Cubic Response Surface Analysis: Investigating Asymmetric and Level-Dependent Congruence Effects With Third-Order Polynomial Models

Congruence hypotheses play a major role in many areas of psychology. They refer to, for example, the consequences of person-environment fit, similarity, or self-other agreement. For example, are people psychologically better adjusted when their self-view is in line with their reputation? A valid statistical approach that can be applied to investigate congruence hypotheses of this kind is quadratic Response Surface Analysis (RSA) in which a second-order polynomial model is fit to the data and appropriately interpreted. However, quadratic RSA does not allow researchers to investigate more precise expectations about a congruence effect. Do the data support an asymmetric congruence effect, in the sense that congruence leads to the highest (or lowest) outcome, but incongruence in one direction (e.g., self-view exceeds reputation) affects the outcome differently than incongruence in the other direction (e.g., self-view falls behind reputation)? Is there a level-dependent congruence effect, such that the amount of congruence is more strongly related to the outcome variable for some levels of the predictors (e.g., high self-view and reputation) than for others (e.g., low self-view and reputation)? Such complex congruence hypotheses have frequently been suggested in the literature, but they could not be investigated because an appropriate statistical approach has yet to be developed. Here, we present analytical strategies, based on third-order polynomial models, that enable users to investigate asymmetric and level-dependent congruence effects, respectively. To facilitate the correct application of the suggested approaches, we provide respective step-by-step guidelines, corresponding R syntax, and illustrative analyses using simulated and real data.