I am running a model examining the relationship between preservice teachers' commitment, their learning, and job satisfaction. I have developed a structural model after conducting exploratory and confirmatory factor analysis (split half method). However, when conducting group analysis using whether student teachers are planning to take a high stake test, I first tested metric invariance by operating on the measurement model and found that the factor structures, means, and variances of the error terms vary. As the software provides posthoc information about locating invariances, I imposed equality constraints on all the parameters that are suggested invariant in the measurement model and began to test group invariances of the structural model. However, the model does not converge at all. Most research studies using SEM and group invariance analysis stop at the measurement model, and among the limited studies testing path eqivaluence, the prerequirement of the equivalence of measurement models is satisfied. What are some suggestions of you if you were in my shoes? I would greatly appreciate?.
Metric equicalence include equivalent item-factor loadings, construct M/Var, as well as equivalence in item error terms. Imposing equality constraints across models and testing for significant decrement in model fit provides evidence for equivalence across these levels.
Matt
I don't understand why you are looking at invariance in the item loadings. In the social sciences, it is usually accepted that some items will be "better" indicators than others. Similarly, in the structural portion of the model, it is highly likely that some on the conceptual variables will have stronger effects than others.
Hi Zhijuan,
metric invariance concerns the equality of factor loadings across groups (not means and variances). For most applications, this is the only thing that is required. With "most situations" I mean that the substantial goal is to test differences in structural effects across groups. Invariance of error variances is necessary when you want to compare standardized reliabilities and invariance of latent variances is required when you want to compare standardized structural effects. Invariance of item intercept is required when you want to compare latent means.
Try to focus on the metric invariance. Perhaps this is all you need and the model converges. In addition, it is relevant that the baseline model (configural invariance) fits in both groups -- both from a substiantial point of view (it makes no sence to test more fine grained hypotheses when the overall model may be misspecified) -- as well a statistical point of view (chi-square differences require the baseline model holds, see Yuan, 2004).
Best,
Holger
Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4-69.
Yuan, K. H. (2004). On chi-square difference and z tests in mean and covariance structure analysis when the base model is misspecified. Educational and Psychological Measurement, 64(5), 737-757. doi:10.1177/0013164404264853
David L Morgan , I agree with you that within-measurement model differences are often empirically happening and that may not be a problem. Psychometrics however, differentiates three basic measurement models that can be tested by within-model invariance test:
a) the congeneric model (all indicators truly measure one underlying factor)
b) essential tau-equivalent model (consists is "a" plus equal loadings)
c) tau-equivalent model (concists of "a", "b" + equal intercepts)
d) parallel model (adds equal error variances).
These models are trivial. For instance, model "b" is required for a reasonable application of Cronbach's alpha. Furthermore, substantial differences between loadings are often a sign of misspecification (given that all were created as exchangeable indicators).
Best,
Holger
Graham, J. M. (2006). Congeneric and (essentially) tau-equivalent estimates of score reliability. Educational and Psychological Measurement, 66(6), 930-944.
Millsap, R. E., & Everson, H. (1991). Confirmatory measurement model comparisons using latent means. Multivariate Behavioral Research, 26(3), 479-497.
I detect a different issue with your approach: The split-half method is considered rather flawed when it comes to CFA. Instead, you should opt for a completely independent sample. Furthermore, scalar invariance is the level of invariance which allows you to compare means across groups. You need at least n=200 in both your groups to get realiable measures. As the RMSEA is influenced by small sample sizes, I would deem changes in the CFI
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