Derivations of Observed Score Equating Methods That Cater to Populations Differing in Ability

For anchor test equating, three linear observed score methods are derived for populations selected to differ in ability. Version A uses the slope, intercept, and standard error of estimate assumptions of selection theory. Version B uses the slope and intercept assumptions and assumes that the equated tests (X, Y) are equally reliable and congeneric. Both employ the synthetic population. Version C requires similar assumptions to Version A but does not use the synthetic population. Each version requires that the correlations of the tests with the selection variable be known. Five further sets of assumptions are made for each model, yielding 15 methods, which are then related to existing equating methods in the literature.