A Strong True-Score Theory, with Applications

A “strong” mathematical model for the relation between observed scores and true scores is developed. This model can be used 1. To estimate the frequency distribution of observed scores that will result when a given test is lengthened. 2. To equate true scores on two tests by the equipercentile method. 3. To estimate the frequencies in the scatterplot between two parallel (nonparallel) tests of the same psychological trait, using only the information in a (the) marginal distribution(s). 4. To estimate the frequency distribution of a test for a group that has taken only a short form of the test (this is useful for obtaining norms). 5. To estimate the effects of selecting individuals on a fallible measure. 6. To effect matching of groups with respect to true score when only a fallible measure is available. 7. To investigate whether two tests really measure the same psychological function when they have a nonlinear relationship. 8. To describe and evaluate the properties of a specific test considered as a measuring instrument. The model has been tested empirically, using it to estimate bivariate distributions from univariate distributions, with good results, as checked by chi-square tests.