Testing fit patterns with polynomial regression models

Fit hypotheses, also labeled ‘congruence’, ‘discrepancy’, or ‘congruity’ hypotheses, contain the notion that an outcome is optimal when two predictor variables match well, while incongruent/discrepant combinations of the predictors lead to a suboptimal outcome. Previous statistical frameworks for analyzing fit hypotheses emphasized the necessity of commensurable scales, which means that both predictors must be measured on the same content dimension and on the same numerical scale. In some research areas, however, it is impossible to achieve scale equivalence, because the predictors have to be measured with different methods, such as explicit attitudes (e.g., questionnaires) and implicit attitudes (e.g., reaction time task). In this paper, I differentiate numerical congruence from fit patterns, a concept that does not depend on the notion of commensurability, and hence can be applied to fit hypotheses with incommensurable scales. Polynomial regression can be used to test for the presence of a fit pattern in empirical data. I propose several new regression models for testing fit patterns which are statistically simpler and conceptually more meaningful than a full polynomial model. An R package is introduced which provides user-friendly functions for the computation, visualization, and model comparison of several fit patterns. An empirical example on implicit/explicit motive fit demonstrates the usage of the new methods.