1. 1crmda.KU.edu Todd D. Little University of Kansas Director, Quantitative Training Program Director, Center for Research Methods and Data Analysis Director, Undergraduate Social and Behavioral Sciences Methodology Minor Member, Developmental Psychology Training Program crmda. KU.edu Colloquium presented 5-24-2012 @ University of Turku, Finland Special Thanks to: Mijke Rhemtulla & Wei Wu Factorial Invariance: Why It's Important and How to Test for It

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

4. Which posits: if the selection process effects only the true score variances of a set of indicators, invariance will hold

5. Classical Measurement Theorem X i = T i + S i + e i Where, X i is a person’s observed score on an item, T i is the 'true' score (i.e., what we hope to measure), S i is the item-specific, yet reliable, component, and e i is random error, or noise. Note that S i and e i are assumed to be normally distributed (with mean of zero) and uncorrelated with each other. And, across all items in a domain, the S i s are uncorrelated with each other, as are the e i s.

6. Selection Theorem on Measurement Theorem X1 = T1 + S1 + e1X2 = T2 + S2 + e2X3 = T3 + S3 + e3X1 = T1 + S1 + e1X2 = T2 + S2 + e2X3 = T3 + S3 + e3 Selection Process

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... Σ = matrix of model-implied indicator variances and covariances Λ = matrix of factor loadings Ψ = matrix of latent variables / common factor variances and covariances Θ = matrix of unique factor variances (i.e., S + e and all covariances are usually 0)

9. The Mean Structures Model where... μ = vector of model-implied indicator means τ = vector of indicator intercepts Λ = matrix of factor loadings α = 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 Configural invariance: Same factor loading pattern across groups, no constraints. Weak (metric) invariance: Factor loadings proportionally equal across groups. Strong (scalar) invariance: Loadings & intercepts proportionally equal across groups. Strict invariance: Add unique variances to be exactly equal across groups.

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).

13. Measured vs. Latent Variables Measured (Manifest) Variables Measured (Manifest) Variables Observable Observable Directly Measurable Directly Measurable A proxy for intended construct A proxy for intended construct Latent Variables Latent Variables The construct of interest The construct of interest Invisible Invisible Must be inferred from measured variables Must be inferred from measured variables Usually ‘Causes’ the measured variables (cf. reflective indicators vs. formative indicators) Usually ‘Causes’ the measured variables (cf. reflective indicators vs. formative indicators) What you wish you could measure directly What you wish you could measure directly

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

15. Manifest vs. Latent Variables X1X1 ξ1ξ1 Ψ 11 X3X3 X2X2 θ 22 θ 11 θ 33 λ 11 λ 21 λ 31

16. Selection Theorem X1X1 Group (Time) 1 Ψ 11 X3X3 X2X2 θ 22 θ 11 θ 33 λ 11 λ 21 λ 31 X1X1 Group (Time) 2 Ψ 11 X3X3 X2X2 θ 22 θ 11 θ 33 λ 11 λ 21 λ 31 Selection Influence

17. Estimating Latent Variables 17 X1X1 ξ1ξ1 Ψ 11 X3X3 X2X2 θ 22 θ 11 θ 33 λ 11 λ 21 λ 31 Implied variance/covariance matrix X1X1 X2X2 X3X3 X1X1 11 y 11 11 + θ 11 X2X2 11 y 11 21 21 y 11 21 + θ 22 X3X3 11 y 11 31 21 y 11 31 31 y 11 31 + θ 33 To solve for the parameters of a latent construct, it is necessary to set a scale (and make sure the parameters are identified)

18. Scale Setting and Identification 18 Three methods of scale-setting (part of identification process) Arbitrary metric methods: Fix the latent variance at 1.0; latent mean at 0 (reference-group method) Fix a loading at 1.0; an indicator’s intercept at 0 (marker-variable method) Non-Arbitrary metric method Constrain the average of loadings to be 1 and the average of intercepts at 0 (effects-coding method; Little, Slegers, & Card, 2006 )

19. 1.Fix the Latent Variance to 1.0 and Latent mean to 0.0) 19 X1X1 ξ1ξ1 1.0* X3X3 X2X2 θ 22 θ 11 θ 33 λ 11 λ 21 λ 31 Implied variance/covariance matrix X1X1 X2X2 X3X3 X1X1 1 11 2 +  11 X2X2 11 1 21 1 21 2 +  22 X3X3 11 1 31 21 1 31 1 31 2 +  33 Three methods of setting scale 1) Fix latent variance (Ψ 11 )

20. 2. Fix a Marker Variable to 1.0 (and its intercept to 0.0) 20 X1X1 ξ1ξ1 Ψ 11 X3X3 X2X2 θ 22 θ 11 θ 33 1.0*λ 21 λ 31 Implied variance/covariance matrix X1X1 X2X2 X3X3 X1X1     X2X2          X3X3             

21. 3. Constrain Loadings to Average 1.0 (and the intercepts to average 0.0) 21 X1X1 ξ1ξ1 Ψ 11 X3X3 X2X2 θ 22 θ 33 λ 11 = 3-λ 21 -λ 31 λ 21 λ 31 Implied variance/covariance matrix X1X1 X2X2 X3X3 X1X1 (3- 21 - 31 )   (3- 21 - 31 ) +  11 X2X2 (3- 21 - 31 )   21 21   21 +  22 X3X3 (3- 21 - 31 )   31 21   31 31   31 +  33 θ 11

22. Configural invariance 12 xx 165432 1*.64.66.71.59.55.57.09.11.07.11.07.06 Group 1: Group 2: 12 xx 165432 1*. 57. 61. 63.59.60.12.10.11.10.07

23. Configural invariance 12 xx 165432 1*.64.76.71.59.55.57.09.11.07.11.07.06 Group 1: Group 2: 12 xx 165432 1*. 57. 51. 63.59.60.12.10.11.10.07

24. Configural invariance 12 -.32 165432 1*.64.66.71.56.55.57.09.11.07.11.07.06 Group 1: Group 2: 12 -.07 165432 1*. 57. 61. 63.59.60.12.10.11.10.07

25. Weak factorial invariance (equate λs across groups) 12 PS(2,1) 165432 1* PS(1,1) PS(2,2) LY(1,1) LY(2,1)LY(3,1)LY(4,2)LY(5,2)LY(6,2) TE(1,1) TE(2,2) TE(3,3)TE(4,4)TE(5,5)TE(6,6) 12 PS(2,1) 165432 ee PS(1,1) PS(2,2) =LY(1,1) =LY(2,1)=LY(3,1)=LY(4,2)=LY(5,2)=LY(6,2) TE(1,1) TE(2,2) TE(3,3)TE(4,4)TE(5,5)TE(6,6) Group 1: Group 2: Note: Variances are now Freed in group 2

26. F: Test of Weak Factorial Invariance PositiveNegative.58.59.64.62.59.61       Model Fit: χ 2 (20, n=759) =49.0; RMSEA=.062 (.040-.084) ; CFI=.99; NNFI=.99 1* Great +Glad + GladUnhappy + Bad Down + Blue Terrible +Sad + SadHappy + Super Cheerful + Good      1.2 2.85    (9.2.1.TwoGroup.Loadings.FactorID)

27. M: Test of Weak Factorial Invariance PositiveNegative 1* 1.02 1.11 1*.95.97 -.03.11.10.11.10.07 Model Fit: χ 2 (20, n=759) =49.0; RMSEA=.062 (.040-.084) ; CFI=.99; NNFI=.99.33.39 Great +Glad + GladUnhappy + Bad Down + Blue Terrible +Sad + SadHappy + Super Cheerful + Good.10.07.11.07.06.41.33 -.12.12.09 (9.2.1.TwoGroup.Loadings.MarkerID)

28. EF: Test of Weak Factorial Invariance PositiveNegative.96.98 1.06 1.03.97 1.00 -.03.11.10.11.10.07 Model Fit: χ 2 (20, n=759) =49.0; RMSEA=.062 (.040-.084) ; CFI=.99; NNFI=.99.36.37 Great +Glad + GladUnhappy + Bad Down + Blue Terrible +Sad + SadHappy + Super Cheerful + Good.10.07.11.07.06.44.31 -.12.12.09 (9.2.1.TwoGroup.Loadings.EffectsID)

29. The results of the two-group model with equality constraints on the corresponding loadings provides a test of proportional equivalence of the loadings: Results Test of Weak Factorial Invariance Nested significance test: (χ 2 (20, n=759) = 49.0) - (χ 2 (16, n=759) = 46.0) = Δ χ 2 (4, n=759) = 3.0, p >.50 The difference in χ 2 is non-significant and therefore the constraints are supported. The loadings are invariant across the two age groups. “Reasonableness” tests: RMSEA: weak invariance =.062 (.040-.084) versus configural =.069 (.046-.093) The two RMSEAs fall within one another’s confidence intervals. CFI: weak invariance =.99 versus configural =.99 The CFIs are virtually identical (one rule of thumb is ΔCFI <=.01 is acceptable). (9.2.TwoGroup. 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. Adding information about means

31. 12 165432 1* AL(1) TY(1)TY(2)TY(3)TY(4) TY(6) TY(5) AL(2) X

32. Adding information about means (9.3.0.TwoGroups.FreeMeans) 12 165432 3.14 3.07 2.99 2.85 3.07 2.98 1.70 1.72 1.53 1.58 1.55 0* Model Fit: χ 2 (20, n=759) = 49.0 (note that model fit does not change) X

33. Strong factorial invariance (aka. loading invariance) – Factor Identification Method (9.3.1.TwoGroups.Intercepts.FactorID) 12 165432 X 3.152.973.081.701.551.54 0* -.16 0*.04 Model Fit: χ 2 (24, n=759) = 58.4, RMSEA =.061 (.041;.081), NNFI =.986, CFI =.989.58.59.64.62.59.61 -.07 -.33 1* 1.22 1* 0.85

34. (9.3.1.TwoGroups.Intercepts.MarkerID) 12 165432 X 0*-.28-.430*-.06-.12 3.15 3.06 1.70 1.72 Model Fit: χ 2 (24, n=759) = 58.4, RMSEA =.061 (.041;.081), NNFI =.986, CFI =.989 1*1.031.111*.95.97 -.03 -.12.33.40.39.33 Strong factorial invariance (aka. loading invariance) – Marker Var. Identification Method

35. (9.3.1.TwoGroups.Intercepts.EffectsID) 12 165432 X.23-.05-.18.06-.00-.06 3.07 2.97 1.59 1.62 Model Fit: χ 2 (24, n=759) = 58.4, RMSEA =.061 (.041;.081), NNFI =.986, CFI =.989.95.981.061.03.971.00 -.03 -.12.36.44.37.31 Strong factorial invariance (aka. loading invariance) – Effects Identification Method

36. Indicator mean = intercept + loading(Latent Mean) i.e., Mean of Y = intercept + slope (X) For Positive Affect then: Group 1 (7 th grade): Group 2 (8 th grade): Y = τ + λ (α) Y = τ + λ (α) 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 How Are the Means Reproduced? __

37. The complete model with means, std’s, and r’s Positive3Negative4 -.07 1* -.32 Positive1Negative2 0* 1.0* (in group 1) 1.0* (in group 1) 1.11 (in group 2).92 (in group 2) X -.16 (z=2.02).04 (z=0.53) 3.152.973.081.701.54.62.59.61.64.58.59 Estimated only in group 2! Group 1 = 0 (9.7.1.Phantom variables.With Means.FactorID) Model Fit: χ 2 (24, n=759) = 58.4, RMSEA =.061 (.041;.081), NNFI =.986, CFI=.989

38. The complete model with means, std’s, and r’s Positive3Negative4 -.07 1* -.32 Positive1Negative2 0*.58 (in group 1).62 (in group 1).64 (in group 2).57 (in group 2) X 3.15 3.06 1.70 1.72 0*-.28-.430*-.12-.06 1*.95.97 1.11 1*1.03 (9.7.2.Phantom variables.With Means.MarkerID) Model Fit: χ 2 (24, n=759) = 58.4, RMSEA =.061 (.041;.081), NNFI =.986, CFI=.989

39. The complete model with means, std’s, and r’s Positive3Negative4 -.07 1* -.32 Positive1Negative2 0*.60 (in group 1).61 (in group 1).67 (in group 2).56 (in group 2) X 3.07 2.97 1.59 1.62.23-.05-.18.06-.06-.00 1.03.97 1.00 1.06.96.98 (9.7.3.Phantom variables.With Means.EffectsID) Model Fit: χ 2 (24, n=759) = 58.4, RMSEA =.061 (.041;.081), NNFI =.986, CFI=.989

40. Cohen’s d =(M 2 – M 1 ) / SD pooled where SD pooled = √[(n 1 Var 1 + n 2 Var 2 )/(n 1 +n 2 )] Effect size of latent mean differences

41. Cohen’s d =(M 2 – M 1 ) / SD pooled where SD pooled = √[(n 1 Var 1 + n 2 Var 2 )/(n 1 +n 2 )] Latent d=(α 2j – α 1j ) / √ ψ pooled where √ ψ pooled = √[(n 1 ψ 1jj + n 2 ψ 2jj )/(n 1 +n 2 )] Effect size of latent mean differences

42. Cohen’s d =(M 2 – M 1 ) / SD pooled where SD pooled = √[(n 1 Var 1 + n 2 Var 2 )/(n 1 +n 2 )] Latent d=(α 2j – α 1j ) / √ ψ pooled where √ ψ pooled = √[(n 1 ψ 1jj + n 2 ψ 2jj )/(n 1 +n 2 )] d positive =(-.16 – 0) / 1.05 where √ ψ pooled = √[(380*1 + 379*1.22)/(380+379)] =-.152 Effect size of latent mean differences

43. 5. Invariance of Variances χ 2 difference test 6. Invariance of Correlations/Covariances χ 2 difference test 3b or 7. Invariance of Latent Means χ 2 difference test 3. Invariance of Intercepts RMSEA/CFI difference Test Comparing parameters across groups 1. Configural Invariance Inter-occular/model fit Test 4. Invariance of Variance/ Covariance Matrix χ 2 difference test 2. Invariance of Loadings RMSEA/CFI difference Test

44. The ‘Null’ Model 44 The standard ‘null’ model assumes that all covariances are zero – only variances are estimated In longitudinal research, a more appropriate ‘null’ model is to assume that the variances of each corresponding indicator are equal at each time point and their means (intercepts) are also equal at each time point (see Widaman & Thompson). In multiple-group settings, a more appropriate ‘null’ model is to assume that the variances of each corresponding indicator are equal across groups and their means are also equal across groups.

45. Byrne, B. M., Shavelson, R. J., & Muthén, B. (1989). Testing for the equivalence of factor covariance and mean structures: The issue of partial measurement invariance. Psychological Bulletin, 105, 456-466. Cheung, G. W., & Rensvold, R. B. (1999). Testing factorial invariance across groups: A reconceptualization and proposed new method. Journal of Management, 25, 1-27. Gonzalez, R., & Griffin, D. (2001). Testing parameters in structural equation modeling: Every “one” matters. Psychological Methods, 6, 258-269. Kaiser, H. F., & Dickman, K. (1962). Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix. Psychometrika, 27, 179-182. Kaplan, D. (1989). Power of the likelihood ratio test in multiple group confirmatory factor analysis under partial measurement invariance. Educational and Psychological Measurement, 49, 579-586. Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13, 59-72. MacCallum, R. C., Roznowski, M., & Necowitz, L. B. (1992). Model modification in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111, 490-504. Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525-543. Steenkamp, J.-B. E. M., & Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90. References 45

46. 46crmda.KU.edu Todd D. Little University of Kansas Director, Quantitative Training Program Director, Center for Research Methods and Data Analysis Director, Undergraduate Social and Behavioral Sciences Methodology Minor Member, Developmental Psychology Training Program crmda. KU.edu Colloquium presented 5-24-2012 @ University of Turku, Finland Special Thanks to: Mijke Rhemtulla & Wei Wu Factorial Invariance: Why It's Important and How to Test for It

47. Update Dr. Todd Little is currently at Texas Tech University Director, Institute for Measurement, Methodology, Analysis and Policy (IMMAP) Director, “Stats Camp” Professor, Educational Psychology and Leadership Email: yhat@ttu.eduyhat@ttu.edu IMMAP (immap.educ.ttu.edu) Stats Camp (Statscamp.org) 47www.Quant.KU.edu