# GEE interaction interpretation

I’m trying to understand the interpretation of interaction terms, specifically in the context of GEE models. I’m familiar with them in conditional models, and am comfortable with marginal effects in the absence of an interaction, but the two together is causing me some trouble. I have illustrated an example below.

Take, for example, a longitudinal study looking at the effect of a drug (treatment) vs a placebo (placebo) on weight in kg (weight, continuous), and how the effect of treatment varies with time in years (time, continuous). If the GEE output was:

Variable | Coefficient (95% CI)

_cons | 70 (50,90)

treatment | -5 (-9,-1)

time | 1.1 (1.05,1.15)

treatment x time | 0.9 (-0.1,1.9)

Without the interaction, I understand that the treatment coefficient would be interpreted as the marginal effect of treatment on weight, as would that for a 1-year increase in time. I am, however, at a loss at to the marginal interpretation of time, treatment or the interaction between them. Any help would be greatly appreciated! Also, I am aware that conditional = marginal in the context of linear models but this is more to demonstrate the point before moving onto more complicated models.

Unfortunately I couldn’t find any questions covering this on the site

The individual coefficients here have the same interpretation (or difficulty in interpretation) as in conditional models with interactions or in generalized models. The differences are the interpretation as marginal instead of conditional associations with outcome for GEEs and, for generalized models, interpretations in terms of the link function associating the linear predictor with outcome. The problem in all model types with interactions is that the individual coefficients for predictors require care in interpretation.

The trick is to remember that everything ends up being coded as numeric, with the following general form for a 2-way interaction between predictors x and z with respect to an outcome y in a generalized linear model with link function g():

$$g(y)=\beta_0' + \beta_1'x+\beta_2'z +\beta_3' xz.$$

Thus the individual coefficient for an interacting predictor represents association with outcome when the other predictor is coded at 0. The interaction coefficient is the extra association with outcome when they both aren't coded at 0.

In this case, if x is treatment, it presumably has value 0 for placebo and 1 for treated. Your treatment coefficient is thus the estimated association of treatment with outcome when time = 0. That might not have a simple interpretation on its own. With your single linear term for time, it's a linear extrapolation to time = 0 from your other time points, in the overall context of the model. If time = 0 is the start of treatment, then a non-zero treatment coefficient might represent a difference in baseline outcome values between treatment groups or a mis-specified model of the association of time with outcome.

The time coefficient is the linear association between outcome and time for the placebo (treatment = 0). The interaction coefficient is the extra linear association of time with outcome with treatment = 1 and, conversely, the extra association of treatment with outcome for each extra unit of time.

In this type of study you typically need to model time more flexibly than with a single simple linear term. Chapter 7 of Frank Harrell's Regression Modeling Strategies covers longitudinal data modeling in some detail. It's mostly from the perspective of a different marginal modeling approach, generalized least squares, but the principles hold for other longitudinal models. The chapter also includes a useful summary table of the strengths and weaknesses of different modeling approaches.

• Amazing! Thank you so much. I think my confusion stemmed from the effects being marginal over the individuals in the study but having to fix time which, for some reason, I was thinking no longer made the interpretation marginal (but rather conditional). I will definitely give Harrell's book a read, thank you for the recommendation! :) Commented Mar 24, 2023 at 14:06
• @Andrew_99 as you move on to more complicated models, look at Harrell's post Unadjusted Odds Ratios are Conditional. That deals with the critical difference between marginal and conditional interpretations of models like binomial or Cox proportional-hazards regressions.
– EdM
Commented Mar 24, 2023 at 14:38