Residual-based Item Parameter Drift (RIPD) Detection Framework

This function computes three RIPD statistics—\(RIPD_{R}\), \(RIPD_{S}\), and \(RIPD_{RS}\)—for each item. \(RIPD_{R}\) captures differences in mean raw residuals between groups, which is typically indicative of uniform item parameter drift (IPD). \(RIPD_{S}\) captures differences in mean squared residuals between groups, reflecting nonuniform IPD. \(RIPD_{RS}\), a combined chi-square-based index, is sensitive to both uniform and nonuniform IPD.

Usage

ripd(x, ...)

# Default S3 method
ripd(
  x,
  data,
  score = NULL,
  group,
  focal.name,
  item.skip = NULL,
  D = 1,
  alpha = 0.05,
  missing = NA,
  purify = FALSE,
  purify.by = c("ripdrs", "ripdr", "ripds"),
  max.iter = 10,
  min.resp = NULL,
  method = "ML",
  range = c(-5, 5),
  norm.prior = c(0, 1),
  nquad = 41,
  weights = NULL,
  ncore = 1,
  verbose = TRUE,
  ...
)

# S3 method for class 'est_irt'
ripd(
  x,
  score = NULL,
  group,
  focal.name,
  item.skip = NULL,
  alpha = 0.05,
  missing = NA,
  purify = FALSE,
  purify.by = c("ripdrs", "ripdr", "ripds"),
  max.iter = 10,
  min.resp = NULL,
  method = "ML",
  range = c(-5, 5),
  norm.prior = c(0, 1),
  nquad = 41,
  weights = NULL,
  ncore = 1,
  verbose = TRUE,
  ...
)

# S3 method for class 'est_item'
ripd(
  x,
  group,
  focal.name,
  item.skip = NULL,
  alpha = 0.05,
  missing = NA,
  purify = FALSE,
  purify.by = c("ripdrs", "ripdr", "ripds"),
  max.iter = 10,
  min.resp = NULL,
  method = "ML",
  range = c(-5, 5),
  norm.prior = c(0, 1),
  nquad = 41,
  weights = NULL,
  ncore = 1,
  verbose = TRUE,
  ...
)

Arguments

x

A data frame containing item metadata (e.g., item parameters, number of categories, IRT model types, etc.); or an object of class est_irt obtained from est_irt(), or est_item from est_item().

See est_irt() or simdat() for more details about the item metadata. This data frame can be easily created using the shape_df() function.

...

Additional arguments passed to the est_score() function.

data

A matrix of examinees' item responses corresponding to the items specified in the x argument. Rows represent examinees and columns represent items.

score

A numeric vector containing examinees' ability estimates (theta values). If not provided, ripd() will estimate ability parameters internally before computing the RIPD statistics. See est_score() for more information on scoring methods. Default is NULL.

group

A numeric or character vector indicating group membership of examinees. The length of vector should the same with the number of rows in the response data matrix.

focal.name

A single numeric or character value specifying the focal group. For instance, given group = c(0, 1, 0, 1, 1) and '1' indicating the focal group, set focal.name = 1.

item.skip

A numeric vector of item indices to exclude from IPD analysis. If NULL, all items are included. Useful for omitting specific items based on prior insights.

D

A scaling constant used in IRT models to make the logistic function closely approximate the normal ogive function. A value of 1.7 is commonly used for this purpose. Default is 1.

alpha

A numeric value specifying the significance level (\(\alpha\)) for hypothesis testing using the RIPD statistics. Default is 0.05.

missing

A value indicating missing values in the response data set. Default is NA.

purify

Logical. Indicates whether to apply a purification procedure. Default is FALSE.

purify.by

A character string specifying which RIPD statistic is used to perform the purification. Available options are "ripdrs" for \(RIPD_{RS}\), "ripdr" for \(RIPD_{R}\), and "ripds" for \(RIPD_{S}\).

max.iter

A positive integer specifying the maximum number of iterations allowed for the purification process. Default is 10.

min.resp

A positive integer specifying the minimum number of valid item responses required from an examinee in order to compute an ability estimate. Default is NULL.

method

A character string indicating the scoring method to use. Available options are:

"ML": Maximum likelihood estimation
"WL": Weighted likelihood estimation (Warm, 1989)
"MAP": Maximum a posteriori estimation (Hambleton et al., 1991)
"EAP": Expected a posteriori estimation (Bock & Mislevy, 1982)

Default is "ML".

range

A numeric vector of length two specifying the lower and upper bounds of the ability scale. This is used for the following scoring methods: "ML", "WL", and "MAP". Default is c(-5, 5).

norm.prior

A numeric vector of length two specifying the mean and standard deviation of the normal prior distribution. These values are used to generate the Gaussian quadrature points and weights. Ignored if method is "ML" or "WL". Default is c(0, 1).

nquad

An integer indicating the number of Gaussian quadrature points to be generated from the normal prior distribution. Used only when method is "EAP". Ignored for "ML", "WL", and "MAP". Default is 41.

weights

A two-column matrix or data frame containing the quadrature points (in the first column) and their corresponding weights (in the second column) for the latent variable prior distribution. The weights and points can be conveniently generated using the function gen.weight().

If NULL and method = "EAP", default quadrature values are generated based on the norm.prior and nquad arguments. Ignored if method is "ML", "WL", or "MAP".

ncore

An integer specifying the number of logical CPU cores to use for parallel processing. Default is 1. See est_score() for details.

verbose

Logical. If TRUE, progress messages from the purification procedure will be displayed; if FALSE, the messages will be suppressed. Default is TRUE.

Value

This function returns a list containing four main components:

no_purify

A list of sub-objects containing the results of IPD analysis without applying a purification procedure. The sub-objects include:

ipd_stat: A data frame summarizing the RIPD analysis results for all items. The columns include: item ID, \(RIPD_{R}\) statistic, standardized \(RIPD_{R}\), \(RIPD_{S}\) statistic, standardized \(RIPD_{S}\), \(RIPD_{RS}\) statistic, p-values for \(RIPD_{R}\), \(RIPD_{S}\), and \(RIPD_{RS}\), sample sizes for the reference and focal groups, and total sample size. Note that \(RIPD_{RS}\) does not have a standardized value because it is a \(\chi^{2}\)-based statistic.
moments: A data frame reporting the first and second moments of the RIPD statistics. The columns include: item ID, mean and standard deviation of \(RIPD_{R}\), mean and standard deviation of \(RIPD_{S}\), and the covariance between \(RIPD_{R}\) and \(RIPD_{S}\).
ipd_item: A list of three numeric vectors identifying items flagged as drifting by each RIPD statistic: \(RIPD_{R}\), \(RIPD_{S}\), and \(RIPD_{RS}\).
score: A numeric vector of ability estimates used to compute the RIPD statistics.

purify

A logical value indicating whether the purification procedure was applied.

with_purify

A list of sub-objects containing the results of IPD analysis with a purification procedure. The sub-objects include:

purify.by: A character string indicating the RIPD statistic used for purification. Possible values are "ripdr", "ripds", and "ripdrs", corresponding to \(RIPD_{R}\), \(RIPD_{S}\), and \(RIPD_{RS}\), respectively.
ipd_stat: A data frame reporting the RIPD analysis results for all items from the final iteration. Same structure as in no_purify, with one additional column indicating the iteration number in which each result was obtained.
moments: A data frame reporting the moments of RIPD statistics from the final iteration. Includes the same columns as in no_purify, with an additional column for the iteration number.
ipd_item: A list of numeric item indices identified as IPD items across iterations.
n.iter: An integer indicating the total number of iterations performed during the purification process.
score: A numeric vector of purified ability estimates used to compute the final RIPD statistics.
complete: A logical value indicating whether the purification process converged. If FALSE, the maximum number of iterations was reached without convergence.

alpha

A numeric value indicating the significance level (\(\alpha\)) used in hypothesis testing for RIPD statistics.

Methods (by class)

ripd(default): Default method for computing the three RIPD statistics using a data frame x that contains item metadata
ripd(est_irt): An object created by the function est_irt().
ripd(est_item): An object created by the function est_item().

References

Lim, H., Choe, E. M., & Han, K. T. (2022). A residual-based differential item functioning detection framework in item response theory. Journal of Educational Measurement, 59(1), 80-104. doi:10.1111/jedm.12313 .

Lim, H., & H, K. T. (2025, April). IRT residual-based approach to detecting item parameter drift in CAT. Paper presented at the annual conference of the National Council on Measurement in Education (NCME), Denver, CO

Author

Hwanggyu Lim hglim83@gmail.com