This article considers the identification conditions of confirmatory factor analysis (CFA) models for ordered categorical outcomes with invariance of different types of parameters across groups. The current practice of invariance testing is to first identify a model with only configural invariance and then test the invariance of parameters based on this identified baseline model. This approach is not optimal because different identification conditions on this baseline model identify the scales of latent continuous responses in different ways. Once an invariance condition is imposed on a parameter, these identification conditions may become restrictions and define statistically non-equivalent models, leading to different conclusions. By analyzing the transformation that leaves the model-implied probabilities of response patterns unchanged, we give identification conditions for models with invariance of different types of parameters without referring to a specific parametrization of the baseline model. Tests based on this approach have the advantage that they do not depend on the specific identification condition chosen for the baseline model.
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This includes the same number of factors, the same loading patterns, and that all nonzero loadings are positive. This assumption is generally satisfied for confirmatory analyses.
Many estimation methods in factor analysis do not assume normality, but because they only fit the covariance and mean structures, the identification conditions mentioned here are still valid.
It is usually good enough to identify a subset of the parameter space as long as its complement has lower dimensions.
Millsap and Yun-Tein (2004) used slightly different notations: the thresholds are \(\nu \) (we use \(\tau \)), the intercepts are \(\tau \) (we use \(\nu \)), groups are indexed in subscript by k (we use superscript (g)), thresholds indexed as \(m=0, 1, \ldots , c+1\) (we use \(k=0, 1, \ldots , K+1\)).
We choose ML because it produces a \(\chi ^2\) distributed statistic if regularity conditions are met.
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Work on this research by the second author was partially supported by the National Institute of Drug Abuse research education program R25DA026-119 (Director: Michael C. Neale) and by grant R01 AG18436 (20112016, Director: Daniel K. Mroczek) from National Institute on Aging, National Institute on Mental Health.
Roger Millsap, whose work inspired this paper, unexpectedly passed away when we were preparing this manuscript. We would like to honor him for his pioneering work in measurement invariance.
Appendix 1: Mplus and OpenMx Codes
Appendix 1: Mplus and OpenMx Codes
Model Specification in Mplus
Model Specification in OpenMx
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Wu, H., Estabrook, R. Identification of Confirmatory Factor Analysis Models of Different Levels of Invariance for Ordered Categorical Outcomes. Psychometrika 81, 1014–1045 (2016). https://doi.org/10.1007/s11336-016-9506-0
- ordered categorical data
- invariance testing
- model identification