Sample size adjustments for varying cluster sizes in cluster randomized trials with binary outcomes analyzed with second-order PQL mixed logistic regression

M.J.J.M. Candel*, G.J.P. van Breukelen

*Corresponding author for this work

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Adjustments of sample size formulas are given for varying cluster sizes in cluster randomized trials with a binary outcome when testing the treatment effect with mixed effects logistic regression using second-order penalized quasi-likelihood estimation (PQL). Starting from first-order marginal quasi-likelihood (MQL) estimation of the treatment effect, the asymptotic relative efficiency of unequal versus equal cluster sizes is derived. A Monte Carlo simulation study shows this asymptotic relative efficiency to be rather accurate for realistic sample sizes, when employing second-order PQL. An approximate, simpler formula is presented to estimate the efficiency loss due to varying cluster sizes when planning a trial. In many cases sampling 14 per cent more clusters is sufficient to repair the efficiency loss due to varying cluster sizes. Since current closed-form formulas for sample size calculation are based on first-order MQL, planning a trial also requires a conversion factor to obtain the variance of the second-order PQL estimator. In a second Monte Carlo study, this conversion factor turned out to be 1.25 at most. 

Original languageEnglish
Pages (from-to)1488-1501
Number of pages14
JournalStatistics in Medicine
Issue number14
Publication statusPublished - 30 Jun 2010


  • cluster randomized trials
  • mixed effects logistic regression
  • quasi-likelihood estimation
  • sample size
  • varying cluster sizes

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