compare mean of latent variable across groups

• compared the means of the latent variables across two groups
• If you want to make statements about mean differences in the LV acrossgroups, you need partial invariance on both the loadings and the intercepts. .Following Steenkamp and Baumgartner, I would suggest these steps:1. impose full metric invariance (loadings equal)--if fit is good, skip to step 3; 2. if fit is poor, see if you can obtain good fit by relaxing equality for just one or two of the 4 items measuring the LV--if fit is still poor, quit; 3. impose full scalar invariance (intercepts equal--if fit is good, you're done); 4. If fit is poor, relax intercept constraints on the same items allowed to differon loadings
• Bollen (1989) suggested that the mean structural differences will be analyzed only if the model of equal form and equal factor loadings reached an acceptable fit.

Reading list

• Alwin, D. F., & Jackson, D. J. (1981). Applications of simultaneous factor analysis to issues of factorial invariance. In D. D. Jackson & E. P. Borgatta(Eds.), Factor analysis and measurement in sociological research: A multidimensional perspective (pp. 249-280). Beverly Hills, CA: Sage.
• Byrne, B.M. & Shavelson, R.J. (1987). Adolescent self-concept:Testing the assumption of equivalent structure across gender. AmericanEducational Research Journal, 24, 365-385.
• Byrne, B.M., Schavelson, R.J., & Muthen, B. (1989). Testing for theequivalence 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.
• Everitt, B. S. (1984). An introduction to latent variable models. New York: Chapman and Hall.
• Hanna, G., & Lei, H. (1985). A longitudinal analysis using the LISRELmodel
  with structured means. Journal of Educational Statistics, 10,161-169.
• Lomax, R. G. (1985). A structural model of public and private schools.Journal of Experimental Education, S3,216-226.
• McGaw, B., & Joreskog, K. G. (1971). Factorial invariance of ability measures in groups differing in intelligence and socioeconomic status. British Journal of Mathematical and Statistical Psychology, 24,154-168.
• Magidson, J. (1977). Toward a causal model approach for adjustingfor preexisting differences in the nonequivalent control groupsituation. Evaluation Quarterly, 1, 399-420.
• Muthen, B.. & Chnstoftersson, A. (1981). Simultaneous factor analysis of dichotomous variables in several groups. Psychometrika, 46,407-419.
• Muthen, B. & Muthen, L. (1998). Mplus.
• Rock, D. A., Werts, C. E., & Flaugher, R. L. (1978). The use of analysis of covariance structures for comparing the psychometric properties of multiple variables across populations. Multivariate Behavioral Research, 13,403-418.
• Sorbom, D. (1974). A general method for studying differences in factor means and factor structure between groups. British Journal of Mathematical and Statistical Psychology, 27, 229-239.
• Sorbom, D. (1982). Structural equation models with structured means. In K. G. Joreskog & H. Wold (Eds.), Systems under direct observation (pp. 183-195). Amsterdam: North-Holland.
• Steenkamp, J-B.E.M. & Baumgartner, H. (1998). Assessing measurement invariance in cross-national consumer research. Journal of Consumer Research, 25, 78-90.
• Tomas, J. M. & Oliver A. (1999). Rosenberg's Self-Esteem Scale: Two factors or method effects. Structural Equation Modeling, 6, 84-98. -- test intercept
• Werts, C.E., Rock, D.A., Linn, R.L., & Joreskog, K.G. (1977).Validating psychometric assumptions within and between severalpopulations. Educational and Psychological Measurement, 37, 863-872.
• Whiteside-Mansell, L, & Corwyn, R. F. (2002). An examination of the Rosenberg self-esteem among adolescents and adults. Educational Psychological Measurement, 62, 1028-1038. -- test intercept
  
  

testing partial invariance

• Look at the modification indices and the parameter estimates to see where the invariance is coming from, and then free up those parameters. Just one parameter could be the reason, or it could be a consistent difference across all parameters. Keep in mind also that with three groups, all three could be different from each other, or the only difference could be between group A and group B. So finding the reason for invariance can be a bit tricky. To give you a better idea, you may want to consider first comparing two groups at a time.

Reading list

• Baumgartner, H., & Steenkamp, J. B. (1998). Multi-group latent variablemodels for varying numbers of items and factors with cross-national andlongitudinal applications. Marketing Letters, 9 (1), 21-35.
• Byrne (1989), A primer of LISREL: Basic applications and programming forconfirmatory factor analytic models.
• Byrne, B.M. & Shavelson, R.J. (1987). Adolescent self-concept:Testing the assumption of equivalent structure across gender. AmericanEducational Research Journal, 24, 365-385.
• Byrne, B. M., Shavelson, R. J., & Methuén, B. (1989). Testing for the equivalence of factor covariance and mean structures: The issue of partial measurement invariance. Psychological Bulletin, 105, 456-466.---Byrne, Shavelson and Muthen (1989), who argue that one can test for latent mean differences even with partial metric/scalar invariance, as long as one item (apart from the referent) is invariant per factor.
• Bryant and Veroff (1984), "Dimensions of subjective mental health inAmerican men and women," Journal of health and social behavior, 116-35
• Cheung, G. W., & Roger B. Rensvold (1999). Testing factorial invariance across groups: A reconceptualization and proposed new method. Journal of Management, 25, 1-27.
• Cheung, G. W. & Roger B. Rensvold. (2000) "Assessing Extreme and Acquiescence Response Sets in Cross-Cultural Research using Structural Equations Modeling", Journal of Cross-Cultural Psychology, 31(2), 187-212.
• Cheung, G. W., & Roger B. Rensvold. (2002). Evaluating Goodness-of-Fit Indices for Testing Measurement Invariance. Structural Equation Modeling Journal, 9(2), 233-255.
• Dolan, C. V., 2000, Investigating Spearman's Hypothesis by means of multi-group confirmatory factor analysis, Multivariate Behavioral Research,35,21--50.
• Gregorich, SE (2006). Do Self-Report Instruments Allow Meaningful Comparisons Across Diverse Population Groups? Testing Measurement Invariance Using the Confirmatory Factor Analysis Framework. Medical Care, Volume 44, Number 11, Suppl 3, p S78-S94.
• Lubke GH, Dolan CV, Kelderman H, Mellengergh GJ (2003) "On the relationship between sources of within- and between-group differences and measurement invariance in the common factor model". Intelligence, 31, 543-566
• Magidson, J. (1977). Toward a causal model approach for adjusting for preexisting differences in the nonequivalent control group situation. Evaluation Quarterly, 1, 399-420.
• Marsh (1994). Confirmatory factor analysis models of factorial invariance:A multifaceted approach. SEM 1(1), 5-34.
• Marsh & Grayson (1994). Longitudinal stability of latent means andindividual differences: A unified approach. SEM 1(2), 317-359.
• Meredith, W. (1993). Measurement invariance, factor analysis, and factorial invariance. Psychometrika, 58,525-543. --- Meredith's 1993 paper on invariance stipulates an even weaker model than the Rasch model. His minimum measurement requirements are invariant loadings and thresholds across groups. But even this weaker model is not accepted as "gold standard" among social scientists because in most cases this level of invariance fails. (Partial invariance is also typicallyaccepted as valid measurement without showing how Meredith was wrong.)---Regarding Meredith, if partial invariance is demonstrated, then the manifest means (and/or variances) based upon the *subset* of invariant items can be defensibly compared across samples. So, supporting cross-sample comparisons based upon a demonstration of partial-invariance does not require one to argue that Meredith was wrong. Meredith's same logic applies to a sub-set rather than the entire set of items. This paper shows (among other things) that equality of loadings, intercepts and residual variances is a requirement of measurement invariance, whereas equality of covariances is not.
• Meredith, W. (1995). Two wrongs may not make a right.Multivariate Behavioral Research, 30, 89-94. debate in Multivariate Behavioral Research. Multivariate Behavioral Research had a special edition on invariance in 1995.
• Millsap, R.E. and Kwok, O.M. (2004). Evaluating the impact of partialfactorial invariance on selection in two populations. PsychologicalMethods, 9, 93-115. The topic of this paper is not "how to" examine partial measurement invariance, but rather how to decide when invariance or lack of invariance matters in the larger scheme of things.
• Reise, S. P., Widaman, K. F., & Pugh, R. H. (1993). Confirmatory factor analysis and item response theory: Two approaches for exploring measurement invariance. Psychological Bulletin, 114, 552-566
• Scholderer J, Grunert KG, Brunso K (2002, in press ??) "A procedure for eliminating additive bias from cross-cultural survey data". Journal of Business Research
• Steenkamp, J-B E. M. and H. Baumgartner (1998), "Assessing Measurement Invariance in Cross-National Consumer Research," Journal of Consumer Research, 25, 78-90.--The paper outlines a practical, sequential procedure for assessing the measurement invariance of scales across groups. We can interpret constructs meaningfully across samples even if only some ofyour measures of each construct are invariant.---- cross-sectional data--Following Steenkamp and Baumgartner (1998), at minimum,you will have to specify invariance of two factor loadings and corresponding item intercepts in order to construct a test of partial 'scalar' invariance. So if you find support for a partial scalar invariance model (i.e., at least two items with invariant factor loadings and item intercepts),you can go ahead and test for differences in latent means. I've checked their proof and agree with their findings. This paper suggests that if only some measures are invariant, then the results might still be interpretable. Following Steenkamp & Baumgartner (1998) & others, scalar invariance is a more stringent test than metric invariance since it additionally constrains the intercepts of the observed variables/items to be equal across groups. Therefore, the intercepts of the observed variables/items should be free to covary across groups when assessing for configural and metric invariance.
• Tanaka, J. S. & Huba, G. J. (1984). Confirmatory hierarchical factor analyses of psychological distress measures. Journal of Personality and Social Psychology, 46, 621-635.
• Tomas, J. M. & Oliver A. (1999). Rosenberg's Self-Esteem Scale: Two factors or method effects. Structural Equation Modeling, 6, 84-98. -- test intercept
• Werts, C.E., Rock, D.A., Linn, R.L., & Joreskog, K.G. (1977).Validating psychometric assumptions within and between several populations. Educational and Psychological Measurement, 37, 863-872.
• Whiteside-Mansell, L., Bradley, R. H., Owen, M., Randolph, S., Cauce, A. M. (In Press). Parenting and children's behavior at 36 months: Equivalence between African Americans and European American mother-child dyads. Parenting: Science and Practice.-- test intercept
• Whiteside-Mansell, L, & Corwyn, R. F. (2002). An examination of the Rosenberg self-esteem among adolescents and adults. Educational Psychological Measurement, 62, 1028-1038. -- test intercept

longitudinal factorial invariance and partial invariance

• Invariant loadings would suggest the same construct is being measured at each time point.
• Regarding loading invariance across time, one approach is to consider partial invariance letting some loadings differ at some time points. This gives a practical approach to testing. For the run with full invariance,the modification indices (in Sorbom's terms) point to which loadings showleast invariance and they can be freed up. Factor variances can still be compared across time when some loadings are noninvariant. When freeing loadings the chi-square presumably drops a lot while the loading values may not change very much (illustrating that the chi-square test has too muchpower), and you can focus on the real question: do the factor varianceparameters of key interest change very much under such alternativespecifications?
• the most appropriate analysis would be to fita multiwave CFA to the data. The multiwaveCFA will have 6 common factors (2 for eachwave). All common factors are allowed to covary.Manifest variable residuals of like items areallowed to covary across time.Invariance tests can be performed by imposingequality constraints on corresponding parametersacross time (e.g., equal factor loadings, equalitem intercepts, equal manifest residual variances). Changes in common factor variation are not athreat to measurement invariance. Changes infactor loadings, item intercepts, and residualitem variation do pose threats to measurementinvariance. Though partial invariance is apossible outcome that should be considered.See Meredith's classic (1993, Psychometrika).Also, Steenkamp JBEM, Baumgartner H (1998,Assessing measurement invariance in cross-national consumer research. JOURNAL OFCONSUMER RESEARCH, 25 (1): 78-90)provide a didactic applied take on Meredith (1993)with some useful additional thoughts on partialmeasurement invariance.The above articles deal with cross-sectional data,but the concepts readily generalize to panel data.Finally, if you find evidence of invariant factor loadingsand item intercepts, at least the factor means can becompared longitudinally, or the manifest means (e.g.,composite scores) can be compared. In my opinion,computing and comparing factor scores introducestoo many problems.
• invariance of mean structures in a single group over time--Marsh, H. W., & Grayson, D. (1994). Longitudinal stability of latent means andindividual differences: A unified approach. Structural Equation Modeling, 1,317-359
• if the same set of itemsare unidimensional with respect to a single common factor at two time points--regardlessof the degree to which the factor loadings arefound to be invariant--then it is likely, but notsure, that substantial *similarity* of constructmeaning exists at both time points. But itsdifficult to understand the nature of thesimilarity--to know how close in meaning thetwo constructs are--and whether a large orsubtle qualitative shift has taken place. I'lladd that a qualitative shift does not rule outsimilarity of constructs--but similarity in thiscontext may only have meaning in the relativesense. As an example, after respondentssomehow become "enlightened" (say, by anintervention or by personal experience) they may,when asked, personally attach the same descriptivelabel to the construct but may think about it verydifferently (e.g.., an attitude that may have becomemore crystallized in the minds of your samplemembers after a shared experience). In thiscase, the items measured qualitatively different,but similar attitudes at the two time points.

Reading list

• Grouzet, FME, Otis, N & Pelletier, L. (2006). Longitudinal cross-gender factorial invariance of the Academic Motivation Scale. SEM: A multidisciplinary journal, 13, 73-98.
• Hoyle, R. H., & Smith, G. T. (1994). Formulating clinical research hypotheses as structural equation models: A conceptual overview. Journal of Consulting and Clinical Psychology, 62, 429-440.
• Pentz, M. A., & Chou, C.-P. (1994). Measurement invariance in longitudinal clinical research assuming change from development and intervention. Journalof Consulting and Clinical Psychology, 62, 450-462.
• Schaie, K.W., Maitland, S.B., Willis, S.L., & Intrieri, R.C. (1998).Longitudinal invariance of adult psychometric ability factor structuresacross 7 years. Psychology & Aging, 13, 8-20.
  

test-retest reliability

• Each CFA factor for a subscale at time 2 can be specified as an endogeneous latent variable, and its corresponding factor at time 1 can be specified as a latent exogenous variable. The standardized coefficient linking these latent variables yield a measure of stability for the latent variables for which measurement errors in the indicators have been corrected. --- from Bagozzi, R. P. & Edwards, J. R. 1998. A general approach for representing constructs in organizational research. Organizational Research Methods, 1(1): 45-87.

uniquess, reliability, squared multiple correltion in CFA

• the square of standardized factor loading gives the reliability for the indicator
• 1 (unity) minus the squared value of the uniqueness (measurement error) yields an estimate of the reliability of the factor, which is also equivalent to the squared multiple correlation coefficient for the factor
• CFA generated reliability estimates reflect the proportion of the variance of an observed variable accounted for by any systematic variance operating among the error terms in addition to that accounted for by the latent variable (Bollen, 1989)
• The explained variance estimates for the items and for the factors are reported as squared mulitple correlations. These values can be interpreted as the reliability of an item as an indicator of its associated latent construct.