This function computes the analytical asymptotic variance-covariance matrices of item parameter estimates for dichotomous and polytomous IRT models, without requiring examinee response data. Given a set of item parameter estimates and the corresponding sample sizes, the function derives the matrices using analytical formulas (e.g., Li & Lissitz, 2004; Thissen & Wainer, 1982). The square roots of the diagonal elements (variances) provide the asymptotic standard errors of the maximum likelihood estimates.
covirt(
x,
D = 1,
nstd = 1000,
pcm.loc = NULL,
norm.prior = c(0, 1),
nquad = 41,
weights = NULL
)A named list with the following two components:
cov: A named list of variance-covariance matrices for item parameter estimates.
Each element corresponds to a single item and contains a square matrix whose
dimensions match the number of estimated parameters for that item. For
dichotomous items, this typically includes slopes and intercepts. For
polytomous items, it includes category difficulty parameters (for PCM) or
both slope and difficulty (or threshold) parameters (for GRM and GPCM).
se: A named list of vectors containing the asymptotic standard errors (ASEs)
of the item parameter estimates, computed as the square roots of the diagonal
elements of each corresponding variance-covariance matrix in cov.
The names of the list elements in both cov and se correspond to the item
identifiers (e.g., item names or labels) as given in the first column of the
input x.
A data frame containing item metadata (e.g., item parameters, number
of categories, IRT model types, etc.). See est_irt() or
simdat() for more details about the item metadata. This data frame
can be easily created using the shape_df() function.
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.
An integer or a vector of integers indicating the sample size(s).
If a vector is provided, its length must match the number of items in the
x argument. Default is 1,000. See Details.
A vector of integers indicating the positions of items
calibrated using the partial credit model (PCM). For PCM items, the
variance-covariance matrices are computed only for the item category
difficulty parameters. Default is NULL. See the Details for more
information.
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 when weights = NULL.
Default is c(0, 1).
An integer indicating the number of Gaussian quadrature points
to be generated from the normal prior distribution. The specified value is
used when weights is not NULL. Default is 41.
An optional two-column data frame or matrix where the first column is the quadrature points (nodes) and the second column is the corresponding weights. This is typically used in quadrature-based IRT analysis.
Hwanggyu Lim hglim83@gmail.com
The standard errors obtained from this analytical approach are generally considered lower bounds of the true standard errors (Thissen & Wainer, 1982). Thus, they may serve as useful approximations for evaluating the precision of item parameter estimates when empirical standard errors are not reported in the literature or research reports.
If the item parameters provided in the x argument were calibrated using
different sample sizes, a corresponding vector of sample sizes must be
specified via the nstd argument. For example, suppose you wish to compute
the variance-covariance matrices of five 3PLM items that were calibrated
using 500, 600, 1,000, 2,000, and 700 examinees, respectively. In this case,
set nstd = c(500, 600, 1000, 2000, 700).
Since the item metadata allows only "GPCM" to denote both the partial
credit model (PCM) and the generalized partial credit model (GPCM), PCM items
must be explicitly identified using the pcm.loc argument. This is necessary
because the category difficulty parameters of PCM items require special
handling when computing variance-covariance matrices. For instance, if you
wish to compute the matrices for five polytomous items and the last two were
calibrated using PCM, then specify pcm.loc = c(4, 5).
Li, Y. & Lissitz, R. (2004). Applications of the analytically derived asymptotic standard errors of item response theory item parameter estimates. Journal of educational measurement, 41(2), 85-117.
Thissen, D. & Wainer, H. (1982). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427-450.
est_irt(), simdat(), shape_df(),
gen.weight()
# Example using a "-prm.txt" file exported from flexMIRT
flex_prm <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
# Select the first two dichotomous items and the last polytomous item
x <- bring.flexmirt(file = flex_prm, "par")$Group1$full_df[c(1:2, 55), ]
# Compute the variance-covariance matrices assuming a sample size of 2,000
covirt(x, D = 1, nstd = 2000, norm.prior = c(0, 1), nquad = 41)
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