The purpose of this study was to investigate population invariance of the true-score linking functions with respect to the ability subgroups (i.e., average-achieving students, low-achieving students, and students with learning disabilities). The mean/mean linking functions for five alternate forms of a curriculum-based math problem solving measure were based on the Rasch model. Most studies of curriculum-based measurement have reported only the reliability and validity of alternate forms of measures. This is necessary but insufficient for establishing alternate forms of curriculum-based measures. It is also necessary to establish equivalency of the forms. The present study was based on data from a previous study that developed equivalent forms of curriculum-based measures using Item Response Theory. The participants in the present study were 1,861 seventh- and eighth-grade students. Equatability indices were used to evaluate population invariance of the Rasch mean/mean linking functions over the ability subgroups. Results indicated that the Rasch mean/mean linking functions were population invariant for the ability subgroups across the five alternate forms. The differences between the linking functions computed on the ability subgroups and the linking function on the whole group were negligible for the five forms. Several implications and recommendations for future studies on population invariance of the linking functions with alternate forms of curriculum-based measures were discussed.

This dissertation investigates the interaction of population invariance, equating assumptions, and equating accuracy with group differences. In addition, matched samples equating methods are considered as a possible way to improve equating accuracy with large group differences. Data from one administration of four mixed-format Advanced Placement (AP) Exams were used to create pseudo old and new forms sharing common items. Population invariance analyses were conducted based on levels of examinee parental education using a single group equating design. Old and new form groups with common item effect sizes (ESs) ranging from 0 to 0.75 were created by sampling examinees based on their level of parental education. Equating was conducted for four common item nonequivalent group design equating methods: frequency estimation, chained equipercentile, IRT true score, and IRT observed score. Additionally, groups with ESs greater than zero were matched using three different matching techniques including exact matching on parental education level and propensity score matching with several other background variables. The accuracy of equating results was evaluated by comparing each equating relationship with an ES greater than zero to the equating relationship where the ES equaled zero. Differences between comparison and criterion equating relationships were quantified using the root expected mean squared difference (REMSD) statistic, classification consistency, and standard errors of equating (SEs). The accuracy of equating results and the adequacy of equating assumptions was compared for unmatched and matched samples. As ES increased, equating results tended to become less accurate and less consistent across equating methods. However, there was relatively little population dependence of equating results, despite large subgroup performance differences. Large differences between criterion and comparison equating relationships appeared to be caused instead by violations of equating assumptions. As group differences increased, the degree to which frequency estimation and chained equipercentile assumptions held decreased. In addition, all four AP Exams showed some evidence of multidimensionality. Because old and new form groups were selected to differ in terms of their respective levels of parental education, the matching methods that included parental education appeared to improve equating accuracy and the degree to which equating assumptions held, at least for very large ESs.