- freely available
Soc. Sci. 2013, 2(2), 91-113; doi:10.3390/socsci2020091
2. Interactions in Linear Models
3. Interactions in Generalized Linear Models
3.1. The Effect of the Link Function
3.2. Illustration of the Link Function Effect
3.3. Interpreting Interaction Effects in Generalized Linear Models
3.4. Reporting GLM Interaction Effects with First Differences
4. A Methodological Controversy in Political Science
5. Higher-Order Interactions
6. An Application to Personal Vote-Earning Attributes
6.1. The First-order Interaction Model
|Model 1||Model 2|
|Estimate||Std. Error||Estimate||Std. Error|
|Null Deviance||1493.6 on 1126 d.f.||1473.3 on 1112 d.f.|
|Residual Deviance||1471.7 on 1123 d.f.||1448.2 on 1105 d.f.|
6.2. The Second-Order Interaction Model
In summary, partial products of variables are correctly interpreted as interactions whatever their level of scaling, whether or not they are correlated, whether or not their means are zero, whether they are observational or experimental in origin, and whether single variables or sets of variables are at issue.
Conflict of Interest
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- 1Unfortunately, the term “main effect" is not very revealing or accurate. The coefficient for a non-multiplicative explanatory variable specification given by a variable that is specified as interacting with others, is the contribution for that variable at zero levels of all of the interacting variables. There is no mathematically implied primacy and a more accurate term would be “non-multiplicative” or “single contribution.” However, it seems unwarranted to try and change standard vernacular used by an overwhelming number of authors, dating from as far back as Fisher and Mackenzie  and Finney . So this term will be used here.
- 2These models use algebraic forms where components such as exponents are estimated (and hence are different from standard polynomial regression, e.g., Fox (, S. 14.2)) in order to make local approximations to continuous, multidimensional response surfaces. For extended discussions, see Box and Draper , Khuri and Cornell , and Cornell and Montgomery .
- 3It is also interesting to note that when the multiplicative interaction term is highly correlated with one of the main effects, it is an indication that the other main effect is not varying much (consider the extreme case where one of the “variables” was the constant!).
- 4Sometimes this is exactly the case and perfectly appropriate. For instance, including gross domestic product (GDP) without separate terms for price and quantity in a model specification implies that price and quantity are important only when considered together. A reader would be quite confused to see price × quantity in this situation as if the constituent parts of GDP had some significance beyond their joint contribution but were not worthy of consideration as individual explanatory variables.
- 5Notice that, according to Carey  and Shugart et al. , list type and district magnitude interact to affect the incentive of the politicians to cultivate a personal vote and this interaction effect makes open lists and close lists influence PVEA in opposite directions. The hypothesized interaction effect between list type and district magnitude is different from the one introduced by the link function. Specifically, the interaction effect introduced through the link function only changes the magnitude of the main effect as can be seen below in Figure 1. However, adding an interaction term might change not only the magnitude but also the direction of the main effect. This is because the effect of the interaction term overwhelms the main effect as we show in Section 6. Therefore, the model presented in Equation (7) is not used to test the theoretical argument but only for the purpose of illustration.
- 6Since the complete effect of the explanatory variables specified as interacting in the model is conditional on the values of the other co-interacting specified variables, the associated standard error is as well. The derivation of the conditional standard errors for the simple case when there are only two explanatory variables is provided by Friedrich (, p. 810) and Jaccard et al. (, p. 27), and an abstract form is given in various textbooks, e.g., Timm .
- 7Freely distributed software for the calculation of conditional standard errors is available as an R package, and further technical derivations, replication data, as well as analyses of related specifications are available at the authors’ webpage.
- 8In their paper, Shugart et al. test for statistical significance of the difference between and , by means of a test, which is statistically significant at . Based on this result, they state that the effect of district magnitude matters for both list types and claim that the slopes for district magnitude in the two list types are statistically distinct from each other. However, the test does not tell us that is positive and statistically significant: the test only shows that is distinct from , which is a less important result.
- 9In fact, the lack of statistical significance of the effects for district magnitude can be observed in Figure 2 in Shugart et al.’s paper, which is reproduced in the left panel of Figure 2. To see that this is true, notice that the predicted probability for open list systems at a given district magnitude is always covered by the 95% confidence intervals within the given range of district magnitudes, which suggests that, in a magnitude range of 5 to 40, the estimated values of are not different from one another. This cannot be observed by only looking at individual coefficients.
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