post_hocPCM {tcl}

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(alpha = 0.05, data, x)

Arguments

alpha

Probability of error of first kind.

data

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

x

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

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_{\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_{\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.

Value

A list of results.

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.

observed global deviation

Observed global deviation from hypothesis to be tested represented by a single number. It is obtained by dividing the test statistic by the informative sample size. The latter does not include persons with minimum or maximum person score.

observed local deviation

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

person score distribution in group 1

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

person score distribution in group 2

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

degrees of freedom

Degrees of freedom df.

noncentrality parameter

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

call

The matched call.

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. (2020). The Gradient Test and its Finite Sample Size Properties in a Conditional Maximum Likelihood and Psychometric Modeling Context. Communications in Statistics-Simulation and Computation, 1-19.

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

## Not run: 
# 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(alpha = 0.05, data = y, x = x)

# > 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
#
# $`observed global deviation`
#     W    LR    RS    GR
# 0.045 0.046 0.045 0.047
#
# $`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
#
# $`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
#
# $`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
#
# $`degrees of freedom`
# [1] 7
#
# $`noncentrality parameter`
#      W     LR     RS     GR
# 11.395 11.818 11.628 11.978
#
# $call
# post_hocPCM(alpha = 0.05, data = y, x = x)

## End(Not run)