title: Relative efficiency and sample size for cluster randomized trials with variable cluster sizes.
abstract: BACKGROUND
  The statistical power of cluster randomized trials depends on two sample size components, the number of clusters per group and the numbers of individuals within clusters (cluster size). Variable cluster sizes are common and this variation alone may have significant impact on study power. Previous approaches have taken this into account by either adjusting total sample size using a designated design effect or adjusting the number of clusters according to an assessment of the relative efficiency of unequal versus equal cluster sizes.
  
  
  OBJECTIVE
  This article defines a relative efficiency of unequal versus equal cluster sizes using noncentrality parameters, investigates properties of this measure, and proposes an approach for adjusting the required sample size accordingly.
  
  
  METHODS
  We focus on comparing two groups with normally distributed outcomes using t-test, and use the noncentrality parameter to define the relative efficiency of unequal versus equal cluster sizes and show that statistical power depends only on this parameter for a given number of clusters. We calculate the sample size required for an unequal cluster sizes trial to have the same power as one with equal cluster sizes.
  
  
  RESULTS
  Relative efficiency based on the noncentrality parameter is straightforward to calculate and easy to interpret. It connects the required mean cluster size directly to the required sample size with equal cluster sizes. Consequently, our approach first determines the sample size requirements with equal cluster sizes for a pre-specified study power and then calculates the required mean cluster size while keeping the number of clusters unchanged. Our approach allows adjustment in mean cluster size alone or simultaneous adjustment in mean cluster size and number of clusters, and is a flexible alternative to and a useful complement to existing methods. Comparison indicated that we have defined a relative efficiency that is greater than the relative efficiency in the literature under some conditions.
  
  
  CONCLUSIONS
  Our measure of relative efficiency might be less than the measure in the literature under some conditions, underestimating the relative efficiency.
  
  
  CONCLUSIONS
  The relative efficiency of unequal versus equal cluster sizes defined using the noncentrality parameter suggests a sample size approach that is a flexible alternative and a useful complement to existing methods.
authors.affiliation: Department of Medicine, School of Medicine, University of Alabama, Birmingham, AL, USA. zyou@mail.dopm.uab.edu
authors.forename: Zhiying
authors.initials: Z
authors.lastname: You
authors.forename: O Dale
authors.initials: OD
authors.lastname: Williams
authors.forename: Inmaculada
authors.initials: I
authors.lastname: Aban
authors.forename: Edmond Kato
authors.initials: EK
authors.lastname: Kabagambe
authors.forename: Hemant K
authors.initials: HK
authors.lastname: Tiwari
authors.forename: Gary
authors.initials: G
authors.lastname: Cutter
mesh_terms: D016000:Cluster Analysis; D006801:Humans; D016011:Normal Distribution; D016032:Randomized Controlled Trials as Topic / Q000379:methods*; D012107:Research Design; D018401:Sample Size