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## Contributions: Abstract## Graphical measurement models based on assumptions whose consequences are directly observable
Because behavioral science measurements are often quite fallible, tests consist of a number of separate items whose responses are combined to obtain a more reliable test score. This, of course, only makes sense if all items measure the same attribute, which implies that their response probabilities should satisfy certain requirements embodied in a, say, common attribute criterion (CAC). To date, a CAC is formulated as a statistical model that explicitly contains a latent variable. However, because the latent variable is unobserved, the precise nature of its relations with its indicators and other variables cannot be determined on empirical grounds. So, one needs a theoretical justification. However, unlike Physics, where theory is almost always cast in mathematical form, the behavioral sciences rarely have such theories. In this paper we explore alternative CACs derived from the manifest conditions such as conditional independence and DeFinetti exchangeability. Measurement models embodying these CACs are formulated in terms of log-linear graphical independence-models that describe the joint distribution of the item responses and the variables of the nomological network around the construct to be measured. It is shown that under certain assumptions, the models may be used to describe metric item scores that may be estimated from the data. The model is a graphical version of Goodman's row-column association model which, in turn, is equivalent to Bock's nominal response model. It is shown that the category scores of the row-column model are proportional to the category discrimination parameters in Bock's model. A particularly interesting feature of the graphical version of that model is that it becomes unnecessary to specify a parametric distribution for the latent underlying variable to identify the model parameters. As a result, measurement models may safely be calibrated using data from populations with unknown latent variable distribution. |