Interpreting Interaction Effects in Generalized Linear Models of Nonlinear Probabilities and Counts

Abstract

Psychology research frequently involves the study of probabilities and counts. These are typically analyzed using generalized linear models (GLMs), which can produce these quantities via nonlinear transformation of model parameters. Interactions are central within many research applications of these models. To date, typical practice in evaluating interactions for probabilities or counts extends directly from linear approaches, in which evidence of an interaction effect is supported by using the product term coefficient between variables of interest. However, unlike linear models, interaction effects in GLMs describing probabilities and counts are not equal to product terms between predictor variables. Instead, interactions may be functions of the predictors of a model, requiring nontraditional approaches for interpreting these effects accurately. Here, we define interactions as change in a marginal effect of one variable as a function of change in another variable, and describe the use of partial derivatives and discrete differences for quantifying these effects. Using guidelines and simulated examples, we then use these approaches to describe how interaction effects should be estimated and interpreted for GLMs on probability and count scales. We conclude with an example using the Adolescent Brain Cognitive Development Study demonstrating how to correctly evaluate interaction effects in a logistic model.

Keywords: Generalized linear modeling; Poisson; interaction; logistic regression; moderation.

Figure 1.

Logistic model regressing differing scales of a dependent variable on a predictor. Note. Gray areas indicate 95% confidence regions.

Figure 2.

Poisson model regressing differing scales of a dependent variable on a predictor. Note. Gray areas indicate 95% confidence regions.

Figure 3.

The relation between $\widehat{\mathbb{E}}[Y\mid x]$ and x₁ across low, median, and high levels of x₂ by biological sex.

Figure 4.

Interaction between x₁ and biological sex as a function of x₁ plotted at low (below median) and high (above median) values of x₂. Note. Black points indicate significant effects and gray points indicate non-significant effects at α = 0.05. This figure illustrates that the interaction effect between x₁ and biological sex varied across observations, such that the effect was larger for observations with high values of the covariate x₂. This highlights that the interaction effect (as well as its statistical significance) may be conditioned on predictors involved in the model. N.S. = Not Significant, Sig. = Significant.

Figure 5.

The relation between $\widehat{\mathbb{E}}[Y\mid x]$ and x₁ across low, median, and high levels of x₂ in a logistic model. Note. This figure represents multiple interaction effects of opposite sign present within the data. We illustrate this is using one-unit rates of change. For instance, in describing one-unit rates of change, $\widehat{\mathbb{E}}[Y\mid x]$ increases by 0.037 as x₁ increases from 1 to 2 when x₂ is at the 75^th percentile, but this increase is over 3 times larger (0.114) when examined at the 25^th percentile of x₂. In contrast, at the lower end of the x₁ range, $\widehat{\mathbb{E}}[Y\mid x]$ increases by 0.160 units as x₁ increases from −2 to −1 at the 75^th percentile of x₂, but the increase is smaller (0.111) at the 25^th percentile of x₂. This illustrates that interactions can have differing signs in the same model resulting from the nonlinear nature of the model.

Figure 6.

The relation between $\widehat{\mathbb{E}}[Y\mid x]$ and x₁ across low, median, and high levels of x₂ in a Poisson model. Note. This figure represents the interaction effect in a Poisson model despite omission of a product term. For instance, the effect of a one-unit increase in x₁ from 0 to 1 on $\widehat{\mathbb{E}}[Y\mid x]$ was greater at higher levels of x₂ (0.023 at the 75^th percentile of x₂) compared to lower levels (0.006 at the 25^th percentile of x₂).

Figure 7.

Empirical bias and mean squared error for ${\widehat{\gamma }}_{12}^2$ and ${\widehat{\beta }}_{12}$ as estimators of ${\gamma }_{12}^2$ across conditions of β₁₂. Note. Points represent values of each estimate averaged across 10,000 simulated datasets for each β₁₂ condition.

Figure 8.

Marginal effects of x₁ plotted against different scalings of $\widehat{Y}$ at low and high values of x₂. Note. This figure illustrates how interaction effects can be of opposite sign depending on the scaling choice for inference. For instance, in the count scale (left-hand side), the curves presented are growing farther apart as x₁ increases, indicating the synergistic interaction on the natural scale. By contrast, on the log-count scale (right hand side), the lines are coming closer together as x₁ increases, indicating the antagonistic interaction on the transformed scale.

Figure 9.

The relation between the predicted probability of alcohol sipping and parental monitoring across low, median, and high levels of school disengagement among Hispanic females by race. Note. Sc.Dis. = School Disengagement.