Cluster Unit Randomized Trials
18. Appendix
Let mij denote the size of the jth cluster assigned to the ith group, i=1,2; j=1,2...k, with 
denoting the total number of subjects in group i, and 
denoting the corresponding value of the overall event rate in this group. Then the standard Pearson chi-square statistic with one degree of freedom may be written as 
Appropriate adjustment of 
for clustering effects requires an estimate of the underlying intracluster correlation coefficient ρ, which, under the null hypothesis of no intervention effect, may be assumed to be constant across intervention groups. The required estimate may be obtained by pooling the observations in both groups and then applying the “analysis of variance approach” described by Donner and Klar (1994). Let MSC and MSW denote the pooled mean square errors between and within groups, respectively. Then defining 
,
We obtain 
, where
and 
The value of is then adjusted by applying a correction factor which depends on both 
and the values of the mij. Letting 
, the adjusted chi-square statistic with one degree of freedom is given by 
. At 
is it clear that 
reduces to while if all clusters are of the same size m, it reduces to 
.
This approach may also be used to construct an approximate confidence interval about 
. Using the notation above, a two sided 95% confidence interval is given by 
. At this expression reduces to the standard confidence interval about a difference between two proportions. However the assumption of a common intracluster correlation coefficient, although guaranteed under the null hypothesis of no intervention effect, may not be appropriate for confidence interval construction. In this case separate estimates of ρ may be used in computing the variance inflation factors C1 and C2.