Correcting an analysis of variance for clustering

Larry V. Hedges*, Christopher H. Rhoads

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


A great deal of educational and social data arises from cluster sampling designs where clusters involve schools, classrooms, or communities. A mistake that is sometimes encountered in the analysis of such data is to ignore the effect of clustering and analyse the data as if it were based on a simple random sample. This typically leads to an overstatement of the precision of results and too liberal conclusions about precision and statistical significance of mean differences. This paper gives simple corrections to the test statistics that would be computed in an analysis of variance if clustering were (incorrectly) ignored. The corrections are multiplicative factors depending on the total sample size, the cluster size, and the intraclass correlation structure. For example, the corrected F statistic has Fisher's F distribution with reduced degrees of freedom. The corrected statistic reduces to the F statistic computed by ignoring clustering when the intraclass correlations are zero. It reduces to the F statistic computed using cluster means when the intraclass correlations are unity, and it is in between otherwise. A similar adjustment to the usual statistic for testing a linear contrast among group means is described.

Original languageEnglish (US)
Pages (from-to)20-37
Number of pages18
JournalBritish Journal of Mathematical and Statistical Psychology
Issue number1
StatePublished - Feb 2011

ASJC Scopus subject areas

  • Arts and Humanities (miscellaneous)
  • Statistics and Probability
  • General Psychology


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