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

TL;DR

This paper introduces the generalized expectile estimating equations (GEEE), a novel method combining GEE and asymmetric least squares to effectively model heteroscedastic longitudinal data across the entire response distribution.

Contribution

The paper proposes GEEE, a new estimator that captures heteroscedasticity and within-subject dependence by estimating effects on response expectiles, extending GEE with asymmetric least squares.

Findings

GEEE is non-biased and efficient in simulations.

GEEE captures heteroscedasticity in real data.

Provides a robust covariance estimator for inference.

Abstract

Generalized estimating equations (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{\textemdash}and therefore do not account for data heterogeneity. Here, we combine the GEE with the asymmetric least squares (expectile) regression to derive a new class of estimators, which we call generalized expectile estimating equations (GEEE). The 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 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.