Estimated estimating equations: Semiparametric inference for clustered/longitudinal data

We introduce a flexible marginal modelling approach for statistical inference for clustered/longitudinal data under minimal assumptions. This estimated estimating equations (EEE) approach is semiparametric and the proposed models are fitted by quasi-likelihood regression, where the unknown marginal means are a function of the fixed-effects linear predictor with unknown smooth link, and variance-covariance is an unknown smooth function of the marginal means. We propose to estimate the nonparametric link and variance-covariance functions via smoothing methods, while the regression parameters are obtained via the estimated estimating equations. These are score equations that contain nonparametric function estimates. The proposed EEE approach is motivated by its flexibility and easy implementation. Moreover, if data follow a generalized linear mixed model (GLMM), with either specified or unspecified distribution of random effects and link function, the proposed model emerges as the corresponding marginal (population-average) version and can be used to obtain inference for the fixed effects in the underlying GLMM, without the need to specify any other components of this GLMM. Among marginal models, the EEE approach provides a flexible alternative to modelling with generalized estimating equations (GEE). Applications of EEE include diagnostics and link selection. The asymptotic distribution of the proposed estimators for the model parameters is derived, enabling statistical inference. Practical illustrations include Poisson modelling of repeated epileptic seizure counts and simulations for clustered binomial responses.