Interaction Effects

Many hypotheses in political science are conditional in nature. Institutional theories typically argue that actors respond differently to similar stimuli when they inhabit different institutional environments. Political culture arguments typically come down to the claim that the ways citizens behave depends on the environment in which they were socialized. Approaches that emphasize strategic interaction identify equilibrium responses that depend on key parameters; when a key parameter is above some critical level, one strategy combination is an equilibrium, while entirely different behavior (as encapsulated in a different comparative static relationship between key variables) is expected when that parameter falls below the critical value. However, many context-conditional claims are tested as if they were unconditional claims. Robert Franzese and Cindy Kam, for example, show that 54% of the articles appearing over a period of 5 years in journals covering three disciplines used statistical methods but only 24% of these used interaction terms in their empirical analysis. The review of the literature by Thomas Brambor, Robert Clark, and Matt Golder is even more discouraging. Examining three leading political science journals over 5 years, they found that more than 90% of the articles that actually specified conditional tests made at least one of the common errors those authors warn about. Some ways to test for interaction effects are presented below.

To get a sense of how multiplicative interaction models allow one to test context-conditional claims, consider the simplest case, where X is hypothesized to be associated with a continuous variable Y in the presence of condition Z but not in its absence. Assume Y and X to be continuous variables and Z is defined such that Z = 1 when the factor in question is present and Z = 0 otherwise. The multiplicative interaction effect approach to testing such a claim is to estimate using ordinary least squares (OLS) regression as follows:

To see how this model can be used to capture a context-conditional claim, examine the case where the condition is absent—that is, when Z = 0. Equation 1 would, therefore become

which simplifies to

Thus, β0 and β1 serve as estimates for the intercept and slope, respectively, in a linear model of the relationship between X and Y in the absence of condition Z. Similarly, Equation 1 allows us to estimate the relationship between these variables when Z is present (i.e., when Z = 1). Once again, substituting the value of the modifying variable into Equation 1, we get

which simplifies to

which shows that Equation 1 allows us to estimate the intercept (β0 + β2) and slope (β1 + β3) of the linear relationship between X and Y in the presence of Z. It is the ability to yield estimates of relationship between X and Y for the case where the hypothesized condition is met as well as when the condition is not met—and the ability to compare those estimates in a straightforward manner—that makes multiplicative interaction effects models a powerful tool for analyzing context-conditional claims.

The promise offered by interactive models has, more often than not, gone unfulfilled as a result of careless application despite the relative simplicity of the technique. Brambor, Clark, and Golder argue that the most common errors found in the literature related to multiplicative interaction models are (a) the failure to include them when appropriate, (b) omission of the individual terms that make up the interaction, (c) failure to calculate the theoretically relevant quantity of interest, and (d) the standard error of the same.

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