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\noindent Statistica Sinica {\bf 15}(2005), 1015-1032
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\vspace{3pc} \normalsize
\centerline{\large\bf QUANTIFYING PQL BIAS IN ESTIMATING}
\vspace{2pt}
\centerline{\large\bf CLUSTER-LEVEL COVARIATE EFFECTS}
\vspace{2pt}
\centerline{\large\bf IN GENERALIZED LINEAR MIXED MODELS}
\vspace{2pt}
\centerline{\large\bf  FOR GROUP-RANDOMIZED TRIALS}
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\author{Scarlett L. Bellamy$^1$, Yi Li$^2$, Xihong Lin$^2$ and Louise M. Ryan$^2$}
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\centerline{\it $^1$University of Pennsylvania and %School of Medicine,
$^2$Harvard School of Public Health}
%\centerline{\it and $^3$University of Michigan} %School of Public Health}
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\begin{quotation}
\noindent {\it Abstract:}
We derive the asymptotic bias and variance of the penalized
quasilikelihood (PQL) estimator of the cluster-level covariate
effect in generalized linear mixed models for group-randomized
trials where the number of clusters $n$ is small and the cluster
size $m$ is large. We show that the asymptotic bias  is
of order $\Order_p(1/m)$ and the asymptotic variance is
of order $O_p(1/n)+O_p\{1/(nm)\}$.
%$\Order_p(n^{-1})$
The practical implication of our
results is that the PQL method works well in settings involving
small numbers of large clusters which are typical in grouped
randomized trials. We illustrate the results using simulation
studies.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Asymptotic bias, asymptotic variance,
generalized linear mixed models, Penalized quasilikelihood.\par
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