| Background:In multicenter clinical trials, center-adjusted analysis have been clearly stated in the relevantly regulatory guidelines on statistical analysis proposed by the United States Food and Drug Administration (FDA), the European Medicines Agency (EMEA) or China Food and Drug Administration (CFDA). The center effect should be always taken into account regardless of its prognostic value.When binominal variable is the endpoint outcome, besides hypothesis testing, confidence interval of rate difference is commonly applied to assess the difference size between the two interventions, especially for widely used non-inferiority design in clinical trials.To construct confidence interval for the rate difference in multicenter clinical trials, the center adjusted method should be considered otherwise it could result in potentially misleading conclusions. Therefore, the statistical analysis should provide the center adjusted confidence interval along with the standard confidence interval for reference.However, estimation methods of confidence interval adjusted by centers for dichotomous variable are rarely used in data handling practice due to the present methods have not yet been widely recognized and the clear guidelines are not available, making some difficulties with demonstrating non-inferiority when rate difference is as the main evaluating indicator. Obviously it is valuable for clinical trial assessment to solve this methodological problem.By now, there have been four methods for confidence interval estimation of rate difference adjusted by centers as follows.Mehrotral & Railkar (2000) proposed minimum risk weight method which takes advantage of CMH and inverse weighting schemes and gives center weights w under the minimum value of the sum of variance and rate difference error. The lower coverage probability is its critical weakness.Yan & Su (2010) proposed Newcombe stratified confidence interval based on the inverse variance weights and stratified Wilson confidence interval of one proportions. This method is difficult to implement and its coverage probability may decline when the number of centers is increased.Ge & Durham et al. (2011) proposed the method to construct rate difference interval based on logistic regression. The main idea is to use logistic regression to include covariates such as centers, and then to estimate the population rate difference and its’variance which is used to construct the center adjusted confidence interval by Delta method. Because of the large sample approximation of Delta method, the coverage probability of this method is slightly lower with small and media sample size.Klingenberg (2014) derives a new confidence interval formula based on the CMH variance estimator method proposed by Greenland & Robins and Sato et al. He uses the corrected CMH weighted rate difference as the midpoint of the confidence interval, and calculates the margin of error with the CMH variance estimator, and then a type of δMid±ME confidence interval is constructed. This method showed good performance except a slightly biased rate difference estimation.Thereby, a newly simple method for confidence interval estimation of rate difference adjusted by a center covariate is desperately needed for clinical trial practice, specially for non-inferiority study.During the procedure of present study, the paper of Klingenberg method was published, so it is taken as one of comparators in this study.Objective:This study aims to construct a novel confidence interval estimation of rate difference adjusted by a dichotomous covariate (ie, center effect), and provide a simple but an effective analytical tool for multicenter clinical trial evaluation.Methods:First, theoretical derivation was conducted for a new confidence interval estimation of rate difference with one dichotomous covariate (ie, center effect). Then, simulation study was applied for comparison between the proposed method and other four methods. Finally, the real examples are provided for validation.(1) Theoretical derivationThe new confidence interval estimation of rate difference with one dichotomous covariate based on Φ adjustment is referred to Φ method. First, the adjustment coefficient Φi for ith center was defined by the ratio of the sum of two proportions in ith center to the sum of two pooled rates. Then, the population rate and its variance with CMH weighting scheme was estimated. Finally, the confidence interval was constructed by Wald method.The adjustment coefficient Ψi is defined as where, pi1, pi2 are the rates of group 1 and group 2 in the ith center respectively. And p1, p2 are the centers pooled rates of group 1 and group 2 respectively.The estimation of population rate difference is followed by θ=∑wiφi(pi1-pi2) where Wi is the CMH weight of the ith center.The estimation of population variance is V(θ)=π1(1-π1)/n1+π2(1-π2)/n2 π1=(∑wipi1+∑wipi2+θ)/2,π2=(∑wipi1+∑wipi2-θ)/2According to the central limit theorem, confidence interval estimation of rate difference with one dichotomous covariate based on Φ adjustment is finally obtained as: θ±z1-a/2(?)(2) Simulation studyMulticenter rate difference sample data is generated using rbinom() function of R software (R 3.1.2 64-bit). Two independent binomial distributions was constructed to generate vector ni1 as sample size and vector xi1 as response cases of group 1 while vector ni2 and xi2 of group 2 in k centers. Then the four vectors is combined as a multicenter simulation sample data.Parameters configuration: Three parameters including the number of centers, the total sample size and two population rates were considered in the generation of sample data. Seven samples (including 8 centers ×12 cases, 12 centers × 12 cases, 16 centers×12 cases, 8 centers × 24 cases, 8 centers×36 cases, 15 centers × 36 cases, 50 centers×24 cases) were same and two were different on sample size across centers in 9 sample size levels by controlling three parameters. The cases number of each group in centers and the pooled number of each group are the same. The population rates of group 1 and group 2 were set to 0.05-0.95 and 0.08-0.98 by 0.1 step delivering 100 combinations.Evaluation criteria:The evaluation of the pros and cons of each method followed two criteria:c) The smaller mean squared error of rate estimation ((θ-θ)2 the better.d) The closer to the nominal level of confidence the coverage probability is, the better.(3) Real examplesSeven representative data sets came from real clinical trials with 3 to 14 centers and the total sample sizes from 68 to 438. The endpoints, such as efficacy, significant efficacy, recovery rate, etc., were binominal variables distributed from 0 to 1 covering all possible proportion range.Results:(1) The simulation studyThe error of rate difference estimation:The error of rate difference estimation given by Φ method was similar to Klingenberg method but smaller than other methods at small sample size levels, that is,25,48 and 72 cases respectively in each pooled group and with population proportions from 0.5 to 0.9. The average error were 0.013,0.0069 and 0.0046 respectively according to the above three samples. The error of rate difference estimation was consistent with the logistic regression and pooled Wald method but slightly higher than other methods when the population rates were closely to 0 or 1.The estimating error of Φ method showed a greater reduction than other methods in the case of center number increased while the sample size in each center kept the same though all the methods got reduced error when the sample size was increased (96,141,144,270 and 600 cases in each group). The ip method had the minimum error in all methods when the number of centers increased to 12 (6 cases a group in each center) and the population rates ranged 0.5-0.9. The Φ and all others except Newcombe method had the same trend when the sample size in centers increased and the center number remained unchanged. The error size of Φ method, pooled Wald method, logistic regression method and Klingenberg method was similar when the sample size increased to 36 cases in each center.Coverage probability of confidence interval:The coverage probability of Φ method was in the vicinity of nominal confidence level if the population rates did not close to 0 or 1. Under the situation with the smallest sample size in simulation (5 centers,25 cases in each pooled group, a total of 50 cases) and the extreme rates (close to 0 or 1), the coverage probability of Φ method declined to an average coverage probability of 93.6% which was slightly away from the nominal level of 95% compared to Klingenberg method (95.23%) and Newcombe method (94.2%), but wss closer to the nominal level compared to the pooled Wald method (92.9%), the minimum risk weight method (86.9%) and logistic regression method (91.1%).The confidence interval estimation by Φ method was closer to the nominal level of confidence interval than pooled Wald method, minimum risk weights method and logistic regression method in addition to the situation of small sample size and the extreme rate, but it’s slightly lower than the Klingenberg method. Corresponding to the pooled group sample size of 48 cases,72 cases,96 cases,144 cases,270 cases and 540 cases, the average coverage probability of confidence interval for the rate difference in all combinations between the two population rates (including extreme values) were 0.944,0.948,0.949,0.948,0.949 and 0.949 respectively which was very close to the nominal level of 0.95.(2) Real examplesThe estimations of rate difference and its confidence intervals for seven efficacy data from clinical trials were given by Φ method and the applicability of Φ method is further validated compared with the results given by other methods.Conclusion:The Φ method proposed by the present study yielded small error of rate difference estimation and made the coverage probability of confidence interval satisfactory when the population rate is not close to 0 or 1. In addition to simply implement, its statistical properties were similar to Klingenberg method and superior to the other four methods. The Φ method can be used to estimate the confidence interval of rate difference with one dichotomous covariate, and it is especially applicable for the confidence interval estimation considering center effect in multicenter clinical trials. |
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