A new GEE method to account for heteroscedasticity using asymmetric least-square regressions

Generalized estimating equations $ ({\rm GEE}) $ (GEE) are widely used to analyze longitudinal data; however, they are not appropriate for heteroscedastic data, because they only estimate regressor effects on the mean response – and therefore do not account for data heterogeneity. Here, we combine the $ {\rm GEE} $ GEE with the asymmetric least squares (expectile) regression to derive a new class of estimators, which we call generalized expectile estimating equations $ ({\rm GEEE}) $ (GEEE). The $ {\rm GEEE} $ GEEE model estimates regressor effects on the expectiles of the response distribution, which provides a detailed view of regressor effects on the entire response distribution. In addition to capturing data heteroscedasticity, the GEEE extends the various working correlation structures to account for within-subject dependence. We derive the asymptotic properties of the $ {\rm GEEE} $ GEEE estimators and propose a robust estimator of its covariance matrix for inference (see our R package, github.com/AmBarry/expectgee). Our simulations show that the GEEE estimator is non-biased and efficient, and our real data analysis shows it captures heteroscedasticity.