Nonlinearity, Multicollinearity and the Probability of Type II Error in Detecting Interaction

The paper analyzes the impact of the inclusion of quadratic terms on the probability of type II error in testing for interaction in the pres-ence of multicollinearity. The analysis focuses on two situations: (a) when the true model includes only linear and interaction terms; and (b) when the true model includes linear, interaction and quadratic terms. The implications of this analysis on the estimation of interaction in multiple regression are discussed. An interaction between two independent variables is said to occur when the impact of one independent variable on the dependent variable depends on the level of another independent variable. When there are two independent variables, X and Z, and one dependent variable, Y, interaction is usually conceptualized in terms of the effect of the product XZ on Y after the linear effects of X and Z are partialled; it is examined by estimating the model: and by testing whether the value of /33 is significantly different from zero. However, examining hypotheses about interaction by estimating model (1) may lead to increased probability of type I error-the error of accepting the hypothesis that an interaction exists (rejecting the hypothesis that an interaction does not exist) when the true model does not include an interaction. Two impor-tant conditions leading to this error are the presence of both multicollinearity between the independent variables and curvilinear (and in particular quadratic) relationships between the independent variables and the dependent variable. That is, if the dquo;truedquo; model is: