A Covariance Estimator for GEE with Improved Small‐Sample Properties

Abstract

In this paper, we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations (GEE). Hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small. Resampling methods, such as the jackknife and bootstrap, have been suggested for covariance estimation when the number of clusters is small. A drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near-zero cell counts caused by resampling. We propose a bias-corrected covariance estimator that avoids this problem. In a small simulation study, we compare the bias-corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10-40 subjects with equal and unequal cluster sizes of 16-64 observations. The bias-corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal, whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2-3 times the nominal level. The methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease.

Covariance estimators for Generalized Estimating Equations (GEE) in longitudinal analysis with small samples

“…Due to the fact that the fitted value μ̂i tends to be closer to Y i than the true value μ i and when sample size is small, in V LZ is biased downward for estimating , and the bias turns to be larger when the sample is much smaller; meanwhile, a greater variability may arise [16,22]. Therefore, the hypothesis testing using V LZ tends to be liberal, and the resulting confidence interval is narrow.…”

“…V MD is another bias-corrected "sandwich" variance estimator proposed by Mancl and DeRouen [22]. Unlike V KC , this estimator does not assume a correctly specified correlation structure, and it is written by (12) with (13) where I i and H ii are defined as the same as V KC .…”

“…Unlike V KC , this estimator does not assume a correctly specified correlation structure, and it is written by (12) with (13) where I i and H ii are defined as the same as V KC . Note that to correct this bias in finite samples, Mancl and DeRouen [22] relied on the following approximate identity but they ignore one term from its first order Taylor's expansion, leading to overcorrection. [26] made a further adjustment on V MD by multiplying a scale factor, which is given by (14) with (15) where η i = I p − N i .…”

Developments in cluster randomized trials and Statistics in Medicine

“…This bias can be removed by dividing the squared error by the diagonal of the 'hat' matrix. It has also been found that using a t-distribution (or a Satterthwaite type degrees of freedom correction) and a jack-knife improves the estimate of the standard error [45][46][47]. Pan and Wall [47] proposed replacing the GEE Wald test by approximate t-or F-tests.…”

“…They state that the GEE approach is asymptotically equivalent to the summary measure approach, and quote Mancl and De Rouen [46] to the effect that using bias-corrected variances can yield valid test sizes even with unequal cluster sizes and with as few as 10 clusters.…”