| Cluster randomized trials (CRT) are experiments in which intact social units (e.g. households, schools, clinics, etc.), rather than independent individuals, are randomly assigned to intervention groups. Analysing CRT without taking the ICC into consideration will underestimate the standard error of the intervention effect, thus inflate the type I error rate, but in a review of a publications involving CRT, we found that most of them ignored the ICC in their analyses; e.g. using independent t tests to analyse CRT. Through mathematical derivation and simulation studies, this research gives the approximate cluster-t-test result when a CRT is incorrectly analysed using independent-t-test, and also obtains the influence of sample size and the ICC on analyzing a CRT for the following aspects.Firstly, we applied mathematical derivation and simulation to study the influence of sample size and ICC on the independent-t-test and the cluster-t-test for a balanced CRT, and also obtain the mathematical relationship between the two tests. We found that when the independent-t-test is not statistically significant, the cluster-t-test is also not statistically significant. We also give the approximate cluster-t-test result when a CRT is incorrectly analyzed using the independent-t-test. Our simulation results indicate that for large cluster sizes, there is a big difference between the independent-t-test and the cluster-t-test even for small ICC (especially when DE> 1.1). And the clusters also has influence on the result, when DE is the same we could expand the clsuers to decrease the difference between the independent-t-test and the cluster-t-test, especially when each condition has more than 10 clusters the difference between the independent-t-test and the cluster-t-test has no change as the clustes expand.Secondly, we used Monte Carlo simulation is used to study the influence of sample size and ICC on the unbalanced CRT. We examined three weighted (cluster size weighted, minimum variance weighted, equal weighted) cluster-t-tests that adjust for the imbalance, and listed the conditions that each weighting method accommodated. The simulation results indicate that (1) the imbalance-ness has minimal impact on the independent-t-test, but the DE has a big influence on the independent-t-test and the cluster-t-test. When DE<1.1, the independent-t-test and the cluster-t-test behave similarly. Imbalance, however, has a big impact on the three weighted cluster-t-test methods, with the minimum variance weighted method being the best no matter how imbalanced the cluster sizes were. (2) When there is little imbalance, all three methods are similar, but if the imbalance grows bigger, the three weighted methods become similar only when DE<1.4, with the cluster size weighted method being slightly better than the equal weighted method. However, when the DE>1.4 the cluster size weighted method performs poorly. (3) The minimum variance weighted method seems to be relatively robust, and (4) when the cluster size is larger than 100, the minimum variance method is almost similar to the equal weighted method.Thirdly, we used Monte Carlo simulation to study the influence of sample size and ICC on the inference of the intervention effect for a clustered randomized intervention study using a mixed effects model. The Wald chi-square test and two different t tests in SAS PROC MIXED are evaluated in terms of their (1) type 1 error and (2) confidence interval coverage. With more than 20 clusters per group, the Wald chi-square test has nominal type 1 error and coverage. The cluster degrees of freedom method is better than the other two methods, but with more than 25 clusters per group, the performances of the three tests become similar. The number of clusters per group is the key factor when inferring about the intervention effect, and caution is needed when choosing the right degrees of freedom for the t test in PROC MIXED.Lastly, Monte Carlo simulation is again used to examine the influence of sample size and ICC on the estimation of the variance of the random effect for a one-way random effects model. SPSS’s MIXED procedure is used to estimate the variance of the random effect using REML. The results show thatρhas the biggest influence: whenρ> 0.5, the estimation is better, but ifρ< 0.09 the estimation worsens. Increasing the number of clusters decreases the median length of the confidence intervals, while increasing the cluster sizes (say>30) improves the coverage of the confidence intervals.Through this research, we can determine how the sample size and ICC influence the independent-t-test and the cluster-t-test, and the how they influence the estimation of the intervention effect (fixed effect) and the variance of the random effect. These results can improve the analysis of CRT and offer advice on how to design CRT. |
