Relating tests given to different samples

Suppose a collection of standard tests is given to all subjects in a random sample, but a different new test is given to each group of subjects in nonoverlapping subsamples. A simple method is developed for displaying the information that the data set contains about the correlational structure of the new tests. This is possible to some extent, even though each subject takes only one new test. The method uses plausible values of the partial correlations among the new tests given the standard tests in order to generate plausible simple correlations among the new tests and plausible multiple correlations between composites of the new tests and the standard tests. The real data example included suggests that the method can be useful in practical problems.     

Estimating the parameters of the latent population distribution

Under consideration is a test battery of binary items. The responses ofn individuals are assumed to follow a Rasch model. It is further assumed that the latent individual parameters are distributed within a given population in accordance with a normal distribution. Methods are then considered for estimating the mean and variance of this latent population distribution. Also considered are methods for checking whether a normal population distribution fits the data. The developed methods are applied to data from an achievement test and from an attitude test.     

The bias function of the maximum likelihood estimate of ability for the dichotomous response level

Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

Multilevel SEM with random slopes in discrete data using the pairwise maximum likelihood

Pairwise maximum likelihood (PML) estimation is a promising method for multilevel models with discrete responses. Multilevel models take into account that units within a cluster tend to be more alike than units from different clusters. The pairwise likelihood is then obtained as the product of bivariate likelihoods for all within-cluster pairs of units and items. In this study, we investigate the PML estimation method with computationally intensive multilevel random intercept and random slope structural equation models (SEM) in discrete data. In pursuing this, we first reconsidered the general ‘wide format’ (WF) approach for SEM models and then extend the WF approach with random slopes. In a small simulation study we the determine accuracy and efficiency of the PML estimation method by varying the sample size (250, 500, 1000, 2000), response scales (two-point, four-point), and data-generating model (mediation model with three random slopes, factor model with one and two random slopes). Overall, results show that the PML estimation method is capable of estimating computationally intensive random intercept and random slopes multilevel models in the SEM framework with discrete data and many (six or more) latent variables with satisfactory accuracy and efficiency. However, the condition with 250 clusters combined with a two-point response scale shows more bias.     

Stepwise variable selection in factor analysis

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain numberp of observed variables (thep-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables isp + 1) and for those by deleting an internal variable (so that the number isp – 1), provided that the number of factors is held constant.A programSEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.The URL of the programSEFA is http://koko15.hus.osaka-u.ac.jp/~harada/factor/stepwise/.     

On the Bock-Aitkin procedure—From an EM algorithm perspective

The relationship between the EM algorithm and the Bock-Aitkin procedure is described with a continuous distribution of ability (latent trait) from an EM-algorithm perspective. Previous work has been restricted to the discrete case from a probit-analysis perspective. The author is grateful to Bradley A. Hanson for valuable discussion and comments. Thanks also go to Terry A. Ackerman, Meichu Fan, Subrata Kundu, and Robert K. Tsutakawa for their help and encouragement in this study.     

On the loss of information in conditional maximum likelihood estimation of item parameters

In item response models of the Rasch type (Fischer & Molenaar, 1995), item parameters are often estimated by the conditional maximum likelihood (CML) method. This paper addresses the loss of information in CML estimation by using the information concept of F-information (Liang, 1983). This concept makes it possible to specify the conditions for no loss of information and to define a quantification of information loss. For the dichotomous Rasch model, the derivations will be given in detail to show the use of the F-information concept for making comparisons for different estimation methods. It is shown that by using CML for item parameter estimation, some information is almost always lost. But compared to JML (joint maximum likelihood) as well as to MML (marginal maximum likelihood) the loss is very small. The reported efficiency in the use of information of CML to JML and to MML in several comparisons is always larger than 93%, and in tests with a length of 20 items or more, larger than 99%.     

The use of item scores and response times to detect examinees who may have benefited from item preknowledge

According to Wollack and Schoenig (2018, The Sage encyclopedia of educational research, measurement, and evaluation. Thousand Oaks, CA: Sage, 260), benefiting from item preknowledge is one of the three broad types of test fraud that occur in educational assessments. We use tools from constrained statistical inference to suggest a new statistic that is based on item scores and response times and can be used to detect examinees who may have benefited from item preknowledge for the case when the set of compromised items is known. The asymptotic distribution of the new statistic under no preknowledge is proved to be a simple mixture of two χ² distributions. We perform a detailed simulation study to show that the Type I error rate of the new statistic is very close to the nominal level and that the power of the new statistic is satisfactory in comparison to that of the existing statistics for detecting item preknowledge based on both item scores and response times. We also include a real data example to demonstrate the usefulness of the suggested statistic.     

Multilevel and Latent Variable Modeling with Composite Links and Exploded Likelihoods

Composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models. Applications considered include survival or duration models, models for rankings, small area estimation with census information, models for ordinal responses, item response models with guessing, randomized response models, unfolding models, latent class models with random effects, multilevel latent class models, models with log-normal latent variables, and zero-inflated Poisson models with random effects. Some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation. We wish to thank The Research Council of Norway for a grant supporting our collaboration.