Marginal and Conditional Treatment-Effect Estimands

Why the Same Trial Can Have Different Treatment Effects

A treatment effect can describe a contrast for patients with specified covariates, or an average contrast for a defined population. These are different estimands, even when they are calculated from the same fitted model. A regression coefficient is not automatically the population-average effect: its meaning depends on the model, the outcome scale, and which covariates are held fixed or averaged over.

This distinction matters in a clinical trial because a conditional result may describe effects for a particular covariate profile, while a marginal result describes an average over the patients represented by a chosen target population. The target population must be stated: changing its covariate distribution can change a marginal effect, particularly when treatment effects vary across covariate values.

The treatment strategies, endpoint, handling of intercurrent events, and target population are part of the estimand, not details to be supplied by the regression model after the fact. Here, the focus is the distinction between conditioning and averaging over baseline covariates. A causal interpretation additionally requires a well-defined treatment, consistency between observed outcomes and the corresponding potential outcomes, and exchangeability and positivity for the relevant treatment comparisons. Random assignment supports exchangeability in expectation; positivity means that each treatment being compared is possible for the covariate profiles in the target population.

To calculate a marginal effect from an outcome model, predict each patient’s outcome under each treatment and average those predictions over the target population. This is often called standardization or g-computation. The same individuals or covariate values must be used for both treatment predictions, so the comparison refers to a common target distribution.

Logistic Regression: Conditional and Marginal Effects

For a binary outcome, a logistic regression model specifies the outcome probability conditional on treatment and covariates. With one binary covariate and no treatment interaction, a model can be written \(\operatorname{logit}\{P(Y=1\mid A,X)\}=\beta_0+\beta_A A+\beta_X X\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). In this model, \(\beta_A\) is a conditional log odds ratio: at a fixed value of \(X\), the treatment odds ratio is \(\exp(\beta_A)\). It is not, in general, the log odds ratio obtained after averaging the outcome probabilities over \(X\).

The marginal probabilities are computed on the probability scale first: \(\mu_a^*=\int P(Y=1\mid A=a,X=x)\,dF^*(x)\). A marginal odds ratio, if that is the chosen contrast, is then formed from these two averaged probabilities. Averaging probabilities and then converting to odds is not the same operation as averaging or retaining conditional odds.

The short derivation is that the conditional model first maps each linear predictor through the expit function, then standardization averages those probabilities. In general, for covariate values \(x\) with weights \(w_x\), \(\sum_x w_x\operatorname{expit}(\beta_0+\beta_A a+\beta_X x)\) does not equal \(\operatorname{expit}\{\beta_0+\beta_A a+\beta_X\sum_x w_xx\}\). Nor does taking the odds of the averaged probability reproduce the conditional odds ratio. The calculation must therefore follow the estimand’s order of operations: predict, average over \(F^*\), and only then apply a contrast defined on the desired scale.

Linear Regression: Interaction and Averaging

For a continuous outcome, consider a linear conditional-mean model with a binary covariate: \(E(Y\mid A,X)=\beta_0+\beta_A A+\beta_X X+\beta_{AX}AX\). At \(X=x\), the conditional mean difference for treatment versus control is \(\beta_A+\beta_{AX}x\). Averaging over a target population with mean covariate value \(E_{F^*}(X)\) gives a marginal mean difference of \(\beta_A+\beta_{AX}E_{F^*}(X)\). Thus, when the interaction is nonzero, the marginal effect depends on the target covariate distribution.

In the special additive linear model with no treatment-by-covariate interaction, the conditional mean difference is the same at every \(x\), namely \(\beta_A\). Averaging a constant over any target distribution leaves it unchanged, so the marginal and conditional mean differences coincide. This equality depends on the outcome scale and model structure; it should not be assumed for a nonlinear model such as logistic regression.

Bayesian Standardization and Practical Checks

In a Bayesian analysis, the model parameters are uncertain, so the standardized estimand is uncertain as well. For each posterior draw of the model parameters, calculate the predicted outcome under each treatment for every target covariate profile, average those predictions using the target-population weights, and then form the specified contrast. The resulting collection of draw-specific estimands represents the posterior distribution of the marginal effect. Do not substitute posterior mean coefficients into a nonlinear prediction formula and treat the result as the posterior mean estimand: in general, transforming posterior means is not equivalent to averaging transformed posterior draws.

Useful diagnostics include checking that target-population weights sum to one, that both treatment predictions use the same covariate profiles, and that predicted outcomes are plausible for those profiles. Examine overlap or positivity: if the target includes covariate patterns with little or no information under one treatment, model-based predictions there depend heavily on extrapolation. Check model fit and functional form, including treatment-by-covariate interactions when scientifically plausible. A model that fits average outcomes but misses important subgroup patterns can produce a misleading standardized estimate.

Standardization is not the only way to estimate a marginal effect. A conditional regression coefficient answers a covariate-specific model question; standardization answers a population-average question by averaging model predictions. Weighting approaches can also target marginal contrasts, but require appropriate weights and their own diagnostics. These approaches should not be treated as interchangeable merely because they use the same trial data. The estimand, target distribution, model assumptions, and effect scale determine what each result means.

A frequent mistake is to report the treatment coefficient from a logistic regression as the population-average treatment effect. That coefficient is a conditional log odds ratio under the specified model; it is not generally a marginal contrast. The correction is to state the target distribution, predict under each treatment for its covariate profiles, average the predictions, and calculate the contrast on the prespecified scale. Another mistake is to average covariates first and predict for an “average patient.” For nonlinear outcome models, that prediction generally differs from averaging predictions across the actual target covariate distribution.