# Testing for the Equivalence of Factor Covariance and Mean Structures: The Issue of Partial Measurement In variance

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4 PARTIAL MEASUREMENT INVARIANCE 459 Table Pattern ofljsrel Parameters for Model Fitting SDQCSC APIGSC SESOSC SDQASC SCAASC SDQESC APIESC SCAESC SDQMSC APIMSC SCAMSC A, X X,, X X X, X04 _0 0 0 X,,, 4_ GSC] ASC ESC * MSCJ 0, SDQGSC APIGSC SESGSC SDQASC SCAASC SDQESC APIESC SCAESC SDQMSC APIMSC SCAMSC 64 a,, o o o o o o o o o o 0 « J S, J M {,, S.oio ' «,,ii Note. A x = factor loading matrix; 9 = factor variance-covariance matrix; 65 = error variance-covariance matrix; J, jj 4 - SC factors ({, = general SC; & = academic SC; 3 = English SC;, = mathematics SC). GSC = general SC: ASC = academic SC; ESC = English SC; MSC = mathematics SC; SDQGSC - Self Description Questionnaire (SDQ)General Self subscale; APIGSC = Affective Perception Inventory (API)Self- Concept subscale; SESOSC = Self-Esteem Scale; SDQASC = SDQ Academic SC subscale; SCAASC = Self-Concept of Ability Scale (SCA); SDQESC = SDQ Verbal SC subscale; APIESC = API English Perceptions subscale; SCAESC = SCA Form B (SC of English ability); SDQMSC = SDQ Mathematics SC subscale; APIMSC = API Mathematics Perceptions subscale; SCAMSC = SCA Form C (SC of mathematics ability). judge model fit solely on the basis of x 2 values (Bentler & Bonett, 980;J6reskog&S6rbom, 985; Marsh etal., 988) or on alternative fit indices (Kaplan, 988; Sobel & Bohrnstedt, 985); rather, assessments should be based on multiple criteria, including "substantive, theoretical and conceptual considerations" (Joreskog, 97, p. 42). Assessment of overall model fit in the present example was based on statistical as well as practical criteria. Statistical indices of fit included the x 2 likelihood ratio test, the \ 2 /dfralio, and the goodness-of-fit index (GFI) and root-mean-square residual (RMR) provided by LISREL; practical indices included the Bentler and Bonett (980) normed index (BBI) and the Tucker and Lewis (973) nonnormed index (TLI). 7 Selection of these indices was based on their widespread use and their usefulness in comparing samples of unequal size (see Marsh et al., 988). Such popularity notwithstanding, we urge readers to be circumspect in their interpretation of the BBI and TLI because both indices derive from comparison with a null model (see Sobel & Bohrnstedt, 985). Furthermore, only the TLI has been shown to be relatively independent of sample size (Marsh et al., 988). To identify sources of misfit within a specified model, a more detailed evaluation of fit was obtained by inspecting the normalized residuals and modification indices (Mis) provided by LISREL. 8 Additionally, we conducted a sensitivity analysis in order to investigate, under alternative specifications, changes in the estimates of important parameters. Finally, we relied heavily on our knowledge of substantive and theoretical research in SC in making final judgments of the adequacy of a particular model in representing the data. Model Fit As is shown in Table 2, the fit of our hypothesized model was poor from a statistical perspective (low track, xin = 60.54; high track, xis = 40.09) and only marginally acceptable from a practical perspective (low track BBI =.89, TLI =.87; high track: BBI =.92, TLI =.89); this model was therefore rejected. 7 The GFI indicates the relative amount of variances and covariances jointly explained by the model; it ranges from zero to.00, with a value close to.00 indicating a good fit The RMR indicates the average discrepancy between elements in the observed and predicted covariance matrices; it ranges from zero to.00, with a value less than.05 being of little practical importance (Sorbom & Joreskog, 982). Interpretations based on the BBI and TLI indicate the percentage of covariance explained by the hypothesized model; a value less than.90 usually means that the model can be improved substantially (Bentler &. Bonett, 980). * An MI may be computed for each constrained parameter and indicates the expected decrease in x 2 if the parameter were to be relaxed; the decrease, however, may actually be higher.

9 464 B. BYRNE, R. SHAVELSON, AND B. MUTHEN Table 5 Maximum Likelihood Estimates and Standard Errors for Self-Concept Facets Low track Across-track equivalencies High track Parameter Estimate SE Estimate SE Estimate SE *i(x,!)»2(x2s)»3<x35) MX«) *5(X«) MW»7(X,5)» 8 (X 8S )»9<X93)»K>(X 0,s) 'llfxll.s) Xj, X 3, XJ2 X,3 X83 XlO.4 X,,.4 x«, X 7 «< «22 «33 «44»5S «66»77»M»» 0 0, 0 0, ««s 0 0,7 0,5 «f««<h,8 T,,(GSC> T2i(ASC) T3, (ESC) Y4, (MSC) fn &Z f33 & S , Note. x?76) OSC = mathematics SC. general self-concept (SC); ASC = academic SC: ESC - English SC; MSC = differences in general SC ( / ) were negligible and not statistically significant. The results demonstrate that, overall, the test for invariant SCs across track based on mean and covariance structures was statistically more powerful than tests based on covariance structures alone. Whereas tests of mvariance based on the latter found academic track differences in mathematics SC (0 44 ) only, this was not so in the analysis that also included mean structures. Significant differences were also found in academic and English SCs. Conclusion Using data based on SC responses for low- and high-track high school students, we demonstrated procedures for (a) establishing a subslantively well-fitting baseline model for each group; (b) conducting sensitivity analyses to assess the stability of a baseline model; (c) determining partial measurement invariance by testing parameters, independently, given findings of noninvariance at the matrix level; and (d) testing for factorial invariance and differences in mean structures, given partially invariant measuring instruments. Throughout the article, we emphasized the exploratory nature of our analyses and noted the limitations of interpretations based on the results. Post hoc analyses with confirmatory covariance structure models are, indeed, problematic; with multiple model respecifications, probability values become meaningless. At this point in time, however, there is simply no direct way to adjust for the probability of either Type I or Type II errors arising from

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