Testing for the equivalence of factor covariance and mean structures: The issue of partial measurement invariance.

TLDR

Abstract

Addresses issues related to partial measurement in variance using a tutorial approach based on the LISREL confirmatory factor analytic model.Specifically, we demonstrate procedures for (a) using "sensitivity analyses" to establish stable and substantively well-fitting baseline models, (b) determining partially invariant measurement parameters, and (c) testing for the invariance of factor covariance and mean structures, given partial measurement invariance.We also show, explicitly, the transformation of parameters from an all-^fto an all-y model specification, for purposes of testing mean structures.These procedures are illustrated with multidimensional self-concept data from low (« = 248) and high (n = 582) academically tracked high school adolescents.An important assumption in testing for mean differences is that the measurement (Drasgow & Kanfer, 1985;Labouvie, 1980; Rock, Werts, & Haugher, 1978) and the structure (Bejar, 1980;Labouvie, 1980; Rock etal., 1978) of the underlying construct are equivalent across groups.One methodological strategy used in testing for this equivalence is the analysis of covariance structures using the LISREL confirmatory factor analytic (CFA) model (Joreskog, 1971).Although a number of empirical investigations and didactic expositions have used this methodology in testing assumptions of factorial invariance for multiple and single parameters, the analyses have been somewhat incomplete.In particular, researchers have not considered the possibility of partial measurement invariance.The primary purpose of this article is to demonstrate the application of CFA in testing for, and with, partial measurement invariance.Specifically, we illustrate (a) testing, independently, for the invariance of factor loading (i.e., measurement) parameters, (b) testing for the invariance of factor variance-covariance (i.e., structural) parameters, given partially invariant factor loadings, and (c) testing for the invariance of factor mean structures. 1 Invariance testing across groups, however, assumes wellfitting single-group models; the problem here is to know when to stop fitting the model.A secondary aim of this article, then, is to demonstrate "sensitivity analyses" that can be used to establish stable and substantively meaningful baseline models. Factorial InvarianceQuestions of factorial invariance focus on the correspondence of factors across different groups in the same study, in separate studies, or in subgroups of the same sample (cf.Alwin & Jackson, 1981).The process centers around two issues: measurement invariance and structural invariance.The measurement issue concerns the invariance of regression intercepts, factor loadings (regression slopes), and error/uniqueness variances.The structural issue addresses the invariance of factor mean and factor variance-covariance structures.Although there are a number of ad hoc methods for comparing factors across independent samples, these procedures were developed primarily for testing the invariance of factors derived from exploratory factor analyses (EFA; see Marsh and Hocevar [1985] and Reynolds and Harding [1983] for reviews).However, Alwin and Jackson (1981) argued that "issues of factorial invariance are not adequately addressed using exploratory factor analysis" (p.250).A methodologically more sophisticated approach is the CFA method originally proposed by Joreskog (1971) and now commercially available to researchers through LISREL VI (Joreskog & Sorbom, 1985) and SPSS X (SPSS Inc., 1986). 2 (For a discussion of the advantages of CFA over EFA, and details regarding application, see Long [1983], Marsh and Hocevar [1985], andWolfle [1981].) LISREL Approach to Testing for Factorial InvarianceAs a prerequisite to testing for factorial invariance, it is convenient to consider a baseline model that is estimated separately for each group.This model represents the most parsimonious, yet substantively most meaningful and best fitting, model to the data.Because the x 2 goodness-of-fit value and its corresponding degrees of freedom are additive, the sum of the x 2 s reflects how