tcl (version 1.0.1)

post_hocPCM: Power analysis of tests of invariance of item parameters between two groups of persons in partial credit model

Description

Returns post hoc power of Wald (W), likelihood ratio (LR), Rao score (RS) and gradient (GR) test given data and probability of error of first kind \(\alpha\). The hypothesis to be tested assumes equal item-category parameters of the partial credit model between two predetermined groups of persons. The alternative states that at least one of the parameters differs between the two groups.

Usage

post_hocPCM(data, splitcr, alpha = 0.05)

Value

A list of results of class tcl_post_hoc.

test

A numeric vector of Wald (W), likelihood ratio (LR), Rao score (RS), and gradient (GR) test statistics.

power

Post hoc power value for each test.

dev_global

Observed global deviation from the hypothesis to be tested, represented by a single number. It is obtained by dividing the test statistic by the informative sample size, which excludes persons with minimum or maximum person scores.

dev_local

CML estimates of free item-category parameters in both groups of persons, representing observed deviation from the hypothesis to be tested locally per item and response category.

score_dist_group1

Relative frequencies of person scores in group 1. Uninformative scores, i.e., minimum and maximum scores, are omitted. Note that the person score distribution also influences the power of the tests.

score_dist_group2

Relative frequencies of person scores in group 2. Uninformative scores, i.e., minimum and maximum scores, are omitted. Note that the person score distribution also influences the power of the tests.

df

Degrees of freedom \(df\).

ncp

Noncentrality parameter \(\lambda\) of the \(\chi^2\) distribution from which power is determined. It equals the observed value of the test statistic.

call

The matched call.

Arguments

data

Data matrix with item responses (in ordered categories starting from 0).

splitcr

A numeric vector of length equal to number of persons that contains zeros and ones indicating group membership of the persons.

alpha

Probability of error of first kind.

Details

The power of the tests (Wald, LR, score, and gradient) is determined from the assumption that the approximate distributions of the four test statistics are from the family of noncentral \(\chi^2\) distributions with \(df\) equal to the number of free item-category parameters in the partial credit model and noncentrality parameter \(\lambda\). In case of evaluating the post hoc power, \(\lambda\) is assumed to be given by the observed value of the test statistic. Given the probability of the error of the first kind \(\alpha\) the post hoc power of the tests can be determined from \(\lambda\). More details about the distributions of the test statistics and the relationship between \(\lambda\), power, and sample size can be found in Draxler and Alexandrowicz (2015).

In particular, let \(q_{1- \alpha}\) be the \(1- \alpha\) quantile of the central \(\chi^2\) distribution with \(df\) equal to the number of free item-category parameters. Then,

$$power = 1 - F_{df, \lambda} (q_{1- \alpha}),$$

where \(F_{df, \lambda}\) is the cumulative distribution function of the noncentral \(\chi^2\) distribution with \(df\) equal to the number of free item-category parameters and \(\lambda\) equal to the observed value of the test statistic.

References

Draxler, C. (2010). Sample Size Determination for Rasch Model Tests. Psychometrika, 75(4), 708–724.Draxler, C., & Alexandrowicz, R. W. (2015). Sample Size Determination Within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model. Psychometrika, 80(4), 897–919.Draxler, C., Kurz, A., & Lemonte, A. J. (2022). The gradient test and its finite sample size properties in a conditional maximum likelihood and psychometric modeling context. Communications in Statistics-Simulation and Computation, 51(6), 3185-3203.Glas, C. A. W., & Verhelst, N. D. (1995a). Testing the Rasch Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch Models: Foundations, Recent Developments, and Applications (pp. 69–95). New York: Springer.Glas, C. A. W., & Verhelst, N. D. (1995b). Tests of Fit for Polytomous Rasch Models. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch Models: Foundations, Recent Developments, and Applications (pp. 325-352). New York: Springer.

See Also

sa_sizePCM, and powerPCM.

Examples

if (FALSE) {
# Numerical example for post hoc power analysis for PCM

y <- eRm::pcmdat2
n <- nrow(y) # sample size
x <- c( rep(0,n/2), rep(1,n/2) ) # binary covariate

res <- post_hocPCM(data = y, splitcr = x, alpha = 0.05)

# > res
# $test
#      W     LR     RS     GR
# 11.395 11.818 11.628 11.978
#
# $power
#     W    LR    RS    GR
# 0.683 0.702 0.694 0.709
#
# $dev_global #`observed global deviation`
#     W    LR    RS    GR
# 0.045 0.046 0.045 0.047
#
# $ dev_local #`observed local deviation`
#        I1-C2 I2-C1 I2-C2  I3-C1  I3-C2  I4-C1  I4-C2
# group1 2.556 0.503 2.573 -2.573 -2.160 -1.272 -0.683
# group2 2.246 0.878 3.135 -1.852 -0.824 -0.494  0.941
#
# $score_dist_group1 #`person score distribution in group 1`
#
#     1     2     3     4     5     6     7
# 0.016 0.097 0.137 0.347 0.121 0.169 0.113
#
# $score_dist_group2 #`person score distribution in group 2`
#
#     1     2     3     4     5     6     7
# 0.015 0.083 0.136 0.280 0.152 0.227 0.106
#
# $df #`degrees of freedom`
# [1] 7
#
# $ncp #`noncentrality parameter`
#      W     LR     RS     GR
# 11.395 11.818 11.628 11.978
#
# $call
# post_hocPCM(alpha = 0.05, data = y, x = x)
}