Optimality of linear factor structures

Factor analysis is often applied in empirical data analysis to explore data structures. Due to its theoretical construction, factor analysis is suitable for the study of linear relationships, and adequacy of a factor analysis solution is often assessed with linear correlation related measures. This paper aims to contribute to literature by examining whether linear factor structures can correspond to multiple requirements simultaneously. Theoretical and simulation results also suggest that under the applied assumptions the examined optimality criteria can not be met simultaneously. These criteria are related to the determinant of the correlation matrix (that should be minimized so that it is close to zero), the determinant of the anti-image correlation matrix (that should be maximized so that it is close to one), and the Kaiser-Meyer-Olkin measure of sampling adequacy (that should be above a predefined minimum value). Results of the analysis highlight the complexity of questions related to the design of quantitative methodology for exploring linear factor structures.