The Mantel‐Haenszel procedure is a noniterative contingency table method for estimating and testing a common two‐factor association parameter in a 2×2×k table. As such it may be used to study “item bias” or differential item functioning in two groups of examinees. This technique is discussed in this context and compared to other related techniques as well as to item response theory methods.

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factor analysis, EM algorithms, maximum likelihood,

We developed an empirical Bayes (EB) enhancement to Mantel‐Haenszel (MH) DIF analysis in which we assume that the MH statistics are normally distributed and that the prior distribution of underlying DIF parameters is also normal. We use the posterior distribution of DIF parameters to make inferences about the item's true DIF status and the posterior predictive distribution to predict the item's future observed status. DIF status is expressed in terms of the probabilities associated with each of the five DIF levels defined by the ETS classification system: C–, B–, A, B+, and C+. The EB methods yield more stable DIF estimates than do conventional methods, especially in small samples, which is advantageous in computer‐adaptive testing. The EB approach may also convey information about DIF stability in a more useful way by representing the state of knowledge about an item's DIF status as probabilistic.

The theory of log‐linear models is developed for multinomially distributed data – i.e., frequency distributions and discrete multivariate distributions. Such models have applications to data smoothing and can be used in test equating. Attention is given to the computations used to find maximum likelihood estimates using Newton's method and to the computation of asymptotic standard errors for the fitted frequencies. Two examples, using real data, are used to illustrate the output of a computer program that implements these ideas.

The well-developed theory of exponential families of distributions is applied to the problem of fitting the univariate histograms and discrete bivariate frequency distributions that often arise in the analysis of test scores. These models are powerful tools for many forms of parametric data smoothing and are particularly well-suited to problems in which there is little or no theory to guide a choice of probability models, e.g., smoothing a distribution to eliminate roughness and zero frequencies in order to equate scores from different tests. Attention is given to efficient computation of the maximum likelihood estimates of the parameters using Newton's Method and to computationally efficient methods for obtaining the asymptotic standard errors of the fitted frequencies and proportions. We discuss tools that can be used to diagnose the quality of the fitted frequencies for both the univariate and the bivariate cases. Five examples, using real data, are used to illustrate the methods of this paper

The Norz-Equivalent-groups Anchor Test (NEAT) design has been in wide use since at least the early 1940s. It involves two populations of test takers, P and Q, and makes use of an anchor test to link them. Two linking methods used for NEAT designs ure those ( a ) based on chain quating and (b) that use the anchor test to post-strati& the distributions of the two operational test scores to a common population (i.e., Tucker equating and frequency estimation). We show that, under digwent sets of assumptions, both methods are observed score equuting methods and we give conditions under which the methods give identical results. In addition, we develop analogues of the Doratis and Holland (2000) RMSD measures of population invariance of equating methods for the NEAT design,for both chain and Vost-stratification equating methods.

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