Population-averaged logistic model for repeated binary outcomes ($\mathrm{logit}\mathbb{E}(Y_{ij}\mid w_{ij})=w_{ij}^T\gamma$)

For independent subjects with repeated Bernoulli observations, specify marginal means $m_{ij}={\mathrm{logit}}^{-1}(w_{ij}^T\gamma )$. A generalized estimating equation uses their derivative matrix $D_i$ and working covariance matrix $V_i$ to solve ${\sum }_i{D}_i^T{V}_i^{-1}(Y_i-m_i)=0$. Subject-level sandwich covariance matrices give robust uncertainty when the mean is correctly specified but the working correlation is not. The coefficients describe population odds ratios; this mean specification does not prescribe a complete joint distribution within a subject.

For monotone missing data, let $R_{ij}$ indicate observation and let ${\rho }_{ij}$ be the product of the conditional retention probabilities through visit $j$. Under sequential missing at random, conditioning on the complete response history makes these retention probabilities depend only on the history already observed at each decision. Therefore $\mathbb{E}(R_{ij}/{\rho }_{ij}\mid Y_i,X_i)=1$. Weighting a working-independence generalized estimating equation by $R_{ij}/{\rho }_{ij}$ restores its mean-zero property. This needs correct retention probabilities, positivity and regularity; uncertainty must account for estimating the weights.