Reporting Conditional And Marginal Results

Why Results Need More Than an Estimate

A number such as 1.8 is not a complete result. It could be an odds ratio conditional on a clinic-specific random effect, a population-averaged odds ratio from a GEE, or a different effect measure altogether. Those quantities answer different questions, even when they arise from the same clustered data.

A useful results statement identifies five things: the model, the estimand (the quantity the analysis targets), the effect scale, the target population and comparison, and the uncertainty measure. This is especially important when comparing GLMM and GEE results. A GLMM usually models an outcome conditional on random effects; a GEE specifies a marginal mean relationship for the outcome given covariates and accounts for within-cluster association separately through a working correlation.

Definition: A conditional effect from a GLMM describes a covariate association conditional on the model's random effects. A marginal effect from a GEE describes the specified mean outcome given covariates, without conditioning on a cluster-specific random effect. The GEE working correlation represents within-cluster association; it does not define the marginal mean by averaging over that association.

State the Model and Its Assumptions

Let \(Y_{ij}\) be the outcome for individual \(i\) in cluster \(j\), and let \(x_{ij}\) be a predictor. For a binary outcome, a random-intercept logistic GLMM can be written as

Y_{ij}\mid b_j,x_{ij}\sim \operatorname{Bernoulli}(p_{ij}),\qquad \operatorname{logit}(p_{ij})=\beta_0+\beta_1x_{ij}+b_j,\qquad b_j\sim N(0,\sigma_b^2).

This specification assumes the conditional Bernoulli distribution and logit mean model are appropriate, clusters are independent, and outcomes are conditionally independent within clusters given the random effects and covariates. The random-intercept distribution is part of the likelihood model. Here, \(\exp(\beta_1)\) is the conditional odds ratio for a one-unit increase in \(x\), holding the random intercept fixed.

For a binary-outcome GEE, the marginal mean is specified directly, for example:

\operatorname{logit}\{E(Y_{ij}\mid x_{ij})\}=\gamma_0+\gamma_1x_{ij}.

The mean model should be correctly specified for the target covariate pattern, and clusters should be independent. A working correlation, such as exchangeable correlation, describes the assumed within-cluster association used in estimation. A sandwich variance estimator can protect large-sample standard errors against misspecification of that working correlation, provided the mean model is correct and there are enough independent clusters for the approximation to be reliable. It does not repair a misspecified mean model.

Reporting principle: Name an estimate by its model and target. For example, write “conditional odds ratio from a logistic random-intercept GLMM” or “marginal odds ratio from a logistic GEE,” rather than simply “odds ratio.”

Why Conditional and Marginal Effects Can Differ

The two models do not generally target the same quantity. In a random-intercept logistic GLMM, the conditional probability at covariate value \(x\) and random effect \(b\) is \(\operatorname{expit}(\beta_0+\beta_1x+b)\). A population mean implied by that GLMM would require integrating this probability over the random-effect distribution. In contrast, a GEE specifies the marginal mean relationship directly; it does not first specify a random-effect distribution.

The link function explains why the GLMM's conditional coefficient cannot simply be relabelled as marginal. In general, integrating a nonlinear mean function and then applying the link does not return the original linear predictor:

\operatorname{logit}\left\{E_b\left[\operatorname{expit}(\beta_0+\beta_1x+b)\right]\right\} \ne \beta_0+\beta_1x.

Thus, the conditional odds ratio \(\exp(\beta_1)\) need not equal a marginal odds ratio. This is not evidence that one model is automatically wrong. The results answer different questions, and a report should not compare them as if they were estimates of one identical estimand.

Worked Example: Reporting a Conditional GLMM Odds Ratio

Suppose a hypothetical study follows patients in 120 clinics and models whether each patient experiences a binary outcome. Treatment is coded \(x=1\) for the new treatment and \(x=0\) for usual care. A logistic random-intercept GLMM estimates \(\hat{\beta}_1=0.693\) with model-based \(SE=0.200\). Assume the conditional mean model and random-intercept assumptions stated above are the intended analysis.

The conditional odds ratio is

\exp(\hat{\beta}_1)=\exp(0.693)=2.00.

A 95% Wald interval is first calculated on the log-odds scale:

0.693\pm1.96(0.200)=0.693\pm0.392=(0.301,\;1.085).

Exponentiating both endpoints gives the interval on the odds-ratio scale:

\left(\exp(0.301),\exp(1.085)\right)=(1.351,\;2.960).

The Wald statistic is \(z=0.693/0.200=3.465\), giving a two-sided p-value of approximately 0.00053. A suitable report is: “In a logistic random-intercept GLMM, treatment was associated with higher conditional odds of the outcome: conditional OR 2.00, 95% model-based Wald CI 1.35 to 2.96, \(p<0.001\). This compares treatment with usual care for patients with the same clinic random intercept, under the fitted model.” The interval concerns a conditional odds ratio, not a probability difference or a marginal odds ratio.

Report a GEE Result as Marginal

A GEE report should identify the marginal mean model, the effect scale, and how uncertainty was estimated. For a logit-link GEE, exponentiating a treatment coefficient gives a marginal odds ratio under that specified mean model. “Marginal” means that the model is for the outcome mean given covariates, not conditional on a cluster-specific random effect. Within-cluster association is handled separately through the working correlation and variance estimation.

Worked Example: Reporting a Marginal GEE Odds Ratio

In a second hypothetical study, patients are enrolled at 80 clinics. A logistic GEE with exchangeable working correlation estimates a treatment coefficient of \(\hat{\gamma}_1=0.405\), with cluster-robust sandwich \(SE=0.150\). Suppose the marginal mean model is appropriate and the 80 independent clinics are adequate for the large-sample sandwich approximation.

The marginal odds ratio is \(\exp(0.405)=1.50\). Its 95% Wald interval is obtained on the coefficient scale and transformed:

0.405\pm1.96(0.150)=0.405\pm0.294=(0.111,\;0.699),

\left(\exp(0.111),\exp(0.699)\right)=(1.117,\;2.012).

The Wald statistic is \(z=0.405/0.150=2.70\), with two-sided \(p\approx0.0069\). A suitable statement is: “A logistic GEE with exchangeable working correlation estimated a marginal treatment odds ratio of 1.50 (95% cluster-robust Wald CI 1.12 to 2.01; \(p=0.007\)). Under the specified marginal mean model, the treatment group had higher odds of the outcome than the usual-care group in the target population represented by the enrolled clinics, adjusting for the stated covariates.” The working correlation accounts for within-clinic association; it is not the mechanism that makes the mean marginal.

Report the Scale and Units Explicitly

Odds ratios and rate ratios are dimensionless, but the underlying outcome and comparison still need to be named. For a continuous outcome with an identity link, a coefficient is instead a mean difference in the outcome's units. For a count outcome with a log link and a follow-up-time offset, exponentiating a coefficient yields a rate ratio. State the time unit and the population to which the modeled mean applies.

Worked Example: Reporting a Marginal GEE Rate Ratio

Consider a hypothetical study of recurrent events observed over different lengths of follow-up in patients from 60 independent clinics. A log-link Poisson GEE includes the log of each patient's observation time as an offset. The treatment coefficient is \(\hat{\gamma}_1=0.262\), with cluster-robust \(SE=0.105\). Assume the log mean model for the event rate is suitable, clusters are independent, and the number of clinics supports the sandwich approximation.

Exponentiating the coefficient gives the marginal rate ratio: \(\exp(0.262)=1.300\). The 95% interval on the coefficient scale is

0.262\pm1.96(0.105)=0.262\pm0.2058=(0.0562,\;0.4678).

After exponentiation, the interval is \((\exp(0.0562),\exp(0.4678))=(1.058,\;1.597)\). The test statistic is \(z=0.262/0.105=2.495\), with two-sided \(p\approx0.0126\). A clear report is: “The log-link Poisson GEE estimated a marginal event-rate ratio of 1.30 for treatment versus usual care (95% cluster-robust Wald CI 1.06 to 1.60; \(p=0.013\)). The modeled event rate was 30% higher in the treatment group, in the target population represented by these clinics, at the same adjustment covariates.” The rate unit is events per unit of follow-up time; the offset determines that unit and should be stated in the full report.

Choose and Name the Uncertainty Measure

A confidence interval should be reported with its level, scale, and variance method. For example, a GLMM may report a model-based Wald interval, a profile-likelihood interval, or a bootstrap interval. A GEE commonly reports a sandwich-based interval, but the exact method should be named. Wald intervals use an approximate normal distribution for the coefficient estimate; transforming both endpoints gives an interval on a ratio scale. A p-value should identify the test, or at least make clear that it is a two-sided Wald test when that is what was used.

A 95% confidence interval is produced by a procedure that, under its assumptions and repeated sampling, covers the target parameter in 95% of repetitions. It does not mean there is a 95% probability that this particular interval contains the fixed parameter. Likewise, a p-value is calculated under a specified null hypothesis; it is not the probability that the null hypothesis is true.

The uncertainty method also has limitations. Wald inference can be unreliable with few independent clusters, sparse binary outcomes, estimates near parameter boundaries, or very large standard errors. A sandwich estimator is not automatically reliable merely because it is called robust: with few clusters its standard error can be biased downward. Check convergence, the number of independent clusters, sparse cells, implausible fitted values, and whether the selected variance method is defensible for the design.

A Practical Reporting Sequence

Name the model.
State the outcome family and link where relevant, and identify a GLMM or GEE. For a GLMM, name the random-effects structure; for a GEE, name the working correlation.

Name the estimand and scale.
Say whether the result is conditional or marginal, and identify the scale, such as a conditional odds ratio, marginal mean difference, or marginal rate ratio.

Define the comparison and population.
State the predictor contrast, adjustment variables or covariate pattern, target population, and outcome units or time scale as appropriate.

Give the estimate and uncertainty.
Report the estimate, interval level and method, and p-value if relevant. Keep the interval on the stated effect scale and name the variance estimator.

Common Reporting Pitfalls

Calling a GLMM coefficient population-averaged. A fixed effect in a GLMM is generally conditional on the random effects. Do not label it marginal unless a marginal estimand was specifically calculated and defined.
Describing GEE as averaging over dependence. A GEE specifies a marginal mean given covariates. The working correlation accounts for within-cluster association separately.
Reporting only “OR = 1.5.” Identify the model, conditional or marginal target, outcome, treatment comparison, population, and uncertainty measure.
Confusing odds with probabilities. An odds ratio of 1.5 does not mean a 50-percentage-point increase in probability. State the odds scale, or provide model-based probabilities if those are the intended quantities.
Overstating causation. A model-based association is not automatically a causal treatment effect. Causal wording requires a suitable design and assumptions beyond the fitted GLMM or GEE.

Key takeaway: A well-reported clustered-data result makes its target unmistakable: model, conditional or marginal estimand, effect scale, population and comparison, and uncertainty method. These details are part of the statistical result, not optional decoration.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions below.

A logistic random-intercept GLMM reports a treatment coefficient. Is its exponentiation automatically a marginal odds ratio?
A GEE uses an exchangeable working correlation. What role does that correlation play, and what defines the marginal mean?
A log-link GEE coefficient is 0.20 with robust standard error 0.10. Calculate its rate ratio and approximate 95% Wald interval.
Give two reasons why “OR = 1.5, \(p=0.02\)” is an incomplete result statement.

Solutions. (1) No. Exponentiation gives a conditional odds ratio under the GLMM, holding the random effect fixed; it is not automatically a marginal odds ratio. (2) The working correlation represents within-cluster association for estimation and variance calculations. The specified mean model for the outcome given covariates defines the marginal mean. (3) The rate ratio is \(\exp(0.20)=1.221\). The coefficient interval is \(0.20\pm1.96(0.10)=(0.004,\;0.396)\), so the rate-ratio interval is \((\exp(0.004),\exp(0.396))=(1.004,\;1.486)\). (4) The statement omits the model and whether the odds ratio is conditional or marginal; it also omits the comparison, target population, interval or other uncertainty method, and possibly the outcome and covariate adjustment.