Multiplicative interaction models are common in the quantitative political science literature. This is so for good reason. Institutional arguments frequently imply that the relationship between political inputs and outcomes varies depending on the institutional context. Moreover, models of strategic interaction typically produce conditional, rather than unconditional, hypotheses. Indeed, it could be argued that any causal claim implies a set of conditions that need to be satisfied before a purported cause is sufficient to bring about its effect. It is little wonder then that conditional hypotheses such as 'an increase in X is associated with an increase in Y when condition Z is met, but not otherwise' are ubiquitous in all fields of political science. It has been well established that the intuition behind conditional hypotheses is captured quite well by multiplicative interaction models. Given the ubiquity and centrality of conditional hypotheses in political science, it is somewhat surprising that the execution of multiplicative interaction models is often flawed and inferential errors are common. In "Understanding Interaction Models: Improving Empirical Analyses" (Political Analysis 14: 63-82), we present several simple suggestions that are easy to implement and that we believe can dramatically improve empirical analyses employing these types of models.
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We strongly encourage analysts to use multiplicative interaction models whenever the hypothesis they want to test is conditional in nature.
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We recommend that scholars include all constitutive terms in their interaction model specifications.
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We remind scholars not to interpret constitutive terms as if they are unconditional marginal effects.
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We urge analysts not to forget to calculate substantively meaningful marginal effects and standard errors.
While these suggestions may seem obvious to some, an examination of three leading political science journals from 1998 to 2002 indicate that our commonsense recommendations are rarely reflected in common practice. In fact, of the 156 articles that we found to employ interaction models, only 16 (10%) actually included all constitutive terms, did not make mistakes interpreting these terms, and calculated substantively meaningful marginal effects and standard errors.
If you find any of the information on this page useful for your research, we would be grateful if you could cite the paper listed above.
Note: We would like to acknowledge a typo in Eq. 12 of our article. Currently, this equation reads "Y = b + b1X + b2Z + b2XZ". This should, of course, read "Y = b + b1X + b2Z + b3XZ". Thanks to Mike Ward for spotting this.
Standard Errors
Below are two tables
illustrating a variety of multiplicative interaction models, marginal effects,
and variances. The tables are based on those in Aiken and West (1991). If
you would like to download these tables as a pdf document, click
here.
Table 1: Marginal
Effects and Variances for Various Interaction Models
(One Modifying Variable and Two Modifying Variables)
Table 2: Marginal Effects and Variances for Various Interaction Models (Quadratic Terms)
Computer Code
As we suggest in our paper, it is often necessary for the analyst to go beyond the typical results table in order to convey quantities of interest such as the marginal effect of X on Y. If the modifying variable is dichotomous, this simply requires the analyst to present four numbers - the marginal effect of X when the modifying variable is 0 and when the modifying variable is one, along with the two corresponding standard errors. If the conditioning variable is continuous the analyst must work a little harder. In our paper, we suggest that a simple figure can succinctly illustrate the marginal effect of X and the corresponding standard errors across the full range of the modifying variable. Below, we provide code in STATA to do this for three types of multiplicative interaction models. It should be easy to adapt this code to deal with other interaction models.
Continous Dependent Variable
with Single Modifying Variable
(Equivalent to Case 1a in Table 1 above)
[Code][Detailed
Explanation of Code]
The example below shows how the marginal effect of temporally-proximate presidential elections on the effective number of electoral parties changes with the number of presidential candidates. The solid sloping line indicates how the marginal effect of temporally proximate presidential elections changes with the number of presidential candidates. 95% confidence intervals around the line allow us to determine the conditions under which presidential elections have a statistically significant effect on the number of electoral parties - they have a statistically significant effect whenever the upper and lower bounds of the confidence interval are both above (or below) the zero line. The figure includes a zero line specifically to help the reader determine when the marginal effect is significant. The marginal effect and the variance necessary to compute confidence intervals are shown in Case 1a in Table 1 above.
Continous dependent variable
with Two Modifying Variables
(Equivalent to Case 4 in Table 1 above)
[Code][Detailed
Explanation of Code] 
Limited dependent variable
with Single Modifying Variable
(A Probit Example)
[Code][Detailed
Explanation of Code]It is almost as easy to use a figure to convey the quantities of interest when the dependent variable is limited instead of continuous. The code presented below can easily be adapted to a variety of limited dependent variables -- dichotomous dependent variables (logit, probit etc.), multichotomous dependent variables (multinomial logit, conditional logit, multinomial probit etc.), ordered dependent variables (ordered probit etc.), duration models (exponential, weibull, generalized gamma etc.). The intuition behind the figures for limited dependent variable models with interaction terms is quite straightforward. Consider an interaction model with a single modifying variable. Typically, the analyst is interested in how some X affects the predicted probability or expected duration of some Y. Let the modifying variable be Z for this example. The code essentially has the following sequence.
- Estimate your model.
- Obtain 1,000 or 10,000 values of the model parameters through simulation and STATA's 'drawnorm' command.
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Calculate the predicted probability or expected duration of Y when X is at some baseline value and the modifying variable Z is at 0 (or its lowest value). You should set any other independent variables at some level of interest - perhaps the mean for continuous variables or the median for dichotomous variables. The specific equation for predicted probabilities or expected durations are available for most models in a several econometric textbooks.
- Calculate the predicted probability or expected duration of Y when X is increased by some value, keeping the modifying variable Z at 0 and all of the other independent variables at the same level as before. The analyst has to decide what a meaningful increase in X would be - this might be a one unit change, some percentage change, or a standard deviation change etc..
- Create a first difference by subtracting the predicted probability calculated in (3) from the predicted probability calculated in (4).
- Repeat steps (3), (4), and (5) for each level of Z.
- Graph the first difference (the effect of an increase in X on Y) across the observed range of Z. Confidence intervals can be created using the 1,000 or 10,000 simulated values of the first differences.
The solid line in the figure below shows how a one unit increase in party system polarization (from its mean) affects the probability of pre-electoral coalition formation across the observed range of electoral threshold. We can see that party system polarization only increases the probability of pre-electoral coalition formation significantly when electoral thresholds are greater than seven.
