Marginalizing Binary Logistic Fits Across Dropout Patterns

From Pattern-Specific Predictions to a Population Probability

A binary logistic model fitted within a dropout pattern gives a probability conditional on that pattern and on specified covariates. A population-level probability answers a different question: among a stated target population, what proportion is predicted to have the outcome? To answer it, combine the pattern-specific probabilities using the target population’s pattern proportions. If covariates are also being standardized, average over the target covariate distribution as well.

This distinction matters because a probability is not an odds value, and neither averaging odds nor averaging logistic linear predictors generally produces the population probability. The weighting scheme is part of the estimand: the same pattern-specific fits can yield different summaries for different target populations.

When the intended target is the actual joint distribution of patterns and covariates, covariate distributions can differ by pattern. In that case the direct law-of-total-probability expression is \(\sum_g\pi_g E[p_g(X)\mid G=g]\). By contrast, using a common \(Q\) across patterns describes a standardized target that gives each pattern the same covariate distribution. State which target is being used; do not silently substitute one for the other.

The pattern-specific linear predictor may include treatment \(A\), covariates, and pattern-specific coefficients, for example \(\eta_g(a,x)=\alpha_g+\beta_g a+\gamma_g^\mathsf{T}x\). The marginal probability for treatment level \(a\), standardized to a common target, is then \(p_Q(a)=\sum_g\pi_g E_Q[\operatorname{expit}\{\eta_g(a,X)\}]\). This calculation averages predicted probabilities, not coefficients.

Why Probabilities, Not Odds, Are Averaged

The reason follows directly from the law of total probability. For a randomly selected member of the target population, the event \(Y=1\) occurs in exactly one of the mutually exclusive patterns. Therefore,

For covariate-standardized predictions, first average within the target covariate distribution and then weight the resulting pattern probabilities. The same identity applies because each component remains a probability of the event. Odds are ratios of event probability to non-event probability, so odds from separate groups do not add in this way.

In general, \(\sum_g\pi_g\{p_g/(1-p_g)\}\) is not the odds corresponding to \(\sum_g\pi_gp_g\). Nor does applying the inverse-logit to an average linear predictor generally recover the average probability: the inverse-logit is nonlinear. These are distinct operations, not alternative algebraic forms of the same marginalization.

Standardizing Over Covariates and Comparing Treatments

When covariates vary, evaluate each pattern-specific model at the target covariate values before averaging. For a finite target sample, \(q_j\) can be \(1/m\) for each of \(m\) records, or prespecified weights that represent the desired target population. Pattern weights \(\pi_g\) represent the target pattern composition. These two sets of weights have different jobs: \(q_j\) standardizes covariates, while \(\pi_g\) combines patterns.

For a treatment comparison, calculate a marginal probability separately for each treatment level using the same target covariate distribution and, when the estimand calls for it, the same target pattern weights. Contrasts such as a risk difference or marginal odds ratio are then formed from those marginal probabilities. A pattern-specific treatment coefficient is a conditional log odds ratio under the model; it is not automatically the marginal log odds ratio.

Assumptions, Diagnostics, and Limitations

The arithmetic of marginalization is exact given its inputs, but the inputs require justification. Before interpreting the result, check the following conditions.

• Target weights are appropriate. Pattern weights must sum to one and represent the stated target population. If they are estimated from data, their estimation uncertainty matters. Do not use sample pattern proportions for a different target population without justification.
• The pattern-specific predictions are meaningful. The logistic model must provide adequate conditional probability predictions for the covariate and treatment values being standardized. Strong nonlinearity, omitted interactions, or extrapolation can make predictions unreliable.
• There is support for the predictions. If a target covariate profile is absent or very rare in a pattern, its pattern-specific prediction may be extrapolated rather than supported by data. Sparse patterns and separation can produce unstable or extreme fitted probabilities.
• The outcome distribution is addressed for every pattern in the target. In a dropout setting, outcomes may not be observed for everyone in every pattern. A fitted pattern-specific probability for an unobserved outcome depends on the modeling and identification assumptions used to obtain it; weighting predictions does not resolve that issue.
• The covariate target is explicit. A common covariate distribution across patterns and each pattern’s own covariate distribution generally define different summaries. Document the selected target and the order of standardization and weighting.

Useful checks include verifying that all weights are nonnegative and sum to one, inspecting pattern-specific predicted probabilities for impossible values or extreme concentration near zero and one, and checking whether the target covariate profiles lie within the support of each pattern’s data. Also compare the reported marginal probability with the direct weighted sum of its components. Disagreement usually indicates a coding or weighting error.

Marginalization does not repair a misspecified logistic model, create information in an empty pattern, or settle assumptions about outcomes that are missing. Nor is a marginal odds ratio generally an average of conditional odds ratios; this reflects the nonlinear odds transformation, even when the calculations are performed correctly. Uncertainty in the fitted models, pattern weights, and standardization distribution should be reflected in intervals or tests. A point estimate alone does not show how precise the population summary is.

Common Mistakes and Practical Alternatives

• Wrong: “Average the pattern odds, then call the result population odds.” Correction: Average probabilities with the target pattern weights, then convert that marginal probability to odds only if odds are the desired scale.
• Wrong: “Average the logistic coefficients and apply the inverse-logit.” Correction: Evaluate each model’s inverse-logit prediction at the target covariates, then average those probabilities.
• Wrong: “The conditional treatment odds ratio is the population treatment odds ratio.” Correction: Compute both treatment-specific marginal probabilities under the stated target, then form the requested marginal contrast.
• Wrong: “The observed pattern proportions are always the right weights.” Correction: Use weights aligned with the estimand and target population; state whether weights are common across arms or arm-specific.

The main alternative summaries answer different questions. A pattern-specific probability describes a particular pattern; a conditional odds ratio compares outcomes within a pattern under the fitted model; and a marginal probability describes a target mixture. Covariate standardization makes the covariate target explicit, while a raw mixture using each pattern’s own covariate distribution retains those pattern-specific covariate compositions. None is universally preferable: report the target and scale that match the question.