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

ABSTRACT

Generalized estimating equations $\left(\mathrm{G}\mathrm{E}\mathrm{E}\right)$ 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 $\mathrm{G}\mathrm{E}\mathrm{E}$ with the asymmetric least squares (expectile) regression to derive a new class of estimators, which we call generalized expectile estimating equations $\left(\mathrm{G}\mathrm{E}\mathrm{E}\mathrm{E}\right)$. The $\mathrm{G}\mathrm{E}\mathrm{E}\mathrm{E}$ 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 $\mathrm{G}\mathrm{E}\mathrm{E}\mathrm{E}$ 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.

KEYWORDS:

• Expectile regression
• quantile regression
• GEE working correlation
• cluster data
• longitudinal data

Disclosure statement

No potential conflict of interest was reported by the author(s).