Disjoint factor analysis with cross-loadings

Disjoint factor analysis (DFA) is a new latent factor model that we proposehere to identify factors that relate to disjoint subsets of variables, thus simplifying theloading matrix structure. Similarly to exploratory factor analysis (EFA), the DFA doesnot hypothesize prior information on the number of factors and on the relevant relationsbetween variables and factors. In DFA the population variance–covariance structure ishypothesized block diagonal after the proper permutation of variables and estimatedby Maximum Likelihood, using an Coordinate Descent type algorithm. Inference onparameters on the number of factors and to confirm the hypothesized simple structureare provided. Properties such as scale equivariance, uniqueness, optimal simplifica-tion of loadings are satisfied by DFA. Relevant cross-loadings are also estimated incase they are detected from the best DFA solution. DFA has also the option to con-strain a variable to load on a pre-specified factor so that the researcher can assume,a priori, some relations between variables and loadings. A simulation study showsperformances of DFA and an application to optimally identify the dimensions of well-being is used to illustrate characteristics of the new methodology. A final discussionconcludes the paper.