The paper identifies and addresses four methodological weaknesses common to most previous studies that have used LISREL confirmatory factor analysis to test for the factorial validity and invariance of a single measuring instrument. Specifically, the paper demonstrates the steps involved in: (1) conducting sensitivity analyses to determine a statistically best-fitting, yet substantively most meaningful baseline model; (2) testing for partial measurement invariance; (3) testing for the invariance of factor variances and covariances, given partial measurement invariance; and (4) testing for the invariance of test item and subscale reliabilities. These procedures are illustrated with item response data from the Perceived Competence Scale for Children from 129 normal and 132 gifted students in grade 5 and 113 normal and 117 gifted students in grade 8 from two public school systems in Ottawa (Ontario). Seven tables present study data. (Author/SLD)