Enhancing Interpretability in Factor Analysis by Means of Mathematical Optimization

Abstract

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors’ interpretability by building a sparse loading matrix. In this paper, we propose an optimization-based procedure to give meaning to the factors arising in EFA by means of an additional set of variables, called explanatory variables, which may include in particular the set of observed variables. A goodness-of-fit criterion is introduced which quantifies the quality of the interpretation given this way. Our methodology also exploits the rotational invariance of EFA to obtain the best orthogonal rotation of the factors, in terms of the goodness-of-fit, but making them match to some of the explanatory variables, thus going beyond traditional rotation methods. Therefore, our approach allows the analyst to interpret the factors not only in terms of the observed variables, but in terms of a broader set of variables. Our experimental results demonstrate how our approach enhances interpretability in EFA, first in an empirical dataset, concerning volumes of reservoirs in California, and second in a synthetic data example.     

Modelling monotonic effects of ordinal predictors in Bayesian regression models

Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the information contained. Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose. We propose a new method for modelling ordinal predictors that applies in situations in which it is reasonable to assume their effects to be monotonic. The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter modelling the normalized differences between categories. This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories. This formulation generalizes to interaction terms as well as multilevel structures. Monotonic effects may be applied not only to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible. In simulation studies we show that the model is well calibrated and, if there is monotonicity present, exhibits predictive performance similar to or even better than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects which allows us to incorporate prior information and to check the assumption of monotonicity. We have implemented this method in the R package brms, so that fitting monotonic effects in a fully Bayesian framework is now straightforward.     

Why are moderators of self-other agreement difficult to establish?

Agreement between the self and other rated personality profiles was studied in two samples involving 11,096 speakers of two languages, Dutch and Estonian, who completed two different personality questionnaires, the NEO-PI-3 and HEXACO-PI-R. An outstanding agreement was achieved in the most occasions: in only 4–6% of dyadic pairs was the correlation between two randomly paired profiles higher than the actually observed correlation between true pairs. As in previous studies, we found that age and sex of participants and length of acquaintance had no significant effect on the level of self-other agreement. However, intimate knowledge helped married and unmarried couples in both samples be more accurate in their personality judgments; family members, in turn, had knowledge that made them more accurate than two people who were just acquaintances or friends. We believe that these outcomes can be explained by the contention that the judgment of another’s personality is a relatively simple task, which is accomplishable for most people most of the time. In other words, because judging another person’s personality is an easy task, we are not able to determine “good targets,” “good judges,” or “good traits.” Perhaps it is only “good information” which determines the closeness of the target-judge relationship, and which has a small but reliable impact on the level of self-other agreement. This explains why it is so difficult to find individual differences in the ability to judge another person’s personality.