# Interactions in Generalized Linear Models: Theoretical Issues and an Application to Personal Vote-Earning Attributes

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## Abstract

**:**

## 1. Introduction

## 2. Interactions in Linear Models

^{1}This takes the following form where ${\beta}_{i}$ is the sample-derived estimate of the unknown population parameter ${\mathbb{B}}_{i}$:

^{2}

^{3}

^{4}Unfortunately, omitting lower-order terms means that common transformations, such as centering and standardization, actually substantively change the coefficient values: violating the principle of invariance [1,18]. In addition, the variance of such an interaction term is interpretable only in the context of the variance of its (now absent) main effect components [26].

## 3. Interactions in Generalized Linear Models

#### 3.1. The Effect of the Link Function

#### 3.2. Illustration of the Link Function Effect

^{5}

**Figure 1.**Probability models for personal vote-earning attributes. The left panel displays a standard linear model and the right panel displays a probit model.

#### 3.3. Interpreting Interaction Effects in Generalized Linear Models

#### 3.4. Reporting GLM Interaction Effects with First Differences

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## 4. A Methodological Controversy in Political Science

## 5. Higher-Order Interactions

## 6. An Application to Personal Vote-Earning Attributes

`native`), which is coded 1 if the legislator is native to the district in which the legislator is nominated, and 0 otherwise.

`open`) and district magnitude (

`logM`), respectively. The former is a dummy variable that takes the value of 1 if the list is open and 0 if it is closed; the latter is the decimal logarithm of district magnitudes. What is of most interest is that list type and district magnitude interact to affect the probability that a legislator exhibits PVEA. Therefore, an interaction term between the list type variable and the district magnitude variable is included in the model.

`rookie`), which is coded 1 if a legislator is at the first term, and 0 otherwise. It is expected that the effect of magnitude and list type are greater for rookie legislators than for veterans. To test this hypothesis, a second-order interaction model is specified.

#### 6.1. The First-order Interaction Model

^{7}The standard error of ${\beta}_{1}+{\beta}_{2}$ conditional on $\mathsf{open}=1$ is about 0.27, which fails to give a statistically significant estimate of the effect of district magnitude for open lists. This result suggests that, in an open list, the probability that legislators exhibit PVEA does not increase with magnitude. While the hypothesized effect of magnitude in closed lists is supported, the one in open lists is not, which is inconsistent with the finding in Shugart et al.

^{8}

Model 1 | Model 2 | |||
---|---|---|---|---|

Estimate | Std. Error | Estimate | Std. Error | |

Constant (${\beta}_{0}$) | 0.74 | 0.14 | 0.69 | 0.18 |

$log\mathsf{M}$ (${\beta}_{1}$) | −0.49 | 0.11 | −0.50 | 0.14 |

$log\mathsf{M}*\mathsf{open}$ (${\beta}_{2}$) | 0.63 | 0.29 | 0.33 | 0.39 |

open (${\beta}_{3}$) | −0.43 | 0.34 | −0.04 | 0.47 |

rookie (${\beta}_{4}$) | 0.11 | 0.26 | ||

$log\mathsf{M}*\mathsf{rookie}$ (${\beta}_{5}$) | 0.02 | 0.20 | ||

$log\mathsf{M}*\mathsf{open}*\mathsf{rookie}$ (${\beta}_{6}$) | 0.36 | 0.56 | ||

$\mathsf{open}*\mathsf{rookie}$ (${\beta}_{7}$) | −0.50 | 0.64 | ||

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. | ||

AIC | 1479.7 | 1464.2 |

^{9}Moreover, comparing the left panel of Figure 2 with the right panel of Figure 1, we observe a remarkable difference between the parameter-based interaction and the link function-based interaction. Specifying an interaction term in the model provides a meaningful interpretation of interactive effects in a theoretically-informed manner.

**Figure 2.**The first-order interaction model. The left panel displays the interactive effect and the right panel displays first differences across districts. That the 95% confidence intervals cover the red line at zero means a lack of statistical significance for first differences.

#### 6.2. The Second-Order Interaction Model

## 7. Conclusions

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.

## Acknowledgements

## Conflict of Interest

## References

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^{1}Unfortunately, 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 [14] and Finney [15]. So this term will be used here.^{2}These models use algebraic forms where components such as exponents are estimated (and hence are different from standard polynomial regression, e.g., Fox ([19], S. 14.2)) in order to make local approximations to continuous, multidimensional response surfaces. For extended discussions, see Box and Draper [20], Khuri and Cornell [21], and Cornell and Montgomery [22].^{3}It is also interesting to note that when the multiplicative interaction term ${X}_{i1}{X}_{i2}$ 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!).^{4}Sometimes 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.^{5}Notice that, according to Carey [41] and Shugart et al. [39], 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.^{6}Since 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 ([1], p. 810) and Jaccard et al. ([45], p. 27), and an abstract form is given in various textbooks, e.g., Timm [46].^{7}Freely 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.^{8}In their paper, Shugart et al. test for statistical significance of the difference between ${\beta}_{1}$ and ${\beta}_{2}$, by means of a ${\chi}^{2}$ test, which is statistically significant at $p<0.05$. 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 ${\chi}^{2}$ test does not tell us that ${\beta}_{1}+{\beta}_{2}$ is positive and statistically significant: the test only shows that ${\beta}_{1}$ is distinct from ${\beta}_{2}$, which is a less important result.^{9}In 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 $Pr\left(\mathsf{native}\right)$ are not different from one another. This cannot be observed by only looking at individual coefficients.

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**MDPI and ACS Style**

Tsai, T.-h.; Gill, J.
Interactions in Generalized Linear Models: Theoretical Issues and an Application to Personal Vote-Earning Attributes. *Soc. Sci.* **2013**, *2*, 91-113.
https://doi.org/10.3390/socsci2020091

**AMA Style**

Tsai T-h, Gill J.
Interactions in Generalized Linear Models: Theoretical Issues and an Application to Personal Vote-Earning Attributes. *Social Sciences*. 2013; 2(2):91-113.
https://doi.org/10.3390/socsci2020091

**Chicago/Turabian Style**

Tsai, Tsung-han, and Jeff Gill.
2013. "Interactions in Generalized Linear Models: Theoretical Issues and an Application to Personal Vote-Earning Attributes" *Social Sciences* 2, no. 2: 91-113.
https://doi.org/10.3390/socsci2020091