Workshop: Factorial Invariance Revisted: Are We There Yet Todd D. Little University of Kansas

2. Comparing Across Groups or Across Time In order to compare constructs across two or more groups OR across two or more time points, the equivalence of measurement must be established. This need is at the heart of the concept of Factorial Invariance. Factorial Invariance is assumed in any cross-group or cross-time comparison SEM is an ideal procedure to test this assumption.

3. Comparing Across Groups or Across Time Meredith provides the definitive rationale for the conditions under which invariance will hold (OR not)�Selection Theorem Note, Pearson originated selection theorem at the turn of the century

7. Levels Of Invariance There are four levels of invariance: 1) Configural invariance - the pattern of fixed & free parameters is the same. 2) Weak factorial invariance - the relative factor loadings are proportionally equal across groups. 3) Strong factorial invariance - the relative indicator means are proportionally equal across groups. 4) Strict factorial invariance - the indicator residuals are exactly equal across groups (this level is not recommended).

8. The Covariance Structures Model where... S = matrix of model-implied indicator variances and covariances L = matrix of factor loadings F = matrix of latent variables / common factor variances and covariances Qd = matrix of unique factor variances (i.e., S + e and all covariances are usually 0) This model is fit to the data because it contains fewer parameters to estimate, yet contains everything we want to know.

9. The Mean Structures Model where... mx = vector of model-implied indicator means tx = vector of indicator intercepts L = matrix of factor loadings a = vector of factor means

10. Factorial Invariance An ideal method for investigating the degree of invariance characterizing an instrument is multiple-group (or multiple-occasion) confirmatory factor analysis; or mean and covariance structures (MACS) models MACS models involve specifying the same factor model in multiple groups (occasions) simultaneously and sequentially imposing a series of cross-group (or occasion) constraints.

11. Some Equations

12. Models and Invariance It is useful to remember that all models are, strictly speaking, incorrect. Invariance models are no exception. "...invariance is a convenient fiction created to help insecure humans make sense out of a universe in which there may be no sense." (Horn, McArdle, & Mason, 1983, p. 186).

14. Manifest vs. Latent Variables �Indicators are our worldly window into the latent space� John R. Nesselroade

15. Manifest vs. Latent Variables

16. Selection Theorem

29. The results of the two-group model with equality constraints on the corresponding loadings provides a test of proportional equivalence of the loadings:

30. When we regress indicators on to constructs we can also estimate the intercept of the indicator. This information can be used to estimate the Latent mean of a construct Equivalence of the loading intercepts across groups is, in fact, a critical criterion to pass in order to say that one has strong factorial invariance.

36. Indicator mean = intercept + loading(Latent Mean) i.e., Mean of Y = intercept + slope (X) For Positive Affect then: Group 1 (7th grade): Group 2 (8th grade): Y = t + ? (a) Y = t + ? (a) 3.14 � 3.15 + .58(0) 3.07 � 3.15 + .58(-.16) = 3.06 2.99 � 2.97 + .59(0) 2.85 � 2.97 + .59(-.16) = 2.88 3.07 � 3.08 + .64(0) 2.97 � 3.08 + .64(-.16) = 2.98 Note: in the raw metric the observed difference would be -.10 3.14 vs. 3.07 = -.07 2.99 vs. 2.85 = -.14 gives an average of -.10 observed 3.07 vs. 2.97 = -.10 ============== i.e. averaging: 3.07 - 2.96 = -.10

40. Cohen�s d = (M2 � M1) / SDpooled where SDpooled = v[(n1Var1 + n2Var2)/(n1+n2)]

41. Cohen�s d = (M2 � M1) / SDpooled where SDpooled = v[(n1Var1 + n2Var2)/(n1+n2)] Latent d = (a2j � a1j) / v?pooled where v?pooled = v[(n1 ?1jj + n2 ?2jj)/(n1+n2)]

42. Cohen�s d = (M2 � M1) / SDpooled where SDpooled = v[(n1Var1 + n2Var2)/(n1+n2)] Latent d = (a2j � a1j) / v?pooled where v?pooled = v[(n1 ?1jj + n2 ?2jj)/(n1+n2)] dpositive = (-.16 � 0) / 1.05 where v?pooled = v[(380*1 + 379*1.22)/(380+379)] = -.152