Evaluating Predictors' Relative Importance Using Bayes Factors in Regression Models

This study presents a Bayesian inference approach to evaluate the relative importance of predictors in regression models. Depending on the interpretation of importance, a number of indices are introduced, such as the standardized regression coefficient, the average squared semipartial correlation, and the dominance analysis measure. Researchers' theories about relative importance are represented by order constrained hypotheses. Support for or against the hypothesis is quantified by the Bayes factor, which can be computed from the prior and posterior distributions of the importance index. As the distributions of the indices are often unknown, we specify prior and posterior distributions for the covariance matrix of all variables in the regression model. The prior and posterior distributions of each importance index can be obtained from the prior and posterior samples of the covariance matrix. Simulation studies are conducted to show different inferences resulting from various importance indices and to investigate the performance of the proposed Bayesian testing approach. The procedure of evaluating relative importance using Bayes factors is illustrated using two real data examples. Translational Abstract Comparing the importance of predictors in regression models answers many questions in psychological researches. For example, whether self-consciousness is the most important factor of the happiness of college students, compared with school study and communication. This study recommends two importance measures: the average squared semipartial correlation and the dominance analysis measure. These measures can be interpreted as the contribution of each predictor to the variation of the outcome variable. More importantly, they can compare grouped or categorical predictors. This study provides a Bayesian inference procedure to evaluate the order of predictors' importance. The relative importance can be represented by an order constrained hypothesis like p1 > p2 > p3 which indicates a theory that the first predictor is more important than the second followed by the third. The Bayes factor is used to quantify the evidence in the data for or against the hypothesis. This helps researchers understand to what extent their hypotheses are supported. The procedure of evaluating relative importance by means of Bayes factors is illustrated by two applied examples. Researchers can either test a confirmatory hypothesis with a specific ordering of importance that comes from a research theory or can conduct exploratory analysis to determine which hypothesis is most supported by the data.