Rival Hypotheses in Linear Structure Modeling: Factor Rotation in Confirmatory Factor Analysis and Latent Path Analysis

After summarizing Algina's (1980) criteria for factor identification in confirmatory factor analysis (CFA) a procedure is given how to determine rotationally underidentified factors by adding further restrictions and how to carry out the rotation to meet the old restrictions together with the additional ones. As CFA may also be part of a more complex structure, for example, in latent path analysis, where CFA usually is employed as measurement model for the latent variables, the next section considers the effects of rotation in the measurement model on the latent path coefficients in the inner part of a latent path model. A last section presents two illustrative examples. The procedure presented may be used in theory based model building to develop an identified model. If there is not enough prior knowledge to make the whole structure identifiable, the procedure may be used in an exploratory way, adding various sets of identifying restrictions in turn to produce alternative models, which may be subject to further investigation.