R. Ligtvoet

Investigating an Invariant Item Ordering for Polytomously Scored Items.

This article discusses the concept of an invariant item ordering (IIO) for polytomously scored items and proposes methods for investigating an IIO in real test data. Method manifest IIO is proposed for assessing whether item response functions intersect. Coefficient HT is defined for polytomously scored items. Given that an IIO holds, coefficient HT expresses the accuracy of the item ordering. Method manifest IIO and coefficient HT are used together to analyze a real data set. Topics for future research are discussed.

Measurement bias detection through factor analysis

Measurement bias is defined as a violation of measurement invariance, which can be investigated through multigroup factor analysis (MGFA), by testing across-group differences in intercepts (uniform bias) and factor loadings (nonuniform bias). Restricted factor analysis (RFA) can also be used to detect measurement bias. To also enable nonuniform bias detection, we extend RFA with latent moderated structures (LMS) or use a random slope parameterization (RSP). In a simulation study we compare the MGFA, RFA/LMS, and RFA/RSP methods in detecting measurement bias, varying type of bias (uniform, nonuniform), type of the variable that violates measurement invariance (dichotomous, continuous), and its relationship with the trait that we want to measure (independent, dependent). For each condition, 500 sets of data are generated and analyzed with each of the three detection methods, in single run and in an iterative procedure. The RFA methods outperform MGFA when the violating variable is continuous.

An Isotonic Partial Credit Model for Ordering Subjects on the Basis of Their Sum Scores

In practice, the sum of the item scores is often used as a basis for comparing subjects. For items that have more than two ordered score categories, only the partial credit model (PCM) and special cases of this model imply that the subjects are stochastically ordered on the common latent variable. However, the PCM is very restrictive with respect to the constraints that it imposes on the data. In this paper, sufficient conditions for the stochastic ordering of subjects by their sum score are obtained. These conditions define the isotonic (nonparametric) PCM model. The isotonic PCM is more flexible than the PCM, which makes it useful for a wider variety of tests. Also, observable properties of the isotonic PCM are derived in the form of inequality constraints. It is shown how to obtain estimates of the score distribution under these constraints by using the Gibbs sampling algorithm. A small simulation study shows that the Bayesian p-values based on the log-likelihood ratio statistic can be used to assess the fit of the isotonic PCM to the data, where model-data fit can be taken as a justification of the use of the sum score to order subjects.

Latent class models for testing monotonicity and invariant item ordering for polytomous items

Two assumptions that are relevant to many applications using item response theory are the assumptions of monotonicity (M) and invariant item ordering (IIO). A latent class model is proposed for ordinal items with inequality constraints on the class-specific item means. This model is used as a tool for testing for violations of M and IIO. A Gibbs sampling scheme is used for estimating the model parameters. It is shown that the deviance information criterion can be used as an overall test of M and IIO, while posterior predictive checks can be used to test these assumptions at the item level. A real data application illustrates a model-fitting strategy for detecting items that violate M and IIO.

Effect of a school library on the reading attitude and reading behaviour in non-western migrant students

There is a lack of clarity as to the effects of school libraries on children with a non-western background in the Netherlands, an educationally disadvantaged group. Using a longitudinal design involving an experimental and a control school, the present study examined whether an integrated library facility in a Dutch primary school has an effect on the reading attitude and reading behaviour of non-western migrant students (n = 140). The results showed no statistically significant effect on the degree in which students think reading is fun. On the other hand, over time, students attending the experimental school considered reading more useful than students visiting the control school. With regard to reading behaviour, no statistically significant effect of the school library was found. However, the school library programme was not implemented in the most optimal form, which may have affected the findings. Reading climate at home was found to be an important predictor of both reading attitude and reading behaviour, stressing the importance of parents as partners for school libraries when it comes to reading promotion.

Model fit after pairwise maximum likelihood

Maximum likelihood factor analysis of discrete data within the structural equation modeling framework rests on the assumption that the observed discrete responses are manifestations of underlying continuous scores that are normally distributed. As maximizing the likelihood of multivariate response patterns is computationally very intensive, the sum of the log–likelihoods of the bivariate response patterns is maximized instead. Little is yet known about how to assess model fit when the analysis is based on such a pairwise maximum likelihood (PML) of two–way contingency tables. We propose new fit criteria for the PML method and conduct a simulation study to evaluate their performance in model selection. With large sample sizes (500 or more), PML performs as well the robust weighted least squares analysis of polychoric correlations.

Remarks and a Correction of Ligtvoet’s Treatment of the Isotonic Partial Credit Model

This note contains some remarks on Ligtvoet’s (Psychometrika, 77:479–494, 2012) treatment of the isotonic partial credit model. The Proposition relating to the observable property MWI is shown to be false.

A Test for Ordinal Measurement Invariance

One problem with the analysis of measurement invariance is the reliance of the analysis on having a parametric model that accurately describes the data. In this paper an ordinal version of the property of measurement invariance is proposed, which relies only on nonparametric restrictions. This property of ordinal measurement invariance provides a coarse (initial) indication of measurement invariance, based on the sum scores. A small example is given to illustrate the procedure for testing the property of ordinal measurement invariance.

A test for using the sum score to obtain a stochastic ordering of subjects

For many psychological test applications, the simple sum score across the items is used to make inferences about subjects. However, most of the item response theory models for psychological test data do not support such usage of the sum score. A simple test is proposed to assess whether the sum score can be used to obtain a stochastic ordering of subjects. This test is based on general (nonparametric) conditions and requires only the estimation of the unconstrained proportions.