What GEE Means by a Population-Averaged Mean
Repeated observations from the same person, clinic, school, or other cluster may be dependent. A generalized estimating equation (GEE) addresses this dependence while specifying the response mean for a target population. The key feature is that the mean is averaged over unobserved cluster effects rather than defined at a fixed value of such an effect.
Let \(Y_{ij}\) be the response for observation \(j\) in cluster \(i\), and let \(X_i\) denote the covariates recorded for that cluster. A GEE mean model relates the expected response to the covariates. Here, “population-averaged” means averaged over unobserved heterogeneity in the target population at the specified covariate values. It does not mean averaged over the distribution of covariates: the mean can still differ across covariate values.
This differs from a subject-specific GLMM mean, which is conditional on latent random effects as well as observed covariates. GEE does not need to introduce a latent effect into its mean equation. Consequently, specifying a GEE mean model is not the same as specifying a probability distribution for all responses jointly: the mean is central, while within-cluster dependence is represented separately for estimation.
Specifying the Mean Model
The link function sets the scale on which the covariates enter the mean. For a binary response, a logit link can model the population-averaged probability. For counts, a log link can model a positive population-averaged mean. For a continuous response, an identity link may model the mean directly. The chosen link and covariates should reflect the scientific question and the possible values of the response.
The model does not assert that each cluster has the same response mean in every realization. Rather, it describes the mean across the target population at the covariate values in the equation. For example, a treatment coefficient describes a contrast between modeled means for different treatment values, with other included covariates held fixed. Its numerical interpretation depends on the link; detailed link-specific interpretations require care.
GEE uses an estimating equation to estimate \(\beta\). For cluster \(i\), collect its responses in \(Y_i\), its means in \(\mu_i\), and define \(D_i=\partial\mu_i/\partial\beta^T\). A working covariance matrix \(V_i\) represents the response variances and an assumed pattern of within-cluster association. The usual estimating equation is:
The covariance structure used in \(V_i\) is called “working” because it supports estimation without requiring that the full joint distribution of a cluster’s responses be correctly specified. It can affect efficiency. With independent clusters and a correctly specified mean, a sandwich variance estimate can provide standard errors that remain valid under some misspecification of the working covariance, provided there are enough independent clusters and the other regularity conditions hold.
Worked Example: A Population-Averaged Binary Mean
Suppose a hypothetical study follows patients in 80 clinics and records whether each patient experiences an event during follow-up. Let \(Y_{ij}=1\) for an event and \(0\) otherwise, and let \(T_{ij}=1\) indicate treatment. Assume the fitted GEE mean model is a logit model with intercept \(-1.386\) and treatment coefficient \(0.693\).
The response mean is the event probability at the recorded covariates, averaged over unobserved clinic-level heterogeneity. With treatment \(T_{ij}=0\), the fitted mean is:
For \(T_{ij}=0\), the linear predictor is \(-1.386\), so \(\mu_{ij}=1/(1+\exp(1.386))\approx 0.200\). For \(T_{ij}=1\), it is \(-1.386+0.693=-0.693\), so \(\mu_{ij}=1/(1+\exp(0.693))\approx 0.333\). Thus, at these covariate values, the model gives population-averaged event probabilities of about 20.0% and 33.3%, respectively.
These are fitted marginal means, not probabilities conditional on a particular clinic’s latent effect. The model’s adequacy depends on a suitable mean structure and link; clinics are the independent clusters, and the 80 clinics—not the total number of patient records—govern the amount of independent information for large-sample inference.
Worked Example: A Population-Averaged Count Mean
Imagine a hypothetical study recording annual event counts for participants in 60 clinics. Each participant has one year of follow-up, and \(T_{ij}\) indicates a program that is either absent or present. Suppose a GEE with a log link gives an intercept of \(\log(1.5)\) and a program coefficient of \(\log(1.4)\).
The modeled mean count for a participant with program status \(T_{ij}\) is:
Without the program, \(T_{ij}=0\), so \(\mu_{ij}=\exp(\log(1.5))=1.5\) events per participant-year. With the program, \(T_{ij}=1\), so:
The two fitted population-averaged means are 1.5 and 2.1 events per participant-year. These are means across the target population at the stated program values, not means conditional on a particular clinic effect. The count distribution and variance pattern should be considered when specifying the analysis; a log-link mean equation alone does not supply a complete joint distribution for all counts.
Worked Example: A Population-Averaged Continuous Mean
Suppose a hypothetical set of students is measured repeatedly in 100 schools, and the response is a test score in points. Let \(V_{ij}=1\) indicate a later visit and \(0\) an initial visit. A GEE identity-link model gives an intercept of 72 points and a visit coefficient of 2.5 points.
With the identity link, the model specifies the mean directly:
At the initial visit, \(V_{ij}=0\), giving a fitted population-averaged mean of \(72+2.5(0)=72\) points. At the later visit, \(V_{ij}=1\), giving \(72+2.5(1)=74.5\) points. The difference between these fitted means is \(74.5-72=2.5\) points.
This model concerns the mean response across the target population at each visit, not the expected score conditional on a school-specific latent effect. A linear mean is sensible only if it is adequate over the covariate range; predicted values should also be checked for plausibility in the score’s possible range.
Why the Estimating Equation Targets the Marginal Mean
The central justification is that the estimating equation has mean zero when the specified marginal mean is correct. Conditional on the observed covariates \(X_i\), correct specification gives \(E(Y_i-\mu_i\mid X_i)=0\). If \(D_i\) and the working covariance \(V_i\) are functions of the covariates and model parameters, then:
Summing these zero-mean contributions over independent clusters yields an estimating equation centered on the population-averaged mean parameter. This argument explains why the working covariance need not equal the true covariance for the equation to target the same mean parameter. A wrong mean model, however, generally means \(E(Y_i-\mu_i\mid X_i)\ne 0\), so this justification fails.
In large samples, the sandwich covariance estimate accounts for variation in the cluster-level estimating contributions. In schematic form it uses an “inverse sensitivity” matrix on both sides of the empirical variability matrix. This corrects standard errors for a misspecified working covariance under suitable conditions; it does not repair a misspecified mean, dependent clusters, or too few independent clusters.
Assumptions, Diagnostics, and Alternatives
A useful GEE analysis requires more than choosing a link. Its main conditions include:
- Independent clusters. Responses may be dependent within a cluster, but the clusters used as independent units should be independent, or their dependence should be addressed by the design and analysis.
- A suitable marginal mean model. The link, covariates, and functional forms should adequately represent \(E(Y_{ij}\mid X_i)\) for the target population.
- Appropriate response support and moments. The mean and variance specification should make sense for the response type, and the required moments should exist.
- Enough independent clusters for large-sample inference. Sandwich standard errors rely on large-sample approximations based on clusters. A large number of observations concentrated in only a few clusters is not equivalent to many independent clusters.
- Suitable handling of missing responses. Missingness can change which observations contribute to the estimating equation; standard GEE does not automatically eliminate bias from informative missingness.
Diagnostics should focus first on the mean: compare observed and fitted responses across important covariate patterns, inspect residual patterns, and check that the link and functional forms are plausible. Also examine whether clusters are genuinely the independent sampling units, how many there are, and whether a few clusters dominate the estimates. Sensitivity to reasonable working-correlation choices can be informative, but similar results under several choices do not prove that the mean model is correct.
The main alternative in this part of the analysis is a GLMM with a conditional mean specified given latent effects. That approach explicitly models cluster heterogeneity and can support cluster-specific questions, but its mean is not automatically the GEE population-averaged mean. GEE is useful when the target is a population-level mean relationship and a full latent-effects distribution is not the primary modeling goal. Neither approach is universally preferable: the estimand and assumptions should guide the choice.
A common mistake is to say that GEE “ignores” clustering because it does not condition on random effects. It does not: the estimating equation uses a working covariance to represent within-cluster association, and inference must respect independent clusters. The correction is to distinguish the marginal mean model from the working dependence structure. The first defines the target mean; the second helps account for correlated observations when estimating it.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- In \(g\{E(Y_{ij}\mid X_i)\}=x_{ij}^{T}\beta\), what is averaged over, and what is conditioned on?
- A binary-response GEE gives a fitted mean of 0.25 for a covariate pattern. Is this a probability conditional on a particular latent clinic effect? Explain.
- For a log-link count model, the linear predictor is \(\log(2)+\log(1.5)T\). Calculate the fitted mean when \(T=0\) and when \(T=1\).
- Why can a sandwich standard error help when the working covariance is wrong, yet fail to fix a wrong mean model?
- A study has 2,000 repeated records from six clinics. Why is the record count alone insufficient to establish reliable large-sample GEE inference?
Solutions. (1) The mean is averaged over unobserved cluster effects and conditioned on the observed covariates \(X_i\). (2) No. It is the covariate-specific population-averaged event probability, not a mean conditional on one clinic effect. (3) At \(T=0\), \(\mu=\exp(\log 2)=2\); at \(T=1\), \(\mu=\exp\{\log 2+\log 1.5\}=2(1.5)=3\). (4) The sandwich can provide robust variance estimation under working-covariance misspecification, but it still depends on the residual mean being correctly modeled; if the mean is wrong, the estimating equation need not be centered at the target parameter. (5) The large-sample approximation relies on independent clusters. Six clusters provide few independent units, regardless of the number of repeated records within them.