GEE after multiple imputation

The motivating application analyzes each completed dataset with a marginal GEE model. This file records why that combination needs a qualification, what longmi claims, and what it refuses to claim.

The workflow

For completed dataset \(m\), the GEE estimator solves

\[ \sum_i D_{im}^{\top} V_{im}^{-1} \left(Y_i^{(m)} - \mu_i(\beta)\right) = 0, \]

yielding \(\widehat\beta^{(m)}\) and a robust (sandwich) complete-data covariance \(U^{(m)}\). The coefficient vectors and covariance matrices are pooled with the multivariate Rubin rules (../algorithms/rubin_pooling.md).

MI followed by GEE is established practice in longitudinal research (Beunckens, Sotto & Molenberghs 2008; Lipsitz, Fitzmaurice & Weiss 2020).

The qualification

A GEE estimator is an estimating-equation (method-of-moments) estimator, not a likelihood-based one. The relevant results:

  • Meng (1994) formalizes congeniality: whether the imputation and analysis procedures can be embedded in a coherent common Bayesian model. Without it, Rubin's variance can be too large or too small.
  • Robins & Wang (2000) derive variance estimators for imputation estimators that remain consistent when the imputation and analysis models are incompatible or misspecified — directly relevant to GEE.
  • Yang & Kim (2016) show Rubin's variance estimator can have asymptotic bias when the complete-data estimator is method-of-moments and congeniality is absent, and propose an over-imputation variance estimator.

longmi therefore does not claim:

any imputation model + any GEE + Rubin pooling is automatically valid.

The defensible claim is:

Standard Rubin pooling is supported when imputations are proper, the imputation model is compatible or congenial with the GEE estimand, and the complete-data sandwich covariance is valid.

What every imputation backend must declare

supported outcome types
assumed missingness mechanism
whether parameter uncertainty is propagated
whether the analysis model is nested in the imputation model
known congeniality conditions
recommended pooling method

This is the ValidityDeclaration contract in ../../src/longmi/contracts.py, surfaced on pooled results via validity_report().

For the motivating count-outcome model, the imputer must contain at least the exposure \(G_i\), time \(t_j\), their interaction \(G_i t_j\), and baseline covariates \(C_i\) — and preferably categorical wave effects with the corresponding exposure-by-wave interactions, since a categorical-wave imputation model contains the linear-time analysis as a restricted special case (A8).

Robust fallback: bootstrap-then-impute

When congeniality is uncertain, the planned robust inference mode (0.2) is bootstrap-then-impute (Bartlett & Hughes 2020):

  1. resample participants (the sampling unit, per A1);
  2. refit the imputation model within the bootstrap sample;
  3. generate imputations inside the bootstrap sample;
  4. refit the complete-data analysis;
  5. obtain inference from the bootstrap distribution.

Bartlett & Hughes find that the order matters: impute-then-bootstrap is generally not valid under uncongeniality or misspecification, whereas appropriate bootstrap-then-impute procedures can be. Nuisance models are refit within each replicate; estimated quantities are never held fixed across replicates.