Estimating Response Weights for GEE

Why Estimate Response Weights?

In a longitudinal study, participants may stop providing outcomes before the study ends. If the probability of remaining observed depends on measured covariates or earlier outcomes, analyzing only observed responses can distort the estimating equation for the marginal mean. Inverse-probability weighting addresses this by giving more influence to observed records that represent records with similar histories but a lower chance of being observed.

The weights depend on response probabilities, which are usually unknown. They must therefore be estimated from the observed study data. This tutorial focuses on specifying those models, calculating the resulting weights, and assessing their stability and uncertainty. The central tradeoff is that a response model rich enough to account for observed predictors of missingness can also produce very small fitted probabilities and highly variable weights.

Sequential Response Probabilities

Let \(i=1,\ldots,n\) index independent clusters and \(t=1,\ldots,T\) index occasions. Write \(R_{it}=1\) if the outcome at occasion \(t\) is observed and \(R_{it}=0\) otherwise. For monotone dropout, a participant observed at \(t\) must have been observed at every earlier occasion. Let \(H_{it}\) denote the information available just before the response at \(t\) is determined, such as baseline covariates, time, and previously observed outcomes.

At the first occasion, the conditioning on prior response is omitted. A common approach estimates each \(q_{it}\) using a logistic regression fitted among participants at risk of dropping out at that occasion. For example, a denominator model may be \( \operatorname{logit}(q_{it})=\alpha_0+\alpha_1 t+\alpha_2^\mathsf{T}X_i+\alpha_3Y_{i,t-1} \), where \(X_i\) contains baseline covariates and \(Y_{i,t-1}\) is included only when it is observed and appropriate to the response history. The model must use only information available by the time response is determined; using an unobserved future outcome as a predictor is not a valid response-probability model.

A stabilized weight replaces the probability one in the numerator with a less history-dependent numerator probability. For example, let \(a_{it}=P(R_{it}=1\mid R_{i,t-1}=1,X_i,t)\), where the numerator conditions on baseline covariates and time but not on prior outcomes. Then \( SW_{it}=\prod_{s=1}^{t}a_{is}/q_{is} \). Both numerator and denominator probabilities are estimated, typically by fitting corresponding logistic regressions in the relevant risk sets. Stabilization can reduce variability, but it does not automatically guarantee the same estimating target for every weighted GEE. The numerator model and the variables in the estimating equation must be chosen so that the intended estimand is preserved. When that condition is not established, the unstabilized weight provides the clearest inverse-probability identity.

What the Inverse-Probability Identity Requires

The basic justification for unstabilized response weights is a conditional expectation identity. It requires a correctly specified response mechanism given the recorded history and positivity: every relevant history must have a nonzero chance of remaining observed. The identity does not assert that the response model is known; it describes what the true probabilities would accomplish.

Proof. Conditional on the complete covariates and outcomes, the probability of remaining observed through \(t\) is the product of the sequential retention probabilities. Under the stated sequential missing-at-random condition, each factor is the probability \(q_{is}\) evaluated at the history that would be observed along the retained path. Hence \( E[R_{it}\mid X_i,Y_i]=P(R_{it}=1\mid X_i,Y_i)=\prod_{s=1}^{t}q_{is}=\pi_{it} \). Because \(\pi_{it}\) is positive and \(g_i\) is fixed when conditioning on complete data, \( E[R_{it}g_i/\pi_{it}\mid X_i,Y_i]=g_i E[R_{it}\mid X_i,Y_i]/\pi_{it}=g_i \). This proves the identity. \(\square\)

Taking expectations on both sides gives \(E[R_{it}g_i/\pi_{it}]=E[g_i]\). In a GEE, \(g_i\) can represent a complete-data estimating contribution, so the identity explains why correct inverse response probabilities can recover the expectation of a contribution that would have been available under complete observation. It does not ensure that a fitted finite-sample weighted equation is stable, or that a misspecified response model will work.

Positivity, Truncation, and the Cost of Extreme Weights

A fitted response probability near zero creates a large inverse weight. Such observations can dominate the weighted estimating equation, increase variance, and make estimates sensitive to small changes in the response model. Inspect the fitted probability distributions by occasion and by relevant histories, along with the weight distribution, its maximum, and the fraction of total weight carried by the largest observations. A high average response rate does not rule out a serious problem in a small subgroup.

One practical response is to truncate weights, for example by replacing weights above a chosen cutoff \(c\) with \(c\), or by capping weights at selected empirical quantiles. This can lower variance, but it changes the estimating equation. The following calculation shows that it can also move the estimate away from the complete-data mean.

A truncation rule should therefore be reported and justified, and the analysis should be compared with results under reasonable alternative cutoffs and without truncation when numerically feasible. Truncation cannot repair a true positivity failure: it limits the numerical influence of the observed records but cannot create information about histories that are never observed.

Uncertainty From Estimated Weights

The response probabilities are estimated from the same study, so the weights are random. Treating them as fixed when calculating uncertainty can omit variation due to fitting the response models. This matters especially when response probabilities are weakly estimated, when some probabilities are small, or when the number of independent clusters is modest.

One route to valid large-sample uncertainty is to regard the outcome-model parameter and response-model parameters as a joint system of estimating equations. If \(\gamma\) denotes the response-model parameters and \(\beta\) the GEE parameter, write the combined equations schematically as \( \sum_i \begin{pmatrix} s_i(\gamma) \\ u_i(\beta,\gamma) \end{pmatrix}=0 \), where \(s_i\) is the response-model score contribution and \(u_i\) is the weighted GEE contribution. A sandwich covariance for this joint system accounts for the fact that estimated \(\gamma\) enters the weights. Its validity requires independent clusters, suitable model regularity and identification, and a correctly specified response model for the weighting argument.

A cluster bootstrap is another practical approach: resample whole clusters, refit every response model in each replicate, recalculate the weights, and refit the weighted GEE. Resampling individual occasions would break within-cluster dependence and would fail to reproduce the analysis. Bootstrap replicates with separation, extreme fitted probabilities, or failed model fits should be monitored rather than silently discarded.

Common Mistakes

• Using only the current-occasion probability. For a response at \(t\), the cumulative probability includes retention at every earlier occasion. Use the product through \(t\).
• Putting unavailable information in the response model. Predictors must be observed by the point at which response is determined. A future or currently missing outcome cannot be used as if it were observed history.
• Assuming stabilization is harmless by definition. A numerator depending on baseline variables can affect the weighted estimating equation. Establish that the numerator and outcome equation preserve the intended estimand.
• Treating truncation as a fix for absent support. Capping weights can trade variance for bias; it cannot recover outcomes from a history with zero response probability.
• Ignoring the fitted weight model in inference. Estimated probabilities add uncertainty. Refit the response models in a cluster bootstrap or account for their estimating equations in a joint sandwich calculation.

The practical goal is not simply to obtain weights with a convenient range. Specify the response history carefully, check that fitted probabilities are plausible and bounded away from zero in relevant strata, and assess how sensitive the GEE is to model choices and any truncation. Report the response models and the weight diagnostics so readers can see what information supports the weighted estimate.