This paper studies whether equating results can be improved if the variable that accounts for all systematic differences between equating populations is identified and used as an anchor in anchor test design or as a variable on which to match equating samples. The sample invariant properties of four anchor test equating methods (Tucker and Levine equally reliable linear models, chained equipercentile, and frequency estimation equipercentile models) were examined under representative, matched-on-equating-test, and matched-on-selection-variable conditions. The selection variable, the variable along which subpopulations differ, was also used as an anchor for the four equating methods and compared to equatings in which the equating test served as the anchor. All equatings were performed with real Scholastic Aptitude Test (SAT) populations or simulated populations. Accuracy criteria were equivalent-groups equipercentile equatings based on old and new form subpopulations of over 115,000 test takers. Results showed that matching on the selection variable improved accuracy over matching on the equating test for all methods. Results with the selection variable as an anchor were good for both the Tucker and frequency estimation methods, but unacceptable for chained equipercentile and Levine results. Two tables, 2 figures, and 16 graphs present analysis data. (Contains 24 references.) (Author/SLD)