QUANTITATIVE METHODS IN PSYCHOLOGY Analysis of Multiplicative Combination Rules When the Causal Variables are Measured With Error

A variety of theories in psychology postulate that the causal variables combine according to a multiplicative rule to determine the value of the dependent variable. To test multiplicative combination rules empirically, applied researchers frequently use an observational method that involves the following procedure: (a) assessment techniques are used to measure the value of each theoretical construct for each individual, (b) product scores are formed by multiplying the measures of the causal variables, and (c) hierarchical regression analysis is used to test the statistical significance of the increment in R contributed by the product term. The purpose of this article is to evaluate the validity of the observational method with respect to two measurement issues: measurement level (i.e., the effects produced by allowing monotonic transformations of the measures), and measurement error (i.e., the effects produced by using unreliable measures of the causal variables). Our evaluation is based on a theoretical distinction between the structural model (the set of equations relating theoretical constructs to each other) and the measurement model (the set of equations relating the theoretical constructs to the observed measures). We conclude that hierarchical regression analysis is inadequate for determining whether the structural model is additive or multiplicative for two reasons. First, an additive structural model may produce multiplicative effects through a nonlinear measurement model. Second, a multiplicative structural model may produce nondetectable multiplicative effects because of multiplicative measurement error. Some alternatives to hierarchical regression analysis are described.