This report builds on and extends existent research on population invariance to new tests and issues. The authors lay the foundation for a deeper understanding of the use of population invariance measures in a wide variety of practical contexts. The invariance of linear, equipercentile and IRT equating methods are examined using data from five testing programs--AP, ACT, the College-Level Examination Program® (CLEP®), LSAT, and the College Board's SAT® (SAT). The five papers in this report address a variety of issues. The SAT paper examines the role of the anchor test in achieving population invariance of linear equatings across male and female subpopulations and test administrations. The AP paper examines IRT models applied to exams with both multiple-choice and constructed-response components. The CLEP paper investigates population invariance of the 1-parameter IRT model applied to testlet-based computerized exams. The LSAT paper extends the application of population invariance methods to subpopulations defined by geographic region, whether examinees applied to law school, and their law school admission status. Finally, the ACT paper examines the population invariance of a science test across different ability groups using IRT true and observed score equating as well as equipercentile equating methods. The papers include: (1) Population Invariance of IRT True Score Equating (Alina A. von Davier and Christine Wilson); (2) Exploring the Population Sensitivity of Linking Functions across Test Administrations Using LSAT Subpopulations (Mei Liu and Paul W. Holland); (3) Invariance of Score Linkings Across Gender Groups for Forms of a Testlet-Based CLEP® Examination (Wen-Ling Yang and Rui Gao); (4) Invariance of Equating Functions across Different Subgroups of Examinees Taking a Science Achievement Test (Qing Yi, Deborah J. Harris, and Xiaohong Gao); and (5) The Role of the Anchor Test in Achieving Population Invariance across Subpopulations and Test Administrations (Neil J. Dorans, Jinghua Liu, and Shelby Hammond). The report concludes with: (1) A Discussion of Population Invariance of Equating (Nancy S. Petersen); and (2) A Discussion of Population Invariance (Robert L. Brennan). Individual papers contain references.