| From the beginning, lead methodologists in psychometrics and quantitative psychology have been well aware of the problems of fitting structural and confirmatory factor models. The question we approach in our research is how to best detect this misfit and how to identify specific sources of misfit by scrutinizing the data at the case level. Since Anscombe's seminal 1973 paper, detecting problems at the case level in ordinary least-squares regression has become the norm in statistical modeling. In contrast, the usual practice in fitting structural and confirmatory factor models has been to only examine misfit at the variable and sufficient statistic level. This practice ignores a small body of literature that has arisen since the early 1990s about diagnostics of case level and case by variable level misfit. An important paper by Bollen and Arminger (1991) and a follow-up paper by Raykov and Penev (1999), have developed theory behind Individual Case Residuals (ICRs). These papers help lay the ground work for more detailed case and case by variable level diagnostics, without discarding traditional variable oriented procedures. Our goal is to demonstrate uses of multivariate techniques, such as robust Mahalanobis distances, biplots and cluster analysis to analyze the multivariate dataset of ICRs and thereby detect sources of data problems with respect to a target model. We hope to encourage researchers to make better use of case level diagnostics among the various classes of latent variable models, especially with the advent of multivariate tools in packages such as R and SAS. |
