What a Marginal GEE Coefficient Describes
A generalized estimating equation (GEE) models the mean response for a population, conditional on specified observed covariates. For a response \(Y_{ij}\) from subject or cluster \(i\) at occasion \(j\), write this mean as \(\mu_{ij}=E[Y_{ij}\mid X_i]\), where \(X_i\) denotes the observed covariates included for that cluster. A marginal mean model specifies \(g(\mu_{ij})=x_{ij}^{\mathsf T}\beta\). The expectation averages over response variability and over unobserved subject characteristics, conditional on the covariates in the model.
Thus, “population-average” does not mean an unadjusted average across everyone, nor does it mean an average over every possible covariate value. It means a conditional mean for the population represented by the specified covariates and sampling target. The coefficient describes how that modeled mean changes with those covariates. The GEE working correlation can help account for dependence when estimating the mean parameters, but it does not turn the marginal mean into a model of each subject’s latent risk or response history.
The link determines the scale of interpretation. With an identity link, a coefficient describes a difference in the modeled population mean per unit change in a covariate. With a log link, it describes a change in the log of that mean. With a logit link for a binary response, it describes a change in the log odds of the population-averaged response probability. These are interpretations of the marginal mean—not statements that every subject has the same response probability or experiences the same change.
Worked Example: Reading a Marginal Logit Coefficient
Suppose a GEE for a binary outcome uses \( \operatorname{logit}\{E[Y_{ij}\mid X_{ij}]\}=\beta_0+\beta_1X_{ij}\), where \(X_{ij}\) is an indicator for exposure. Let the fitted exposure coefficient be \(\hat\beta_1=0.40\).
The population-average odds ratio comparing the exposed and unexposed covariate profiles is \(\exp(0.40)\approx1.492\). The modeled odds are therefore about 1.49 times as large for the exposed profile. This is a comparison of marginal odds conditional on the specified covariates.
It is not correct to say that the exposure increases each person’s probability by 0.40. The coefficient is on the log-odds scale, and the corresponding change in probability depends on the baseline probability. For instance, if the unexposed marginal probability is \(0.20\), its odds are \(0.20/0.80=0.25\). Multiplying by 1.492 gives exposed odds \(0.373\), corresponding to probability \(0.373/(1+0.373)\approx0.272\), a difference of about \(0.072\). At a different baseline probability, the probability difference would be different.
Even this probability comparison is for the modeled population mean. It does not say which exposed individuals will have the outcome, or that every individual’s probability changes by the same amount.
Why Marginal and Subject-Specific Effects Can Differ
A subject-specific model conditions on a subject-level characteristic, often an unobserved random effect. A marginal model averages over that characteristic. When the link is nonlinear, averaging and applying the inverse link generally do not commute. Consequently, a coefficient in a conditional model need not equal the corresponding coefficient in a marginal GEE, even when both models use the same covariate.
The distinction is not a defect in either model: the parameters answer different questions. A conditional coefficient compares subjects held at the same random-effect value. A marginal coefficient compares covariate-defined groups after averaging across the distribution of those effects. If the scientific question concerns an individual’s conditional response process, a marginal coefficient alone does not supply the answer.
Worked Example: A Conditional Odds Ratio Is Not the Marginal Odds Ratio
Let \(X\) be binary and let a subject-specific logistic model be \(P(Y=1\mid X,U)=\operatorname{expit}(\beta_0+\beta_1X+U)\), where \(U\) equals \(-\log 3\) or \(+\log 3\), each with probability \(1/2\), independently of \(X\). Set \(\beta_0=0\) and \(\beta_1=\log 2\).
At either fixed value of \(U\), increasing \(X\) from 0 to 1 multiplies the conditional odds by \(\exp(\beta_1)=2\). For \(X=0\), the two conditional probabilities are \(\operatorname{expit}(-\log3)=1/4\) and \(\operatorname{expit}(\log3)=3/4\). Averaging gives the marginal probability \(P(Y=1\mid X=0)=(1/4+3/4)/2=1/2\), with marginal odds 1.
For \(X=1\), the linear predictors are \(\log2-\log3=\log(2/3)\) and \(\log2+\log3=\log6\). The corresponding probabilities are \(2/5\) and \(6/7\), so the marginal probability is \[ \{2/5+6/7\}/2=\{14/35+30/35\}/2=22/35. \] The marginal odds are \((22/35)/(13/35)=22/13\), giving marginal odds ratio \(22/13\approx1.692\), not 2. The difference arises because the marginal odds are calculated after averaging the probabilities over \(U\); averaging does not preserve the conditional odds ratio.
This example shows why a marginal GEE logit coefficient should not be reported as a subject-specific log-odds effect. Recovering a conditional effect requires additional modeling assumptions about the latent heterogeneity and its distribution.
There are special cases in which conditional and marginal slopes do agree. For example, an identity-link model with an additive random intercept independent of the covariates preserves the slope after averaging. This is a useful contrast, not a general property of GEE.
Proof. By iterated expectation and the assumed conditional mean, \[ E[Y\mid X]=E[E[Y\mid X,U]\mid X] =E[\alpha+\beta X+U\mid X] =\alpha+\beta X+E[U\mid X]. \] Independence gives \(E[U\mid X]=E[U]\), which is finite by the integrability assumption. Hence \(E[Y\mid X]=\alpha+E[U]+\beta X\), with slope \(\beta\). If \(E[U]=0\), this reduces to \(\alpha+\beta X\). \(\square\)
Independence and the additive identity-link form matter. If the random effect’s distribution changes with \(X\), then \(E[U\mid X]\) need not be constant and the marginal slope can differ. Nor does this result justify transferring the same interpretation to a nonlinear link.
Worked Example: When an Identity-Link Slope Is Preserved
Suppose a continuous outcome follows \(E[Y\mid X,U]=10+3X+U\), with \(E[U]=0\) and \(U\) independent of \(X\). At \(X=0\), the conditional mean given \(U\) is \(10+U\); at \(X=2\), it is \(16+U\). Averaging over \(U\), the marginal means are 10 and 16, respectively.
The marginal mean difference is \(16-10=6\) across a two-unit increase in \(X\), or \(6/2=3\) per unit. This matches the conditional slope. The calculation does not show that every subject’s observed outcome rises by 3: \(Y\) includes response variation, and the equality concerns expected means under the stated assumptions.
Why a Marginal Mean Is Not an Individual Prediction
A GEE mean can give a valid prediction of \(E[Y_{ij}\mid X_i]\) under its mean-model assumptions. That is a prediction of an average response for subjects with the specified covariates. It is not a prediction of a particular subject’s next outcome conditional on their latent traits or past responses. Such a prediction requires a model for the relevant conditional distribution, including how past outcomes relate to future outcomes and, where relevant, subject-specific heterogeneity.
A mean model also does not uniquely determine a joint response distribution. Even knowing all marginal means and pairwise covariances may be insufficient to determine how a subject’s future outcome depends on a longer response history. The following construction makes that limitation explicit.
Worked Example: Same Means and Pairwise Covariances, Different Conditional Prediction
Consider three binary responses \(Y_1,Y_2,Y_3\). In Model A, all eight possible response patterns have probability \(1/8\), so the responses are independent Bernoulli variables with probability \(1/2\). In Model B, assign a response pattern \(y=(y_1,y_2,y_3)\) probability \(1/8+(1/16)(-1)^{y_1+y_2+y_3}\). Every probability is either \(3/16\) or \(1/16\), so this is a valid distribution.
For either model, each response has probability \(1/2\): among patterns with a given \(Y_j=1\), the alternating signs in Model B cancel in pairs. Each pair \((Y_j,Y_k)\) also has all four values with probability \(1/4\), because, for each fixed pair, summing the added terms over the third response gives zero. Thus both models have means \(1/2\), pairwise covariance zero, and pairwise correlation zero.
Yet the conditional prediction after observing \(Y_1=Y_2=1\) differs. In Model A, \(P(Y_3=1\mid Y_1=Y_2=1)=1/2\) by independence. In Model B, \(P(Y_1=Y_2=Y_3=1)=1/16\), while \(P(Y_1=Y_2=1)=1/4\). Therefore \(P(Y_3=1\mid Y_1=Y_2=1)=(1/16)/(1/4)=1/4\).
The marginal means and pairwise covariances do not distinguish these conditional predictions. This does not mean that a GEE cannot estimate a marginal mean; it shows why mean parameters, even accompanied by pairwise association information, do not by themselves specify individual response histories.
Scientific Questions a Marginal GEE Cannot Answer Alone
A marginal association can be useful for describing how average response differs across covariate profiles. It cannot, without additional design assumptions and models, answer every scientific question that might motivate the study.
- Which particular subjects will respond? A mean probability describes a group average, not a ranking or a subject-specific risk.
- What is the effect for a given individual? A population-average contrast does not identify individual treatment effects or the distribution of those effects.
- Is the association causal? A GEE coefficient is not causal merely because it adjusts for recorded covariates. A causal interpretation requires appropriate assumptions about treatment assignment, confounding, consistency, positivity, and the target population.
- What would happen under an unobserved response history? A marginal mean model does not, on its own, specify the full conditional distribution needed to answer this question.
- What target population does the coefficient represent? Interpretation depends on the study’s sampling and weighting target. In clustered data, the population represented by the estimating procedure must be stated rather than assumed to be every individual or every cluster equally.
These are boundaries of interpretation, not reasons to avoid GEE. A well-defined marginal estimand is often exactly what a policy or public-health question needs—for example, the difference in average outcome under two population-level exposure conditions. The key is to align the claim with the mean model and the target population, and to use a different modeling strategy when the question is explicitly subject-specific.
Common Mistake: Calling a Marginal Coefficient an Individual Effect
Wrong reasoning: “The GEE exposure coefficient is 0.40, so exposure raises every subject’s log odds by 0.40.” This treats a population-average coefficient as if it were conditional on each subject’s latent characteristics. In nonlinear models, the example above shows that the marginal and conditional odds ratios can differ even when the conditional model is correctly specified.
Correction: State the modeled scale and target: “The fitted marginal log odds differ by 0.40 between these covariate profiles,” or, after exponentiation, “the population-average odds ratio is about 1.49.” If the question concerns an individual’s conditional effect, prediction after a personal history, or causal effects, specify the additional assumptions and model needed to address it. Do not infer those quantities from the marginal coefficient alone.
Check Your Understanding
Answer each question by identifying the estimand and the extra information, if any, that would be needed.
- In a logit-link GEE, what does exponentiating a binary exposure coefficient describe, and what does it not describe?
- Why can a conditional logistic-regression coefficient differ from a marginal GEE coefficient after averaging over a random intercept?
- Under what assumptions does an additive random intercept preserve an identity-link slope after marginalization?
- What is the difference between predicting a conditional population mean and predicting a particular subject’s next response?
- Can a marginal association from GEE be called causal without further assumptions? Explain briefly.