As we noted in Chapter 10, the standard mathematical representation of a positional analysis frequently uses blockmodels to describe and study the equivalence classes (or positions) determined by a set of measured relations. Recall that a blockmodel consists of a partition of the actors in N into positions and a statement of how the positions relate to each other. The adequacy of this construct can be studied with the methods presented in this chapter. We have also noted that these representations of equivalences can also be found by using hierarchical clustering and multidimensional scaling.

A researcher must determine how well a blockmodel, or another mathematical representation of the positions among the actors, “fits” a given network data set. Such tasks are usually called goodness-of-fit problems in statistics, and we will present several goodness-of-fit indices here, all of which are designed to measure the fit of a blockmodel to a given network data set.

There have been two main approaches to this goodness-of-fit task in the literature. The first uses a standard data analytic technique of comparing the observed data set (in this case, the R sociomatrices X1, X2, …, XR) to the predicted data set, which is based on the blockmodel to be evaluated. A number of measures for this comparison have been presented in the literature; here, we discuss several of them. Unfortunately, there is little consensus or agreement on such statistics.