Mediation and moderation in distance-based regression with triangular fuzzy data: simulation evidence and an application to academic performance

We study mediation and moderation in the distance-based regression model for triangular fuzzy responses, in which the center and transformed spreads of the response are linear in the components of fuzzy and crisp covariates. With a common design, componentwise least squares minimizes the unconstrained quadratic extension of the Yang--Ko distance on the transformed scale, for any choice of its shape parameters. Direct, indirect and total effects are defined separately for the center and the spreads of the response as plug-in contrasts on the fitted equations; when the mediator spreads are transformed and carry part of the indirect association, the product-of-coefficients rule no longer holds for the center. Moderation is modeled through triangular interaction regressors and tested with a Freedman--Lane permutation test. In simulations calibrated on real data, percentile bootstrap intervals for the indirect effects had coverage between 0.905 and 0.975, the empirical size of the indirect-effect tests did not exceed 0.085, and ignoring the paths through the mediator spreads biased the indirect effect on the center by 16 to 20% at every sample size. The fuzzy model predicted better than a centers-only model only from n=250. In data on 91 university students collected in 2014, the indirect contrast through a course indicator accounted for about half of the model-based association between prior knowledge and the center of academic performance, the interactions of prior knowledge with sex and with motivation remained after a Holm adjustment, and the spreads of the covariates did not improve leave-one-out prediction.