Fixed factor analysis with clustered factor score constraint

In the fixed factor model for factor analysis (FA), common factor scores are treated as fixed parameters. However, they cannot be estimated jointly with the other parameters, since the maximum likelihood (ML) for the model diverges to infinity. In order to avoid the divergence so that all parameters can be jointly estimated, we propose a constrained fixed factor model. Here, observations are classified into clusters, with each cluster characterized by an equivalent factor score. The ML procedure with the proposed model is named fixed clustered FA (FCFA). An iterative algorithm for FCFA is developed, which provides the ML estimates of the factor loadings, unique variances, the classification of observations into clusters, and the cluster factor scores. This FCFA can be viewed as the FA version of Reduced K-means analysis (RKM), in which the principal components are extracted while clustering observations. We compare FCFA, RKM, and a related procedure called Factorial K-means analysis (FKM). We also provide real data examples, which show that FCFA outperforms RKM and FKM in terms of classification accuracy. This result is attributed to the unique variances in FCFA. In other words, the error variances are allowed to be unique to the corresponding variables.