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Internal measures of differential functioning of items and tests (DHFIT) based on item response theory (IRT) are proposed. Within the DFIT context, the new differential test functioning (DTF) index leads to two new measures of differential item functioning (DIF) with the following properties: (1) The compensatory DIF (CDIF) indexes for all items in a test sum to the DTF index for that test and, unlike current DIF procedures, the CDIF index for an item does not assume that the other items in the test are unbi ased ; (2) the noncompensatory DIF (NCDIF) index, which assumes that the other items in the test are unbiased, is comparable to some of the IRT-based DIP indexes; and (3) COIF and NCDIF, as well as DTF, are equally valid for polytomous and multidimensional IRT models. Monte carlo study results, comparing these indexes with Lord's χ2 test, the signed area measure, and the unsigned area measure, demonstrate that the DFIT framework is accu rate in assessing DTF, COIF, and NCDIF.

A lognormal model for the response times of a person on a set of test items is investigated. The model has a parameter structure analogous to the twoparameter logistic response models in item response theory, with a parameter for the speed of each person as well as parameters for the time intensity and discriminating power of each item. It is shown how these parameters can be estimated by a Markov chain Monte Carlo method (Gibbs sampler). The method was used to analyze response times for the adaptive version of a test from the Armed Services Vocational Aptitude Battery. The same data set was used to test the validity of the model against a normal model using posterior predictive checks on the response times. The lognormal model showed an excellent fit to the data, whereas the normal model seemed unable to allow for a characteristic skewness of the response time distributions. The addition of an equality constraint on the discrimination parameters led only to a slight loss of fit. The potential use of the model for improving the daily practice of testing is indicated.

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