In a simple randomized clinical trial, the use of unequal allocation ratios, particularly the allocation ratio of 3:1, will significantly reduce the power of study for detecting significance difference between two treatments [13–15]. To our knowledge, few published studies investigated the impact of within-center and among-centers inequality on the statistical properties of the tests of homogeneity of odds ratios in multicenter clinical trials [1, 3, 4]. As illustrated in Tables 3, 4 and 5, the type I error rate of the three homogeneity tests is approximately close to the nominal level of 0.05 except for LR when K = 4. Since the results show that these tests have almost the same type I error rate, power comparisons are possible. As compared with the equal sample size design, the power of the LR, BD and DL tests will decrease if the same total sample size, which can be allocated equally within one center or among centers, is allocated unequally. In this case, the power ranking of the tests was BD≥DL≥LR. It is worth mentioning that, as compared with within-center inequality, among-centers inequality has stronger adverse effect on the power of the homogeneity tests. Despite the use of different tests, these findings are inconsistent with those of Paul, who reported the adverse effect of within-center inequality to be stronger .
Also, this paper shows how to use a mixed logistic model to test homogeneity of odds ratios in multicenter trials. In Model 1, there are two types of homogeneity: homogeneity of odds ratios among centers and homogeneity of centers. However, removing the center-by-treatment interaction from Model 1 leads to a model which can only be used to test homogeneity of centers. This model, which has been previously discussed by Gao, assumes that the odds ratios are constant over centers . Therefore, it should not be used to generate data for comparing the tests of heterogeneity of odds ratios. Furthermore, the power of the three tests of homogeneity increases more when we increase the number of K and n
compared to when we increase the number of K and
. This result is in agreement with previous studies which have evaluated the influence of K and n
on the power of the homogeneity tests [5, 17–19]. Nevertheless, our simulation study shows that the degree of among-centers heterogeneity,
, has little or no effect on the power of the three tests of homogeneity, except for DL when
and sample size is small.
In addition, it is noteworthy that we used the DL statistic calculated from the one-way random effects model, which has approximately a chi-square distribution. However, Biggerstaff and Jackson  have calculated the exact distribution and power of the well-known Q statistic based on the same random effects model, which can be used for testing homogeneity of odds ratios and be compared with the tests used in the present study.
In conclusion, of the three tests of homogeneity, the BD seems to be the most appealing with regard to its statistical properties: its type I error rate is close to the nominal level and its power is greater than that of DL and LR. Moreover, it has the advantage of simplicity of calculation and is recommended by a number of authors [1, 4–6]. However, one limitation of BD test is that it has low power when the sample size within each center is small, even if the number of centers is large [1, 2]. Nevertheless, despite having low power under small number of centers and its complexity, Model 1 has its own advantages. Firstly, when the centers are a random sample themselves, the LR test from the Model 1 enables inferences to extend to the population of centers. Secondly, a further consideration is that common odds ratio can be estimated from the fixed part of the Model 1, even when the odds ratios are not homogeneous. Thirdly, in each center, Model 1 provides a predicted log-odds ratio that shrinks the sample value toward the mean. This is especially useful when the sample size in a center is small and the ordinary sample odds ratio has a large standard error . In addition, the mixed logistic model described in this study will potentially be applicable to meta-analysis studies.
It is clear that, based on Model 1, the odds ratio in the kth center, as given in the appendix 4, is exp(β + b
), which can be written as C × exp(b
) where C = exp(β) .This indicates that the odds ratio in each center is absolutely independent of α and u
. Indeed, the odds ratios are affected by b
, and β has the same effect on odds ratio in all centers. Hence, to generate heterogeneous odds ratios among centers, the fixed simulation parameters, ie α and β, can be chosen arbitrarily.
It should be noted that, although using unequal sample size designs in multicenter clinical trials reduces both the power of the study and the power of the homogeneity tests, a substantial reduction in the total cost of the trial will compensate for the reduction in the power of the statistical tests [14, 15]. Finally, further research is warranted to investigate the influence of the number of centers, unequal sample size design, sparseness and also deviation from normal assumption of the random effects on the robustness and accuracy of the estimates of the fixed and random parameters of the Model 1.