LOCATING THE STRUCTURAL ZEROS FOR INTERNAL ANCHOR TESTS: INCLUDING THE CASE OF ROUNDED FORMULA SCORES

Abstract

There are test-equating situations in which it may be appropriate to fit a loglinear or other type of probability model to the joint distribution of a total score on a test and a score on part of that test. For anchor test designs, this situation arises for internal anchor tests, which are embedded within the total test. Similarly, a part-whole relationship arises between two scores when a few test items are dropped from a test and a single group design is used to equate the scores of the full test to the part that remains after those items are deleted. In these part-whole situations, the resulting bivariate frequency distribution will exhibit structural zeros due to the fact that some scores on the total test are impossible for specific values on the partial test. Without knowing where the structural zeros are, it is impossible to distinguish them from zero frequencies that are simply due to size of the sample (i.e., sampling zeros). When probability models are estimated for these joint distributions, it is usual to require the models to assign positive probability to the sampling zeros but to avoid assigning positive probability to the structural zeros. To do this, it is important to be able to locate where the structural zeros are in the bivariate distribution. When the scores on the tests are consecutive integers, it is easy to determine the location of the structural zeros. This report gives a solution to the problem of locating the structural zeros that arise for a class of bivariate distributions that includes both number-right scores and formula scores that have been rounded to integer values. The result for rounded formula scores is a simple alteration of the case where all the scores are consecutive integers.