Earlier, this chapter reviewed how the impact of clustering on sample size requirements can be accounted for by incorporating the value of the “variance inflation factor” (VIF) into standard sample size formulas. A similar approach may be taken to adjusting for cluster effects when analyses are conducted at the individual level, with sample estimates of the VIF now incorporated into standard test statistics. Attention in this section will be focused mainly on binary outcome data, which tend to arise more frequently in cluster randomization trials than continuous, count and time-to-event outcomes. Detailed discussion of statistical methods that can be applied to a variety of outcome variables arising in CRTs may be found in Donner and Klar, 2000, Chapters 6-8.
To illustrate one such approach, consider a trial evaluating the effect of tailored general practice guidelines on the proportion of patients with benign prostatic hyperplasia (BPH) that remained under specialist care at 12 months post-randomization (Mollison et al., 2000). Of main interest here is a comparison of event rates observed on 150 patients contributed by 23 experimental group practices to event rates observed on 142 patients contributed by 26 control group practices. Median cluster sizes in this completely randomized trial were 6 and 3.5 in the experimental and control groups, respectively, a difference that can reasonably be attributed to chance.
66 (44%) patients in the experimental group were still under specialist care at 12 months as compared to 77 (54.2%) patients in the control group. Application of the standard Pearson chi-square test with one degree of freedom to these data yields x2p =3.05(p=.08), indicating a difference that is statistically significant at the 10% level. However this test fails to account for the similarity of responses (clustering) among patients belonging to the same practice, and therefore overstates the true level of significance. We therefore compute the “adjusted chi-square statistic” x2A, obtained by dividing x2p by an appropriate estimate of VIF (Donner and Klar, 1994). Application of this procedure, based on an estimated value of ρ given by 0.077, yields x2A =1.684(p=.19, one degree of freedom), a result no longer statistically significant at any conventional level. Algebraic formulas for all results presented in this example are given in the Appendix.