046157573  0.20979661         NA         NA
#> 26 G2I14    2  3PLM 1.4329862  0.2158522639  0.12986846         NA         NA
#> 27 G2I15    2  3PLM 1.5763766  0.0515443558  0.27114790         NA         NA
#> 28 G2I16    2  3PLM 1.7692947  0.7418363904  0.21979684         NA         NA
#> 29 G2I17    2  3PLM 2.1059175 -1.2085830250  0.20089939         NA         NA
#> 30 G2I18    2  3PLM 1.7861465  0.6306602727  0.14248762         NA         NA
#> 31 G2I19    2  3PLM 1.1236963  0.0462753636  0.21071941         NA         NA
#> 32 G2I20    2  3PLM 1.6957930  2.1576947259  0.16005055         NA         NA
#> 33 G2I21    2  3PLM 2.4882719  1.5329597736  0.11756098         NA         NA
#> 34 G2I22    2  3PLM 1.4964966  0.2397808198  0.21725633         NA         NA
#> 35 G2I23    2  3PLM 2.1906673  0.4452744277  0.28388427         NA         NA
#> 36 G2I24    2  3PLM 1.4315517  0.3567207429  0.07277537         NA         NA
#> 37 G2I25    2  3PLM 1.8049099  1.3706377057  0.07249339         NA         NA
#> 38 G2I26    2  3PLM 1.8993430 -0.8763222582  0.24753546         NA         NA
#> 39 G2I27    2  3PLM 0.9922602 -0.5809395811  0.20058057         NA         NA
#> 40 G2I28    5   GRM 1.1428941 -0.3726286557  0.13986303  0.7760909  1.4399545
#> 41  C2I1    2  3PLM 1.0724788 -0.2925185560  0.12706645         NA         NA
#> 42  C2I2    2  3PLM 1.0026458  1.2329410896  0.06038371         NA         NA
#> 43  C2I3    2  3PLM 1.6298126  1.3717691195  0.12593191         NA         NA
#> 44  C2I4    2  3PLM 1.5539481  0.2386833733  0.18401402         NA         NA
#> 45  C2I5    2  3PLM 1.4284843 -0.0690446128  0.11886473         NA         NA
#> 46  C2I6    2  3PLM 2.0356386 -0.0609918387  0.04627366         NA         NA
#> 47  C2I7    2  3PLM 1.5601571  0.0009650238  0.17678863         NA         NA
#> 48  C2I8    2  3PLM 1.5324423  1.2675649741  0.18822170         NA         NA
#> 49  C2I9    2  3PLM 2.1652174 -1.0147828245  0.14241930         NA         NA
#> 50 C2I10    2  3PLM 1.5756487 -1.3546338562  0.26239714         NA         NA
#> 
#> $group$Group3
#>       id cats model     par.1         par.2      par.3     par.4    par.5
#> 1   C2I1    2  3PLM 1.0724788 -0.2925185560 0.12706645        NA       NA
#> 2   C2I2    2  3PLM 1.0026458  1.2329410896 0.06038371        NA       NA
#> 3   C2I3    2  3PLM 1.6298126  1.3717691195 0.12593191        NA       NA
#> 4   C2I4    2  3PLM 1.5539481  0.2386833733 0.18401402        NA       NA
#> 5   C2I5    2  3PLM 1.4284843 -0.0690446128 0.11886473        NA       NA
#> 6   C2I6    2  3PLM 2.0356386 -0.0609918387 0.04627366        NA       NA
#> 7   C2I7    2  3PLM 1.5601571  0.0009650238 0.17678863        NA       NA
#> 8   C2I8    2  3PLM 1.5324423  1.2675649741 0.18822170        NA       NA
#> 9   C2I9    2  3PLM 2.1652174 -1.0147828245 0.14241930        NA       NA
#> 10 C2I10    2  3PLM 1.5756487 -1.3546338562 0.26239714        NA       NA
#> 11  G3I1    2  3PLM 1.5177028 -1.1012930860 0.16029848        NA       NA
#> 12  G3I2    2  3PLM 0.7684745 -0.5958229695 0.19239503        NA       NA
#> 13  G3I3    2  3PLM 1.1511024  0.0760256826 0.16953254        NA       NA
#> 14  G3I4    2  3PLM 1.3078975  1.7075770987 0.22441977        NA       NA
#> 15  G3I5    2  3PLM 0.7934450 -1.0028686365 0.28801113        NA       NA
#> 16  G3I6    2  3PLM 1.1835586 -1.2995355416 0.32753494        NA       NA
#> 17  G3I7    2  3PLM 1.3696476  0.2557793128 0.14270858        NA       NA
#> 18  G3I8    2  3PLM 1.9236418 -0.0843458748 0.14793716        NA       NA
#> 19  G3I9    2  3PLM 1.1300723 -1.1978738749 0.21206463        NA       NA
#> 20 G3I10    2  3PLM 1.5126694  0.9703312346 0.29033962        NA       NA
#> 21 G3I11    2  3PLM 0.9021794  0.7968831683 0.08388012        NA       NA
#> 22 G3I12    2  3PLM 1.4214420 -0.7766765557 0.25975672        NA       NA
#> 23 G3I13    2  3PLM 1.1082496 -1.0983147672 0.23504140        NA       NA
#> 24 G3I14    2  3PLM 1.3797649  0.3022772316 0.16690037        NA       NA
#> 25 G3I15    2  3PLM 1.2876135 -0.0573943519 0.19245588        NA       NA
#> 26 G3I16    2  3PLM 1.5824294  0.7372872341 0.20217816        NA       NA
#> 27 G3I17    2  3PLM 1.6058953 -1.5522762053 0.18645088        NA       NA
#> 28 G3I18    2  3PLM 1.3800008  0.7080448816 0.11881653        NA       NA
#> 29 G3I19    2  3PLM 0.9933705  0.0911167346 0.16557109        NA       NA
#> 30 G3I20    2  3PLM 1.1645170  2.2678742079 0.11821851        NA       NA
#> 31 G3I21    2  3PLM 3.0616274  1.6301700614 0.14040136        NA       NA
#> 32 G3I22    2  3PLM 1.1712500  0.0856353364 0.10276551        NA       NA
#> 33 G3I23    2  3PLM 1.7915979  0.3172563768 0.15045425        NA       NA
#> 34 G3I24    2  3PLM 1.1259988  0.3187456987 0.05486984        NA       NA
#> 35 G3I25    2  3PLM 1.3848088  1.3476897612 0.05145709        NA       NA
#> 36 G3I26    2  3PLM 1.6627860 -0.9983631667 0.21786458        NA       NA
#> 37 G3I27    2  3PLM 0.9242740 -0.6199576206 0.18433956        NA       NA
#> 38 G3I28    5   GRM 0.9426521 -0.3795334501 0.14801722 0.8117698 1.551592
#> 
#> 

# Extract the standard error estimates
getirt(fit.1, what = "se.est")
#> $overall
#>        id cats model      par.1      par.2      par.3      par.4      par.5
#> 1    C1I1    2  3PLM 0.17989519 0.16035224 0.04893744         NA         NA
#> 2    C1I2    2  3PLM 0.14509575 0.09689007 0.05855244         NA         NA
#> 3    C1I3    2  3PLM 0.12333025 0.13013772 0.04568686         NA         NA
#> 4    C1I4    2  3PLM 0.12539953 0.21610221 0.06984577         NA         NA
#> 5    C1I5    2  3PLM 0.09159813 0.22015093 0.06797074         NA         NA
#> 6    C1I6    2  3PLM 0.13215821 0.04030740 0.01860230         NA         NA
#> 7    C1I7    2  3PLM 0.13398772 0.08891913 0.03307573         NA         NA
#> 8    C1I8    2  3PLM 0.12334991 0.13614025 0.04646945         NA         NA
#> 9    C1I9    2  3PLM 0.12236592 0.18495675 0.05639681         NA         NA
#> 10  C1I10    2  3PLM 0.11760294 0.08417251 0.03717477         NA         NA
#> 11   G1I1    2  3PLM 0.10655826 0.20217054 0.06672572         NA         NA
#> 12   G1I2    2  3PLM 0.13694638 0.13626565 0.03909387         NA         NA
#> 13   G1I3    2  3PLM 0.26266869 0.09156215 0.02556423         NA         NA
#> 14   G1I4    2  3PLM 0.21896209 0.11821220 0.04253288         NA         NA
#> 15   G1I5    2  3PLM 0.14227648 0.12915817 0.05186212         NA         NA
#> 16   G1I6    2  3PLM 0.18452247 0.05538028 0.02621436         NA         NA
#> 17   G1I7    2  3PLM 0.16049350 0.12171798 0.04882272         NA         NA
#> 18   G1I8    2  3PLM 0.46550649 0.06951993 0.01895467         NA         NA
#> 19   G1I9    2  3PLM 0.26711886 0.11407326 0.06322653         NA         NA
#> 20  G1I10    2  3PLM 0.12545580 0.22750745 0.09762660         NA         NA
#> 21  G1I11    2  3PLM 0.15057675 0.16486816 0.07867466         NA         NA
#> 22  G1I12    2  3PLM 0.09367336 0.31303395 0.08422275         NA         NA
#> 23  G1I13    2  3PLM 0.14115709 0.20788070 0.06872976         NA         NA
#> 24  G1I14    2  3PLM 0.38235203 0.15053141 0.02485445         NA         NA
#> 25  G1I15    2  3PLM 0.09621277 0.29800051 0.09181670         NA         NA
#> 26  G1I16    2  3PLM 0.11062833 0.26571952 0.09735725         NA         NA
#> 27  G1I17    2  3PLM 0.13151860 0.15421189 0.05197027         NA         NA
#> 28  G1I18    2  3PLM 0.20611213 0.08024248 0.03657769         NA         NA
#> 29  G1I19    2  3PLM 0.12685072 0.19375886 0.08485688         NA         NA
#> 30  G1I20    2  3PLM 0.18089581 0.18707909 0.05966072         NA         NA
#> 31  G1I21    2  3PLM 0.13455458 0.14996962 0.04703101         NA         NA
#> 32  G1I22    2  3PLM 0.21911500 0.15737913 0.06424983         NA         NA
#> 33  G1I23    2  3PLM 0.15053755 0.22982042 0.09322036         NA         NA
#> 34  G1I24    2  3PLM 0.19679685 0.09411360 0.03689301         NA         NA
#> 35  G1I25    2  3PLM 0.18243227 0.12257416 0.04913825         NA         NA
#> 36  G1I26    2  3PLM 0.25542584 0.07703704 0.02893084         NA         NA
#> 37  G1I27    2  3PLM 0.17691915 0.20386267 0.10127163         NA         NA
#> 38  G1I28    2  3PLM 0.16553187 0.09754762 0.03666743         NA         NA
#> 39  G1I29    2  3PLM 0.09087413 0.17161992 0.05512983         NA         NA
#> 40  G1I30    2  3PLM 0.28799644 0.24787264 0.02955840         NA         NA
#> 41  G1I31    2  3PLM 0.48866407 0.09220125 0.01439037         NA         NA
#> 42  G1I32    2  3PLM 0.12263290 0.15178218 0.05400259         NA         NA
#> 43  G1I33    2  3PLM 0.17530464 0.08877908 0.03732072         NA         NA
#> 44  G1I34    2  3PLM 0.13610866 0.09412723 0.03674769         NA         NA
#> 45  G1I35    2  3PLM 0.17137910 0.08344182 0.02039465         NA         NA
#> 46  G1I36    2  3PLM 0.14523548 0.19046168 0.08649362         NA         NA
#> 47  G1I37    2  3PLM 0.14678991 0.27080023 0.08750606         NA         NA
#> 48  G1I38    5   GRM 0.05824336 0.05464783 0.05036679 0.06337710 0.08439014
#> 49  C1I11    5   GRM 0.04768216 0.09429091 0.06737581 0.04635165 0.03448717
#> 50  C1I12    5   GRM 0.04132681 0.05629429 0.04082063 0.04311500 0.05653026
#> 51   G2I1    2  3PLM 0.19350221 0.17643375 0.09757730         NA         NA
#> 52   G2I2    2  3PLM 0.11489091 0.30524136 0.09207477         NA         NA
#> 53   G2I3    2  3PLM 0.13653157 0.20117207 0.07586750         NA         NA
#> 54   G2I4    2  3PLM 0.29245623 0.09333564 0.03808894         NA         NA
#> 55   G2I5    2  3PLM 0.10481807 0.39787167 0.08903051         NA         NA
#> 56   G2I6    2  3PLM 0.14518031 0.29678044 0.09513691         NA         NA
#> 57   G2I7    2  3PLM 0.13785758 0.12130037 0.05478446         NA         NA
#> 58   G2I8    2  3PLM 0.19269491 0.08072746 0.05145801         NA         NA
#> 59   G2I9    2  3PLM 0.13251099 0.27721615 0.09193314         NA         NA
#> 60  G2I10    2  3PLM 0.28338384 0.11486665 0.05004309         NA         NA
#> 61  G2I11    2  3PLM 0.12959812 0.14260521 0.05017878         NA         NA
#> 62  G2I12    2  3PLM 0.19165147 0.18551525 0.09840785         NA         NA
#> 63  G2I13    2  3PLM 0.13361413 0.24090086 0.09247457         NA         NA
#> 64  G2I14    2  3PLM 0.14344036 0.11234266 0.05306700         NA         NA
#> 65  G2I15    2  3PLM 0.20783033 0.17092977 0.07863958         NA         NA
#> 66  G2I16    2  3PLM 0.24141540 0.09487270 0.04614521         NA         NA
#> 67  G2I17    2  3PLM 0.21812356 0.14792918 0.08902957         NA         NA
#> 68  G2I18    2  3PLM 0.20825056 0.08176892 0.04319252         NA         NA
#> 69  G2I19    2  3PLM 0.14974281 0.21807451 0.08279427         NA         NA
#> 70  G2I20    2  3PLM 0.47112164 0.18433168 0.02351512         NA         NA
#> 71  G2I21    2  3PLM 0.37699918 0.05953101 0.01720186         NA         NA
#> 72  G2I22    2  3PLM 0.18888770 0.14944483 0.06872351         NA         NA
#> 73  G2I23    2  3PLM 0.28077865 0.09153703 0.04827635         NA         NA
#> 74  G2I24    2  3PLM 0.12130175 0.07457793 0.03386516         NA         NA
#> 75  G2I25    2  3PLM 0.22582407 0.06157614 0.02227878         NA         NA
#> 76  G2I26    2  3PLM 0.20134542 0.16552101 0.09828502         NA         NA
#> 77  G2I27    2  3PLM 0.11980495 0.24841774 0.08769248         NA         NA
#> 78  G2I28    5   GRM 0.07259152 0.06782688 0.04972658 0.04738506 0.07009125
#> 79   C2I1    2  3PLM 0.07967461 0.10986098 0.03524967         NA         NA
#> 80   C2I2    2  3PLM 0.08692171 0.06717781 0.01809171         NA         NA
#> 81   C2I3    2  3PLM 0.15623585 0.05282304 0.01408261         NA         NA
#> 82   C2I4    2  3PLM 0.11189913 0.06186328 0.02306786         NA         NA
#> 83   C2I5    2  3PLM 0.09565798 0.06810561 0.02521438         NA         NA
#> 84   C2I6    2  3PLM 0.10326422 0.03696659 0.01351802         NA         NA
#> 85   C2I7    2  3PLM 0.10574626 0.06606876 0.02489720         NA         NA
#> 86   C2I8    2  3PLM 0.15231896 0.05723105 0.01673332         NA         NA
#> 87   C2I9    2  3PLM 0.13437617 0.07069516 0.03095926         NA         NA
#> 88  C2I10    2  3PLM 0.12209895 0.14135303 0.05284333         NA         NA
#> 89   G3I1    2  3PLM 0.14295950 0.12757467 0.04598806         NA         NA
#> 90   G3I2    2  3PLM 0.09806822 0.27491495 0.06605199         NA         NA
#> 91   G3I3    2  3PLM 0.12223986 0.11810373 0.03455361         NA         NA
#> 92   G3I4    2  3PLM 0.23674338 0.12489657 0.02042690         NA         NA
#> 93   G3I5    2  3PLM 0.11027073 0.35844706 0.08124875         NA         NA
#> 94   G3I6    2  3PLM 0.15212744 0.25219491 0.07173387         NA         NA
#> 95   G3I7    2  3PLM 0.13192537 0.08582984 0.02599794         NA         NA
#> 96   G3I8    2  3PLM 0.17753632 0.06568100 0.02302363         NA         NA
#> 97   G3I9    2  3PLM 0.12073992 0.20628990 0.06412558         NA         NA
#> 98  G3I10    2  3PLM 0.21325449 0.09460878 0.02268460         NA         NA
#> 99  G3I11    2  3PLM 0.10223560 0.11256805 0.02810817         NA         NA
#> 100 G3I12    2  3PLM 0.16809784 0.14982785 0.04805076         NA         NA
#> 101 G3I13    2  3PLM 0.12969261 0.21859906 0.06476116         NA         NA
#> 102 G3I14    2  3PLM 0.14184270 0.08769788 0.02633227         NA         NA
#> 103 G3I15    2  3PLM 0.13462872 0.11103389 0.03394777         NA         NA
#> 104 G3I16    2  3PLM 0.17691571 0.07758799 0.02083063         NA         NA
#> 105 G3I17    2  3PLM 0.15688465 0.15428884 0.05926460         NA         NA
#> 106 G3I18    2  3PLM 0.14382506 0.07697407 0.02111022         NA         NA
#> 107 G3I19    2  3PLM 0.11393846 0.14464817 0.04000362         NA         NA
#> 108 G3I20    2  3PLM 0.23541976 0.19741351 0.01684581         NA         NA
#> 109 G3I21    2  3PLM 0.57479199 0.06270311 0.01022201         NA         NA
#> 110 G3I22    2  3PLM 0.11463786 0.09814469 0.02978440         NA         NA
#> 111 G3I23    2  3PLM 0.16694784 0.06445158 0.01994484         NA         NA
#> 112 G3I24    2  3PLM 0.09417036 0.07931539 0.02063766         NA         NA
#> 113 G3I25    2  3PLM 0.14627533 0.07996861 0.01264067         NA         NA
#> 114 G3I26    2  3PLM 0.16594755 0.12414916 0.04434197         NA         NA
#> 115 G3I27    2  3PLM 0.10437934 0.20932778 0.05657496         NA         NA
#> 116 G3I28    5   GRM 0.04771674 0.05965450 0.05957007 0.07120741 0.09365933
#> 
#> $group
#> $group$Group1
#>       id cats model      par.1      par.2      par.3      par.4      par.5
#> 1   C1I1    2  3PLM 0.17989519 0.16035224 0.04893744         NA         NA
#> 2   C1I2    2  3PLM 0.14509575 0.09689007 0.05855244         NA         NA
#> 3   C1I3    2  3PLM 0.12333025 0.13013772 0.04568686         NA         NA
#> 4   C1I4    2  3PLM 0.12539953 0.21610221 0.06984577         NA         NA
#> 5   C1I5    2  3PLM 0.09159813 0.22015093 0.06797074         NA         NA
#> 6   C1I6    2  3PLM 0.13215821 0.04030740 0.01860230         NA         NA
#> 7   C1I7    2  3PLM 0.13398772 0.08891913 0.03307573         NA         NA
#> 8   C1I8    2  3PLM 0.12334991 0.13614025 0.04646945         NA         NA
#> 9   C1I9    2  3PLM 0.12236592 0.18495675 0.05639681         NA         NA
#> 10 C1I10    2  3PLM 0.11760294 0.08417251 0.03717477         NA         NA
#> 11  G1I1    2  3PLM 0.10655826 0.20217054 0.06672572         NA         NA
#> 12  G1I2    2  3PLM 0.13694638 0.13626565 0.03909387         NA         NA
#> 13  G1I3    2  3PLM 0.26266869 0.09156215 0.02556423         NA         NA
#> 14  G1I4    2  3PLM 0.21896209 0.11821220 0.04253288         NA         NA
#> 15  G1I5    2  3PLM 0.14227648 0.12915817 0.05186212         NA         NA
#> 16  G1I6    2  3PLM 0.18452247 0.05538028 0.02621436         NA         NA
#> 17  G1I7    2  3PLM 0.16049350 0.12171798 0.04882272         NA         NA
#> 18  G1I8    2  3PLM 0.46550649 0.06951993 0.01895467         NA         NA
#> 19  G1I9    2  3PLM 0.26711886 0.11407326 0.06322653         NA         NA
#> 20 G1I10    2  3PLM 0.12545580 0.22750745 0.09762660         NA         NA
#> 21 G1I11    2  3PLM 0.15057675 0.16486816 0.07867466         NA         NA
#> 22 G1I12    2  3PLM 0.09367336 0.31303395 0.08422275         NA         NA
#> 23 G1I13    2  3PLM 0.14115709 0.20788070 0.06872976         NA         NA
#> 24 G1I14    2  3PLM 0.38235203 0.15053141 0.02485445         NA         NA
#> 25 G1I15    2  3PLM 0.09621277 0.29800051 0.09181670         NA         NA
#> 26 G1I16    2  3PLM 0.11062833 0.26571952 0.09735725         NA         NA
#> 27 G1I17    2  3PLM 0.13151860 0.15421189 0.05197027         NA         NA
#> 28 G1I18    2  3PLM 0.20611213 0.08024248 0.03657769         NA         NA
#> 29 G1I19    2  3PLM 0.12685072 0.19375886 0.08485688         NA         NA
#> 30 G1I20    2  3PLM 0.18089581 0.18707909 0.05966072         NA         NA
#> 31 G1I21    2  3PLM 0.13455458 0.14996962 0.04703101         NA         NA
#> 32 G1I22    2  3PLM 0.21911500 0.15737913 0.06424983         NA         NA
#> 33 G1I23    2  3PLM 0.15053755 0.22982042 0.09322036         NA         NA
#> 34 G1I24    2  3PLM 0.19679685 0.09411360 0.03689301         NA         NA
#> 35 G1I25    2  3PLM 0.18243227 0.12257416 0.04913825         NA         NA
#> 36 G1I26    2  3PLM 0.25542584 0.07703704 0.02893084         NA         NA
#> 37 G1I27    2  3PLM 0.17691915 0.20386267 0.10127163         NA         NA
#> 38 G1I28    2  3PLM 0.16553187 0.09754762 0.03666743         NA         NA
#> 39 G1I29    2  3PLM 0.09087413 0.17161992 0.05512983         NA         NA
#> 40 G1I30    2  3PLM 0.28799644 0.24787264 0.02955840         NA         NA
#> 41 G1I31    2  3PLM 0.48866407 0.09220125 0.01439037         NA         NA
#> 42 G1I32    2  3PLM 0.12263290 0.15178218 0.05400259         NA         NA
#> 43 G1I33    2  3PLM 0.17530464 0.08877908 0.03732072         NA         NA
#> 44 G1I34    2  3PLM 0.13610866 0.09412723 0.03674769         NA         NA
#> 45 G1I35    2  3PLM 0.17137910 0.08344182 0.02039465         NA         NA
#> 46 G1I36    2  3PLM 0.14523548 0.19046168 0.08649362         NA         NA
#> 47 G1I37    2  3PLM 0.14678991 0.27080023 0.08750606         NA         NA
#> 48 G1I38    5   GRM 0.05824336 0.05464783 0.05036679 0.06337710 0.08439014
#> 49 C1I11    5   GRM 0.04768216 0.09429091 0.06737581 0.04635165 0.03448717
#> 50 C1I12    5   GRM 0.04132681 0.05629429 0.04082063 0.04311500 0.05653026
#> 
#> $group$Group2
#>       id cats model      par.1      par.2      par.3      par.4      par.5
#> 1   C1I1    2  3PLM 0.17989519 0.16035224 0.04893744         NA         NA
#> 2   C1I2    2  3PLM 0.14509575 0.09689007 0.05855244         NA         NA
#> 3   C1I3    2  3PLM 0.12333025 0.13013772 0.04568686         NA         NA
#> 4   C1I4    2  3PLM 0.12539953 0.21610221 0.06984577         NA         NA
#> 5   C1I5    2  3PLM 0.09159813 0.22015093 0.06797074         NA         NA
#> 6   C1I6    2  3PLM 0.13215821 0.04030740 0.01860230         NA         NA
#> 7   C1I7    2  3PLM 0.13398772 0.08891913 0.03307573         NA         NA
#> 8   C1I8    2  3PLM 0.12334991 0.13614025 0.04646945         NA         NA
#> 9   C1I9    2  3PLM 0.12236592 0.18495675 0.05639681         NA         NA
#> 10 C1I10    2  3PLM 0.11760294 0.08417251 0.03717477         NA         NA
#> 11 C1I11    5   GRM 0.04768216 0.09429091 0.06737581 0.04635165 0.03448717
#> 12 C1I12    5   GRM 0.04132681 0.05629429 0.04082063 0.04311500 0.05653026
#> 13  G2I1    2  3PLM 0.19350221 0.17643375 0.09757730         NA         NA
#> 14  G2I2    2  3PLM 0.11489091 0.30524136 0.09207477         NA         NA
#> 15  G2I3    2  3PLM 0.13653157 0.20117207 0.07586750         NA         NA
#> 16  G2I4    2  3PLM 0.29245623 0.09333564 0.03808894         NA         NA
#> 17  G2I5    2  3PLM 0.10481807 0.39787167 0.08903051         NA         NA
#> 18  G2I6    2  3PLM 0.14518031 0.29678044 0.09513691         NA         NA
#> 19  G2I7    2  3PLM 0.13785758 0.12130037 0.05478446         NA         NA
#> 20  G2I8    2  3PLM 0.19269491 0.08072746 0.05145801         NA         NA
#> 21  G2I9    2  3PLM 0.13251099 0.27721615 0.09193314         NA         NA
#> 22 G2I10    2  3PLM 0.28338384 0.11486665 0.05004309         NA         NA
#> 23 G2I11    2  3PLM 0.12959812 0.14260521 0.05017878         NA         NA
#> 24 G2I12    2  3PLM 0.19165147 0.18551525 0.09840785         NA         NA
#> 25 G2I13    2  3PLM 0.13361413 0.24090086 0.09247457         NA         NA
#> 26 G2I14    2  3PLM 0.14344036 0.11234266 0.05306700         NA         NA
#> 27 G2I15    2  3PLM 0.20783033 0.17092977 0.07863958         NA         NA
#> 28 G2I16    2  3PLM 0.24141540 0.09487270 0.04614521         NA         NA
#> 29 G2I17    2  3PLM 0.21812356 0.14792918 0.08902957         NA         NA
#> 30 G2I18    2  3PLM 0.20825056 0.08176892 0.04319252         NA         NA
#> 31 G2I19    2  3PLM 0.14974281 0.21807451 0.08279427         NA         NA
#> 32 G2I20    2  3PLM 0.47112164 0.18433168 0.02351512         NA         NA
#> 33 G2I21    2  3PLM 0.37699918 0.05953101 0.01720186         NA         NA
#> 34 G2I22    2  3PLM 0.18888770 0.14944483 0.06872351         NA         NA
#> 35 G2I23    2  3PLM 0.28077865 0.09153703 0.04827635         NA         NA
#> 36 G2I24    2  3PLM 0.12130175 0.07457793 0.03386516         NA         NA
#> 37 G2I25    2  3PLM 0.22582407 0.06157614 0.02227878         NA         NA
#> 38 G2I26    2  3PLM 0.20134542 0.16552101 0.09828502         NA         NA
#> 39 G2I27    2  3PLM 0.11980495 0.24841774 0.08769248         NA         NA
#> 40 G2I28    5   GRM 0.07259152 0.06782688 0.04972658 0.04738506 0.07009125
#> 41  C2I1    2  3PLM 0.07967461 0.10986098 0.03524967         NA         NA
#> 42  C2I2    2  3PLM 0.08692171 0.06717781 0.01809171         NA         NA
#> 43  C2I3    2  3PLM 0.15623585 0.05282304 0.01408261         NA         NA
#> 44  C2I4    2  3PLM 0.11189913 0.06186328 0.02306786         NA         NA
#> 45  C2I5    2  3PLM 0.09565798 0.06810561 0.02521438         NA         NA
#> 46  C2I6    2  3PLM 0.10326422 0.03696659 0.01351802         NA         NA
#> 47  C2I7    2  3PLM 0.10574626 0.06606876 0.02489720         NA         NA
#> 48  C2I8    2  3PLM 0.15231896 0.05723105 0.01673332         NA         NA
#> 49  C2I9    2  3PLM 0.13437617 0.07069516 0.03095926         NA         NA
#> 50 C2I10    2  3PLM 0.12209895 0.14135303 0.05284333         NA         NA
#> 
#> $group$Group3
#>       id cats model      par.1      par.2      par.3      par.4      par.5
#> 1   C2I1    2  3PLM 0.07967461 0.10986098 0.03524967         NA         NA
#> 2   C2I2    2  3PLM 0.08692171 0.06717781 0.01809171         NA         NA
#> 3   C2I3    2  3PLM 0.15623585 0.05282304 0.01408261         NA         NA
#> 4   C2I4    2  3PLM 0.11189913 0.06186328 0.02306786         NA         NA
#> 5   C2I5    2  3PLM 0.09565798 0.06810561 0.02521438         NA         NA
#> 6   C2I6    2  3PLM 0.10326422 0.03696659 0.01351802         NA         NA
#> 7   C2I7    2  3PLM 0.10574626 0.06606876 0.02489720         NA         NA
#> 8   C2I8    2  3PLM 0.15231896 0.05723105 0.01673332         NA         NA
#> 9   C2I9    2  3PLM 0.13437617 0.07069516 0.03095926         NA         NA
#> 10 C2I10    2  3PLM 0.12209895 0.14135303 0.05284333         NA         NA
#> 11  G3I1    2  3PLM 0.14295950 0.12757467 0.04598806         NA         NA
#> 12  G3I2    2  3PLM 0.09806822 0.27491495 0.06605199         NA         NA
#> 13  G3I3    2  3PLM 0.12223986 0.11810373 0.03455361         NA         NA
#> 14  G3I4    2  3PLM 0.23674338 0.12489657 0.02042690         NA         NA
#> 15  G3I5    2  3PLM 0.11027073 0.35844706 0.08124875         NA         NA
#> 16  G3I6    2  3PLM 0.15212744 0.25219491 0.07173387         NA         NA
#> 17  G3I7    2  3PLM 0.13192537 0.08582984 0.02599794         NA         NA
#> 18  G3I8    2  3PLM 0.17753632 0.06568100 0.02302363         NA         NA
#> 19  G3I9    2  3PLM 0.12073992 0.20628990 0.06412558         NA         NA
#> 20 G3I10    2  3PLM 0.21325449 0.09460878 0.02268460         NA         NA
#> 21 G3I11    2  3PLM 0.10223560 0.11256805 0.02810817         NA         NA
#> 22 G3I12    2  3PLM 0.16809784 0.14982785 0.04805076         NA         NA
#> 23 G3I13    2  3PLM 0.12969261 0.21859906 0.06476116         NA         NA
#> 24 G3I14    2  3PLM 0.14184270 0.08769788 0.02633227         NA         NA
#> 25 G3I15    2  3PLM 0.13462872 0.11103389 0.03394777         NA         NA
#> 26 G3I16    2  3PLM 0.17691571 0.07758799 0.02083063         NA         NA
#> 27 G3I17    2  3PLM 0.15688465 0.15428884 0.05926460         NA         NA
#> 28 G3I18    2  3PLM 0.14382506 0.07697407 0.02111022         NA         NA
#> 29 G3I19    2  3PLM 0.11393846 0.14464817 0.04000362         NA         NA
#> 30 G3I20    2  3PLM 0.23541976 0.19741351 0.01684581         NA         NA
#> 31 G3I21    2  3PLM 0.57479199 0.06270311 0.01022201         NA         NA
#> 32 G3I22    2  3PLM 0.11463786 0.09814469 0.02978440         NA         NA
#> 33 G3I23    2  3PLM 0.16694784 0.06445158 0.01994484         NA         NA
#> 34 G3I24    2  3PLM 0.09417036 0.07931539 0.02063766         NA         NA
#> 35 G3I25    2  3PLM 0.14627533 0.07996861 0.01264067         NA         NA
#> 36 G3I26    2  3PLM 0.16594755 0.12414916 0.04434197         NA         NA
#> 37 G3I27    2  3PLM 0.10437934 0.20932778 0.05657496         NA         NA
#> 38 G3I28    5   GRM 0.04771674 0.05965450 0.05957007 0.07120741 0.09365933
#> 
#> 

# Extract the group-level parameter estimates (i.e., scale parameters)
getirt(fit.1, what = "group.par")
#> $Group1
#>           mu sigma2 sigma
#> estimates  0      1     1
#> se        NA     NA    NA
#> 
#> $Group2
#>                   mu    sigma2      sigma
#> estimates 0.48545465 0.5575252 0.74667607
#> se        0.01669618 0.0176349 0.01180894
#> 
#> $Group3
#>                    mu     sigma2      sigma
#> estimates -0.36960845 1.95177274 1.39705860
#> se         0.03123918 0.06173591 0.02209496
#> 

# Extract the posterior latent ability distributions for each group
getirt(fit.1, what = "weights")
#> $Group1
#>    theta       weight
#> 1  -6.00 2.596404e-24
#> 2  -5.75 6.661461e-12
#> 3  -5.50 6.687303e-11
#> 4  -5.25 3.455143e-10
#> 5  -5.00 1.240179e-09
#> 6  -4.75 3.547688e-09
#> 7  -4.50 8.892960e-09
#> 8  -4.25 2.132090e-08
#> 9  -4.00 5.399524e-08
#> 10 -3.75 1.635844e-07
#> 11 -3.50 6.878012e-07
#> 12 -3.25 4.538677e-06
#> 13 -3.00 4.536928e-05
#> 14 -2.75 4.969458e-04
#> 15 -2.50 3.578851e-03
#> 16 -2.25 1.233388e-02
#> 17 -2.00 2.221829e-02
#> 18 -1.75 2.756719e-02
#> 19 -1.50 3.067494e-02
#> 20 -1.25 3.699025e-02
#> 21 -1.00 5.137755e-02
#> 22 -0.75 7.176369e-02
#> 23 -0.50 8.884787e-02
#> 24 -0.25 1.006945e-01
#> 25  0.00 1.009843e-01
#> 26  0.25 9.093334e-02
#> 27  0.50 8.369774e-02
#> 28  0.75 8.097105e-02
#> 29  1.00 7.169684e-02
#> 30  1.25 5.091446e-02
#> 31  1.50 3.038547e-02
#> 32  1.75 1.784466e-02
#> 33  2.00 1.099557e-02
#> 34  2.25 6.559709e-03
#> 35  2.50 3.586983e-03
#> 36  2.75 1.890639e-03
#> 37  3.00 1.057103e-03
#> 38  3.25 6.683050e-04
#> 39  3.50 4.688643e-04
#> 40  3.75 3.327412e-04
#> 41  4.00 2.150724e-04
#> 42  4.25 1.183608e-04
#> 43  4.50 5.413395e-05
#> 44  4.75 2.065077e-05
#> 45  5.00 6.683841e-06
#> 46  5.25 1.876663e-06
#> 47  5.50 4.678457e-07
#> 48  5.75 1.058922e-07
#> 49  6.00 2.221172e-08
#> 
#> $Group2
#>    theta       weight
#> 1  -6.00 4.462701e-59
#> 2  -5.75 5.478200e-57
#> 3  -5.50 1.250041e-54
#> 4  -5.25 5.639732e-52
#> 5  -5.00 5.211160e-49
#> 6  -4.75 9.838326e-46
#> 7  -4.50 3.615812e-42
#> 8  -4.25 2.347659e-38
#> 9  -4.00 2.341231e-34
#> 10 -3.75 3.033389e-30
#> 11 -3.50 4.296681e-26
#> 12 -3.25 5.685725e-22
#> 13 -3.00 6.113539e-18
#> 14 -2.75 4.561112e-14
#> 15 -2.50 1.824133e-10
#> 16 -2.25 2.490398e-07
#> 17 -2.00 6.146621e-05
#> 18 -1.75 1.684077e-03
#> 19 -1.50 7.306548e-03
#> 20 -1.25 1.587374e-02
#> 21 -1.00 2.333185e-02
#> 22 -0.75 1.961168e-02
#> 23 -0.50 3.881370e-02
#> 24 -0.25 1.156738e-01
#> 25  0.00 8.052857e-02
#> 26  0.25 8.668950e-02
#> 27  0.50 1.901804e-01
#> 28  0.75 1.359231e-01
#> 29  1.00 8.624615e-02
#> 30  1.25 8.381817e-02
#> 31  1.50 5.935414e-02
#> 32  1.75 3.231143e-02
#> 33  2.00 1.436083e-02
#> 34  2.25 3.834694e-03
#> 35  2.50 1.089214e-03
#> 36  2.75 7.241950e-04
#> 37  3.00 9.441983e-04
#> 38  3.25 1.046937e-03
#> 39  3.50 5.018765e-04
#> 40  3.75 8.379032e-05
#> 41  4.00 5.539575e-06
#> 42  4.25 1.915245e-07
#> 43  4.50 4.643638e-09
#> 44  4.75 1.007048e-10
#> 45  5.00 2.322071e-12
#> 46  5.25 6.327049e-14
#> 47  5.50 2.144754e-15
#> 48  5.75 9.160693e-17
#> 49  6.00 4.870649e-18
#> 
#> $Group3
#>    theta       weight
#> 1  -6.00 7.731298e-07
#> 2  -5.75 3.145368e-06
#> 3  -5.50 1.179526e-05
#> 4  -5.25 4.054836e-05
#> 5  -5.00 1.268999e-04
#> 6  -4.75 3.584830e-04
#> 7  -4.50 9.049299e-04
#> 8  -4.25 2.018717e-03
#> 9  -4.00 3.937687e-03
#> 10 -3.75 6.669793e-03
#> 11 -3.50 9.833099e-03
#> 12 -3.25 1.285721e-02
#> 13 -3.00 1.552609e-02
#> 14 -2.75 1.830649e-02
#> 15 -2.50 2.198421e-02
#> 16 -2.25 2.658226e-02
#> 17 -2.00 3.040018e-02
#> 18 -1.75 3.230747e-02
#> 19 -1.50 3.635519e-02
#> 20 -1.25 5.077580e-02
#> 21 -1.00 7.272466e-02
#> 22 -0.75 6.898185e-02
#> 23 -0.50 5.513930e-02
#> 24 -0.25 6.620643e-02
#> 25  0.00 8.981610e-02
#> 26  0.25 8.120609e-02
#> 27  0.50 5.730756e-02
#> 28  0.75 4.971849e-02
#> 29  1.00 4.623885e-02
#> 30  1.25 3.634235e-02
#> 31  1.50 3.452349e-02
#> 32  1.75 3.537759e-02
#> 33  2.00 2.023739e-02
#> 34  2.25 7.344692e-03
#> 35  2.50 2.991667e-03
#> 36  2.75 1.765572e-03
#> 37  3.00 1.415897e-03
#> 38  3.25 1.272697e-03
#> 39  3.50 1.056410e-03
#> 40  3.75 7.149887e-04
#> 41  4.00 3.785780e-04
#> 42  4.25 1.595304e-04
#> 43  4.50 5.584698e-05
#> 44  4.75 1.699530e-05
#> 45  5.00 4.664197e-06
#> 46  5.25 1.184031e-06
#> 47  5.50 2.823534e-07
#> 48  5.75 6.377004e-08
#> 49  6.00 1.368772e-08
#> 

# 1-(2). Alternatively, the same parameter estimation can be performed by
# inserting a list of item metadata for the groups into the 'x' argument.
# If the item metadata contains item parameters to be used as starting values,
# set 'use.startval = TRUE'.
# Also, specify the groups in which the ability distribution scales
# will be freely estimated using their group names.
free.group <- group.name[2:3]
fit.2 <-
  est_mg(
    x = x, data = data, group.name = group.name, D = 1,
    free.group = free.group, use.gprior = TRUE,
    gprior = list(dist = "beta", params = c(5, 16)),
    group.mean = 0, group.var = 1, EmpHist = TRUE, use.startval = TRUE,
    Etol = 0.001, MaxE = 500
  )
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -159736.5073, Max-Change: 0.434608
 EM iteration: 2, Loglike: -159610.2190, Max-Change: 0.165418
 EM iteration: 3, Loglike: -159601.3164, Max-Change: 0.088375
 EM iteration: 4, Loglike: -159596.5635, Max-Change: 0.054733
 EM iteration: 5, Loglike: -159593.6124, Max-Change: 0.040346
 EM iteration: 6, Loglike: -159591.6852, Max-Change: 0.035828
 EM iteration: 7, Loglike: -159590.3654, Max-Change: 0.030999
 EM iteration: 8, Loglike: -159589.4215, Max-Change: 0.026538
 EM iteration: 9, Loglike: -159588.7193, Max-Change: 0.022633
 EM iteration: 10, Loglike: -159588.1782, Max-Change: 0.019291
 EM iteration: 11, Loglike: -159587.7475, Max-Change: 0.016548
 EM iteration: 12, Loglike: -159587.3947, Max-Change: 0.014582
 EM iteration: 13, Loglike: -159587.0984, Max-Change: 0.012898
 EM iteration: 14, Loglike: -159586.8441, Max-Change: 0.011459
 EM iteration: 15, Loglike: -159586.6218, Max-Change: 0.010231
 EM iteration: 16, Loglike: -159586.4248, Max-Change: 0.009182
 EM iteration: 17, Loglike: -159586.2479, Max-Change: 0.008283
 EM iteration: 18, Loglike: -159586.0876, Max-Change: 0.007509
 EM iteration: 19, Loglike: -159585.9412, Max-Change: 0.006838
 EM iteration: 20, Loglike: -159585.8067, Max-Change: 0.006252
 EM iteration: 21, Loglike: -159585.6824, Max-Change: 0.005735
 EM iteration: 22, Loglike: -159585.5673, Max-Change: 0.005275
 EM iteration: 23, Loglike: -159585.4607, Max-Change: 0.004862
 EM iteration: 24, Loglike: -159585.3616, Max-Change: 0.004488
 EM iteration: 25, Loglike: -159585.2689, Max-Change: 0.004146
 EM iteration: 26, Loglike: -159585.1819, Max-Change: 0.003832
 EM iteration: 27, Loglike: -159585.0999, Max-Change: 0.003542
 EM iteration: 28, Loglike: -159585.0226, Max-Change: 0.003272
 EM iteration: 29, Loglike: -159584.9497, Max-Change: 0.00302
 EM iteration: 30, Loglike: -159584.8808, Max-Change: 0.002784
 EM iteration: 31, Loglike: -159584.8156, Max-Change: 0.002563
 EM iteration: 32, Loglike: -159584.7538, Max-Change: 0.002355
 EM iteration: 33, Loglike: -159584.6951, Max-Change: 0.00216
 EM iteration: 34, Loglike: -159584.6391, Max-Change: 0.001976
 EM iteration: 35, Loglike: -159584.5858, Max-Change: 0.001804
 EM iteration: 36, Loglike: -159584.5348, Max-Change: 0.001641
 EM iteration: 37, Loglike: -159584.4860, Max-Change: 0.001489
 EM iteration: 38, Loglike: -159584.4392, Max-Change: 0.001351
 EM iteration: 39, Loglike: -159584.3942, Max-Change: 0.00126
 EM iteration: 40, Loglike: -159584.3510, Max-Change: 0.001174
 EM iteration: 41, Loglike: -159584.3092, Max-Change: 0.001092
 EM iteration: 42, Loglike: -159584.2690, Max-Change: 0.001035
 EM iteration: 43, Loglike: -159584.2300, Max-Change: 0.001006
 EM iteration: 44, Loglike: -159584.1923, Max-Change: 0.000979 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 9.09 seconds. 

# Summary of the estimation
summary(fit.2)
#> 
#> Call:
#> est_mg(x = x, data = data, group.name = group.name, D = 1, free.group = free.group, 
#>     use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 
#>         16)), group.mean = 0, group.var = 1, EmpHist = TRUE, 
#>     use.startval = TRUE, Etol = 0.001, MaxE = 500)
#> 
#> Summary of the Data 
#>  Number of Items: 
#>   Overall: 116 unique items 
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of Cases: 
#>   Overall: 6000
#>   By group: 2000(Group1), 2000(Group2), 2000(Group3)
#> 
#> 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: 500
#>    - Item parameters: 358
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 142
#>  Number of fixed items: 
#>   Overall: 0
#>   By group: 0(Group1), 0(Group2), 0(Group3)
#>  Number of E-step cycles completed: 44
#>  Maximum parameter change: 0.0009786731
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 8.37
#>  Standard error computation: 0.21
#>  Total computation: 9.09
#> 
#> 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: 
#>   Overall: 319168.4
#>   By group: 120345.749(Group1), 113945.701(Group2), 84876.932(Group3)
#> 
#>  Akaike Information Criterion (AIC): 320168.4
#>  Bayesian Information Criterion (BIC): 323518.1
#>  Item Parameters (Overall): 
#>         id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1     C1I1     2   3PLM   0.91  0.18   1.38  0.16   0.28  0.05     NA    NA
#> 2     C1I2     2   3PLM   2.14  0.15  -0.95  0.10   0.19  0.06     NA    NA
#> 3     C1I3     2   3PLM   1.07  0.12   0.63  0.13   0.18  0.04     NA    NA
#> 4     C1I4     2   3PLM   1.09  0.13  -0.14  0.21   0.29  0.07     NA    NA
#> 5     C1I5     2   3PLM   0.88  0.09  -0.12  0.22   0.18  0.07     NA    NA
#> 6     C1I6     2   3PLM   1.93  0.13   0.61  0.04   0.09  0.02     NA    NA
#> 7     C1I7     2   3PLM   1.12  0.14   1.12  0.09   0.15  0.03     NA    NA
#> 8     C1I8     2   3PLM   0.95  0.12   0.90  0.13   0.16  0.05     NA    NA
#> 9     C1I9     2   3PLM   0.91  0.13   0.66  0.18   0.21  0.06     NA    NA
#> 10   C1I10     2   3PLM   1.50  0.12   0.16  0.08   0.16  0.04     NA    NA
#> 11    G1I1     2   3PLM   0.97  0.11  -0.43  0.20   0.16  0.07     NA    NA
#> 12    G1I2     2   3PLM   0.90  0.14   1.21  0.13   0.11  0.04     NA    NA
#> 13    G1I3     2   3PLM   1.55  0.27   1.30  0.09   0.19  0.03     NA    NA
#> 14    G1I4     2   3PLM   1.58  0.22   0.30  0.12   0.30  0.04     NA    NA
#> 15    G1I5     2   3PLM   1.36  0.14  -0.15  0.13   0.16  0.05     NA    NA
#> 16    G1I6     2   3PLM   2.18  0.18   0.03  0.06   0.09  0.03     NA    NA
#> 17    G1I7     2   3PLM   1.48  0.16  -0.04  0.12   0.20  0.05     NA    NA
#> 18    G1I8     2   3PLM   2.57  0.47   1.18  0.07   0.32  0.02     NA    NA
#> 19    G1I9     2   3PLM   2.43  0.27  -0.90  0.11   0.25  0.06     NA    NA
#> 20   G1I10     2   3PLM   1.26  0.13  -1.73  0.24   0.24  0.10     NA    NA
#> 21   G1I11     2   3PLM   1.55  0.15  -1.08  0.17   0.21  0.08     NA    NA
#> 22   G1I12     2   3PLM   0.76  0.10  -0.78  0.32   0.20  0.08     NA    NA
#> 23   G1I13     2   3PLM   1.06  0.14  -0.10  0.20   0.21  0.07     NA    NA
#> 24   G1I14     2   3PLM   1.52  0.38   1.74  0.15   0.30  0.02     NA    NA
#> 25   G1I15     2   3PLM   0.87  0.10  -1.41  0.30   0.22  0.09     NA    NA
#> 26   G1I16     2   3PLM   1.05  0.11  -1.93  0.27   0.23  0.10     NA    NA
#> 27   G1I17     2   3PLM   1.07  0.13   0.28  0.15   0.16  0.05     NA    NA
#> 28   G1I18     2   3PLM   2.13  0.21  -0.06  0.08   0.24  0.04     NA    NA
#> 29   G1I19     2   3PLM   1.33  0.13  -1.36  0.20   0.21  0.09     NA    NA
#> 30   G1I20     2   3PLM   1.04  0.18   0.54  0.18   0.23  0.06     NA    NA
#> 31   G1I21     2   3PLM   0.93  0.14   0.78  0.15   0.13  0.05     NA    NA
#> 32   G1I22     2   3PLM   1.82  0.22  -0.59  0.15   0.37  0.06     NA    NA
#> 33   G1I23     2   3PLM   1.32  0.15  -1.13  0.23   0.26  0.09     NA    NA
#> 34   G1I24     2   3PLM   1.68  0.20   0.34  0.09   0.23  0.04     NA    NA
#> 35   G1I25     2   3PLM   1.62  0.18  -0.09  0.12   0.26  0.05     NA    NA
#> 36   G1I26     2   3PLM   1.94  0.26   0.67  0.08   0.26  0.03     NA    NA
#> 37   G1I27     2   3PLM   1.63  0.19  -1.50  0.21   0.28  0.10     NA    NA
#> 38   G1I28     2   3PLM   1.36  0.17   0.58  0.10   0.15  0.04     NA    NA
#> 39   G1I29     2   3PLM   0.93  0.09  -0.35  0.17   0.13  0.06     NA    NA
#> 40   G1I30     2   3PLM   1.06  0.29   2.24  0.25   0.17  0.03     NA    NA
#> 41   G1I31     2   3PLM   2.47  0.49   1.62  0.09   0.18  0.01     NA    NA
#> 42   G1I32     2   3PLM   1.12  0.12  -0.05  0.15   0.15  0.05     NA    NA
#> 43   G1I33     2   3PLM   1.64  0.18   0.20  0.09   0.16  0.04     NA    NA
#> 44   G1I34     2   3PLM   1.35  0.14   0.26  0.09   0.11  0.04     NA    NA
#> 45   G1I35     2   3PLM   1.36  0.17   1.29  0.08   0.07  0.02     NA    NA
#> 46   G1I36     2   3PLM   1.45  0.15  -1.20  0.19   0.23  0.09     NA    NA
#> 47   G1I37     2   3PLM   1.08  0.15  -0.54  0.27   0.28  0.09     NA    NA
#> 48   G1I38     5    GRM   1.07  0.06  -0.35  0.05   0.23  0.05   0.87  0.06
#> 49   C1I11     5    GRM   1.19  0.05  -2.18  0.10  -1.43  0.07  -0.72  0.05
#> 50   C1I12     5    GRM   0.92  0.04  -0.66  0.06   0.05  0.04   0.70  0.04
#> 51    G2I1     2   3PLM   1.79  0.20  -0.81  0.17   0.24  0.10     NA    NA
#> 52    G2I2     2   3PLM   0.86  0.12  -0.40  0.30   0.21  0.09     NA    NA
#> 53    G2I3     2   3PLM   1.11  0.14   0.12  0.20   0.18  0.08     NA    NA
#> 54    G2I4     2   3PLM   1.53  0.30   1.46  0.09   0.18  0.04     NA    NA
#> 55    G2I5     2   3PLM   0.73  0.11  -1.60  0.39   0.20  0.09     NA    NA
#> 56    G2I6     2   3PLM   1.11  0.15  -1.54  0.29   0.22  0.09     NA    NA
#> 57    G2I7     2   3PLM   1.35  0.14   0.29  0.12   0.13  0.05     NA    NA
#> 58    G2I8     2   3PLM   2.26  0.20  -0.05  0.08   0.14  0.05     NA    NA
#> 59    G2I9     2   3PLM   1.10  0.13  -1.51  0.27   0.21  0.09     NA    NA
#> 60   G2I10     2   3PLM   1.68  0.29   0.90  0.11   0.29  0.05     NA    NA
#> 61   G2I11     2   3PLM   0.98  0.13   0.88  0.14   0.11  0.05     NA    NA
#> 62   G2I12     2   3PLM   1.71  0.19  -0.64  0.18   0.26  0.10     NA    NA
#> 63   G2I13     2   3PLM   1.19  0.14  -1.25  0.24   0.21  0.09     NA    NA
#> 64   G2I14     2   3PLM   1.46  0.15   0.25  0.11   0.13  0.05     NA    NA
#> 65   G2I15     2   3PLM   1.60  0.21   0.08  0.17   0.27  0.08     NA    NA
#> 66   G2I16     2   3PLM   1.80  0.25   0.76  0.09   0.22  0.05     NA    NA
#> 67   G2I17     2   3PLM   2.14  0.22  -1.16  0.15   0.20  0.09     NA    NA
#> 68   G2I18     2   3PLM   1.82  0.21   0.65  0.08   0.14  0.04     NA    NA
#> 69   G2I19     2   3PLM   1.15  0.15   0.08  0.21   0.21  0.08     NA    NA
#> 70   G2I20     2   3PLM   1.72  0.48   2.16  0.18   0.16  0.02     NA    NA
#> 71   G2I21     2   3PLM   2.52  0.38   1.54  0.06   0.12  0.02     NA    NA
#> 72   G2I22     2   3PLM   1.52  0.19   0.27  0.15   0.22  0.07     NA    NA
#> 73   G2I23     2   3PLM   2.22  0.28   0.47  0.09   0.28  0.05     NA    NA
#> 74   G2I24     2   3PLM   1.46  0.12   0.38  0.07   0.07  0.03     NA    NA
#> 75   G2I25     2   3PLM   1.84  0.23   1.38  0.06   0.07  0.02     NA    NA
#> 76   G2I26     2   3PLM   1.93  0.20  -0.83  0.16   0.25  0.10     NA    NA
#> 77   G2I27     2   3PLM   1.01  0.12  -0.54  0.24   0.20  0.09     NA    NA
#> 78   G2I28     5    GRM   1.16  0.07  -0.33  0.07   0.17  0.05   0.80  0.05
#> 79    C2I1     2   3PLM   1.10  0.08  -0.26  0.11   0.12  0.04     NA    NA
#> 80    C2I2     2   3PLM   1.02  0.09   1.24  0.07   0.06  0.02     NA    NA
#> 81    C2I3     2   3PLM   1.66  0.16   1.38  0.05   0.13  0.01     NA    NA
#> 82    C2I4     2   3PLM   1.59  0.11   0.27  0.06   0.18  0.02     NA    NA
#> 83    C2I5     2   3PLM   1.46  0.10  -0.03  0.07   0.12  0.03     NA    NA
#> 84    C2I6     2   3PLM   2.08  0.11  -0.02  0.04   0.05  0.01     NA    NA
#> 85    C2I7     2   3PLM   1.59  0.11   0.03  0.07   0.18  0.03     NA    NA
#> 86    C2I8     2   3PLM   1.56  0.16   1.28  0.06   0.19  0.02     NA    NA
#> 87    C2I9     2   3PLM   2.22  0.14  -0.95  0.07   0.14  0.03     NA    NA
#> 88   C2I10     2   3PLM   1.61  0.12  -1.30  0.14   0.25  0.05     NA    NA
#> 89    G3I1     2   3PLM   1.56  0.15  -1.04  0.12   0.16  0.05     NA    NA
#> 90    G3I2     2   3PLM   0.79  0.10  -0.55  0.27   0.19  0.07     NA    NA
#> 91    G3I3     2   3PLM   1.18  0.13   0.11  0.12   0.17  0.04     NA    NA
#> 92    G3I4     2   3PLM   1.34  0.24   1.71  0.12   0.22  0.02     NA    NA
#> 93    G3I5     2   3PLM   0.81  0.11  -0.97  0.35   0.28  0.08     NA    NA
#> 94    G3I6     2   3PLM   1.20  0.15  -1.26  0.25   0.31  0.08     NA    NA
#> 95    G3I7     2   3PLM   1.40  0.14   0.29  0.08   0.14  0.03     NA    NA
#> 96    G3I8     2   3PLM   1.97  0.18  -0.04  0.06   0.15  0.02     NA    NA
#> 97    G3I9     2   3PLM   1.16  0.12  -1.14  0.20   0.21  0.07     NA    NA
#> 98   G3I10     2   3PLM   1.55  0.22   0.99  0.09   0.29  0.02     NA    NA
#> 99   G3I11     2   3PLM   0.93  0.10   0.81  0.11   0.08  0.03     NA    NA
#> 100  G3I12     2   3PLM   1.46  0.17  -0.72  0.15   0.26  0.05     NA    NA
#> 101  G3I13     2   3PLM   1.14  0.13  -1.04  0.21   0.23  0.07     NA    NA
#> 102  G3I14     2   3PLM   1.41  0.15   0.33  0.09   0.17  0.03     NA    NA
#> 103  G3I15     2   3PLM   1.32  0.14  -0.02  0.11   0.19  0.03     NA    NA
#> 104  G3I16     2   3PLM   1.62  0.18   0.76  0.08   0.20  0.02     NA    NA
#> 105  G3I17     2   3PLM   1.66  0.16  -1.47  0.15   0.18  0.06     NA    NA
#> 106  G3I18     2   3PLM   1.41  0.15   0.73  0.08   0.12  0.02     NA    NA
#> 107  G3I19     2   3PLM   1.02  0.12   0.12  0.14   0.16  0.04     NA    NA
#> 108  G3I20     2   3PLM   1.19  0.24   2.25  0.19   0.12  0.02     NA    NA
#> 109  G3I21     2   3PLM   3.09  0.57   1.63  0.06   0.14  0.01     NA    NA
#> 110  G3I22     2   3PLM   1.20  0.12   0.12  0.10   0.10  0.03     NA    NA
#> 111  G3I23     2   3PLM   1.84  0.17   0.35  0.06   0.15  0.02     NA    NA
#> 112  G3I24     2   3PLM   1.16  0.10   0.35  0.08   0.05  0.02     NA    NA
#> 113  G3I25     2   3PLM   1.42  0.15   1.35  0.08   0.05  0.01     NA    NA
#> 114  G3I26     2   3PLM   1.70  0.17  -0.94  0.12   0.21  0.05     NA    NA
#> 115  G3I27     2   3PLM   0.95  0.11  -0.57  0.20   0.18  0.06     NA    NA
#> 116  G3I28     5    GRM   0.97  0.05  -0.32  0.06   0.19  0.06   0.83  0.07
#>      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      NA    NA
#> 40      NA    NA
#> 41      NA    NA
#> 42      NA    NA
#> 43      NA    NA
#> 44      NA    NA
#> 45      NA    NA
#> 46      NA    NA
#> 47      NA    NA
#> 48    1.43  0.08
#> 49   -0.10  0.03
#> 50    1.26  0.06
#> 51      NA    NA
#> 52      NA    NA
#> 53      NA    NA
#> 54      NA    NA
#> 55      NA    NA
#> 56      NA    NA
#> 57      NA    NA
#> 58      NA    NA
#> 59      NA    NA
#> 60      NA    NA
#> 61      NA    NA
#> 62      NA    NA
#> 63      NA    NA
#> 64      NA    NA
#> 65      NA    NA
#> 66      NA    NA
#> 67      NA    NA
#> 68      NA    NA
#> 69      NA    NA
#> 70      NA    NA
#> 71      NA    NA
#> 72      NA    NA
#> 73      NA    NA
#> 74      NA    NA
#> 75      NA    NA
#> 76      NA    NA
#> 77      NA    NA
#> 78    1.45  0.07
#> 79      NA    NA
#> 80      NA    NA
#> 81      NA    NA
#> 82      NA    NA
#> 83      NA    NA
#> 84      NA    NA
#> 85      NA    NA
#> 86      NA    NA
#> 87      NA    NA
#> 88      NA    NA
#> 89      NA    NA
#> 90      NA    NA
#> 91      NA    NA
#> 92      NA    NA
#> 93      NA    NA
#> 94      NA    NA
#> 95      NA    NA
#> 96      NA    NA
#> 97      NA    NA
#> 98      NA    NA
#> 99      NA    NA
#> 100     NA    NA
#> 101     NA    NA
#> 102     NA    NA
#> 103     NA    NA
#> 104     NA    NA
#> 105     NA    NA
#> 106     NA    NA
#> 107     NA    NA
#> 108     NA    NA
#> 109     NA    NA
#> 110     NA    NA
#> 111     NA    NA
#> 112     NA    NA
#> 113     NA    NA
#> 114     NA    NA
#> 115     NA    NA
#> 116   1.55  0.09
#>  Group Parameters: 
#>                      mu  sigma2  sigma
#> estimate(Group1)   0.00    1.00   1.00
#> se(Group1)           NA      NA     NA
#> estimate(Group2)   0.51    0.54   0.73
#> se(Group2)         0.02    0.02   0.01
#> estimate(Group3)  -0.31    1.80   1.34
#> se(Group3)         0.03    0.06   0.02
#> 

## ------------------------------------------------------------------------------
# 2. MG calibration with FIPC using simMG data
#  - Details:
#    (a) Fix the parameters of the common items between the groups
#        (i.e., items C1I1 - C1I12 between Groups 1 and 2, and
#        items C2I1 - C2I10 between Groups 2 and 3)
#    (b) Freely estimate the means and variances of the ability
#        distributions for all three groups
## ------------------------------------------------------------------------------
# 2-(1). Freely estimate the means and variances for all three groups
# Set all three groups as free groups in which the scales
# of the ability distributions are to be freely estimated
free.group <- 1:3 # or use 'free.group <- group.name'

# Specify the locations of items to be fixed in each group's metadata
# For Group 1: C1I1 - C1I12 are located in rows 1-10 and 49-50
# For Group 2: C1I1 - C1I12 are in rows 1-12, and
#              C2I1 - C2I10 are in rows 41-50
# For Group 3: C2I1 - C2I10 are in rows 1-10
fix.loc <- list(
  c(1:10, 49:50),
  c(1:12, 41:50),
  c(1:10)
)

# Estimate IRT parameters using MG-FIPC:
# When FIPC is implemented, item metadata for all groups
# must be provided via the 'x' argument.
# For fixed items, their item parameters must be specified
# in the metadata. For non-fixed items, any placeholder values
# can be used in the metadata.
# Also set fipc = TRUE and fipc.method = "MEM"
fit.3 <-
  est_mg(
    x = x, data = data, group.name = group.name, D = 1,
    free.group = free.group, use.gprior = TRUE,
    gprior = list(dist = "beta", params = c(5, 16)),
    EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE,
    fipc.method = "MEM", fix.loc = fix.loc
  )
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -56756.7548, Max-Change: 3.554166
 EM iteration: 2, Loglike: -160424.0878, Max-Change: 0.780368
 EM iteration: 3, Loglike: -159755.3559, Max-Change: 0.276071
 EM iteration: 4, Loglike: -159686.2158, Max-Change: 0.151073
 EM iteration: 5, Loglike: -159657.5520, Max-Change: 0.102849
 EM iteration: 6, Loglike: -159642.1127, Max-Change: 0.080257
 EM iteration: 7, Loglike: -159633.1411, Max-Change: 0.066343
 EM iteration: 8, Loglike: -159627.6844, Max-Change: 0.055853
 EM iteration: 9, Loglike: -159624.2459, Max-Change: 0.047172
 EM iteration: 10, Loglike: -159622.0120, Max-Change: 0.039781
 EM iteration: 11, Loglike: -159620.5195, Max-Change: 0.033468
 EM iteration: 12, Loglike: -159619.4950, Max-Change: 0.028101
 EM iteration: 13, Loglike: -159618.7723, Max-Change: 0.023565
 EM iteration: 14, Loglike: -159618.2481, Max-Change: 0.019752
 EM iteration: 15, Loglike: -159617.8570, Max-Change: 0.016845
 EM iteration: 16, Loglike: -159617.5566, Max-Change: 0.014864
 EM iteration: 17, Loglike: -159617.3191, Max-Change: 0.01318
 EM iteration: 18, Loglike: -159617.1261, Max-Change: 0.011751
 EM iteration: 19, Loglike: -159616.9650, Max-Change: 0.010538
 EM iteration: 20, Loglike: -159616.8273, Max-Change: 0.009507
 EM iteration: 21, Loglike: -159616.7072, Max-Change: 0.008626
 EM iteration: 22, Loglike: -159616.6005, Max-Change: 0.007869
 EM iteration: 23, Loglike: -159616.5042, Max-Change: 0.007214
 EM iteration: 24, Loglike: -159616.4163, Max-Change: 0.006643
 EM iteration: 25, Loglike: -159616.3352, Max-Change: 0.006141
 EM iteration: 26, Loglike: -159616.2596, Max-Change: 0.005695
 EM iteration: 27, Loglike: -159616.1888, Max-Change: 0.005295
 EM iteration: 28, Loglike: -159616.1221, Max-Change: 0.004934
 EM iteration: 29, Loglike: -159616.0589, Max-Change: 0.004606
 EM iteration: 30, Loglike: -159615.9989, Max-Change: 0.004306
 EM iteration: 31, Loglike: -159615.9416, Max-Change: 0.00403
 EM iteration: 32, Loglike: -159615.8868, Max-Change: 0.003774
 EM iteration: 33, Loglike: -159615.8343, Max-Change: 0.003537
 EM iteration: 34, Loglike: -159615.7838, Max-Change: 0.003317
 EM iteration: 35, Loglike: -159615.7352, Max-Change: 0.003111
 EM iteration: 36, Loglike: -159615.6884, Max-Change: 0.002919
 EM iteration: 37, Loglike: -159615.6431, Max-Change: 0.00274
 EM iteration: 38, Loglike: -159615.5994, Max-Change: 0.002572
 EM iteration: 39, Loglike: -159615.5570, Max-Change: 0.002416
 EM iteration: 40, Loglike: -159615.5159, Max-Change: 0.002269
 EM iteration: 41, Loglike: -159615.4761, Max-Change: 0.002131
 EM iteration: 42, Loglike: -159615.4373, Max-Change: 0.002003
 EM iteration: 43, Loglike: -159615.3997, Max-Change: 0.001882
 EM iteration: 44, Loglike: -159615.3630, Max-Change: 0.00177
 EM iteration: 45, Loglike: -159615.3274, Max-Change: 0.001665
 EM iteration: 46, Loglike: -159615.2926, Max-Change: 0.001566
 EM iteration: 47, Loglike: -159615.2587, Max-Change: 0.001474
 EM iteration: 48, Loglike: -159615.2256, Max-Change: 0.001388
 EM iteration: 49, Loglike: -159615.1933, Max-Change: 0.001308
 EM iteration: 50, Loglike: -159615.1617, Max-Change: 0.001232
 EM iteration: 51, Loglike: -159615.1309, Max-Change: 0.001162
 EM iteration: 52, Loglike: -159615.1007, Max-Change: 0.001097
 EM iteration: 53, Loglike: -159615.0711, Max-Change: 0.001035
 EM iteration: 54, Loglike: -159615.0422, Max-Change: 0.000978 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 9.24 seconds. 

# Summary of the estimation
summary(fit.3)
#> 
#> Call:
#> est_mg(x = x, data = data, group.name = group.name, D = 1, free.group = free.group, 
#>     use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 
#>         16)), EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Summary of the Data 
#>  Number of Items: 
#>   Overall: 116 unique items 
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of Cases: 
#>   Overall: 6000
#>   By group: 2000(Group1), 2000(Group2), 2000(Group3)
#> 
#> 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: 432
#>    - Item parameters: 288
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 144
#>  Number of fixed items: 
#>   Overall: 22
#>   By group: 12(Group1), 22(Group2), 10(Group3)
#>  Number of E-step cycles completed: 54
#>  Maximum parameter change: 0.0009779691
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 8.44
#>  Standard error computation: 0.16
#>  Total computation: 9.24
#> 
#> 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: 
#>   Overall: 319230.1
#>   By group: 120354.311(Group1), 113985.392(Group2), 84890.382(Group3)
#> 
#>  Akaike Information Criterion (AIC): 320094.1
#>  Bayesian Information Criterion (BIC): 322988.3
#>  Item Parameters (Overall): 
#>         id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1     C1I1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2     C1I2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3     C1I3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4     C1I4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5     C1I5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6     C1I6     2   3PLM   1.70    NA   0.63    NA   0.07    NA     NA    NA
#> 7     C1I7     2   3PLM   0.91    NA   1.02    NA   0.12    NA     NA    NA
#> 8     C1I8     2   3PLM   0.84    NA   0.80    NA   0.11    NA     NA    NA
#> 9     C1I9     2   3PLM   0.85    NA   0.85    NA   0.26    NA     NA    NA
#> 10   C1I10     2   3PLM   1.53    NA   0.09    NA   0.14    NA     NA    NA
#> 11    G1I1     2   3PLM   0.96  0.10  -0.46  0.20   0.16  0.07     NA    NA
#> 12    G1I2     2   3PLM   0.88  0.13   1.21  0.14   0.10  0.04     NA    NA
#> 13    G1I3     2   3PLM   1.49  0.26   1.31  0.09   0.18  0.03     NA    NA
#> 14    G1I4     2   3PLM   1.52  0.21   0.27  0.12   0.30  0.04     NA    NA
#> 15    G1I5     2   3PLM   1.33  0.14  -0.18  0.13   0.16  0.05     NA    NA
#> 16    G1I6     2   3PLM   2.15  0.18   0.01  0.05   0.08  0.03     NA    NA
#> 17    G1I7     2   3PLM   1.45  0.16  -0.07  0.12   0.20  0.05     NA    NA
#> 18    G1I8     2   3PLM   2.50  0.46   1.18  0.07   0.32  0.02     NA    NA
#> 19    G1I9     2   3PLM   2.40  0.26  -0.93  0.11   0.24  0.06     NA    NA
#> 20   G1I10     2   3PLM   1.26  0.13  -1.75  0.23   0.24  0.10     NA    NA
#> 21   G1I11     2   3PLM   1.55  0.15  -1.11  0.16   0.21  0.08     NA    NA
#> 22   G1I12     2   3PLM   0.75  0.09  -0.81  0.31   0.19  0.08     NA    NA
#> 23   G1I13     2   3PLM   1.04  0.14  -0.13  0.21   0.20  0.07     NA    NA
#> 24   G1I14     2   3PLM   1.46  0.37   1.77  0.15   0.30  0.03     NA    NA
#> 25   G1I15     2   3PLM   0.86  0.09  -1.44  0.29   0.21  0.09     NA    NA
#> 26   G1I16     2   3PLM   1.05  0.11  -1.95  0.27   0.23  0.10     NA    NA
#> 27   G1I17     2   3PLM   1.04  0.13   0.25  0.15   0.15  0.05     NA    NA
#> 28   G1I18     2   3PLM   2.08  0.20  -0.09  0.08   0.24  0.04     NA    NA
#> 29   G1I19     2   3PLM   1.32  0.13  -1.38  0.19   0.20  0.08     NA    NA
#> 30   G1I20     2   3PLM   1.01  0.17   0.51  0.19   0.22  0.06     NA    NA
#> 31   G1I21     2   3PLM   0.92  0.13   0.77  0.15   0.13  0.05     NA    NA
#> 32   G1I22     2   3PLM   1.77  0.21  -0.65  0.16   0.35  0.06     NA    NA
#> 33   G1I23     2   3PLM   1.31  0.15  -1.17  0.23   0.25  0.09     NA    NA
#> 34   G1I24     2   3PLM   1.64  0.19   0.32  0.10   0.23  0.04     NA    NA
#> 35   G1I25     2   3PLM   1.58  0.18  -0.13  0.12   0.25  0.05     NA    NA
#> 36   G1I26     2   3PLM   1.88  0.25   0.66  0.08   0.25  0.03     NA    NA
#> 37   G1I27     2   3PLM   1.63  0.18  -1.54  0.20   0.27  0.10     NA    NA
#> 38   G1I28     2   3PLM   1.33  0.16   0.57  0.10   0.15  0.04     NA    NA
#> 39   G1I29     2   3PLM   0.92  0.09  -0.37  0.17   0.12  0.05     NA    NA
#> 40   G1I30     2   3PLM   1.00  0.27   2.30  0.25   0.17  0.03     NA    NA
#> 41   G1I31     2   3PLM   2.39  0.46   1.64  0.09   0.18  0.01     NA    NA
#> 42   G1I32     2   3PLM   1.10  0.12  -0.08  0.15   0.15  0.05     NA    NA
#> 43   G1I33     2   3PLM   1.60  0.17   0.18  0.09   0.16  0.04     NA    NA
#> 44   G1I34     2   3PLM   1.33  0.13   0.24  0.09   0.11  0.04     NA    NA
#> 45   G1I35     2   3PLM   1.33  0.17   1.30  0.08   0.07  0.02     NA    NA
#> 46   G1I36     2   3PLM   1.44  0.14  -1.23  0.19   0.22  0.09     NA    NA
#> 47   G1I37     2   3PLM   1.05  0.14  -0.60  0.27   0.27  0.09     NA    NA
#> 48   G1I38     5    GRM   1.06  0.06  -0.37  0.05   0.22  0.05   0.87  0.06
#> 49   C1I11     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 50   C1I12     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#> 51    G2I1     2   3PLM   1.74  0.19  -0.87  0.17   0.23  0.10     NA    NA
#> 52    G2I2     2   3PLM   0.83  0.11  -0.45  0.31   0.21  0.09     NA    NA
#> 53    G2I3     2   3PLM   1.06  0.13   0.08  0.20   0.18  0.07     NA    NA
#> 54    G2I4     2   3PLM   1.45  0.28   1.50  0.10   0.18  0.04     NA    NA
#> 55    G2I5     2   3PLM   0.70  0.10  -1.68  0.40   0.20  0.09     NA    NA
#> 56    G2I6     2   3PLM   1.08  0.14  -1.60  0.29   0.22  0.09     NA    NA
#> 57    G2I7     2   3PLM   1.30  0.13   0.27  0.12   0.13  0.05     NA    NA
#> 58    G2I8     2   3PLM   2.18  0.19  -0.09  0.08   0.14  0.05     NA    NA
#> 59    G2I9     2   3PLM   1.07  0.13  -1.57  0.28   0.21  0.09     NA    NA
#> 60   G2I10     2   3PLM   1.59  0.27   0.90  0.12   0.29  0.05     NA    NA
#> 61   G2I11     2   3PLM   0.93  0.12   0.88  0.15   0.11  0.05     NA    NA
#> 62   G2I12     2   3PLM   1.66  0.18  -0.70  0.18   0.25  0.10     NA    NA
#> 63   G2I13     2   3PLM   1.16  0.13  -1.31  0.24   0.21  0.09     NA    NA
#> 64   G2I14     2   3PLM   1.39  0.14   0.22  0.11   0.13  0.05     NA    NA
#> 65   G2I15     2   3PLM   1.53  0.20   0.03  0.17   0.26  0.08     NA    NA
#> 66   G2I16     2   3PLM   1.70  0.23   0.75  0.10   0.21  0.05     NA    NA
#> 67   G2I17     2   3PLM   2.11  0.22  -1.20  0.14   0.20  0.09     NA    NA
#> 68   G2I18     2   3PLM   1.72  0.20   0.64  0.08   0.14  0.04     NA    NA
#> 69   G2I19     2   3PLM   1.10  0.14   0.05  0.22   0.21  0.08     NA    NA
#> 70   G2I20     2   3PLM   1.62  0.45   2.24  0.19   0.16  0.02     NA    NA
#> 71   G2I21     2   3PLM   2.39  0.36   1.59  0.06   0.12  0.02     NA    NA
#> 72   G2I22     2   3PLM   1.43  0.17   0.22  0.15   0.20  0.07     NA    NA
#> 73   G2I23     2   3PLM   2.09  0.26   0.44  0.10   0.28  0.05     NA    NA
#> 74   G2I24     2   3PLM   1.40  0.12   0.37  0.08   0.07  0.03     NA    NA
#> 75   G2I25     2   3PLM   1.74  0.22   1.42  0.06   0.07  0.02     NA    NA
#> 76   G2I26     2   3PLM   1.89  0.20  -0.88  0.16   0.24  0.10     NA    NA
#> 77   G2I27     2   3PLM   0.98  0.12  -0.59  0.25   0.20  0.09     NA    NA
#> 78   G2I28     5    GRM   1.12  0.07  -0.37  0.07   0.15  0.05   0.80  0.05
#> 79    C2I1     2   3PLM   0.97    NA  -0.46    NA   0.05    NA     NA    NA
#> 80    C2I2     2   3PLM   0.85    NA   1.18    NA   0.01    NA     NA    NA
#> 81    C2I3     2   3PLM   1.43    NA   1.41    NA   0.10    NA     NA    NA
#> 82    C2I4     2   3PLM   1.48    NA   0.18    NA   0.17    NA     NA    NA
#> 83    C2I5     2   3PLM   1.27    NA  -0.23    NA   0.03    NA     NA    NA
#> 84    C2I6     2   3PLM   2.02    NA  -0.09    NA   0.01    NA     NA    NA
#> 85    C2I7     2   3PLM   1.37    NA  -0.13    NA   0.10    NA     NA    NA
#> 86    C2I8     2   3PLM   1.67    NA   1.25    NA   0.19    NA     NA    NA
#> 87    C2I9     2   3PLM   2.28    NA  -1.01    NA   0.10    NA     NA    NA
#> 88   C2I10     2   3PLM   1.42    NA  -1.65    NA   0.11    NA     NA    NA
#> 89    G3I1     2   3PLM   1.60  0.14  -1.04  0.12   0.14  0.05     NA    NA
#> 90    G3I2     2   3PLM   0.79  0.09  -0.62  0.25   0.16  0.07     NA    NA
#> 91    G3I3     2   3PLM   1.16  0.12   0.07  0.12   0.16  0.04     NA    NA
#> 92    G3I4     2   3PLM   1.30  0.24   1.72  0.12   0.22  0.02     NA    NA
#> 93    G3I5     2   3PLM   0.80  0.10  -1.09  0.33   0.24  0.09     NA    NA
#> 94    G3I6     2   3PLM   1.18  0.13  -1.38  0.24   0.26  0.08     NA    NA
#> 95    G3I7     2   3PLM   1.37  0.13   0.26  0.09   0.13  0.03     NA    NA
#> 96    G3I8     2   3PLM   1.95  0.18  -0.07  0.06   0.14  0.02     NA    NA
#> 97    G3I9     2   3PLM   1.18  0.11  -1.16  0.19   0.18  0.07     NA    NA
#> 98   G3I10     2   3PLM   1.50  0.21   0.98  0.10   0.29  0.02     NA    NA
#> 99   G3I11     2   3PLM   0.91  0.10   0.80  0.11   0.08  0.03     NA    NA
#> 100  G3I12     2   3PLM   1.44  0.17  -0.78  0.15   0.23  0.06     NA    NA
#> 101  G3I13     2   3PLM   1.15  0.13  -1.06  0.20   0.21  0.07     NA    NA
#> 102  G3I14     2   3PLM   1.38  0.14   0.31  0.09   0.16  0.03     NA    NA
#> 103  G3I15     2   3PLM   1.29  0.13  -0.06  0.11   0.18  0.04     NA    NA
#> 104  G3I16     2   3PLM   1.58  0.18   0.75  0.08   0.20  0.02     NA    NA
#> 105  G3I17     2   3PLM   1.72  0.15  -1.46  0.13   0.16  0.06     NA    NA
#> 106  G3I18     2   3PLM   1.37  0.14   0.71  0.08   0.11  0.02     NA    NA
#> 107  G3I19     2   3PLM   0.99  0.11   0.07  0.15   0.15  0.04     NA    NA
#> 108  G3I20     2   3PLM   1.14  0.23   2.30  0.20   0.12  0.02     NA    NA
#> 109  G3I21     2   3PLM   2.97  0.55   1.66  0.06   0.14  0.01     NA    NA
#> 110  G3I22     2   3PLM   1.19  0.11   0.10  0.10   0.10  0.03     NA    NA
#> 111  G3I23     2   3PLM   1.79  0.17   0.33  0.06   0.15  0.02     NA    NA
#> 112  G3I24     2   3PLM   1.15  0.09   0.34  0.08   0.05  0.02     NA    NA
#> 113  G3I25     2   3PLM   1.38  0.15   1.36  0.08   0.05  0.01     NA    NA
#> 114  G3I26     2   3PLM   1.72  0.17  -0.96  0.12   0.20  0.05     NA    NA
#> 115  G3I27     2   3PLM   0.95  0.10  -0.61  0.20   0.16  0.06     NA    NA
#> 116  G3I28     5    GRM   1.00  0.05  -0.31  0.05   0.19  0.05   0.82  0.07
#>      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      NA    NA
#> 40      NA    NA
#> 41      NA    NA
#> 42      NA    NA
#> 43      NA    NA
#> 44      NA    NA
#> 45      NA    NA
#> 46      NA    NA
#> 47      NA    NA
#> 48    1.44  0.08
#> 49   -0.12    NA
#> 50    1.25    NA
#> 51      NA    NA
#> 52      NA    NA
#> 53      NA    NA
#> 54      NA    NA
#> 55      NA    NA
#> 56      NA    NA
#> 57      NA    NA
#> 58      NA    NA
#> 59      NA    NA
#> 60      NA    NA
#> 61      NA    NA
#> 62      NA    NA
#> 63      NA    NA
#> 64      NA    NA
#> 65      NA    NA
#> 66      NA    NA
#> 67      NA    NA
#> 68      NA    NA
#> 69      NA    NA
#> 70      NA    NA
#> 71      NA    NA
#> 72      NA    NA
#> 73      NA    NA
#> 74      NA    NA
#> 75      NA    NA
#> 76      NA    NA
#> 77      NA    NA
#> 78    1.48  0.07
#> 79      NA    NA
#> 80      NA    NA
#> 81      NA    NA
#> 82      NA    NA
#> 83      NA    NA
#> 84      NA    NA
#> 85      NA    NA
#> 86      NA    NA
#> 87      NA    NA
#> 88      NA    NA
#> 89      NA    NA
#> 90      NA    NA
#> 91      NA    NA
#> 92      NA    NA
#> 93      NA    NA
#> 94      NA    NA
#> 95      NA    NA
#> 96      NA    NA
#> 97      NA    NA
#> 98      NA    NA
#> 99      NA    NA
#> 100     NA    NA
#> 101     NA    NA
#> 102     NA    NA
#> 103     NA    NA
#> 104     NA    NA
#> 105     NA    NA
#> 106     NA    NA
#> 107     NA    NA
#> 108     NA    NA
#> 109     NA    NA
#> 110     NA    NA
#> 111     NA    NA
#> 112     NA    NA
#> 113     NA    NA
#> 114     NA    NA
#> 115     NA    NA
#> 116   1.52  0.09
#>  Group Parameters: 
#>                      mu  sigma2  sigma
#> estimate(Group1)  -0.01    1.01   1.01
#> se(Group1)         0.02    0.03   0.02
#> estimate(Group2)   0.50    0.58   0.76
#> se(Group2)         0.02    0.02   0.01
#> estimate(Group3)  -0.28    1.66   1.29
#> se(Group3)         0.03    0.05   0.02
#> 

# Extract the item parameter estimates
getirt(fit.3, what = "par.est")
#> $overall
#>        id cats model     par.1       par.2       par.3      par.4     par.5
#> 1    C1I1    2  3PLM 0.7600000  1.46000000  0.26000000         NA        NA
#> 2    C1I2    2  3PLM 1.9200000 -1.05000000  0.18000000         NA        NA
#> 3    C1I3    2  3PLM 0.9300000  0.39000000  0.10000000         NA        NA
#> 4    C1I4    2  3PLM 1.0500000 -0.41000000  0.20000000         NA        NA
#> 5    C1I5    2  3PLM 0.8700000 -0.12000000  0.16000000         NA        NA
#> 6    C1I6    2  3PLM 1.7000000  0.63000000  0.07000000         NA        NA
#> 7    C1I7    2  3PLM 0.9100000  1.02000000  0.12000000         NA        NA
#> 8    C1I8    2  3PLM 0.8400000  0.80000000  0.11000000         NA        NA
#> 9    C1I9    2  3PLM 0.8500000  0.85000000  0.26000000         NA        NA
#> 10  C1I10    2  3PLM 1.5300000  0.09000000  0.14000000         NA        NA
#> 11   G1I1    2  3PLM 0.9601094 -0.46323480  0.15743926         NA        NA
#> 12   G1I2    2  3PLM 0.8792351  1.21120501  0.10365721         NA        NA
#> 13   G1I3    2  3PLM 1.4922037  1.31254694  0.18433436         NA        NA
#> 14   G1I4    2  3PLM 1.5246246  0.26709705  0.29751014         NA        NA
#> 15   G1I5    2  3PLM 1.3343885 -0.18476888  0.15642097         NA        NA
#> 16   G1I6    2  3PLM 2.1457296  0.01122821  0.08373745         NA        NA
#> 17   G1I7    2  3PLM 1.4492421 -0.07232467  0.19831474         NA        NA
#> 18   G1I8    2  3PLM 2.4981366  1.18142115  0.32459740         NA        NA
#> 19   G1I9    2  3PLM 2.4036619 -0.93389393  0.24090792         NA        NA
#> 20  G1I10    2  3PLM 1.2603565 -1.75076139  0.23712596         NA        NA
#> 21  G1I11    2  3PLM 1.5482444 -1.11447546  0.20564573         NA        NA
#> 22  G1I12    2  3PLM 0.7501881 -0.81274221  0.19483606         NA        NA
#> 23  G1I13    2  3PLM 1.0391850 -0.13129451  0.20305864         NA        NA
#> 24  G1I14    2  3PLM 1.4641086  1.76817734  0.29587746         NA        NA
#> 25  G1I15    2  3PLM 0.8642507 -1.44005202  0.21246953         NA        NA
#> 26  G1I16    2  3PLM 1.0481967 -1.94936219  0.22996558         NA        NA
#> 27  G1I17    2  3PLM 1.0413440  0.24963719  0.15297903         NA        NA
#> 28  G1I18    2  3PLM 2.0821164 -0.08782853  0.23783714         NA        NA
#> 29  G1I19    2  3PLM 1.3246385 -1.38457105  0.20248019         NA        NA
#> 30  G1I20    2  3PLM 1.0078270  0.50901778  0.22362359         NA        NA
#> 31  G1I21    2  3PLM 0.9153677  0.76786397  0.13229024         NA        NA
#> 32  G1I22    2  3PLM 1.7709516 -0.64567759  0.35461144         NA        NA
#> 33  G1I23    2  3PLM 1.3100376 -1.16619310  0.25418714         NA        NA
#> 34  G1I24    2  3PLM 1.6392138  0.31576653  0.22824910         NA        NA
#> 35  G1I25    2  3PLM 1.5820253 -0.12733170  0.25023679         NA        NA
#> 36  G1I26    2  3PLM 1.8799489  0.66003504  0.25368918         NA        NA
#> 37  G1I27    2  3PLM 1.6294671 -1.53964203  0.27165399         NA        NA
#> 38  G1I28    2  3PLM 1.3331814  0.56546916  0.14913136         NA        NA
#> 39  G1I29    2  3PLM 0.9154613 -0.37004169  0.12429485         NA        NA
#> 40  G1I30    2  3PLM 1.0026860  2.29722730  0.16842602         NA        NA
#> 41  G1I31    2  3PLM 2.3881154  1.64223594  0.18163836         NA        NA
#> 42  G1I32    2  3PLM 1.1031428 -0.07825039  0.14531435         NA        NA
#> 43  G1I33    2  3PLM 1.5998724  0.17672610  0.15654349         NA        NA
#> 44  G1I34    2  3PLM 1.3266408  0.24185668  0.10791955         NA        NA
#> 45  G1I35    2  3PLM 1.3307122  1.29621629  0.06712955         NA        NA
#> 46  G1I36    2  3PLM 1.4375780 -1.23256008  0.21920779         NA        NA
#> 47  G1I37    2  3PLM 1.0523257 -0.59860569  0.26551312         NA        NA
#> 48  G1I38    5   GRM 1.0557817 -0.36687770  0.21653854  0.8654794  1.435633
#> 49  C1I11    5   GRM 1.2300000 -2.08000000 -1.35000000 -0.7100000 -0.120000
#> 50  C1I12    5   GRM 0.8800000 -0.76000000 -0.01000000  0.6700000  1.250000
#> 51   G2I1    2  3PLM 1.7426027 -0.86693468  0.23241989         NA        NA
#> 52   G2I2    2  3PLM 0.8271656 -0.44864390  0.21072500         NA        NA
#> 53   G2I3    2  3PLM 1.0594812  0.08411544  0.17868700         NA        NA
#> 54   G2I4    2  3PLM 1.4496890  1.49538158  0.17845021         NA        NA
#> 55   G2I5    2  3PLM 0.7000088 -1.68320349  0.19753124         NA        NA
#> 56   G2I6    2  3PLM 1.0845772 -1.60171524  0.21553624         NA        NA
#> 57   G2I7    2  3PLM 1.2984963  0.26873160  0.12783879         NA        NA
#> 58   G2I8    2  3PLM 2.1768681 -0.08931817  0.13528274         NA        NA
#> 59   G2I9    2  3PLM 1.0731850 -1.56763064  0.20570964         NA        NA
#> 60  G2I10    2  3PLM 1.5904952  0.90286013  0.28755659         NA        NA
#> 61  G2I11    2  3PLM 0.9314018  0.88217244  0.11179900         NA        NA
#> 62  G2I12    2  3PLM 1.6623481 -0.69600414  0.24746346         NA        NA
#> 63  G2I13    2  3PLM 1.1643479 -1.30658561  0.20913476         NA        NA
#> 64  G2I14    2  3PLM 1.3949743  0.22018889  0.12742097         NA        NA
#> 65  G2I15    2  3PLM 1.5256012  0.03274459  0.25930675         NA        NA
#> 66  G2I16    2  3PLM 1.6958062  0.75377859  0.21478247         NA        NA
#> 67  G2I17    2  3PLM 2.1138538 -1.19916943  0.19757122         NA        NA
#> 68  G2I18    2  3PLM 1.7193451  0.64117712  0.13803059         NA        NA
#> 69  G2I19    2  3PLM 1.0971580  0.04569867  0.20768248         NA        NA
#> 70  G2I20    2  3PLM 1.6230735  2.23612968  0.15952981         NA        NA
#> 71  G2I21    2  3PLM 2.3884669  1.58525725  0.11741983         NA        NA
#> 72  G2I22    2  3PLM 1.4332531  0.22129279  0.20450739         NA        NA
#> 73  G2I23    2  3PLM 2.0941037  0.44296609  0.27618451         NA        NA
#> 74  G2I24    2  3PLM 1.3973606  0.36942946  0.07288178         NA        NA
#> 75  G2I25    2  3PLM 1.7363207  1.41508375  0.07191836         NA        NA
#> 76  G2I26    2  3PLM 1.8858048 -0.88197041  0.23993266         NA        NA
#> 77  G2I27    2  3PLM 0.9754090 -0.59038894  0.19672791         NA        NA
#> 78  G2I28    5   GRM 1.1216203 -0.37176001  0.15040063  0.7990619  1.476485
#> 79   C2I1    2  3PLM 0.9700000 -0.46000000  0.05000000         NA        NA
#> 80   C2I2    2  3PLM 0.8500000  1.18000000  0.01000000         NA        NA
#> 81   C2I3    2  3PLM 1.4300000  1.41000000  0.10000000         NA        NA
#> 82   C2I4    2  3PLM 1.4800000  0.18000000  0.17000000         NA        NA
#> 83   C2I5    2  3PLM 1.2700000 -0.23000000  0.03000000         NA        NA
#> 84   C2I6    2  3PLM 2.0200000 -0.09000000  0.01000000         NA        NA
#> 85   C2I7    2  3PLM 1.3700000 -0.13000000  0.10000000         NA        NA
#> 86   C2I8    2  3PLM 1.6700000  1.25000000  0.19000000         NA        NA
#> 87   C2I9    2  3PLM 2.2800000 -1.01000000  0.10000000         NA        NA
#> 88  C2I10    2  3PLM 1.4200000 -1.65000000  0.11000000         NA        NA
#> 89   G3I1    2  3PLM 1.5973952 -1.04293518  0.14221272         NA        NA
#> 90   G3I2    2  3PLM 0.7882034 -0.61710502  0.16466619         NA        NA
#> 91   G3I3    2  3PLM 1.1574588  0.07488161  0.15704670         NA        NA
#> 92   G3I4    2  3PLM 1.3010656  1.72470571  0.22305233         NA        NA
#> 93   G3I5    2  3PLM 0.8030039 -1.08728779  0.23797751         NA        NA
#> 94   G3I6    2  3PLM 1.1828479 -1.37600012  0.26017944         NA        NA
#> 95   G3I7    2  3PLM 1.3732288  0.26144126  0.13453648         NA        NA
#> 96   G3I8    2  3PLM 1.9483776 -0.06609774  0.14303723         NA        NA
#> 97   G3I9    2  3PLM 1.1769474 -1.16424114  0.18147943         NA        NA
#> 98  G3I10    2  3PLM 1.4966087  0.98263824  0.28748102         NA        NA
#> 99  G3I11    2  3PLM 0.9143147  0.79757776  0.07731952         NA        NA
#> 100 G3I12    2  3PLM 1.4413693 -0.77635289  0.23363873         NA        NA
#> 101 G3I13    2  3PLM 1.1547205 -1.06358212  0.20951365         NA        NA
#> 102 G3I14    2  3PLM 1.3766468  0.30600869  0.15853392         NA        NA
#> 103 G3I15    2  3PLM 1.2913722 -0.05844768  0.17903882         NA        NA
#> 104 G3I16    2  3PLM 1.5776924  0.75034401  0.19938501         NA        NA
#> 105 G3I17    2  3PLM 1.7191587 -1.46022828  0.15855847         NA        NA
#> 106 G3I18    2  3PLM 1.3690901  0.71412800  0.11276074         NA        NA
#> 107 G3I19    2  3PLM 0.9937729  0.07222966  0.14756849         NA        NA
#> 108 G3I20    2  3PLM 1.1444553  2.29572750  0.11629624         NA        NA
#> 109 G3I21    2  3PLM 2.9714475  1.65685452  0.14005235         NA        NA
#> 110 G3I22    2  3PLM 1.1948944  0.10009253  0.09517841         NA        NA
#> 111 G3I23    2  3PLM 1.7915379  0.32938893  0.14617082         NA        NA
#> 112 G3I24    2  3PLM 1.1522431  0.33724431  0.05080719         NA        NA
#> 113 G3I25    2  3PLM 1.3842436  1.36390607  0.04998934         NA        NA
#> 114 G3I26    2  3PLM 1.7163634 -0.96107355  0.19794829         NA        NA
#> 115 G3I27    2  3PLM 0.9521484 -0.60866897  0.16408973         NA        NA
#> 116 G3I28    5   GRM 0.9955303 -0.31405937  0.18617231  0.8178505  1.523770
#> 
#> $group
#> $group$Group1
#>       id cats model     par.1       par.2       par.3      par.4     par.5
#> 1   C1I1    2  3PLM 0.7600000  1.46000000  0.26000000         NA        NA
#> 2   C1I2    2  3PLM 1.9200000 -1.05000000  0.18000000         NA        NA
#> 3   C1I3    2  3PLM 0.9300000  0.39000000  0.10000000         NA        NA
#> 4   C1I4    2  3PLM 1.0500000 -0.41000000  0.20000000         NA        NA
#> 5   C1I5    2  3PLM 0.8700000 -0.12000000  0.16000000         NA        NA
#> 6   C1I6    2  3PLM 1.7000000  0.63000000  0.07000000         NA        NA
#> 7   C1I7    2  3PLM 0.9100000  1.02000000  0.12000000         NA        NA
#> 8   C1I8    2  3PLM 0.8400000  0.80000000  0.11000000         NA        NA
#> 9   C1I9    2  3PLM 0.8500000  0.85000000  0.26000000         NA        NA
#> 10 C1I10    2  3PLM 1.5300000  0.09000000  0.14000000         NA        NA
#> 11  G1I1    2  3PLM 0.9601094 -0.46323480  0.15743926         NA        NA
#> 12  G1I2    2  3PLM 0.8792351  1.21120501  0.10365721         NA        NA
#> 13  G1I3    2  3PLM 1.4922037  1.31254694  0.18433436         NA        NA
#> 14  G1I4    2  3PLM 1.5246246  0.26709705  0.29751014         NA        NA
#> 15  G1I5    2  3PLM 1.3343885 -0.18476888  0.15642097         NA        NA
#> 16  G1I6    2  3PLM 2.1457296  0.01122821  0.08373745         NA        NA
#> 17  G1I7    2  3PLM 1.4492421 -0.07232467  0.19831474         NA        NA
#> 18  G1I8    2  3PLM 2.4981366  1.18142115  0.32459740         NA        NA
#> 19  G1I9    2  3PLM 2.4036619 -0.93389393  0.24090792         NA        NA
#> 20 G1I10    2  3PLM 1.2603565 -1.75076139  0.23712596         NA        NA
#> 21 G1I11    2  3PLM 1.5482444 -1.11447546  0.20564573         NA        NA
#> 22 G1I12    2  3PLM 0.7501881 -0.81274221  0.19483606         NA        NA
#> 23 G1I13    2  3PLM 1.0391850 -0.13129451  0.20305864         NA        NA
#> 24 G1I14    2  3PLM 1.4641086  1.76817734  0.29587746         NA        NA
#> 25 G1I15    2  3PLM 0.8642507 -1.44005202  0.21246953         NA        NA
#> 26 G1I16    2  3PLM 1.0481967 -1.94936219  0.22996558         NA        NA
#> 27 G1I17    2  3PLM 1.0413440  0.24963719  0.15297903         NA        NA
#> 28 G1I18    2  3PLM 2.0821164 -0.08782853  0.23783714         NA        NA
#> 29 G1I19    2  3PLM 1.3246385 -1.38457105  0.20248019         NA        NA
#> 30 G1I20    2  3PLM 1.0078270  0.50901778  0.22362359         NA        NA
#> 31 G1I21    2  3PLM 0.9153677  0.76786397  0.13229024         NA        NA
#> 32 G1I22    2  3PLM 1.7709516 -0.64567759  0.35461144         NA        NA
#> 33 G1I23    2  3PLM 1.3100376 -1.16619310  0.25418714         NA        NA
#> 34 G1I24    2  3PLM 1.6392138  0.31576653  0.22824910         NA        NA
#> 35 G1I25    2  3PLM 1.5820253 -0.12733170  0.25023679         NA        NA
#> 36 G1I26    2  3PLM 1.8799489  0.66003504  0.25368918         NA        NA
#> 37 G1I27    2  3PLM 1.6294671 -1.53964203  0.27165399         NA        NA
#> 38 G1I28    2  3PLM 1.3331814  0.56546916  0.14913136         NA        NA
#> 39 G1I29    2  3PLM 0.9154613 -0.37004169  0.12429485         NA        NA
#> 40 G1I30    2  3PLM 1.0026860  2.29722730  0.16842602         NA        NA
#> 41 G1I31    2  3PLM 2.3881154  1.64223594  0.18163836         NA        NA
#> 42 G1I32    2  3PLM 1.1031428 -0.07825039  0.14531435         NA        NA
#> 43 G1I33    2  3PLM 1.5998724  0.17672610  0.15654349         NA        NA
#> 44 G1I34    2  3PLM 1.3266408  0.24185668  0.10791955         NA        NA
#> 45 G1I35    2  3PLM 1.3307122  1.29621629  0.06712955         NA        NA
#> 46 G1I36    2  3PLM 1.4375780 -1.23256008  0.21920779         NA        NA
#> 47 G1I37    2  3PLM 1.0523257 -0.59860569  0.26551312         NA        NA
#> 48 G1I38    5   GRM 1.0557817 -0.36687770  0.21653854  0.8654794  1.435633
#> 49 C1I11    5   GRM 1.2300000 -2.08000000 -1.35000000 -0.7100000 -0.120000
#> 50 C1I12    5   GRM 0.8800000 -0.76000000 -0.01000000  0.6700000  1.250000
#> 
#> $group$Group2
#>       id cats model     par.1       par.2       par.3      par.4     par.5
#> 1   C1I1    2  3PLM 0.7600000  1.46000000  0.26000000         NA        NA
#> 2   C1I2    2  3PLM 1.9200000 -1.05000000  0.18000000         NA        NA
#> 3   C1I3    2  3PLM 0.9300000  0.39000000  0.10000000         NA        NA
#> 4   C1I4    2  3PLM 1.0500000 -0.41000000  0.20000000         NA        NA
#> 5   C1I5    2  3PLM 0.8700000 -0.12000000  0.16000000         NA        NA
#> 6   C1I6    2  3PLM 1.7000000  0.63000000  0.07000000         NA        NA
#> 7   C1I7    2  3PLM 0.9100000  1.02000000  0.12000000         NA        NA
#> 8   C1I8    2  3PLM 0.8400000  0.80000000  0.11000000         NA        NA
#> 9   C1I9    2  3PLM 0.8500000  0.85000000  0.26000000         NA        NA
#> 10 C1I10    2  3PLM 1.5300000  0.09000000  0.14000000         NA        NA
#> 11 C1I11    5   GRM 1.2300000 -2.08000000 -1.35000000 -0.7100000 -0.120000
#> 12 C1I12    5   GRM 0.8800000 -0.76000000 -0.01000000  0.6700000  1.250000
#> 13  G2I1    2  3PLM 1.7426027 -0.86693468  0.23241989         NA        NA
#> 14  G2I2    2  3PLM 0.8271656 -0.44864390  0.21072500         NA        NA
#> 15  G2I3    2  3PLM 1.0594812  0.08411544  0.17868700         NA        NA
#> 16  G2I4    2  3PLM 1.4496890  1.49538158  0.17845021         NA        NA
#> 17  G2I5    2  3PLM 0.7000088 -1.68320349  0.19753124         NA        NA
#> 18  G2I6    2  3PLM 1.0845772 -1.60171524  0.21553624         NA        NA
#> 19  G2I7    2  3PLM 1.2984963  0.26873160  0.12783879         NA        NA
#> 20  G2I8    2  3PLM 2.1768681 -0.08931817  0.13528274         NA        NA
#> 21  G2I9    2  3PLM 1.0731850 -1.56763064  0.20570964         NA        NA
#> 22 G2I10    2  3PLM 1.5904952  0.90286013  0.28755659         NA        NA
#> 23 G2I11    2  3PLM 0.9314018  0.88217244  0.11179900         NA        NA
#> 24 G2I12    2  3PLM 1.6623481 -0.69600414  0.24746346         NA        NA
#> 25 G2I13    2  3PLM 1.1643479 -1.30658561  0.20913476         NA        NA
#> 26 G2I14    2  3PLM 1.3949743  0.22018889  0.12742097         NA        NA
#> 27 G2I15    2  3PLM 1.5256012  0.03274459  0.25930675         NA        NA
#> 28 G2I16    2  3PLM 1.6958062  0.75377859  0.21478247         NA        NA
#> 29 G2I17    2  3PLM 2.1138538 -1.19916943  0.19757122         NA        NA
#> 30 G2I18    2  3PLM 1.7193451  0.64117712  0.13803059         NA        NA
#> 31 G2I19    2  3PLM 1.0971580  0.04569867  0.20768248         NA        NA
#> 32 G2I20    2  3PLM 1.6230735  2.23612968  0.15952981         NA        NA
#> 33 G2I21    2  3PLM 2.3884669  1.58525725  0.11741983         NA        NA
#> 34 G2I22    2  3PLM 1.4332531  0.22129279  0.20450739         NA        NA
#> 35 G2I23    2  3PLM 2.0941037  0.44296609  0.27618451         NA        NA
#> 36 G2I24    2  3PLM 1.3973606  0.36942946  0.07288178         NA        NA
#> 37 G2I25    2  3PLM 1.7363207  1.41508375  0.07191836         NA        NA
#> 38 G2I26    2  3PLM 1.8858048 -0.88197041  0.23993266         NA        NA
#> 39 G2I27    2  3PLM 0.9754090 -0.59038894  0.19672791         NA        NA
#> 40 G2I28    5   GRM 1.1216203 -0.37176001  0.15040063  0.7990619  1.476485
#> 41  C2I1    2  3PLM 0.9700000 -0.46000000  0.05000000         NA        NA
#> 42  C2I2    2  3PLM 0.8500000  1.18000000  0.01000000         NA        NA
#> 43  C2I3    2  3PLM 1.4300000  1.41000000  0.10000000         NA        NA
#> 44  C2I4    2  3PLM 1.4800000  0.18000000  0.17000000         NA        NA
#> 45  C2I5    2  3PLM 1.2700000 -0.23000000  0.03000000         NA        NA
#> 46  C2I6    2  3PLM 2.0200000 -0.09000000  0.01000000         NA        NA
#> 47  C2I7    2  3PLM 1.3700000 -0.13000000  0.10000000         NA        NA
#> 48  C2I8    2  3PLM 1.6700000  1.25000000  0.19000000         NA        NA
#> 49  C2I9    2  3PLM 2.2800000 -1.01000000  0.10000000         NA        NA
#> 50 C2I10    2  3PLM 1.4200000 -1.65000000  0.11000000         NA        NA
#> 
#> $group$Group3
#>       id cats model     par.1       par.2      par.3     par.4   par.5
#> 1   C2I1    2  3PLM 0.9700000 -0.46000000 0.05000000        NA      NA
#> 2   C2I2    2  3PLM 0.8500000  1.18000000 0.01000000        NA      NA
#> 3   C2I3    2  3PLM 1.4300000  1.41000000 0.10000000        NA      NA
#> 4   C2I4    2  3PLM 1.4800000  0.18000000 0.17000000        NA      NA
#> 5   C2I5    2  3PLM 1.2700000 -0.23000000 0.03000000        NA      NA
#> 6   C2I6    2  3PLM 2.0200000 -0.09000000 0.01000000        NA      NA
#> 7   C2I7    2  3PLM 1.3700000 -0.13000000 0.10000000        NA      NA
#> 8   C2I8    2  3PLM 1.6700000  1.25000000 0.19000000        NA      NA
#> 9   C2I9    2  3PLM 2.2800000 -1.01000000 0.10000000        NA      NA
#> 10 C2I10    2  3PLM 1.4200000 -1.65000000 0.11000000        NA      NA
#> 11  G3I1    2  3PLM 1.5973952 -1.04293518 0.14221272        NA      NA
#> 12  G3I2    2  3PLM 0.7882034 -0.61710502 0.16466619        NA      NA
#> 13  G3I3    2  3PLM 1.1574588  0.07488161 0.15704670        NA      NA
#> 14  G3I4    2  3PLM 1.3010656  1.72470571 0.22305233        NA      NA
#> 15  G3I5    2  3PLM 0.8030039 -1.08728779 0.23797751        NA      NA
#> 16  G3I6    2  3PLM 1.1828479 -1.37600012 0.26017944        NA      NA
#> 17  G3I7    2  3PLM 1.3732288  0.26144126 0.13453648        NA      NA
#> 18  G3I8    2  3PLM 1.9483776 -0.06609774 0.14303723        NA      NA
#> 19  G3I9    2  3PLM 1.1769474 -1.16424114 0.18147943        NA      NA
#> 20 G3I10    2  3PLM 1.4966087  0.98263824 0.28748102        NA      NA
#> 21 G3I11    2  3PLM 0.9143147  0.79757776 0.07731952        NA      NA
#> 22 G3I12    2  3PLM 1.4413693 -0.77635289 0.23363873        NA      NA
#> 23 G3I13    2  3PLM 1.1547205 -1.06358212 0.20951365        NA      NA
#> 24 G3I14    2  3PLM 1.3766468  0.30600869 0.15853392        NA      NA
#> 25 G3I15    2  3PLM 1.2913722 -0.05844768 0.17903882        NA      NA
#> 26 G3I16    2  3PLM 1.5776924  0.75034401 0.19938501        NA      NA
#> 27 G3I17    2  3PLM 1.7191587 -1.46022828 0.15855847        NA      NA
#> 28 G3I18    2  3PLM 1.3690901  0.71412800 0.11276074        NA      NA
#> 29 G3I19    2  3PLM 0.9937729  0.07222966 0.14756849        NA      NA
#> 30 G3I20    2  3PLM 1.1444553  2.29572750 0.11629624        NA      NA
#> 31 G3I21    2  3PLM 2.9714475  1.65685452 0.14005235        NA      NA
#> 32 G3I22    2  3PLM 1.1948944  0.10009253 0.09517841        NA      NA
#> 33 G3I23    2  3PLM 1.7915379  0.32938893 0.14617082        NA      NA
#> 34 G3I24    2  3PLM 1.1522431  0.33724431 0.05080719        NA      NA
#> 35 G3I25    2  3PLM 1.3842436  1.36390607 0.04998934        NA      NA
#> 36 G3I26    2  3PLM 1.7163634 -0.96107355 0.19794829        NA      NA
#> 37 G3I27    2  3PLM 0.9521484 -0.60866897 0.16408973        NA      NA
#> 38 G3I28    5   GRM 0.9955303 -0.31405937 0.18617231 0.8178505 1.52377
#> 
#> 

# Extract the standard error estimates
getirt(fit.3, what = "se.est")
#> $overall
#>        id cats model      par.1      par.2      par.3      par.4      par.5
#> 1    C1I1    2  3PLM         NA         NA         NA         NA         NA
#> 2    C1I2    2  3PLM         NA         NA         NA         NA         NA
#> 3    C1I3    2  3PLM         NA         NA         NA         NA         NA
#> 4    C1I4    2  3PLM         NA         NA         NA         NA         NA
#> 5    C1I5    2  3PLM         NA         NA         NA         NA         NA
#> 6    C1I6    2  3PLM         NA         NA         NA         NA         NA
#> 7    C1I7    2  3PLM         NA         NA         NA         NA         NA
#> 8    C1I8    2  3PLM         NA         NA         NA         NA         NA
#> 9    C1I9    2  3PLM         NA         NA         NA         NA         NA
#> 10  C1I10    2  3PLM         NA         NA         NA         NA         NA
#> 11   G1I1    2  3PLM 0.10426168 0.19914118 0.06583145         NA         NA
#> 12   G1I2    2  3PLM 0.13011827 0.13758723 0.03866179         NA         NA
#> 13   G1I3    2  3PLM 0.25521536 0.09288148 0.02598262         NA         NA
#> 14   G1I4    2  3PLM 0.21216024 0.12139692 0.04369469         NA         NA
#> 15   G1I5    2  3PLM 0.13737157 0.12762451 0.05144114         NA         NA
#> 16   G1I6    2  3PLM 0.17837536 0.05370898 0.02614114         NA         NA
#> 17   G1I7    2  3PLM 0.15675133 0.12128691 0.04842894         NA         NA
#> 18   G1I8    2  3PLM 0.46148303 0.06925484 0.01882411         NA         NA
#> 19   G1I9    2  3PLM 0.25992134 0.10854364 0.06176901         NA         NA
#> 20  G1I10    2  3PLM 0.12525002 0.22773298 0.09804119         NA         NA
#> 21  G1I11    2  3PLM 0.14906877 0.16092483 0.07743255         NA         NA
#> 22  G1I12    2  3PLM 0.09144445 0.31119356 0.08351947         NA         NA
#> 23  G1I13    2  3PLM 0.13678756 0.20744666 0.06854745         NA         NA
#> 24  G1I14    2  3PLM 0.36846246 0.15351046 0.02514203         NA         NA
#> 25  G1I15    2  3PLM 0.09478648 0.29483636 0.09102878         NA         NA
#> 26  G1I16    2  3PLM 0.10986557 0.26721069 0.09771458         NA         NA
#> 27  G1I17    2  3PLM 0.12646970 0.15451738 0.05176849         NA         NA
#> 28  G1I18    2  3PLM 0.20058125 0.08013120 0.03656393         NA         NA
#> 29  G1I19    2  3PLM 0.12663826 0.19277821 0.08479494         NA         NA
#> 30  G1I20    2  3PLM 0.17409361 0.19113321 0.06033487         NA         NA
#> 31  G1I21    2  3PLM 0.13256306 0.15087630 0.04704321         NA         NA
#> 32  G1I22    2  3PLM 0.21340356 0.15629299 0.06435608         NA         NA
#> 33  G1I23    2  3PLM 0.14898851 0.22567950 0.09214904         NA         NA
#> 34  G1I24    2  3PLM 0.19188388 0.09525298 0.03753622         NA         NA
#> 35  G1I25    2  3PLM 0.17877681 0.12332508 0.04964582         NA         NA
#> 36  G1I26    2  3PLM 0.25183655 0.07738128 0.02912676         NA         NA
#> 37  G1I27    2  3PLM 0.17771713 0.20258618 0.10060823         NA         NA
#> 38  G1I28    2  3PLM 0.16142785 0.09763951 0.03660579         NA         NA
#> 39  G1I29    2  3PLM 0.08893330 0.17133724 0.05492922         NA         NA
#> 40  G1I30    2  3PLM 0.26897282 0.25352870 0.03089404         NA         NA
#> 41  G1I31    2  3PLM 0.46169958 0.09359258 0.01431405         NA         NA
#> 42  G1I32    2  3PLM 0.11905116 0.15068461 0.05352540         NA         NA
#> 43  G1I33    2  3PLM 0.17012521 0.08891040 0.03755055         NA         NA
#> 44  G1I34    2  3PLM 0.13288626 0.09359115 0.03655936         NA         NA
#> 45  G1I35    2  3PLM 0.16791769 0.08361518 0.02044036         NA         NA
#> 46  G1I36    2  3PLM 0.14309878 0.18590021 0.08516936         NA         NA
#> 47  G1I37    2  3PLM 0.14194404 0.26869625 0.08697510         NA         NA
#> 48  G1I38    5   GRM 0.05705443 0.05322210 0.04926777 0.06290244 0.08428075
#> 49  C1I11    5   GRM         NA         NA         NA         NA         NA
#> 50  C1I12    5   GRM         NA         NA         NA         NA         NA
#> 51   G2I1    2  3PLM 0.18701977 0.17226079 0.09591121         NA         NA
#> 52   G2I2    2  3PLM 0.11055188 0.30549386 0.09111811         NA         NA
#> 53   G2I3    2  3PLM 0.12900982 0.20135385 0.07488982         NA         NA
#> 54   G2I4    2  3PLM 0.28006193 0.09631836 0.03881020         NA         NA
#> 55   G2I5    2  3PLM 0.10228622 0.40358987 0.08893723         NA         NA
#> 56   G2I6    2  3PLM 0.14204281 0.29399636 0.09476517         NA         NA
#> 57   G2I7    2  3PLM 0.13098448 0.12235808 0.05447273         NA         NA
#> 58   G2I8    2  3PLM 0.18554592 0.08066166 0.05149244         NA         NA
#> 59   G2I9    2  3PLM 0.13004997 0.27643093 0.09157031         NA         NA
#> 60  G2I10    2  3PLM 0.27183179 0.12027438 0.05121721         NA         NA
#> 61  G2I11    2  3PLM 0.12329519 0.14644859 0.05005698         NA         NA
#> 62  G2I12    2  3PLM 0.18329666 0.18051784 0.09670036         NA         NA
#> 63  G2I13    2  3PLM 0.13036275 0.23937423 0.09227547         NA         NA
#> 64  G2I14    2  3PLM 0.13691011 0.11354387 0.05286438         NA         NA
#> 65  G2I15    2  3PLM 0.19565359 0.17450556 0.08012343         NA         NA
#> 66  G2I16    2  3PLM 0.22938735 0.09966642 0.04736991         NA         NA
#> 67  G2I17    2  3PLM 0.21703924 0.14435249 0.08787073         NA         NA
#> 68  G2I18    2  3PLM 0.19775206 0.08387898 0.04368134         NA         NA
#> 69  G2I19    2  3PLM 0.14342944 0.22082957 0.08261270         NA         NA
#> 70  G2I20    2  3PLM 0.45488499 0.19197090 0.02387855         NA         NA
#> 71  G2I21    2  3PLM 0.35929005 0.05999541 0.01724135         NA         NA
#> 72  G2I22    2  3PLM 0.17462991 0.15301616 0.06937934         NA         NA
#> 73  G2I23    2  3PLM 0.26460348 0.09594883 0.04984224         NA         NA
#> 74  G2I24    2  3PLM 0.11632156 0.07551221 0.03397862         NA         NA
#> 75  G2I25    2  3PLM 0.21505543 0.06239024 0.02239864         NA         NA
#> 76  G2I26    2  3PLM 0.19522559 0.16121604 0.09680686         NA         NA
#> 77  G2I27    2  3PLM 0.11542196 0.24707543 0.08656505         NA         NA
#> 78  G2I28    5   GRM 0.07020432 0.06705796 0.04891269 0.04705132 0.07019674
#> 79   C2I1    2  3PLM         NA         NA         NA         NA         NA
#> 80   C2I2    2  3PLM         NA         NA         NA         NA         NA
#> 81   C2I3    2  3PLM         NA         NA         NA         NA         NA
#> 82   C2I4    2  3PLM         NA         NA         NA         NA         NA
#> 83   C2I5    2  3PLM         NA         NA         NA         NA         NA
#> 84   C2I6    2  3PLM         NA         NA         NA         NA         NA
#> 85   C2I7    2  3PLM         NA         NA         NA         NA         NA
#> 86   C2I8    2  3PLM         NA         NA         NA         NA         NA
#> 87   C2I9    2  3PLM         NA         NA         NA         NA         NA
#> 88  C2I10    2  3PLM         NA         NA         NA         NA         NA
#> 89   G3I1    2  3PLM 0.13986318 0.11556266 0.04828949         NA         NA
#> 90   G3I2    2  3PLM 0.08945397 0.25055681 0.06577738         NA         NA
#> 91   G3I3    2  3PLM 0.12166153 0.12145104 0.03777657         NA         NA
#> 92   G3I4    2  3PLM 0.23742580 0.12346854 0.02115715         NA         NA
#> 93   G3I5    2  3PLM 0.09747072 0.33207410 0.08571279         NA         NA
#> 94   G3I6    2  3PLM 0.13415716 0.24170387 0.08325843         NA         NA
#> 95   G3I7    2  3PLM 0.13182139 0.08662754 0.02782581         NA         NA
#> 96   G3I8    2  3PLM 0.17855395 0.06374914 0.02406454         NA         NA
#> 97   G3I9    2  3PLM 0.11323201 0.18515456 0.06633175         NA         NA
#> 98  G3I10    2  3PLM 0.21267233 0.09577230 0.02369318         NA         NA
#> 99  G3I11    2  3PLM 0.09929092 0.11096928 0.02888083         NA         NA
#> 100 G3I12    2  3PLM 0.16578402 0.15209966 0.05508812         NA         NA
#> 101 G3I13    2  3PLM 0.12510673 0.20472281 0.06955329         NA         NA
#> 102 G3I14    2  3PLM 0.14193927 0.08926242 0.02823279         NA         NA
#> 103 G3I15    2  3PLM 0.13431761 0.11420069 0.03726663         NA         NA
#> 104 G3I16    2  3PLM 0.17708800 0.07754598 0.02161000         NA         NA
#> 105 G3I17    2  3PLM 0.14820688 0.12802058 0.05977877         NA         NA
#> 106 G3I18    2  3PLM 0.14295569 0.07697678 0.02235150         NA         NA
#> 107 G3I19    2  3PLM 0.11089570 0.14924845 0.04377532         NA         NA
#> 108 G3I20    2  3PLM 0.23479290 0.19832103 0.01759904         NA         NA
#> 109 G3I21    2  3PLM 0.54769558 0.06208590 0.01022861         NA         NA
#> 110 G3I22    2  3PLM 0.11351503 0.09624776 0.03123112         NA         NA
#> 111 G3I23    2  3PLM 0.16621787 0.06369671 0.02083676         NA         NA
#> 112 G3I24    2  3PLM 0.09162888 0.07576239 0.02065526         NA         NA
#> 113 G3I25    2  3PLM 0.14521791 0.07831223 0.01301757         NA         NA
#> 114 G3I26    2  3PLM 0.16527458 0.12057791 0.04950403         NA         NA
#> 115 G3I27    2  3PLM 0.09992189 0.19902502 0.05904246         NA         NA
#> 116 G3I28    5   GRM 0.04870285 0.05360023 0.05471689 0.06669018 0.08789166
#> 
#> $group
#> $group$Group1
#>       id cats model      par.1      par.2      par.3      par.4      par.5
#> 1   C1I1    2  3PLM         NA         NA         NA         NA         NA
#> 2   C1I2    2  3PLM         NA         NA         NA         NA         NA
#> 3   C1I3    2  3PLM         NA         NA         NA         NA         NA
#> 4   C1I4    2  3PLM         NA         NA         NA         NA         NA
#> 5   C1I5    2  3PLM         NA         NA         NA         NA         NA
#> 6   C1I6    2  3PLM         NA         NA         NA         NA         NA
#> 7   C1I7    2  3PLM         NA         NA         NA         NA         NA
#> 8   C1I8    2  3PLM         NA         NA         NA         NA         NA
#> 9   C1I9    2  3PLM         NA         NA         NA         NA         NA
#> 10 C1I10    2  3PLM         NA         NA         NA         NA         NA
#> 11  G1I1    2  3PLM 0.10426168 0.19914118 0.06583145         NA         NA
#> 12  G1I2    2  3PLM 0.13011827 0.13758723 0.03866179         NA         NA
#> 13  G1I3    2  3PLM 0.25521536 0.09288148 0.02598262         NA         NA
#> 14  G1I4    2  3PLM 0.21216024 0.12139692 0.04369469         NA         NA
#> 15  G1I5    2  3PLM 0.13737157 0.12762451 0.05144114         NA         NA
#> 16  G1I6    2  3PLM 0.17837536 0.05370898 0.02614114         NA         NA
#> 17  G1I7    2  3PLM 0.15675133 0.12128691 0.04842894         NA         NA
#> 18  G1I8    2  3PLM 0.46148303 0.06925484 0.01882411         NA         NA
#> 19  G1I9    2  3PLM 0.25992134 0.10854364 0.06176901         NA         NA
#> 20 G1I10    2  3PLM 0.12525002 0.22773298 0.09804119         NA         NA
#> 21 G1I11    2  3PLM 0.14906877 0.16092483 0.07743255         NA         NA
#> 22 G1I12    2  3PLM 0.09144445 0.31119356 0.08351947         NA         NA
#> 23 G1I13    2  3PLM 0.13678756 0.20744666 0.06854745         NA         NA
#> 24 G1I14    2  3PLM 0.36846246 0.15351046 0.02514203         NA         NA
#> 25 G1I15    2  3PLM 0.09478648 0.29483636 0.09102878         NA         NA
#> 26 G1I16    2  3PLM 0.10986557 0.26721069 0.09771458         NA         NA
#> 27 G1I17    2  3PLM 0.12646970 0.15451738 0.05176849         NA         NA
#> 28 G1I18    2  3PLM 0.20058125 0.08013120 0.03656393         NA         NA
#> 29 G1I19    2  3PLM 0.12663826 0.19277821 0.08479494         NA         NA
#> 30 G1I20    2  3PLM 0.17409361 0.19113321 0.06033487         NA         NA
#> 31 G1I21    2  3PLM 0.13256306 0.15087630 0.04704321         NA         NA
#> 32 G1I22    2  3PLM 0.21340356 0.15629299 0.06435608         NA         NA
#> 33 G1I23    2  3PLM 0.14898851 0.22567950 0.09214904         NA         NA
#> 34 G1I24    2  3PLM 0.19188388 0.09525298 0.03753622         NA         NA
#> 35 G1I25    2  3PLM 0.17877681 0.12332508 0.04964582         NA         NA
#> 36 G1I26    2  3PLM 0.25183655 0.07738128 0.02912676         NA         NA
#> 37 G1I27    2  3PLM 0.17771713 0.20258618 0.10060823         NA         NA
#> 38 G1I28    2  3PLM 0.16142785 0.09763951 0.03660579         NA         NA
#> 39 G1I29    2  3PLM 0.08893330 0.17133724 0.05492922         NA         NA
#> 40 G1I30    2  3PLM 0.26897282 0.25352870 0.03089404         NA         NA
#> 41 G1I31    2  3PLM 0.46169958 0.09359258 0.01431405         NA         NA
#> 42 G1I32    2  3PLM 0.11905116 0.15068461 0.05352540         NA         NA
#> 43 G1I33    2  3PLM 0.17012521 0.08891040 0.03755055         NA         NA
#> 44 G1I34    2  3PLM 0.13288626 0.09359115 0.03655936         NA         NA
#> 45 G1I35    2  3PLM 0.16791769 0.08361518 0.02044036         NA         NA
#> 46 G1I36    2  3PLM 0.14309878 0.18590021 0.08516936         NA         NA
#> 47 G1I37    2  3PLM 0.14194404 0.26869625 0.08697510         NA         NA
#> 48 G1I38    5   GRM 0.05705443 0.05322210 0.04926777 0.06290244 0.08428075
#> 49 C1I11    5   GRM         NA         NA         NA         NA         NA
#> 50 C1I12    5   GRM         NA         NA         NA         NA         NA
#> 
#> $group$Group2
#>       id cats model      par.1      par.2      par.3      par.4      par.5
#> 1   C1I1    2  3PLM         NA         NA         NA         NA         NA
#> 2   C1I2    2  3PLM         NA         NA         NA         NA         NA
#> 3   C1I3    2  3PLM         NA         NA         NA         NA         NA
#> 4   C1I4    2  3PLM         NA         NA         NA         NA         NA
#> 5   C1I5    2  3PLM         NA         NA         NA         NA         NA
#> 6   C1I6    2  3PLM         NA         NA         NA         NA         NA
#> 7   C1I7    2  3PLM         NA         NA         NA         NA         NA
#> 8   C1I8    2  3PLM         NA         NA         NA         NA         NA
#> 9   C1I9    2  3PLM         NA         NA         NA         NA         NA
#> 10 C1I10    2  3PLM         NA         NA         NA         NA         NA
#> 11 C1I11    5   GRM         NA         NA         NA         NA         NA
#> 12 C1I12    5   GRM         NA         NA         NA         NA         NA
#> 13  G2I1    2  3PLM 0.18701977 0.17226079 0.09591121         NA         NA
#> 14  G2I2    2  3PLM 0.11055188 0.30549386 0.09111811         NA         NA
#> 15  G2I3    2  3PLM 0.12900982 0.20135385 0.07488982         NA         NA
#> 16  G2I4    2  3PLM 0.28006193 0.09631836 0.03881020         NA         NA
#> 17  G2I5    2  3PLM 0.10228622 0.40358987 0.08893723         NA         NA
#> 18  G2I6    2  3PLM 0.14204281 0.29399636 0.09476517         NA         NA
#> 19  G2I7    2  3PLM 0.13098448 0.12235808 0.05447273         NA         NA
#> 20  G2I8    2  3PLM 0.18554592 0.08066166 0.05149244         NA         NA
#> 21  G2I9    2  3PLM 0.13004997 0.27643093 0.09157031         NA         NA
#> 22 G2I10    2  3PLM 0.27183179 0.12027438 0.05121721         NA         NA
#> 23 G2I11    2  3PLM 0.12329519 0.14644859 0.05005698         NA         NA
#> 24 G2I12    2  3PLM 0.18329666 0.18051784 0.09670036         NA         NA
#> 25 G2I13    2  3PLM 0.13036275 0.23937423 0.09227547         NA         NA
#> 26 G2I14    2  3PLM 0.13691011 0.11354387 0.05286438         NA         NA
#> 27 G2I15    2  3PLM 0.19565359 0.17450556 0.08012343         NA         NA
#> 28 G2I16    2  3PLM 0.22938735 0.09966642 0.04736991         NA         NA
#> 29 G2I17    2  3PLM 0.21703924 0.14435249 0.08787073         NA         NA
#> 30 G2I18    2  3PLM 0.19775206 0.08387898 0.04368134         NA         NA
#> 31 G2I19    2  3PLM 0.14342944 0.22082957 0.08261270         NA         NA
#> 32 G2I20    2  3PLM 0.45488499 0.19197090 0.02387855         NA         NA
#> 33 G2I21    2  3PLM 0.35929005 0.05999541 0.01724135         NA         NA
#> 34 G2I22    2  3PLM 0.17462991 0.15301616 0.06937934         NA         NA
#> 35 G2I23    2  3PLM 0.26460348 0.09594883 0.04984224         NA         NA
#> 36 G2I24    2  3PLM 0.11632156 0.07551221 0.03397862         NA         NA
#> 37 G2I25    2  3PLM 0.21505543 0.06239024 0.02239864         NA         NA
#> 38 G2I26    2  3PLM 0.19522559 0.16121604 0.09680686         NA         NA
#> 39 G2I27    2  3PLM 0.11542196 0.24707543 0.08656505         NA         NA
#> 40 G2I28    5   GRM 0.07020432 0.06705796 0.04891269 0.04705132 0.07019674
#> 41  C2I1    2  3PLM         NA         NA         NA         NA         NA
#> 42  C2I2    2  3PLM         NA         NA         NA         NA         NA
#> 43  C2I3    2  3PLM         NA         NA         NA         NA         NA
#> 44  C2I4    2  3PLM         NA         NA         NA         NA         NA
#> 45  C2I5    2  3PLM         NA         NA         NA         NA         NA
#> 46  C2I6    2  3PLM         NA         NA         NA         NA         NA
#> 47  C2I7    2  3PLM         NA         NA         NA         NA         NA
#> 48  C2I8    2  3PLM         NA         NA         NA         NA         NA
#> 49  C2I9    2  3PLM         NA         NA         NA         NA         NA
#> 50 C2I10    2  3PLM         NA         NA         NA         NA         NA
#> 
#> $group$Group3
#>       id cats model      par.1      par.2      par.3      par.4      par.5
#> 1   C2I1    2  3PLM         NA         NA         NA         NA         NA
#> 2   C2I2    2  3PLM         NA         NA         NA         NA         NA
#> 3   C2I3    2  3PLM         NA         NA         NA         NA         NA
#> 4   C2I4    2  3PLM         NA         NA         NA         NA         NA
#> 5   C2I5    2  3PLM         NA         NA         NA         NA         NA
#> 6   C2I6    2  3PLM         NA         NA         NA         NA         NA
#> 7   C2I7    2  3PLM         NA         NA         NA         NA         NA
#> 8   C2I8    2  3PLM         NA         NA         NA         NA         NA
#> 9   C2I9    2  3PLM         NA         NA         NA         NA         NA
#> 10 C2I10    2  3PLM         NA         NA         NA         NA         NA
#> 11  G3I1    2  3PLM 0.13986318 0.11556266 0.04828949         NA         NA
#> 12  G3I2    2  3PLM 0.08945397 0.25055681 0.06577738         NA         NA
#> 13  G3I3    2  3PLM 0.12166153 0.12145104 0.03777657         NA         NA
#> 14  G3I4    2  3PLM 0.23742580 0.12346854 0.02115715         NA         NA
#> 15  G3I5    2  3PLM 0.09747072 0.33207410 0.08571279         NA         NA
#> 16  G3I6    2  3PLM 0.13415716 0.24170387 0.08325843         NA         NA
#> 17  G3I7    2  3PLM 0.13182139 0.08662754 0.02782581         NA         NA
#> 18  G3I8    2  3PLM 0.17855395 0.06374914 0.02406454         NA         NA
#> 19  G3I9    2  3PLM 0.11323201 0.18515456 0.06633175         NA         NA
#> 20 G3I10    2  3PLM 0.21267233 0.09577230 0.02369318         NA         NA
#> 21 G3I11    2  3PLM 0.09929092 0.11096928 0.02888083         NA         NA
#> 22 G3I12    2  3PLM 0.16578402 0.15209966 0.05508812         NA         NA
#> 23 G3I13    2  3PLM 0.12510673 0.20472281 0.06955329         NA         NA
#> 24 G3I14    2  3PLM 0.14193927 0.08926242 0.02823279         NA         NA
#> 25 G3I15    2  3PLM 0.13431761 0.11420069 0.03726663         NA         NA
#> 26 G3I16    2  3PLM 0.17708800 0.07754598 0.02161000         NA         NA
#> 27 G3I17    2  3PLM 0.14820688 0.12802058 0.05977877         NA         NA
#> 28 G3I18    2  3PLM 0.14295569 0.07697678 0.02235150         NA         NA
#> 29 G3I19    2  3PLM 0.11089570 0.14924845 0.04377532         NA         NA
#> 30 G3I20    2  3PLM 0.23479290 0.19832103 0.01759904         NA         NA
#> 31 G3I21    2  3PLM 0.54769558 0.06208590 0.01022861         NA         NA
#> 32 G3I22    2  3PLM 0.11351503 0.09624776 0.03123112         NA         NA
#> 33 G3I23    2  3PLM 0.16621787 0.06369671 0.02083676         NA         NA
#> 34 G3I24    2  3PLM 0.09162888 0.07576239 0.02065526         NA         NA
#> 35 G3I25    2  3PLM 0.14521791 0.07831223 0.01301757         NA         NA
#> 36 G3I26    2  3PLM 0.16527458 0.12057791 0.04950403         NA         NA
#> 37 G3I27    2  3PLM 0.09992189 0.19902502 0.05904246         NA         NA
#> 38 G3I28    5   GRM 0.04870285 0.05360023 0.05471689 0.06669018 0.08789166
#> 
#> 

# Extract the group parameter estimates (i.e., scale parameters)
getirt(fit.3, what = "group.par")
#> $Group1
#>                    mu     sigma2     sigma
#> estimates -0.01450724 1.01249024 1.0062257
#> se         0.02249989 0.03202576 0.0159138
#> 
#> $Group2
#>                   mu     sigma2      sigma
#> estimates 0.50413571 0.58171603 0.76270311
#> se        0.01705456 0.01840008 0.01206241
#> 
#> $Group3
#>                    mu     sigma2      sigma
#> estimates -0.28118672 1.66447266 1.29014443
#> se         0.02884851 0.05264841 0.02040408
#> 

# Extract the posterior latent ability distributions of the groups
getirt(fit.3, what = "weights")
#> $Group1
#>    theta       weight
#> 1  -6.00 7.662190e-12
#> 2  -5.75 2.451684e-11
#> 3  -5.50 7.119151e-11
#> 4  -5.25 1.891374e-10
#> 5  -5.00 4.676083e-10
#> 6  -4.75 1.108652e-09
#> 7  -4.50 2.644856e-09
#> 8  -4.25 6.812933e-09
#> 9  -4.00 2.081206e-08
#> 10 -3.75 8.401450e-08
#> 11 -3.50 4.930076e-07
#> 12 -3.25 4.326576e-06
#> 13 -3.00 5.074487e-05
#> 14 -2.75 5.872308e-04
#> 15 -2.50 4.332804e-03
#> 16 -2.25 1.444351e-02
#> 17 -2.00 2.242166e-02
#> 18 -1.75 2.444472e-02
#> 19 -1.50 2.775462e-02
#> 20 -1.25 3.781543e-02
#> 21 -1.00 5.911397e-02
#> 22 -0.75 7.686054e-02
#> 23 -0.50 8.228516e-02
#> 24 -0.25 1.049100e-01
#> 25  0.00 1.078960e-01
#> 26  0.25 8.138445e-02
#> 27  0.50 7.575408e-02
#> 28  0.75 8.485785e-02
#> 29  1.00 8.007142e-02
#> 30  1.25 4.680550e-02
#> 31  1.50 2.165959e-02
#> 32  1.75 1.619189e-02
#> 33  2.00 1.539816e-02
#> 34  2.25 8.170377e-03
#> 35  2.50 2.384944e-03
#> 36  2.75 7.954930e-04
#> 37  3.00 5.389228e-04
#> 38  3.25 6.906744e-04
#> 39  3.50 9.912521e-04
#> 40  3.75 9.167267e-04
#> 41  4.00 3.910055e-04
#> 42  4.25 7.029895e-05
#> 43  4.50 5.775249e-06
#> 44  4.75 2.563307e-07
#> 45  5.00 7.455139e-09
#> 46  5.25 1.703871e-10
#> 47  5.50 3.561747e-12
#> 48  5.75 7.640892e-14
#> 49  6.00 1.820327e-15
#> 
#> $Group2
#>    theta       weight
#> 1  -6.00 3.894816e-89
#> 2  -5.75 3.951306e-84
#> 3  -5.50 7.764024e-79
#> 4  -5.25 2.603004e-73
#> 5  -5.00 1.291279e-67
#> 6  -4.75 8.143647e-62
#> 7  -4.50 5.597431e-56
#> 8  -4.25 3.620629e-50
#> 9  -4.00 1.945094e-44
#> 10 -3.75 7.996023e-39
#> 11 -3.50 2.475576e-33
#> 12 -3.25 6.096104e-28
#> 13 -3.00 1.277570e-22
#> 14 -2.75 2.100737e-17
#> 15 -2.50 1.722718e-12
#> 16 -2.25 2.851606e-08
#> 17 -2.00 3.328725e-05
#> 18 -1.75 1.652181e-03
#> 19 -1.50 7.124060e-03
#> 20 -1.25 1.323762e-02
#> 21 -1.00 2.233184e-02
#> 22 -0.75 2.551256e-02
#> 23 -0.50 4.082653e-02
#> 24 -0.25 1.029622e-01
#> 25  0.00 9.278880e-02
#> 26  0.25 8.719857e-02
#> 27  0.50 1.665702e-01
#> 28  0.75 1.473383e-01
#> 29  1.00 8.777312e-02
#> 30  1.25 7.526865e-02
#> 31  1.50 6.301574e-02
#> 32  1.75 3.752019e-02
#> 33  2.00 1.769110e-02
#> 34  2.25 6.091144e-03
#> 35  2.50 1.740485e-03
#> 36  2.75 7.205189e-04
#> 37  3.00 5.733371e-04
#> 38  3.25 6.627171e-04
#> 39  3.50 6.883348e-04
#> 40  3.75 4.588606e-04
#> 41  4.00 1.743136e-04
#> 42  4.25 3.899078e-05
#> 43  4.50 5.702020e-06
#> 44  4.75 6.165462e-07
#> 45  5.00 5.519581e-08
#> 46  5.25 4.482665e-09
#> 47  5.50 3.531305e-10
#> 48  5.75 2.821138e-11
#> 49  6.00 2.345691e-12
#> 
#> $Group3
#>    theta       weight
#> 1  -6.00 6.924473e-08
#> 2  -5.75 2.943789e-07
#> 3  -5.50 1.166017e-06
#> 4  -5.25 4.289972e-06
#> 5  -5.00 1.460129e-05
#> 6  -4.75 4.573324e-05
#> 7  -4.50 1.309612e-04
#> 8  -4.25 3.402905e-04
#> 9  -4.00 7.962462e-04
#> 10 -3.75 1.668503e-03
#> 11 -3.50 3.132224e-03
#> 12 -3.25 5.329452e-03
#> 13 -3.00 8.445236e-03
#> 14 -2.75 1.295753e-02
#> 15 -2.50 1.981722e-02
#> 16 -2.25 2.965113e-02
#> 17 -2.00 3.982099e-02
#> 18 -1.75 4.375602e-02
#> 19 -1.50 4.160532e-02
#> 20 -1.25 4.561268e-02
#> 21 -1.00 6.646806e-02
#> 22 -0.75 8.026361e-02
#> 23 -0.50 6.427858e-02
#> 24 -0.25 6.407623e-02
#> 25  0.00 8.542032e-02
#> 26  0.25 8.548725e-02
#> 27  0.50 6.053382e-02
#> 28  0.75 4.722574e-02
#> 29  1.00 4.420511e-02
#> 30  1.25 3.945090e-02
#> 31  1.50 3.541132e-02
#> 32  1.75 3.265182e-02
#> 33  2.00 2.150663e-02
#> 34  2.25 9.341872e-03
#> 35  2.50 3.769655e-03
#> 36  2.75 1.886082e-03
#> 37  3.00 1.256293e-03
#> 38  3.25 1.030125e-03
#> 39  3.50 8.997076e-04
#> 40  3.75 7.280932e-04
#> 41  4.00 4.987633e-04
#> 42  4.25 2.792318e-04
#> 43  4.50 1.282712e-04
#> 44  4.75 4.948786e-05
#> 45  5.00 1.649488e-05
#> 46  5.25 4.873378e-06
#> 47  5.50 1.302434e-06
#> 48  5.75 3.195413e-07
#> 49  6.00 7.269746e-08
#> 

# 2-(2). Alternatively, MG-FIPC can be implemented by specifying the
# IDs of the items to be fixed using the 'fix.id' argument.
# Provide a character vector of fixed item IDs to 'fix.id'
fix.id <- c(paste0("C1I", 1:12), paste0("C2I", 1:10))
fit.4 <-
  est_mg(
    x = x, data = data, group.name = group.name, D = 1,
    free.group = free.group, use.gprior = TRUE,
    gprior = list(dist = "beta", params = c(5, 16)),
    EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE,
    fipc.method = "MEM", fix.id = fix.id
  )
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -56756.7548, Max-Change: 3.554166
 EM iteration: 2, Loglike: -160424.0878, Max-Change: 0.780368
 EM iteration: 3, Loglike: -159755.3559, Max-Change: 0.276071
 EM iteration: 4, Loglike: -159686.2158, Max-Change: 0.151073
 EM iteration: 5, Loglike: -159657.5520, Max-Change: 0.102849
 EM iteration: 6, Loglike: -159642.1127, Max-Change: 0.080257
 EM iteration: 7, Loglike: -159633.1411, Max-Change: 0.066343
 EM iteration: 8, Loglike: -159627.6844, Max-Change: 0.055853
 EM iteration: 9, Loglike: -159624.2459, Max-Change: 0.047172
 EM iteration: 10, Loglike: -159622.0120, Max-Change: 0.039781
 EM iteration: 11, Loglike: -159620.5195, Max-Change: 0.033468
 EM iteration: 12, Loglike: -159619.4950, Max-Change: 0.028101
 EM iteration: 13, Loglike: -159618.7723, Max-Change: 0.023565
 EM iteration: 14, Loglike: -159618.2481, Max-Change: 0.019752
 EM iteration: 15, Loglike: -159617.8570, Max-Change: 0.016845
 EM iteration: 16, Loglike: -159617.5566, Max-Change: 0.014864
 EM iteration: 17, Loglike: -159617.3191, Max-Change: 0.01318
 EM iteration: 18, Loglike: -159617.1261, Max-Change: 0.011751
 EM iteration: 19, Loglike: -159616.9650, Max-Change: 0.010538
 EM iteration: 20, Loglike: -159616.8273, Max-Change: 0.009507
 EM iteration: 21, Loglike: -159616.7072, Max-Change: 0.008626
 EM iteration: 22, Loglike: -159616.6005, Max-Change: 0.007869
 EM iteration: 23, Loglike: -159616.5042, Max-Change: 0.007214
 EM iteration: 24, Loglike: -159616.4163, Max-Change: 0.006643
 EM iteration: 25, Loglike: -159616.3352, Max-Change: 0.006141
 EM iteration: 26, Loglike: -159616.2596, Max-Change: 0.005695
 EM iteration: 27, Loglike: -159616.1888, Max-Change: 0.005295
 EM iteration: 28, Loglike: -159616.1221, Max-Change: 0.004934
 EM iteration: 29, Loglike: -159616.0589, Max-Change: 0.004606
 EM iteration: 30, Loglike: -159615.9989, Max-Change: 0.004306
 EM iteration: 31, Loglike: -159615.9416, Max-Change: 0.00403
 EM iteration: 32, Loglike: -159615.8868, Max-Change: 0.003774
 EM iteration: 33, Loglike: -159615.8343, Max-Change: 0.003537
 EM iteration: 34, Loglike: -159615.7838, Max-Change: 0.003317
 EM iteration: 35, Loglike: -159615.7352, Max-Change: 0.003111
 EM iteration: 36, Loglike: -159615.6884, Max-Change: 0.002919
 EM iteration: 37, Loglike: -159615.6431, Max-Change: 0.00274
 EM iteration: 38, Loglike: -159615.5994, Max-Change: 0.002572
 EM iteration: 39, Loglike: -159615.5570, Max-Change: 0.002416
 EM iteration: 40, Loglike: -159615.5159, Max-Change: 0.002269
 EM iteration: 41, Loglike: -159615.4761, Max-Change: 0.002131
 EM iteration: 42, Loglike: -159615.4373, Max-Change: 0.002003
 EM iteration: 43, Loglike: -159615.3997, Max-Change: 0.001882
 EM iteration: 44, Loglike: -159615.3630, Max-Change: 0.00177
 EM iteration: 45, Loglike: -159615.3274, Max-Change: 0.001665
 EM iteration: 46, Loglike: -159615.2926, Max-Change: 0.001566
 EM iteration: 47, Loglike: -159615.2587, Max-Change: 0.001474
 EM iteration: 48, Loglike: -159615.2256, Max-Change: 0.001388
 EM iteration: 49, Loglike: -159615.1933, Max-Change: 0.001308
 EM iteration: 50, Loglike: -159615.1617, Max-Change: 0.001232
 EM iteration: 51, Loglike: -159615.1309, Max-Change: 0.001162
 EM iteration: 52, Loglike: -159615.1007, Max-Change: 0.001097
 EM iteration: 53, Loglike: -159615.0711, Max-Change: 0.001035
 EM iteration: 54, Loglike: -159615.0422, Max-Change: 0.000978 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 9.22 seconds. 

# Summary of the estimation
summary(fit.4)
#> 
#> Call:
#> est_mg(x = x, data = data, group.name = group.name, D = 1, free.group = free.group, 
#>     use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 
#>         16)), EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE, 
#>     fipc.method = "MEM", fix.id = fix.id)
#> 
#> Summary of the Data 
#>  Number of Items: 
#>   Overall: 116 unique items 
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of Cases: 
#>   Overall: 6000
#>   By group: 2000(Group1), 2000(Group2), 2000(Group3)
#> 
#> 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: 432
#>    - Item parameters: 288
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 144
#>  Number of fixed items: 
#>   Overall: 22
#>   By group: 12(Group1), 22(Group2), 10(Group3)
#>  Number of E-step cycles completed: 54
#>  Maximum parameter change: 0.0009779691
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 8.47
#>  Standard error computation: 0.16
#>  Total computation: 9.22
#> 
#> 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: 
#>   Overall: 319230.1
#>   By group: 120354.311(Group1), 113985.392(Group2), 84890.382(Group3)
#> 
#>  Akaike Information Criterion (AIC): 320094.1
#>  Bayesian Information Criterion (BIC): 322988.3
#>  Item Parameters (Overall): 
#>         id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1     C1I1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2     C1I2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3     C1I3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4     C1I4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5     C1I5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6     C1I6     2   3PLM   1.70    NA   0.63    NA   0.07    NA     NA    NA
#> 7     C1I7     2   3PLM   0.91    NA   1.02    NA   0.12    NA     NA    NA
#> 8     C1I8     2   3PLM   0.84    NA   0.80    NA   0.11    NA     NA    NA
#> 9     C1I9     2   3PLM   0.85    NA   0.85    NA   0.26    NA     NA    NA
#> 10   C1I10     2   3PLM   1.53    NA   0.09    NA   0.14    NA     NA    NA
#> 11    G1I1     2   3PLM   0.96  0.10  -0.46  0.20   0.16  0.07     NA    NA
#> 12    G1I2     2   3PLM   0.88  0.13   1.21  0.14   0.10  0.04     NA    NA
#> 13    G1I3     2   3PLM   1.49  0.26   1.31  0.09   0.18  0.03     NA    NA
#> 14    G1I4     2   3PLM   1.52  0.21   0.27  0.12   0.30  0.04     NA    NA
#> 15    G1I5     2   3PLM   1.33  0.14  -0.18  0.13   0.16  0.05     NA    NA
#> 16    G1I6     2   3PLM   2.15  0.18   0.01  0.05   0.08  0.03     NA    NA
#> 17    G1I7     2   3PLM   1.45  0.16  -0.07  0.12   0.20  0.05     NA    NA
#> 18    G1I8     2   3PLM   2.50  0.46   1.18  0.07   0.32  0.02     NA    NA
#> 19    G1I9     2   3PLM   2.40  0.26  -0.93  0.11   0.24  0.06     NA    NA
#> 20   G1I10     2   3PLM   1.26  0.13  -1.75  0.23   0.24  0.10     NA    NA
#> 21   G1I11     2   3PLM   1.55  0.15  -1.11  0.16   0.21  0.08     NA    NA
#> 22   G1I12     2   3PLM   0.75  0.09  -0.81  0.31   0.19  0.08     NA    NA
#> 23   G1I13     2   3PLM   1.04  0.14  -0.13  0.21   0.20  0.07     NA    NA
#> 24   G1I14     2   3PLM   1.46  0.37   1.77  0.15   0.30  0.03     NA    NA
#> 25   G1I15     2   3PLM   0.86  0.09  -1.44  0.29   0.21  0.09     NA    NA
#> 26   G1I16     2   3PLM   1.05  0.11  -1.95  0.27   0.23  0.10     NA    NA
#> 27   G1I17     2   3PLM   1.04  0.13   0.25  0.15   0.15  0.05     NA    NA
#> 28   G1I18     2   3PLM   2.08  0.20  -0.09  0.08   0.24  0.04     NA    NA
#> 29   G1I19     2   3PLM   1.32  0.13  -1.38  0.19   0.20  0.08     NA    NA
#> 30   G1I20     2   3PLM   1.01  0.17   0.51  0.19   0.22  0.06     NA    NA
#> 31   G1I21     2   3PLM   0.92  0.13   0.77  0.15   0.13  0.05     NA    NA
#> 32   G1I22     2   3PLM   1.77  0.21  -0.65  0.16   0.35  0.06     NA    NA
#> 33   G1I23     2   3PLM   1.31  0.15  -1.17  0.23   0.25  0.09     NA    NA
#> 34   G1I24     2   3PLM   1.64  0.19   0.32  0.10   0.23  0.04     NA    NA
#> 35   G1I25     2   3PLM   1.58  0.18  -0.13  0.12   0.25  0.05     NA    NA
#> 36   G1I26     2   3PLM   1.88  0.25   0.66  0.08   0.25  0.03     NA    NA
#> 37   G1I27     2   3PLM   1.63  0.18  -1.54  0.20   0.27  0.10     NA    NA
#> 38   G1I28     2   3PLM   1.33  0.16   0.57  0.10   0.15  0.04     NA    NA
#> 39   G1I29     2   3PLM   0.92  0.09  -0.37  0.17   0.12  0.05     NA    NA
#> 40   G1I30     2   3PLM   1.00  0.27   2.30  0.25   0.17  0.03     NA    NA
#> 41   G1I31     2   3PLM   2.39  0.46   1.64  0.09   0.18  0.01     NA    NA
#> 42   G1I32     2   3PLM   1.10  0.12  -0.08  0.15   0.15  0.05     NA    NA
#> 43   G1I33     2   3PLM   1.60  0.17   0.18  0.09   0.16  0.04     NA    NA
#> 44   G1I34     2   3PLM   1.33  0.13   0.24  0.09   0.11  0.04     NA    NA
#> 45   G1I35     2   3PLM   1.33  0.17   1.30  0.08   0.07  0.02     NA    NA
#> 46   G1I36     2   3PLM   1.44  0.14  -1.23  0.19   0.22  0.09     NA    NA
#> 47   G1I37     2   3PLM   1.05  0.14  -0.60  0.27   0.27  0.09     NA    NA
#> 48   G1I38     5    GRM   1.06  0.06  -0.37  0.05   0.22  0.05   0.87  0.06
#> 49   C1I11     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 50   C1I12     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#> 51    G2I1     2   3PLM   1.74  0.19  -0.87  0.17   0.23  0.10     NA    NA
#> 52    G2I2     2   3PLM   0.83  0.11  -0.45  0.31   0.21  0.09     NA    NA
#> 53    G2I3     2   3PLM   1.06  0.13   0.08  0.20   0.18  0.07     NA    NA
#> 54    G2I4     2   3PLM   1.45  0.28   1.50  0.10   0.18  0.04     NA    NA
#> 55    G2I5     2   3PLM   0.70  0.10  -1.68  0.40   0.20  0.09     NA    NA
#> 56    G2I6     2   3PLM   1.08  0.14  -1.60  0.29   0.22  0.09     NA    NA
#> 57    G2I7     2   3PLM   1.30  0.13   0.27  0.12   0.13  0.05     NA    NA
#> 58    G2I8     2   3PLM   2.18  0.19  -0.09  0.08   0.14  0.05     NA    NA
#> 59    G2I9     2   3PLM   1.07  0.13  -1.57  0.28   0.21  0.09     NA    NA
#> 60   G2I10     2   3PLM   1.59  0.27   0.90  0.12   0.29  0.05     NA    NA
#> 61   G2I11     2   3PLM   0.93  0.12   0.88  0.15   0.11  0.05     NA    NA
#> 62   G2I12     2   3PLM   1.66  0.18  -0.70  0.18   0.25  0.10     NA    NA
#> 63   G2I13     2   3PLM   1.16  0.13  -1.31  0.24   0.21  0.09     NA    NA
#> 64   G2I14     2   3PLM   1.39  0.14   0.22  0.11   0.13  0.05     NA    NA
#> 65   G2I15     2   3PLM   1.53  0.20   0.03  0.17   0.26  0.08     NA    NA
#> 66   G2I16     2   3PLM   1.70  0.23   0.75  0.10   0.21  0.05     NA    NA
#> 67   G2I17     2   3PLM   2.11  0.22  -1.20  0.14   0.20  0.09     NA    NA
#> 68   G2I18     2   3PLM   1.72  0.20   0.64  0.08   0.14  0.04     NA    NA
#> 69   G2I19     2   3PLM   1.10  0.14   0.05  0.22   0.21  0.08     NA    NA
#> 70   G2I20     2   3PLM   1.62  0.45   2.24  0.19   0.16  0.02     NA    NA
#> 71   G2I21     2   3PLM   2.39  0.36   1.59  0.06   0.12  0.02     NA    NA
#> 72   G2I22     2   3PLM   1.43  0.17   0.22  0.15   0.20  0.07     NA    NA
#> 73   G2I23     2   3PLM   2.09  0.26   0.44  0.10   0.28  0.05     NA    NA
#> 74   G2I24     2   3PLM   1.40  0.12   0.37  0.08   0.07  0.03     NA    NA
#> 75   G2I25     2   3PLM   1.74  0.22   1.42  0.06   0.07  0.02     NA    NA
#> 76   G2I26     2   3PLM   1.89  0.20  -0.88  0.16   0.24  0.10     NA    NA
#> 77   G2I27     2   3PLM   0.98  0.12  -0.59  0.25   0.20  0.09     NA    NA
#> 78   G2I28     5    GRM   1.12  0.07  -0.37  0.07   0.15  0.05   0.80  0.05
#> 79    C2I1     2   3PLM   0.97    NA  -0.46    NA   0.05    NA     NA    NA
#> 80    C2I2     2   3PLM   0.85    NA   1.18    NA   0.01    NA     NA    NA
#> 81    C2I3     2   3PLM   1.43    NA   1.41    NA   0.10    NA     NA    NA
#> 82    C2I4     2   3PLM   1.48    NA   0.18    NA   0.17    NA     NA    NA
#> 83    C2I5     2   3PLM   1.27    NA  -0.23    NA   0.03    NA     NA    NA
#> 84    C2I6     2   3PLM   2.02    NA  -0.09    NA   0.01    NA     NA    NA
#> 85    C2I7     2   3PLM   1.37    NA  -0.13    NA   0.10    NA     NA    NA
#> 86    C2I8     2   3PLM   1.67    NA   1.25    NA   0.19    NA     NA    NA
#> 87    C2I9     2   3PLM   2.28    NA  -1.01    NA   0.10    NA     NA    NA
#> 88   C2I10     2   3PLM   1.42    NA  -1.65    NA   0.11    NA     NA    NA
#> 89    G3I1     2   3PLM   1.60  0.14  -1.04  0.12   0.14  0.05     NA    NA
#> 90    G3I2     2   3PLM   0.79  0.09  -0.62  0.25   0.16  0.07     NA    NA
#> 91    G3I3     2   3PLM   1.16  0.12   0.07  0.12   0.16  0.04     NA    NA
#> 92    G3I4     2   3PLM   1.30  0.24   1.72  0.12   0.22  0.02     NA    NA
#> 93    G3I5     2   3PLM   0.80  0.10  -1.09  0.33   0.24  0.09     NA    NA
#> 94    G3I6     2   3PLM   1.18  0.13  -1.38  0.24   0.26  0.08     NA    NA
#> 95    G3I7     2   3PLM   1.37  0.13   0.26  0.09   0.13  0.03     NA    NA
#> 96    G3I8     2   3PLM   1.95  0.18  -0.07  0.06   0.14  0.02     NA    NA
#> 97    G3I9     2   3PLM   1.18  0.11  -1.16  0.19   0.18  0.07     NA    NA
#> 98   G3I10     2   3PLM   1.50  0.21   0.98  0.10   0.29  0.02     NA    NA
#> 99   G3I11     2   3PLM   0.91  0.10   0.80  0.11   0.08  0.03     NA    NA
#> 100  G3I12     2   3PLM   1.44  0.17  -0.78  0.15   0.23  0.06     NA    NA
#> 101  G3I13     2   3PLM   1.15  0.13  -1.06  0.20   0.21  0.07     NA    NA
#> 102  G3I14     2   3PLM   1.38  0.14   0.31  0.09   0.16  0.03     NA    NA
#> 103  G3I15     2   3PLM   1.29  0.13  -0.06  0.11   0.18  0.04     NA    NA
#> 104  G3I16     2   3PLM   1.58  0.18   0.75  0.08   0.20  0.02     NA    NA
#> 105  G3I17     2   3PLM   1.72  0.15  -1.46  0.13   0.16  0.06     NA    NA
#> 106  G3I18     2   3PLM   1.37  0.14   0.71  0.08   0.11  0.02     NA    NA
#> 107  G3I19     2   3PLM   0.99  0.11   0.07  0.15   0.15  0.04     NA    NA
#> 108  G3I20     2   3PLM   1.14  0.23   2.30  0.20   0.12  0.02     NA    NA
#> 109  G3I21     2   3PLM   2.97  0.55   1.66  0.06   0.14  0.01     NA    NA
#> 110  G3I22     2   3PLM   1.19  0.11   0.10  0.10   0.10  0.03     NA    NA
#> 111  G3I23     2   3PLM   1.79  0.17   0.33  0.06   0.15  0.02     NA    NA
#> 112  G3I24     2   3PLM   1.15  0.09   0.34  0.08   0.05  0.02     NA    NA
#> 113  G3I25     2   3PLM   1.38  0.15   1.36  0.08   0.05  0.01     NA    NA
#> 114  G3I26     2   3PLM   1.72  0.17  -0.96  0.12   0.20  0.05     NA    NA
#> 115  G3I27     2   3PLM   0.95  0.10  -0.61  0.20   0.16  0.06     NA    NA
#> 116  G3I28     5    GRM   1.00  0.05  -0.31  0.05   0.19  0.05   0.82  0.07
#>      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      NA    NA
#> 40      NA    NA
#> 41      NA    NA
#> 42      NA    NA
#> 43      NA    NA
#> 44      NA    NA
#> 45      NA    NA
#> 46      NA    NA
#> 47      NA    NA
#> 48    1.44  0.08
#> 49   -0.12    NA
#> 50    1.25    NA
#> 51      NA    NA
#> 52      NA    NA
#> 53      NA    NA
#> 54      NA    NA
#> 55      NA    NA
#> 56      NA    NA
#> 57      NA    NA
#> 58      NA    NA
#> 59      NA    NA
#> 60      NA    NA
#> 61      NA    NA
#> 62      NA    NA
#> 63      NA    NA
#> 64      NA    NA
#> 65      NA    NA
#> 66      NA    NA
#> 67      NA    NA
#> 68      NA    NA
#> 69      NA    NA
#> 70      NA    NA
#> 71      NA    NA
#> 72      NA    NA
#> 73      NA    NA
#> 74      NA    NA
#> 75      NA    NA
#> 76      NA    NA
#> 77      NA    NA
#> 78    1.48  0.07
#> 79      NA    NA
#> 80      NA    NA
#> 81      NA    NA
#> 82      NA    NA
#> 83      NA    NA
#> 84      NA    NA
#> 85      NA    NA
#> 86      NA    NA
#> 87      NA    NA
#> 88      NA    NA
#> 89      NA    NA
#> 90      NA    NA
#> 91      NA    NA
#> 92      NA    NA
#> 93      NA    NA
#> 94      NA    NA
#> 95      NA    NA
#> 96      NA    NA
#> 97      NA    NA
#> 98      NA    NA
#> 99      NA    NA
#> 100     NA    NA
#> 101     NA    NA
#> 102     NA    NA
#> 103     NA    NA
#> 104     NA    NA
#> 105     NA    NA
#> 106     NA    NA
#> 107     NA    NA
#> 108     NA    NA
#> 109     NA    NA
#> 110     NA    NA
#> 111     NA    NA
#> 112     NA    NA
#> 113     NA    NA
#> 114     NA    NA
#> 115     NA    NA
#> 116   1.52  0.09
#>  Group Parameters: 
#>                      mu  sigma2  sigma
#> estimate(Group1)  -0.01    1.01   1.01
#> se(Group1)         0.02    0.03   0.02
#> estimate(Group2)   0.50    0.58   0.76
#> se(Group2)         0.02    0.02   0.01
#> estimate(Group3)  -0.28    1.66   1.29
#> se(Group3)         0.03    0.05   0.02
#> 

## ------------------------------------------------------------------------------
# 3. MG calibration with FIPC using simMG data
#    (Estimate group parameters only)
#  - Details:
#    (a) Fix all item parameters across all three groups
#    (b) Freely estimate the means and variances of the ability
#        distributions for all three groups
## ------------------------------------------------------------------------------
# 3-(1). Freely estimate the means and variances for all three groups
# Set all three groups as free groups in which the scales
# of the ability distributions will be freely estimated
free.group <- 1:3 # or use 'free.group <- group.name'

# Specify the locations of all fixed items in each group's metadata
fix.loc <- list(1:50, 1:50, 1:38)

# Estimate group parameters only using MG-FIPC
fit.5 <-
  est_mg(
    x = x, data = data, group.name = group.name, D = 1,
    free.group = free.group, use.gprior = TRUE,
    gprior = list(dist = "beta", params = c(5, 16)),
    EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE,
    fipc.method = "MEM", fix.loc = fix.loc
  )
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -159735.7617, Max-Change: 0.476403
 EM iteration: 2, Loglike: -159719.0182, Max-Change: 0.138031
 EM iteration: 3, Loglike: -159716.3699, Max-Change: 0.04617
 EM iteration: 4, Loglike: -159715.2802, Max-Change: 0.018371
 EM iteration: 5, Loglike: -159714.6363, Max-Change: 0.009115
 EM iteration: 6, Loglike: -159714.1843, Max-Change: 0.005329
 EM iteration: 7, Loglike: -159713.8343, Max-Change: 0.003463
 EM iteration: 8, Loglike: -159713.5471, Max-Change: 0.002427
 EM iteration: 9, Loglike: -159713.3033, Max-Change: 0.00181
 EM iteration: 10, Loglike: -159713.0914, Max-Change: 0.001421
 EM iteration: 11, Loglike: -159712.9045, Max-Change: 0.001165
 EM iteration: 12, Loglike: -159712.7377, Max-Change: 0.000987 
#> Estimation is finished in 0.98 seconds. 

# Summary of the estimation
summary(fit.5)
#> 
#> Call:
#> est_mg(x = x, data = data, group.name = group.name, D = 1, free.group = free.group, 
#>     use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 
#>         16)), EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = fix.loc)
#> 
#> Summary of the Data 
#>  Number of Items: 
#>   Overall: 116 unique items 
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of Cases: 
#>   Overall: 6000
#>   By group: 2000(Group1), 2000(Group2), 2000(Group3)
#> 
#> 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: 144
#>    - Item parameters: 0
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 144
#>  Number of fixed items: 
#>   Overall: 116
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of E-step cycles completed: 12
#>  Maximum parameter change: 0.0009867474
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 0.46
#>  Standard error computation: 
#>  Total computation: 0.98
#> 
#> 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: 
#>   Overall: 319425.5
#>   By group: 120445.354(Group1), 114033.356(Group2), 84946.765(Group3)
#> 
#>  Akaike Information Criterion (AIC): 319713.5
#>  Bayesian Information Criterion (BIC): 320678.2
#>  Item Parameters (Overall): 
#>         id  cats  model  par.1  se.1  par.2  se.2  par.3  se.3  par.4  se.4
#> 1     C1I1     2   3PLM   0.76    NA   1.46    NA   0.26    NA     NA    NA
#> 2     C1I2     2   3PLM   1.92    NA  -1.05    NA   0.18    NA     NA    NA
#> 3     C1I3     2   3PLM   0.93    NA   0.39    NA   0.10    NA     NA    NA
#> 4     C1I4     2   3PLM   1.05    NA  -0.41    NA   0.20    NA     NA    NA
#> 5     C1I5     2   3PLM   0.87    NA  -0.12    NA   0.16    NA     NA    NA
#> 6     C1I6     2   3PLM   1.70    NA   0.63    NA   0.07    NA     NA    NA
#> 7     C1I7     2   3PLM   0.91    NA   1.02    NA   0.12    NA     NA    NA
#> 8     C1I8     2   3PLM   0.84    NA   0.80    NA   0.11    NA     NA    NA
#> 9     C1I9     2   3PLM   0.85    NA   0.85    NA   0.26    NA     NA    NA
#> 10   C1I10     2   3PLM   1.53    NA   0.09    NA   0.14    NA     NA    NA
#> 11    G1I1     2   3PLM   1.00    NA  -0.46    NA   0.13    NA     NA    NA
#> 12    G1I2     2   3PLM   0.88    NA   1.18    NA   0.09    NA     NA    NA
#> 13    G1I3     2   3PLM   1.46    NA   1.41    NA   0.18    NA     NA    NA
#> 14    G1I4     2   3PLM   1.51    NA   0.18    NA   0.25    NA     NA    NA
#> 15    G1I5     2   3PLM   1.30    NA  -0.23    NA   0.11    NA     NA    NA
#> 16    G1I6     2   3PLM   2.05    NA  -0.09    NA   0.05    NA     NA    NA
#> 17    G1I7     2   3PLM   1.40    NA  -0.13    NA   0.18    NA     NA    NA
#> 18    G1I8     2   3PLM   1.70    NA   1.25    NA   0.27    NA     NA    NA
#> 19    G1I9     2   3PLM   2.31    NA  -1.01    NA   0.18    NA     NA    NA
#> 20   G1I10     2   3PLM   1.45    NA  -1.65    NA   0.19    NA     NA    NA
#> 21   G1I11     2   3PLM   1.63    NA  -1.19    NA   0.12    NA     NA    NA
#> 22   G1I12     2   3PLM   0.83    NA  -0.68    NA   0.20    NA     NA    NA
#> 23   G1I13     2   3PLM   0.98    NA  -0.26    NA   0.13    NA     NA    NA
#> 24   G1I14     2   3PLM   1.14    NA   1.68    NA   0.25    NA     NA    NA
#> 25   G1I15     2   3PLM   0.79    NA  -1.39    NA   0.26    NA     NA    NA
#> 26   G1I16     2   3PLM   1.09    NA  -1.85    NA   0.17    NA     NA    NA
#> 27   G1I17     2   3PLM   1.17    NA   0.07    NA   0.13    NA     NA    NA
#> 28   G1I18     2   3PLM   2.15    NA  -0.09    NA   0.21    NA     NA    NA
#> 29   G1I19     2   3PLM   1.28    NA  -1.38    NA   0.20    NA     NA    NA
#> 30   G1I20     2   3PLM   1.35    NA   0.82    NA   0.32    NA     NA    NA
#> 31   G1I21     2   3PLM   0.82    NA   0.71    NA   0.08    NA     NA    NA
#> 32   G1I22     2   3PLM   1.52    NA  -0.89    NA   0.26    NA     NA    NA
#> 33   G1I23     2   3PLM   1.27    NA  -1.31    NA   0.19    NA     NA    NA
#> 34   G1I24     2   3PLM   1.31    NA   0.19    NA   0.16    NA     NA    NA
#> 35   G1I25     2   3PLM   1.47    NA  -0.14    NA   0.23    NA     NA    NA
#> 36   G1I26     2   3PLM   1.47    NA   0.64    NA   0.23    NA     NA    NA
#> 37   G1I27     2   3PLM   1.76    NA  -1.53    NA   0.16    NA     NA    NA
#> 38   G1I28     2   3PLM   1.44    NA   0.54    NA   0.14    NA     NA    NA
#> 39   G1I29     2   3PLM   0.98    NA  -0.37    NA   0.13    NA     NA    NA
#> 40   G1I30     2   3PLM   0.99    NA   2.37    NA   0.16    NA     NA    NA
#> 41   G1I31     2   3PLM   2.27    NA   1.62    NA   0.18    NA     NA    NA
#> 42   G1I32     2   3PLM   1.23    NA  -0.07    NA   0.13    NA     NA    NA
#> 43   G1I33     2   3PLM   1.64    NA   0.17    NA   0.18    NA     NA    NA
#> 44   G1I34     2   3PLM   1.21    NA   0.24    NA   0.08    NA     NA    NA
#> 45   G1I35     2   3PLM   1.32    NA   1.34    NA   0.08    NA     NA    NA
#> 46   G1I36     2   3PLM   1.74    NA  -1.00    NA   0.25    NA     NA    NA
#> 47   G1I37     2   3PLM   0.97    NA  -0.73    NA   0.22    NA     NA    NA
#> 48   G1I38     5    GRM   1.14    NA  -0.37    NA   0.22    NA   0.85    NA
#> 49   C1I11     5    GRM   1.23    NA  -2.08    NA  -1.35    NA  -0.71    NA
#> 50   C1I12     5    GRM   0.88    NA  -0.76    NA  -0.01    NA   0.67    NA
#> 51    G2I1     2   3PLM   1.60    NA  -1.19    NA   0.04    NA     NA    NA
#> 52    G2I2     2   3PLM   0.80    NA  -0.68    NA   0.12    NA     NA    NA
#> 53    G2I3     2   3PLM   0.95    NA  -0.26    NA   0.05    NA     NA    NA
#> 54    G2I4     2   3PLM   1.11    NA   1.68    NA   0.17    NA     NA    NA
#> 55    G2I5     2   3PLM   0.76    NA  -1.39    NA   0.18    NA     NA    NA
#> 56    G2I6     2   3PLM   1.06    NA  -1.85    NA   0.09    NA     NA    NA
#> 57    G2I7     2   3PLM   1.14    NA   0.07    NA   0.05    NA     NA    NA
#> 58    G2I8     2   3PLM   2.12    NA  -0.09    NA   0.13    NA     NA    NA
#> 59    G2I9     2   3PLM   1.25    NA  -1.38    NA   0.12    NA     NA    NA
#> 60   G2I10     2   3PLM   1.32    NA   0.82    NA   0.24    NA     NA    NA
#> 61   G2I11     2   3PLM   0.79    NA   0.71    NA   0.01    NA     NA    NA
#> 62   G2I12     2   3PLM   1.49    NA  -0.89    NA   0.18    NA     NA    NA
#> 63   G2I13     2   3PLM   1.24    NA  -1.31    NA   0.11    NA     NA    NA
#> 64   G2I14     2   3PLM   1.28    NA   0.19    NA   0.08    NA     NA    NA
#> 65   G2I15     2   3PLM   1.44    NA  -0.14    NA   0.15    NA     NA    NA
#> 66   G2I16     2   3PLM   1.44    NA   0.64    NA   0.15    NA     NA    NA
#> 67   G2I17     2   3PLM   1.73    NA  -1.53    NA   0.08    NA     NA    NA
#> 68   G2I18     2   3PLM   1.41    NA   0.54    NA   0.06    NA     NA    NA
#> 69   G2I19     2   3PLM   0.95    NA  -0.37    NA   0.05    NA     NA    NA
#> 70   G2I20     2   3PLM   0.96    NA   2.37    NA   0.08    NA     NA    NA
#> 71   G2I21     2   3PLM   2.24    NA   1.62    NA   0.10    NA     NA    NA
#> 72   G2I22     2   3PLM   1.20    NA  -0.07    NA   0.05    NA     NA    NA
#> 73   G2I23     2   3PLM   1.61    NA   0.17    NA   0.10    NA     NA    NA
#> 74   G2I24     2   3PLM   1.18    NA   0.24    NA   0.01    NA     NA    NA
#> 75   G2I25     2   3PLM   1.29    NA   1.34    NA   0.01    NA     NA    NA
#> 76   G2I26     2   3PLM   1.71    NA  -1.00    NA   0.17    NA     NA    NA
#> 77   G2I27     2   3PLM   0.94    NA  -0.73    NA   0.14    NA     NA    NA
#> 78   G2I28     5    GRM   1.11    NA  -0.37    NA   0.14    NA   0.78    NA
#> 79    C2I1     2   3PLM   0.97    NA  -0.46    NA   0.05    NA     NA    NA
#> 80    C2I2     2   3PLM   0.85    NA   1.18    NA   0.01    NA     NA    NA
#> 81    C2I3     2   3PLM   1.43    NA   1.41    NA   0.10    NA     NA    NA
#> 82    C2I4     2   3PLM   1.48    NA   0.18    NA   0.17    NA     NA    NA
#> 83    C2I5     2   3PLM   1.27    NA  -0.23    NA   0.03    NA     NA    NA
#> 84    C2I6     2   3PLM   2.02    NA  -0.09    NA   0.01    NA     NA    NA
#> 85    C2I7     2   3PLM   1.37    NA  -0.13    NA   0.10    NA     NA    NA
#> 86    C2I8     2   3PLM   1.67    NA   1.25    NA   0.19    NA     NA    NA
#> 87    C2I9     2   3PLM   2.28    NA  -1.01    NA   0.10    NA     NA    NA
#> 88   C2I10     2   3PLM   1.42    NA  -1.65    NA   0.11    NA     NA    NA
#> 89    G3I1     2   3PLM   1.55    NA  -1.12    NA   0.07    NA     NA    NA
#> 90    G3I2     2   3PLM   0.75    NA  -0.61    NA   0.15    NA     NA    NA
#> 91    G3I3     2   3PLM   0.90    NA  -0.19    NA   0.08    NA     NA    NA
#> 92    G3I4     2   3PLM   1.06    NA   1.75    NA   0.20    NA     NA    NA
#> 93    G3I5     2   3PLM   0.71    NA  -1.32    NA   0.21    NA     NA    NA
#> 94    G3I6     2   3PLM   1.01    NA  -1.78    NA   0.12    NA     NA    NA
#> 95    G3I7     2   3PLM   1.09    NA   0.14    NA   0.08    NA     NA    NA
#> 96    G3I8     2   3PLM   2.07    NA  -0.02    NA   0.16    NA     NA    NA
#> 97    G3I9     2   3PLM   1.20    NA  -1.31    NA   0.15    NA     NA    NA
#> 98   G3I10     2   3PLM   1.27    NA   0.89    NA   0.27    NA     NA    NA
#> 99   G3I11     2   3PLM   0.74    NA   0.78    NA   0.04    NA     NA    NA
#> 100  G3I12     2   3PLM   1.44    NA  -0.82    NA   0.21    NA     NA    NA
#> 101  G3I13     2   3PLM   1.19    NA  -1.24    NA   0.14    NA     NA    NA
#> 102  G3I14     2   3PLM   1.23    NA   0.26    NA   0.11    NA     NA    NA
#> 103  G3I15     2   3PLM   1.39    NA  -0.07    NA   0.18    NA     NA    NA
#> 104  G3I16     2   3PLM   1.39    NA   0.71    NA   0.18    NA     NA    NA
#> 105  G3I17     2   3PLM   1.68    NA  -1.46    NA   0.11    NA     NA    NA
#> 106  G3I18     2   3PLM   1.36    NA   0.61    NA   0.09    NA     NA    NA
#> 107  G3I19     2   3PLM   0.90    NA  -0.30    NA   0.08    NA     NA    NA
#> 108  G3I20     2   3PLM   0.91    NA   2.44    NA   0.11    NA     NA    NA
#> 109  G3I21     2   3PLM   2.19    NA   1.69    NA   0.13    NA     NA    NA
#> 110  G3I22     2   3PLM   1.15    NA   0.00    NA   0.08    NA     NA    NA
#> 111  G3I23     2   3PLM   1.56    NA   0.24    NA   0.13    NA     NA    NA
#> 112  G3I24     2   3PLM   1.13    NA   0.31    NA   0.04    NA     NA    NA
#> 113  G3I25     2   3PLM   1.24    NA   1.41    NA   0.04    NA     NA    NA
#> 114  G3I26     2   3PLM   1.66    NA  -0.93    NA   0.20    NA     NA    NA
#> 115  G3I27     2   3PLM   0.89    NA  -0.66    NA   0.17    NA     NA    NA
#> 116  G3I28     5    GRM   1.06    NA  -0.30    NA   0.17    NA   0.78    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      NA    NA
#> 40      NA    NA
#> 41      NA    NA
#> 42      NA    NA
#> 43      NA    NA
#> 44      NA    NA
#> 45      NA    NA
#> 46      NA    NA
#> 47      NA    NA
#> 48    1.38    NA
#> 49   -0.12    NA
#> 50    1.25    NA
#> 51      NA    NA
#> 52      NA    NA
#> 53      NA    NA
#> 54      NA    NA
#> 55      NA    NA
#> 56      NA    NA
#> 57      NA    NA
#> 58      NA    NA
#> 59      NA    NA
#> 60      NA    NA
#> 61      NA    NA
#> 62      NA    NA
#> 63      NA    NA
#> 64      NA    NA
#> 65      NA    NA
#> 66      NA    NA
#> 67      NA    NA
#> 68      NA    NA
#> 69      NA    NA
#> 70      NA    NA
#> 71      NA    NA
#> 72      NA    NA
#> 73      NA    NA
#> 74      NA    NA
#> 75      NA    NA
#> 76      NA    NA
#> 77      NA    NA
#> 78    1.44    NA
#> 79      NA    NA
#> 80      NA    NA
#> 81      NA    NA
#> 82      NA    NA
#> 83      NA    NA
#> 84      NA    NA
#> 85      NA    NA
#> 86      NA    NA
#> 87      NA    NA
#> 88      NA    NA
#> 89      NA    NA
#> 90      NA    NA
#> 91      NA    NA
#> 92      NA    NA
#> 93      NA    NA
#> 94      NA    NA
#> 95      NA    NA
#> 96      NA    NA
#> 97      NA    NA
#> 98      NA    NA
#> 99      NA    NA
#> 100     NA    NA
#> 101     NA    NA
#> 102     NA    NA
#> 103     NA    NA
#> 104     NA    NA
#> 105     NA    NA
#> 106     NA    NA
#> 107     NA    NA
#> 108     NA    NA
#> 109     NA    NA
#> 110     NA    NA
#> 111     NA    NA
#> 112     NA    NA
#> 113     NA    NA
#> 114     NA    NA
#> 115     NA    NA
#> 116   1.39    NA
#>  Group Parameters: 
#>                      mu  sigma2  sigma
#> estimate(Group1)   0.01    0.96   0.98
#> se(Group1)         0.02    0.03   0.02
#> estimate(Group2)   0.52    0.60   0.78
#> se(Group2)         0.02    0.02   0.01
#> estimate(Group3)  -0.28    1.63   1.28
#> se(Group3)         0.03    0.05   0.02
#> 

# Extract the group parameter estimates (i.e., scale parameters)
getirt(fit.5, what = "group.par")
#> $Group1
#>                    mu     sigma2      sigma
#> estimates 0.005021632 0.96136149 0.98049044
#> se        0.021924433 0.03040852 0.01550679
#> 
#> $Group2
#>                   mu     sigma2      sigma
#> estimates 0.51580977 0.60138258 0.77548861
#> se        0.01734045 0.01902214 0.01226462
#> 
#> $Group3
#>                    mu     sigma2      sigma
#> estimates -0.27507220 1.63376235 1.27818714
#> se         0.02858113 0.05167702 0.02021497
#> 

## ------------------------------------------------------------------------------
# 4. MG calibration with FIPC using simMG data
#    (Fix only the unique items of Group 1)
#  - Details:
#    (a) Fix item parameters of the unique items in Group 1 only
#    (b) Constrain the common items across groups to have
#        the same item parameters (i.e., C1I1 - C1I12 between
#        Groups 1 and 2, and C2I1 - C2I10 between Groups 2 and 3)
#    (c) Freely estimate the means and variances of the ability
#        distributions for all three groups
## ------------------------------------------------------------------------------
# 4-(1). Freely estimate the means and variances for all three groups
# Set all three groups as free groups in which the scales
# of the ability distributions will be freely estimated
free.group <- group.name # or use 'free.group <- 1:3'

# Specify the item IDs of the unique items in Group 1 to be fixed using
# the `fix.id` argument.
fix.id <- paste0("G1I", 1:38)

# Alternatively, use the 'fix.loc' argument as
# 'fix.loc = list(11:48, NULL, NULL)'

# Estimate IRT parameters using MG-FIPC
fit.6 <-
  est_mg(
    x = x, data = data, group.name = group.name, D = 1,
    free.group = free.group, use.gprior = TRUE,
    gprior = list(dist = "beta", params = c(5, 16)),
    EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE,
    fipc.method = "MEM", fix.loc = NULL, fix.id = fix.id
  )
#> Parsing input... 
#> Estimating item parameters... 
#> 
 EM iteration: 1, Loglike: -43183.7384, Max-Change: 432.57651
 EM iteration: 2, Loglike: -171691.1579, Max-Change: 4362.10210
 EM iteration: 3, Loglike: -167102.4773, Max-Change: 21693.47046
 EM iteration: 4, Loglike: -161125.0380, Max-Change: 0.614652
 EM iteration: 5, Loglike: -160697.8762, Max-Change: 14592.39778
 EM iteration: 6, Loglike: -160607.1937, Max-Change: 0.223703
 EM iteration: 7, Loglike: -160564.3124, Max-Change: 41702.94716
 EM iteration: 8, Loglike: -160537.2472, Max-Change: 0.114086
 EM iteration: 9, Loglike: -160522.5563, Max-Change: 0.085219
 EM iteration: 10, Loglike: -160512.8240, Max-Change: 0.065365
 EM iteration: 11, Loglike: -160505.9120, Max-Change: 0.05449
 EM iteration: 12, Loglike: -160500.7123, Max-Change: 0.048557
 EM iteration: 13, Loglike: -160496.6128, Max-Change: 0.043318
 EM iteration: 14, Loglike: -160493.2569, Max-Change: 0.038756
 EM iteration: 15, Loglike: -160490.4265, Max-Change: 0.034813
 EM iteration: 16, Loglike: -160487.9825, Max-Change: 0.031416
 EM iteration: 17, Loglike: -160485.8323, Max-Change: 0.028489
 EM iteration: 18, Loglike: -160483.9129, Max-Change: 0.025964
 EM iteration: 19, Loglike: -160482.1797, Max-Change: 0.024175
 EM iteration: 20, Loglike: -160480.6001, Max-Change: 0.02304
 EM iteration: 21, Loglike: -160479.1504, Max-Change: 0.021983
 EM iteration: 22, Loglike: -160477.8119, Max-Change: 0.021228
 EM iteration: 23, Loglike: -160476.5708, Max-Change: 0.02051
 EM iteration: 24, Loglike: -160475.4156, Max-Change: 0.019825
 EM iteration: 25, Loglike: -160474.3374, Max-Change: 0.019175
 EM iteration: 26, Loglike: -160473.3286, Max-Change: 0.018559
 EM iteration: 27, Loglike: -160472.3828, Max-Change: 0.017975
 EM iteration: 28, Loglike: -160471.4949, Max-Change: 0.017422
 EM iteration: 29, Loglike: -160470.6599, Max-Change: 0.016898
 EM iteration: 30, Loglike: -160469.8740, Max-Change: 0.01640
 EM iteration: 31, Loglike: -160469.1334, Max-Change: 0.015925
 EM iteration: 32, Loglike: -160468.4349, Max-Change: 0.015472
 EM iteration: 33, Loglike: -160467.7756, Max-Change: 0.015038
 EM iteration: 34, Loglike: -160467.1528, Max-Change: 0.014623
 EM iteration: 35, Loglike: -160466.5640, Max-Change: 0.014223
 EM iteration: 36, Loglike: -160466.0071, Max-Change: 0.013838
 EM iteration: 37, Loglike: -160465.4800, Max-Change: 0.013467
 EM iteration: 38, Loglike: -160464.9808, Max-Change: 0.013108
 EM iteration: 39, Loglike: -160464.5076, Max-Change: 0.012761
 EM iteration: 40, Loglike: -160464.0590, Max-Change: 0.012425
 EM iteration: 41, Loglike: -160463.6333, Max-Change: 0.012099
 EM iteration: 42, Loglike: -160463.2291, Max-Change: 0.011782
 EM iteration: 43, Loglike: -160462.8451, Max-Change: 0.011475
 EM iteration: 44, Loglike: -160462.4799, Max-Change: 0.011176
 EM iteration: 45, Loglike: -160462.1326, Max-Change: 0.010886
 EM iteration: 46, Loglike: -160461.8019, Max-Change: 0.010604
 EM iteration: 47, Loglike: -160461.4867, Max-Change: 0.01033
 EM iteration: 48, Loglike: -160461.1864, Max-Change: 0.010063
 EM iteration: 49, Loglike: -160460.8997, Max-Change: 0.009804
 EM iteration: 50, Loglike: -160460.6260, Max-Change: 0.009552
 EM iteration: 51, Loglike: -160460.3646, Max-Change: 0.009306
 EM iteration: 52, Loglike: -160460.1149, Max-Change: 0.009068
 EM iteration: 53, Loglike: -160459.8761, Max-Change: 0.008836
 EM iteration: 54, Loglike: -160459.6479, Max-Change: 0.00861
 EM iteration: 55, Loglike: -160459.4298, Max-Change: 0.008391
 EM iteration: 56, Loglike: -160459.2214, Max-Change: 0.008178
 EM iteration: 57, Loglike: -160459.0222, Max-Change: 0.007971
 EM iteration: 58, Loglike: -160458.8319, Max-Change: 0.007769
 EM iteration: 59, Loglike: -160458.6500, Max-Change: 0.007574
 EM iteration: 60, Loglike: -160458.4761, Max-Change: 0.007383
 EM iteration: 61, Loglike: -160458.3098, Max-Change: 0.007199
 EM iteration: 62, Loglike: -160458.1507, Max-Change: 0.007019
 EM iteration: 63, Loglike: -160457.9985, Max-Change: 0.006845
 EM iteration: 64, Loglike: -160457.8527, Max-Change: 0.006676
 EM iteration: 65, Loglike: -160457.7131, Max-Change: 0.006512
 EM iteration: 66, Loglike: -160457.5792, Max-Change: 0.006353
 EM iteration: 67, Loglike: -160457.4509, Max-Change: 0.006198
 EM iteration: 68, Loglike: -160457.3278, Max-Change: 0.006048
 EM iteration: 69, Loglike: -160457.2096, Max-Change: 0.005902
 EM iteration: 70, Loglike: -160457.0963, Max-Change: 0.00576
 EM iteration: 71, Loglike: -160456.9874, Max-Change: 0.005623
 EM iteration: 72, Loglike: -160456.8827, Max-Change: 0.005489
 EM iteration: 73, Loglike: -160456.7822, Max-Change: 0.00536
 EM iteration: 74, Loglike: -160456.6856, Max-Change: 0.005234
 EM iteration: 75, Loglike: -160456.5926, Max-Change: 0.005112
 EM iteration: 76, Loglike: -160456.5032, Max-Change: 0.004994
 EM iteration: 77, Loglike: -160456.4172, Max-Change: 0.004879
 EM iteration: 78, Loglike: -160456.3342, Max-Change: 0.004767
 EM iteration: 79, Loglike: -160456.2546, Max-Change: 0.004658
 EM iteration: 80, Loglike: -160456.1778, Max-Change: 0.004553
 EM iteration: 81, Loglike: -160456.1038, Max-Change: 0.004451
 EM iteration: 82, Loglike: -160456.0324, Max-Change: 0.004351
 EM iteration: 83, Loglike: -160455.9636, Max-Change: 0.004254
 EM iteration: 84, Loglike: -160455.8973, Max-Change: 0.00416
 EM iteration: 85, Loglike: -160455.8332, Max-Change: 0.004069
 EM iteration: 86, Loglike: -160455.7714, Max-Change: 0.00398
 EM iteration: 87, Loglike: -160455.7117, Max-Change: 0.003893
 EM iteration: 88, Loglike: -160455.6541, Max-Change: 0.003809
 EM iteration: 89, Loglike: -160455.5985, Max-Change: 0.003727
 EM iteration: 90, Loglike: -160455.5447, Max-Change: 0.003648
 EM iteration: 91, Loglike: -160455.4926, Max-Change: 0.00357
 EM iteration: 92, Loglike: -160455.4424, Max-Change: 0.003495
 EM iteration: 93, Loglike: -160455.3937, Max-Change: 0.003421
 EM iteration: 94, Loglike: -160455.3466, Max-Change: 0.00335
 EM iteration: 95, Loglike: -160455.3011, Max-Change: 0.00328
 EM iteration: 96, Loglike: -160455.2570, Max-Change: 0.003212
 EM iteration: 97, Loglike: -160455.2143, Max-Change: 0.003146
 EM iteration: 98, Loglike: -160455.1729, Max-Change: 0.003082
 EM iteration: 99, Loglike: -160455.1328, Max-Change: 0.003019
 EM iteration: 100, Loglike: -160455.0939, Max-Change: 0.002958
 EM iteration: 101, Loglike: -160455.0563, Max-Change: 0.002898
 EM iteration: 102, Loglike: -160455.0197, Max-Change: 0.00284
 EM iteration: 103, Loglike: -160454.9843, Max-Change: 0.002784
 EM iteration: 104, Loglike: -160454.9499, Max-Change: 0.002728
 EM iteration: 105, Loglike: -160454.9164, Max-Change: 0.002674
 EM iteration: 106, Loglike: -160454.8840, Max-Change: 0.002622
 EM iteration: 107, Loglike: -160454.8525, Max-Change: 0.00257
 EM iteration: 108, Loglike: -160454.8218, Max-Change: 0.00252
 EM iteration: 109, Loglike: -160454.7920, Max-Change: 0.002471
 EM iteration: 110, Loglike: -160454.7631, Max-Change: 0.002424
 EM iteration: 111, Loglike: -160454.7350, Max-Change: 0.002377
 EM iteration: 112, Loglike: -160454.7076, Max-Change: 0.002332
 EM iteration: 113, Loglike: -160454.6810, Max-Change: 0.002287
 EM iteration: 114, Loglike: -160454.6550, Max-Change: 0.002244
 EM iteration: 115, Loglike: -160454.6297, Max-Change: 0.002201
 EM iteration: 116, Loglike: -160454.6052, Max-Change: 0.00216
 EM iteration: 117, Loglike: -160454.5812, Max-Change: 0.00212
 EM iteration: 118, Loglike: -160454.5578, Max-Change: 0.00208
 EM iteration: 119, Loglike: -160454.5350, Max-Change: 0.002041
 EM iteration: 120, Loglike: -160454.5129, Max-Change: 0.002004
 EM iteration: 121, Loglike: -160454.4912, Max-Change: 0.001967
 EM iteration: 122, Loglike: -160454.4701, Max-Change: 0.001931
 EM iteration: 123, Loglike: -160454.4494, Max-Change: 0.001895
 EM iteration: 124, Loglike: -160454.4293, Max-Change: 0.001861
 EM iteration: 125, Loglike: -160454.4096, Max-Change: 0.001827
 EM iteration: 126, Loglike: -160454.3904, Max-Change: 0.001794
 EM iteration: 127, Loglike: -160454.3717, Max-Change: 0.001762
 EM iteration: 128, Loglike: -160454.3534, Max-Change: 0.00173
 EM iteration: 129, Loglike: -160454.3355, Max-Change: 0.001699
 EM iteration: 130, Loglike: -160454.3179, Max-Change: 0.001669
 EM iteration: 131, Loglike: -160454.3008, Max-Change: 0.00164
 EM iteration: 132, Loglike: -160454.2841, Max-Change: 0.001611
 EM iteration: 133, Loglike: -160454.2676, Max-Change: 0.001582
 EM iteration: 134, Loglike: -160454.2516, Max-Change: 0.001555
 EM iteration: 135, Loglike: -160454.2358, Max-Change: 0.001528
 EM iteration: 136, Loglike: -160454.2204, Max-Change: 0.001501
 EM iteration: 137, Loglike: -160454.2054, Max-Change: 0.001475
 EM iteration: 138, Loglike: -160454.1905, Max-Change: 0.00145
 EM iteration: 139, Loglike: -160454.1761, Max-Change: 0.001425
 EM iteration: 140, Loglike: -160454.1619, Max-Change: 0.00140
 EM iteration: 141, Loglike: -160454.1480, Max-Change: 0.001377
 EM iteration: 142, Loglike: -160454.1343, Max-Change: 0.001353
 EM iteration: 143, Loglike: -160454.1209, Max-Change: 0.00133
 EM iteration: 144, Loglike: -160454.1078, Max-Change: 0.001308
 EM iteration: 145, Loglike: -160454.0948, Max-Change: 0.001286
 EM iteration: 146, Loglike: -160454.0822, Max-Change: 0.001265
 EM iteration: 147, Loglike: -160454.0697, Max-Change: 0.001244
 EM iteration: 148, Loglike: -160454.0575, Max-Change: 0.001223
 EM iteration: 149, Loglike: -160454.0455, Max-Change: 0.001203
 EM iteration: 150, Loglike: -160454.0337, Max-Change: 0.001183
 EM iteration: 151, Loglike: -160454.0221, Max-Change: 0.001164
 EM iteration: 152, Loglike: -160454.0107, Max-Change: 0.001145
 EM iteration: 153, Loglike: -160453.9995, Max-Change: 0.001126
 EM iteration: 154, Loglike: -160453.9885, Max-Change: 0.001108
 EM iteration: 155, Loglike: -160453.9776, Max-Change: 0.00109
 EM iteration: 156, Loglike: -160453.9670, Max-Change: 0.001072
 EM iteration: 157, Loglike: -160453.9565, Max-Change: 0.001055
 EM iteration: 158, Loglike: -160453.9461, Max-Change: 0.001038
 EM iteration: 159, Loglike: -160453.9359, Max-Change: 0.001022
 EM iteration: 160, Loglike: -160453.9259, Max-Change: 0.001006
 EM iteration: 161, Loglike: -160453.9160, Max-Change: 0.00099 
#> Warning: Convergence criteria are not satisfied. 
#> Computing item parameter var-covariance matrix... 
#> Estimation is finished in 60.13 seconds. 

# Summary of the estimation
summary(fit.6)
#> 
#> Call:
#> est_mg(x = x, data = data, group.name = group.name, D = 1, free.group = free.group, 
#>     use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 
#>         16)), EmpHist = TRUE, Etol = 0.001, MaxE = 500, fipc = TRUE, 
#>     fipc.method = "MEM", fix.loc = NULL, fix.id = fix.id)
#> 
#> Summary of the Data 
#>  Number of Items: 
#>   Overall: 116 unique items 
#>   By group: 50(Group1), 50(Group2), 38(Group3)
#>  Number of Cases: 
#>   Overall: 6000
#>   By group: 2000(Group1), 2000(Group2), 2000(Group3)
#> 
#> 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: 386
#>    - Item parameters: 242
#>    - Latent distribution parameters (mean and variance): 0
#>    - Latent distribution weights (empirical histogram): 144
#>  Number of fixed items: 
#>   Overall: 38
#>   By group: 38(Group1), 0(Group2), 0(Group3)
#>  Number of E-step cycles completed: 161
#>  Maximum parameter change: 0.0009899273
#> 
#> Processing time (in seconds) 
#>  EM algorithm: 59.39
#>  Standard error computation: 0.13
#>  Total computation: 60.13
#> 
#> Convergence and Stability of Solution 
#>  First-order test: Convergence criteria are not satisfied.
#>  Second-order test: Information matrix of item parameter estimates is positive definite.
#>  Computation of variance-covariance matrix: 
#>   Variance-covariance matrix of item parameter estimates is obtainable.
#> 
#> Summary of Estimation Results 
#>  -2loglikelihood: 
#>   Overall: 320907.8
#>   By group: 120433.219(Group1), 114532.038(Group2), 85942.571(Group3)
#> 
#>  Akaike Information Criterion (AIC): 321679.8
#>  Bayesian Information Criterion (BIC): 324265.8
#>  Item Parameters (Overall): 
#>         id  cats  model  par.1  se.1     par.2      se.2  par.3  se.3  par.4
#> 1     C1I1     2   3PLM   0.88  0.17      1.37      0.17   0.27  0.05     NA
#> 2     C1I2     2   3PLM   2.17  0.14     -0.97      0.09   0.17  0.06     NA
#> 3     C1I3     2   3PLM   1.05  0.12      0.63      0.13   0.18  0.05     NA
#> 4     C1I4     2   3PLM   1.05  0.12     -0.24      0.22   0.25  0.07     NA
#> 5     C1I5     2   3PLM   0.87  0.09     -0.16      0.21   0.17  0.07     NA
#> 6     C1I6     2   3PLM   1.87  0.13      0.61      0.04   0.08  0.02     NA
#> 7     C1I7     2   3PLM   1.07  0.13      1.12      0.09   0.14  0.03     NA
#> 8     C1I8     2   3PLM   0.93  0.12      0.90      0.14   0.16  0.05     NA
#> 9     C1I9     2   3PLM   0.89  0.12      0.63      0.19   0.20  0.06     NA
#> 10   C1I10     2   3PLM   1.46  0.11      0.13      0.09   0.14  0.04     NA
#> 11    G1I1     2   3PLM   1.00    NA     -0.46        NA   0.13    NA     NA
#> 12    G1I2     2   3PLM   0.88    NA      1.18        NA   0.09    NA     NA
#> 13    G1I3     2   3PLM   1.46    NA      1.41        NA   0.18    NA     NA
#> 14    G1I4     2   3PLM   1.51    NA      0.18        NA   0.25    NA     NA
#> 15    G1I5     2   3PLM   1.30    NA     -0.23        NA   0.11    NA     NA
#> 16    G1I6     2   3PLM   2.05    NA     -0.09        NA   0.05    NA     NA
#> 17    G1I7     2   3PLM   1.40    NA     -0.13        NA   0.18    NA     NA
#> 18    G1I8     2   3PLM   1.70    NA      1.25        NA   0.27    NA     NA
#> 19    G1I9     2   3PLM   2.31    NA     -1.01        NA   0.18    NA     NA
#> 20   G1I10     2   3PLM   1.45    NA     -1.65        NA   0.19    NA     NA
#> 21   G1I11     2   3PLM   1.63    NA     -1.19        NA   0.12    NA     NA
#> 22   G1I12     2   3PLM   0.83    NA     -0.68        NA   0.20    NA     NA
#> 23   G1I13     2   3PLM   0.98    NA     -0.26        NA   0.13    NA     NA
#> 24   G1I14     2   3PLM   1.14    NA      1.68        NA   0.25    NA     NA
#> 25   G1I15     2   3PLM   0.79    NA     -1.39        NA   0.26    NA     NA
#> 26   G1I16     2   3PLM   1.09    NA     -1.85        NA   0.17    NA     NA
#> 27   G1I17     2   3PLM   1.17    NA      0.07        NA   0.13    NA     NA
#> 28   G1I18     2   3PLM   2.15    NA     -0.09        NA   0.21    NA     NA
#> 29   G1I19     2   3PLM   1.28    NA     -1.38        NA   0.20    NA     NA
#> 30   G1I20     2   3PLM   1.35    NA      0.82        NA   0.32    NA     NA
#> 31   G1I21     2   3PLM   0.82    NA      0.71        NA   0.08    NA     NA
#> 32   G1I22     2   3PLM   1.52    NA     -0.89        NA   0.26    NA     NA
#> 33   G1I23     2   3PLM   1.27    NA     -1.31        NA   0.19    NA     NA
#> 34   G1I24     2   3PLM   1.31    NA      0.19        NA   0.16    NA     NA
#> 35   G1I25     2   3PLM   1.47    NA     -0.14        NA   0.23    NA     NA
#> 36   G1I26     2   3PLM   1.47    NA      0.64        NA   0.23    NA     NA
#> 37   G1I27     2   3PLM   1.76    NA     -1.53        NA   0.16    NA     NA
#> 38   G1I28     2   3PLM   1.44    NA      0.54        NA   0.14    NA     NA
#> 39   G1I29     2   3PLM   0.98    NA     -0.37        NA   0.13    NA     NA
#> 40   G1I30     2   3PLM   0.99    NA      2.37        NA   0.16    NA     NA
#> 41   G1I31     2   3PLM   2.27    NA      1.62        NA   0.18    NA     NA
#> 42   G1I32     2   3PLM   1.23    NA     -0.07        NA   0.13    NA     NA
#> 43   G1I33     2   3PLM   1.64    NA      0.17        NA   0.18    NA     NA
#> 44   G1I34     2   3PLM   1.21    NA      0.24        NA   0.08    NA     NA
#> 45   G1I35     2   3PLM   1.32    NA      1.34        NA   0.08    NA     NA
#> 46   G1I36     2   3PLM   1.74    NA     -1.00        NA   0.25    NA     NA
#> 47   G1I37     2   3PLM   0.97    NA     -0.73        NA   0.22    NA     NA
#> 48   G1I38     5    GRM   1.14    NA     -0.37        NA   0.22    NA   0.85
#> 49   C1I11     5    GRM   1.21  0.05     -2.13      0.09  -1.39  0.06  -0.70
#> 50   C1I12     5    GRM   0.92  0.04     -0.66      0.05   0.05  0.04   0.70
#> 51    G2I1     2   3PLM   1.76  0.19     -0.85      0.17   0.22  0.09     NA
#> 52    G2I2     2   3PLM   0.00  0.09  21770.29  99999.00   0.21  0.09     NA
#> 53    G2I3     2   3PLM   1.08  0.14      0.12      0.20   0.18  0.08     NA
#> 54    G2I4     2   3PLM   1.53  0.30      1.47      0.09   0.18  0.04     NA
#> 55    G2I5     2   3PLM   0.00  0.09  15000.26  99999.00   0.21  0.09     NA
#> 56    G2I6     2   3PLM   1.07  0.15     -1.61      0.31   0.22  0.10     NA
#> 57    G2I7     2   3PLM   1.33  0.14      0.29      0.12   0.13  0.05     NA
#> 58    G2I8     2   3PLM   2.24  0.20     -0.06      0.08   0.14  0.05     NA
#> 59    G2I9     2   3PLM   1.08  0.13     -1.54      0.28   0.21  0.09     NA
#> 60   G2I10     2   3PLM   1.77  0.30      0.95      0.10   0.31  0.05     NA
#> 61   G2I11     2   3PLM   0.96  0.13      0.89      0.15   0.12  0.05     NA
#> 62   G2I12     2   3PLM   0.00  0.11  20016.75  99999.00   0.21  0.09     NA
#> 63   G2I13     2   3PLM   1.18  0.13     -1.27      0.24   0.21  0.09     NA
#> 64   G2I14     2   3PLM   1.41  0.14      0.23      0.11   0.12  0.05     NA
#> 65   G2I15     2   3PLM   1.58  0.21      0.09      0.17   0.28  0.08     NA
#> 66   G2I16     2   3PLM   1.79  0.25      0.78      0.09   0.22  0.05     NA
#> 67   G2I17     2   3PLM   2.07  0.22     -1.20      0.15   0.20  0.09     NA
#> 68   G2I18     2   3PLM   1.76  0.21      0.66      0.08   0.14  0.04     NA
#> 69   G2I19     2   3PLM   1.11  0.14      0.05      0.22   0.20  0.08     NA
#> 70   G2I20     2   3PLM   1.75  0.48      2.17      0.18   0.16  0.02     NA
#> 71   G2I21     2   3PLM   2.52  0.38      1.56      0.06   0.12  0.02     NA
#> 72   G2I22     2   3PLM   1.52  0.20      0.29      0.15   0.23  0.07     NA
#> 73   G2I23     2   3PLM   2.20  0.29      0.49      0.09   0.29  0.05     NA
#> 74   G2I24     2   3PLM   1.42  0.12      0.38      0.07   0.07  0.03     NA
#> 75   G2I25     2   3PLM   1.80  0.23      1.40      0.06   0.07  0.02     NA
#> 76   G2I26     2   3PLM   1.91  0.20     -0.84      0.17   0.25  0.10     NA
#> 77   G2I27     2   3PLM   1.00  0.12     -0.53      0.26   0.21  0.09     NA
#> 78   G2I28     5    GRM   1.14  0.07     -0.35      0.07   0.17  0.05   0.81
#> 79    C2I1     2   3PLM   1.08  0.08     -0.25      0.11   0.13  0.03     NA
#> 80    C2I2     2   3PLM   0.99  0.09      1.27      0.07   0.06  0.02     NA
#> 81    C2I3     2   3PLM   1.61  0.16      1.41      0.05   0.13  0.01     NA
#> 82    C2I4     2   3PLM   1.55  0.11      0.28      0.06   0.19  0.02     NA
#> 83    C2I5     2   3PLM   1.42  0.10     -0.03      0.07   0.12  0.03     NA
#> 84    C2I6     2   3PLM   2.06  0.11     -0.02      0.04   0.05  0.01     NA
#> 85    C2I7     2   3PLM   1.56  0.11      0.04      0.07   0.18  0.02     NA
#> 86    C2I8     2   3PLM   1.51  0.15      1.30      0.06   0.19  0.02     NA
#> 87    C2I9     2   3PLM   2.10  0.13     -1.03      0.07   0.13  0.03     NA
#> 88   C2I10     2   3PLM   0.00  0.04  42123.37  99999.00   0.21  0.09     NA
#> 89    G3I1     2   3PLM   1.50  0.14     -1.07      0.13   0.17  0.05     NA
#> 90    G3I2     2   3PLM   0.00  0.05    299.41  99999.00   0.24  0.10     NA
#> 91    G3I3     2   3PLM   1.10  0.12      0.07      0.12   0.16  0.03     NA
#> 92    G3I4     2   3PLM   1.32  0.24      1.75      0.12   0.23  0.02     NA
#> 93    G3I5     2   3PLM   0.00  0.05   4466.26  99999.00   0.21  0.09     NA
#> 94    G3I6     2   3PLM   1.17  0.15     -1.27      0.25   0.33  0.07     NA
#> 95    G3I7     2   3PLM   1.36  0.13      0.29      0.09   0.14  0.03     NA
#> 96    G3I8     2   3PLM   1.85  0.17     -0.07      0.07   0.14  0.02     NA
#> 97    G3I9     2   3PLM   1.12  0.12     -1.18      0.20   0.22  0.06     NA
#> 98   G3I10     2   3PLM   1.54  0.22      1.02      0.09   0.29  0.02     NA
#> 99   G3I11     2   3PLM   0.89  0.10      0.84      0.11   0.09  0.03     NA
#> 100  G3I12     2   3PLM   1.42  0.17     -0.73      0.15   0.27  0.05     NA
#> 101  G3I13     2   3PLM   1.13  0.13     -1.01      0.21   0.26  0.06     NA
#> 102  G3I14     2   3PLM   1.34  0.14      0.34      0.09   0.17  0.03     NA
#> 103  G3I15     2   3PLM   1.30  0.14      0.01      0.11   0.20  0.03     NA
#> 104  G3I16     2   3PLM   1.59  0.18      0.79      0.08   0.21  0.02     NA
#> 105  G3I17     2   3PLM   1.55  0.15     -1.59      0.15   0.17  0.06     NA
#> 106  G3I18     2   3PLM   1.36  0.14      0.74      0.08   0.12  0.02     NA
#> 107  G3I19     2   3PLM   0.98  0.11      0.13      0.15   0.17  0.04     NA
#> 108  G3I20     2   3PLM   1.15  0.23      2.32      0.20   0.12  0.02     NA
#> 109  G3I21     2   3PLM   3.02  0.56      1.68      0.06   0.14  0.01     NA
#> 110  G3I22     2   3PLM   1.13  0.11      0.09      0.10   0.09  0.03     NA
#> 111  G3I23     2   3PLM   1.77  0.17      0.35      0.07   0.15  0.02     NA
#> 112  G3I24     2   3PLM   1.12  0.09      0.35      0.08   0.06  0.02     NA
#> 113  G3I25     2   3PLM   1.36  0.15      1.40      0.08   0.05  0.01     NA
#> 114  G3I26     2   3PLM   1.61  0.16     -1.01      0.13   0.21  0.05     NA
#> 115  G3I27     2   3PLM   0.91  0.10     -0.62      0.21   0.18  0.06     NA
#> 116  G3I28     5    GRM   0.93  0.05     -0.36      0.06   0.18  0.06   0.86
#>      se.4  par.5  se.5
#> 1      NA     NA    NA
#> 2      NA     NA    NA
#> 3      NA     NA    NA
#> 4      NA     NA    NA
#> 5      NA     NA    NA
#> 6      NA     NA    NA
#> 7      NA     NA    NA
#> 8      NA     NA    NA
#> 9      NA     NA    NA
#> 10     NA     NA    NA
#> 11     NA     NA    NA
#> 12     NA     NA    NA
#> 13     NA     NA    NA
#> 14     NA     NA    NA
#> 15     NA     NA    NA
#> 16     NA     NA    NA
#> 17     NA     NA    NA
#> 18     NA     NA    NA
#> 19     NA     NA    NA
#> 20     NA     NA    NA
#> 21     NA     NA    NA
#> 22     NA     NA    NA
#> 23     NA     NA    NA
#> 24     NA     NA    NA
#> 25     NA     NA    NA
#> 26     NA     NA    NA
#> 27     NA     NA    NA
#> 28     NA     NA    NA
#> 29     NA     NA    NA
#> 30     NA     NA    NA
#> 31     NA     NA    NA
#> 32     NA     NA    NA
#> 33     NA     NA    NA
#> 34     NA     NA    NA
#> 35     NA     NA    NA
#> 36     NA     NA    NA
#> 37     NA     NA    NA
#> 38     NA     NA    NA
#> 39     NA     NA    NA
#> 40     NA     NA    NA
#> 41     NA     NA    NA
#> 42     NA     NA    NA
#> 43     NA     NA    NA
#> 44     NA     NA    NA
#> 45     NA     NA    NA
#> 46     NA     NA    NA
#> 47     NA     NA    NA
#> 48     NA   1.38    NA
#> 49   0.04  -0.09  0.03
#> 50   0.04   1.27  0.06
#> 51     NA     NA    NA
#> 52     NA     NA    NA
#> 53     NA     NA    NA
#> 54     NA     NA    NA
#> 55     NA     NA    NA
#> 56     NA     NA    NA
#> 57     NA     NA    NA
#> 58     NA     NA    NA
#> 59     NA     NA    NA
#> 60     NA     NA    NA
#> 61     NA     NA    NA
#> 62     NA     NA    NA
#> 63     NA     NA    NA
#> 64     NA     NA    NA
#> 65     NA     NA    NA
#> 66     NA     NA    NA
#> 67     NA     NA    NA
#> 68     NA     NA    NA
#> 69     NA     NA    NA
#> 70     NA     NA    NA
#> 71     NA     NA    NA
#> 72     NA     NA    NA
#> 73     NA     NA    NA
#> 74     NA     NA    NA
#> 75     NA     NA    NA
#> 76     NA     NA    NA
#> 77     NA     NA    NA
#> 78   0.05   1.47  0.07
#> 79     NA     NA    NA
#> 80     NA     NA    NA
#> 81     NA     NA    NA
#> 82     NA     NA    NA
#> 83     NA     NA    NA
#> 84     NA     NA    NA
#> 85     NA     NA    NA
#> 86     NA     NA    NA
#> 87     NA     NA    NA
#> 88     NA     NA    NA
#> 89     NA     NA    NA
#> 90     NA     NA    NA
#> 91     NA     NA    NA
#> 92     NA     NA    NA
#> 93     NA     NA    NA
#> 94     NA     NA    NA
#> 95     NA     NA    NA
#> 96     NA     NA    NA
#> 97     NA     NA    NA
#> 98     NA     NA    NA
#> 99     NA     NA    NA
#> 100    NA     NA    NA
#> 101    NA     NA    NA
#> 102    NA     NA    NA
#> 103    NA     NA    NA
#> 104    NA     NA    NA
#> 105    NA     NA    NA
#> 106    NA     NA    NA
#> 107    NA     NA    NA
#> 108    NA     NA    NA
#> 109    NA     NA    NA
#> 110    NA     NA    NA
#> 111    NA     NA    NA
#> 112    NA     NA    NA
#> 113    NA     NA    NA
#> 114    NA     NA    NA
#> 115    NA     NA    NA
#> 116  0.07   1.61  0.10
#>  Group Parameters: 
#>                      mu  sigma2  sigma
#> estimate(Group1)   0.01    0.96   0.98
#> se(Group1)         0.02    0.03   0.02
#> estimate(Group2)   0.51    0.56   0.75
#> se(Group2)         0.02    0.02   0.01
#> estimate(Group3)  -0.36    2.11   1.45
#> se(Group3)         0.03    0.07   0.02
#> 

# Extract the group parameter estimates (i.e., scale parameters)
getirt(fit.6, what = "group.par")
#> $Group1
#>                    mu     sigma2      sigma
#> estimates 0.008398804 0.95861049 0.97908656
#> se        0.021893041 0.03032151 0.01548459
#> 
#> $Group2
#>                  mu     sigma2     sigma
#> estimates 0.5138033 0.55799390 0.7469899
#> se        0.0167032 0.01764973 0.0118139
#> 
#> $Group3
#>                    mu     sigma2      sigma
#> estimates -0.36200936 2.10778116 1.45181995
#> se         0.03246368 0.06667056 0.02296103
#> 

# }