Models for residual dependencies
The models discussed in the previous chapters recognize the clustered structure of data one is confronted with most often in psychometrics (i.e., items within persons). The within-person dependencies arising from this clustering are handled through a random effect or latent variable for person p, denoted as θ p . In some cases, there are several major sources of individual differences, and they have to be accounted for by more than one random effect (see Chapter 8 on multidimensionality). Conditional on these random effects, the responses to the different items in the data set should be independent — this requirement is called conditional independence or local (stochastic) independence. However, it appears that in many applications, not all dependence between the responses can be explained by the random effects one assumed to underly the responses. In those cases, it is said that there remain some residual dependencies not accounted for by the model, a phenomenon also denoted as local item dependencies (LIDs). Situations in which residual dependencies may occur are ample. Consider for instance the case where items of a reading test can be subdivided into groups of items each sharing the same reading passage. Data from a test with reading passages may show more dependencies than can be accounted for alone by a single reading ability dimension.
KeywordsProbit Model Conditional Independence Item Parameter Recursive Model Polytomous Item
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