An alternative approach to evaluation of poolability for stability studies

The current method for pooling the data from different batches or factors, suggested by ICH Q1E guidance, is to use analysis of covariance (ANCOVA) for test interaction between slopes and intercepts and factors. Failure to reject the null hypothesis of equality of slopes and equality of intercepts, however, does not prove that slopes and intercepts from different levels of factors are the same, and the data can be pooled for estimation of shelf life. In addition, the ANCOVA approach uses indirect parameters of intercepts and slopes in the regression model for assessment of poolability. The hypothesis for poolability is then formulated on the basis of the concept of equivalence for the means among the distributions of the quantitative attributes at a particular time point. Methods based on the intersection-union procedure are proposed to test the hypothesis of equivalence. A large simulation study was conducted to empirically investigate the size and power of the proposed method for the bracketing and matrixing designs given in the ICH Q1D guidance. Simulation results show that the proposed method can adequately control the size and provides sufficient power when the number of factors considered is fewer than three. A numerical example using the published data illustrates the proposed method.     

Statistical assessment of between batch stability equivalence

A statistical method for testing the equivalence between batches regarding their stability is proposed. This method is based on the statistical linear model making use of a set of dummy variables to code the different batches. The method gives us the point estimates of the slope and zero intercept of one batch, and the differences and the corresponding confidence intervals with the remaining batches. In a second step, zero intercepts and slopes are estimated for all the batches. Stability equivalence assessment is based on the comparison of the confidence intervals for the differences between batches with the maximum difference allowable. The main advantages of this method are the possibility to compare several batches, to disclose the equivalence stability criteria from the statistical hypothesis about the equality between slopes, and the joint estimated of the residual variance whatever the decision to pool or not the data from different batches. This method is illustrated with two data set; the first one, previously published by other authors, involved six batches; the second data set include two batches and arose in a stability study of a commercial human insulin conducted in our laboratory.     

Extensions to the ICH Q1E Approach for the Statistical Evaluation of Stability Data for a Large Number of Lots

Purpose

The ICH Q1E is the existing guideline that provides a statistical approach for analyzing stability data to evaluate/propose a product shelf life and in general for many other scenarios that are suited for statistical analysis of slopes and intercepts for three lots. However, the Q1E approach and criteria may not be appropriate for scenarios with more than three lots. To bridge the gap, this work proposes two supplemental approaches for stability data evaluation with greater than three lots.

Methods

The first extension focuses on the poolability test. By exploring the mathematical principles underlying the test, power curves for various numbers of lots are plotted and the appropriate intersection point is identified to align power performance. The second extension addresses the worst profile criteria when failing the poolability test. Order statistics and other statistical techniques are explored to characterize the relative performance of lots within their population.

Results

The significance level for poolability tests is suggested to be reduced accordingly, as the number of lots increases. If the poolability criteria are not met, a “conservative” batch based on order statistics rather than the worst case is recommended as a reasonable estimate of the intrinsic stability change rate for a large number of lots.

Conclusions

This work offers two supplementary extensions for the current ICH Q1E approaches when the number of lots is greater than three to identify the intrinsic stability profile of a product that is representative of future performance. The proposed extensions do not change Q1E principles and deliver similar levels of statistical performance as implied by the Q1E approach.

     

Shelf life determination based on equivalence assessment

In a regular analysis of covariance (ANCOVA) approach to stability analysis, the decision for pooling data from different batches plays a key role in the determination of the shelf life of the drug product. Conventionally, the decision to pool data for the estimate of slope and intercept of common or individual regression lines is made by "no evidence to reject the null hypothesis of no difference." With typically limited observations, a significance level of much higher than 0.05 was recommended for the pooling tests in order to avoid inflation of type-I error rate of the shelf life testing. This logic of the pooling test decision making discouraged the use of replicates to improve power of testing and precision of estimation. The concept of pooling by equivalence test was originally proposed by Ruberg and Hsu in their 1990 article "Multiple comparison procedures for pooling batches in stability studies" Such a concept has evolved to pooling batches based on the shelf life equivalence test by Yoshioka et al. in their 1996 article "Power of analysis of variance for assessing batch-variation of stability data of pharmaceuticals." In this article, an approximation test of shelf life equivalence and a test of chemical value equivalence for the data pooling decision are proposed as an alternative to the conventional ANCOVA approach.     

Comparison of statistical models adjusting for baseline in the analysis of parallel-group thorough QT/QTc studies

The ICH E14 guidance recommends the use of a time-matched baseline, while others recommend alternative baseline definitions including a day-averaged baseline. In this article we consider six models adjusting for baselines. We derive the explicit covariances and compare their power under various conditions. Simulation results are provided. We conclude that type I error rates are controlled. However, one model outperforms the others on statistical power under certain conditions. In general, the analysis of covariance (ANCOVA) model using a day-averaged baseline is preferred. If the time-matched baseline has to be used as per requests from regulatory agencies, the analysis by time point using ANCOVA model should be recommended.     

Comparison of Statistical Models Adjusting for Baseline in the Analysis of Parallel-Group Thorough QT/QTc Studies

The ICH E14 guidance recommends the use of a time-matched baseline, while others recommend alternative baseline definitions including a day-averaged baseline. In this article we consider six models adjusting for baselines. We derive the explicit covariances and compare their power under various conditions. Simulation results are provided. We conclude that type I error rates are controlled. However, one model outperforms the others on statistical power under certain conditions. In general, the analysis of covariance (ANCOVA) model using a day-averaged baseline is preferred. If the time-matched baseline has to be used as per requests from regulatory agencies, the analysis by time point using ANCOVA model should be recommended.     

Stratified tests, stratified slopes, and random effects models for clinical trials with missing data

Because missing observations may affect the size and power of statistical tests of equality, various analytical techniques explicitly or implicitly condition the analysis on the amount of information available per person. We illustrate the difference between stratifying a slope estimate and stratifying a test statistic based on slopes. We compare a nonparametric version of the latter approach with the parametric tests available from SAS Proc Mixed. Power and size of these two approaches are considered under different parametric settings, distributions, and missing data mechanisms.     

STRATIFIED TESTS,STRATIFIED SLOPES,AND RANDOM EFFECTS MODELS FOR CLINICAL TRIALS WITH MISSING DATA

Because missing observations may affect the size and power of statistical tests of equality, various analytical techniques explicitly or implicitly condition the analysis on the amount of information available per person. We illustrate the difference between stratifying a slope estimate and stratifying a test statistic based on slopes. We compare a nonparametric version of the latter approach with the parametric tests available from SAS Proc Mixed. Power and size of these two approaches are considered under different parametric settings, distributions, and missing data mechanisms.     

Tests for equivalence based on odds ratio for matched-pair design

Currently, methods for evaluation of equivalence under a matched-pair design use either difference in proportions or relative risk as measures of risk association. However, these measures of association are only for cross-sectional studies or prospective investigations, such as clinical trials and they cannot be applied to retrospective research such as case-control studies. As a result, under a matched-pair design, we propose the use of the conditional odds ratio for assessment of equivalence in both prospective and retrospective research. We suggest the use of the asymptotic confidence interval of the conditional odds ratio for evaluation of equivalence. In addition, a score test based on the restricted maximum likelihood estimator (RMLE) is derived to test the hypothesis of equivalence under a matched-pair design. On the other hand, a sample size formula is also provided. A simulation study was conducted to empirically investigate the size and power of the proposed procedures. Simulation results show that the score test not only adequately controls the Type I error but it can also provide sufficient power. A numerical example illustrates the proposed methods.     

Tests for Equivalence Based on Odds Ratio for Matched-Pair Design

Currently, methods for evaluation of equivalence under a matched-pair design use either difference in proportions or relative risk as measures of risk association. However, these measures of association are only for cross-sectional studies or prospective investigations, such as clinical trials and they cannot be applied to retrospective research such as case-control studies. As a result, under a matched-pair design, we propose the use of the conditional odds ratio for assessment of equivalence in both prospective and retrospective research. We suggest the use of the asymptotic confidence interval of the conditional odds ratio for evaluation of equivalence. In addition, a score test based on the restricted maximum likelihood estimator (RMLE) is derived to test the hypothesis of equivalence under a matched-pair design. On the other hand, a sample size formula is also provided. A simulation study was conducted to empirically investigate the size and power of the proposed procedures. Simulation results show that the score test not only adequately controls the Type I error but it can also provide sufficient power. A numerical example illustrates the proposed methods.     

Testing for positive control activity in a thorough QTc study

The ICH E14 guidance (ICH, 2005) recommend that a concurrent positive control should be included in a thorough QTc clinical trial to validate the study. The ICH E14 guidance (ICH, 2005) state that "The positive control should have an effect on the mean QTc interval of about 5 ms (i.e., an effect that is close to the QTc effect that represents the threshold of regulatory concern, around 5 ms)". This task may be carried out through some statistical tests. The current practice is to test at each time point where QT measurements are collected. This method is usually not efficient. In this article, I discuss two types of statistical procedures. The first one is a local statistical test to make a time-point-specific claim, i.e., to claim a mild QTc effect due to the positive control at some specific time points. A different approach, named as a global test, is also proposed, to make a general claim that the mean difference of the positive control and placebo after baseline adjustment will be about 5 ms without specifying at which time points. An example will be used to illustrate how to apply the two procedures. How to best allocate sample size in a parallel QTc study is also discussed in this paper.     

非劣性/等效性试验的样本含量估计及统计推断

就近年应用逐渐增多的非劣性/等效性试验中涉及的一些关键统计学问题进行详细介绍，其中包括设计过程中的非劣性/等效性界值的确定、样本含量的估计方法和统计推断过程中的检验假设建立、检验统计量计算以及可信区间计算方法。结合7个有针对性的应用实例有助于对相关事项的理解和在非劣性/等效性试验时进行参照。     

A simple formula for sample size calculation in equivalence studies

Bioequivalence and clinical equivalence can be claimed based on the two one-sided test approach or the confidence interval approach. Consequently the power function of the equivalence test can be derived from either noncentral t-distribution or central t-distribution. The sample size is then determined from the power function either by numerical method or closed formulas. In this paper, we propose a simple formula for sample size calculation based on central t-distribution. The proposed formula has better properties than those currently available and it can be easily applied in all equivalence studies.     

Validation Testing in Thorough QT/QTc Clinical Trials∗

Bioequivalence and clinical equivalence can be claimed based on the two one-sided test approach or the confidence interval approach. Consequently the power function of the equivalence test can be derived from either noncentral t-distribution or central t-distribution. The sample size is then determined from the power function either by numerical method or closed formulas. In this paper, we propose a simple formula for sample size calculation based on central t-distribution. The proposed formula has better properties than those currently available and it can be easily applied in all equivalence studies.     

Evaluating the Performance of the ICH Guidelines for Shelf Life Estimation

The goal of shelf life estimation is to determine the storage time during which the entire product meets specification with acceptably high probability. The estimated shelf life should be “applicable to all future batches” (ICH Q1E, International Conference on Harmonization, 2003b). There is compelling evidence of issues with the International Conference on Harmonization (ICH) guidelines for shelf life estimation. Issues include fixed batch effects, poolability tests, and confidence intervals for the mean. Two conclusions from evaluating the ICH procedure are that batch effects should be random and that focus should be on a quantile. A procedure is needed that combines random batches with the ICH objective of estimating the minimum batch shelf life.     

Analysis of time-to-event data with nonuniform patient entry and loss to follow-up under a two-stage seamless adaptive design with weibull distribution

In the pharmaceutical industry, a two-stage seamless adaptive design that combines two separate independent clinical trials into a single clinical study is commonly employed in clinical research and development. In practice, in the interest of shortening the development process, it is not uncommon to consider study endpoints with different treatment durations at different stages (Chow and Chang, 2006 ; Maca et al., 2006 ). In this study, our attention is placed on the case where the study endpoints of interest are time-to-event data where the durations at the two stages are different with nonuniform patient entry and losses to follow-up or dropouts. Test statistics for the final analysis based on the combined data are developed under various hypotheses for testing equality, superiority, noninferiority, and equivalence. In addition, formulas for sample size calculation and allocation between the two stages based on the proposed test statistic are derived.     

Mixed noninferiority margin and statistical tests in active controlled trials

In an active controlled noninferiority trial without a placebo arm, one of the major considerations is the selection of the noninferiority margin. Although the ICH E10 guideline provides general principles for the selection of appropriate noninferiority margins, there are no established rules or gold standards for the selection of noninferiority margins in active control trials. Hung et al. (2003) proposed a margin selection based on relative risk. However, with relative risk, it is difficult to adjust for covariates. On the other hand, Chow and Shao (2006) proposed a method for selecting noninferiority margins based on treatment difference. The determination of noninferiority margin based on either a test for treatment difference or a test for relative risk would be critical. In this paper, we propose a method for noninferiority testing with the use of a mixed null hypothesis. The mixed null hypothesis consists of a margin based on treatment difference and a margin based on relative risk. Both noninferiority margins will simultaneously satisfy the principles as described in the ICH E10 guideline. Statistical tests for mixed noninferiority margin are also derived. An example concerning the efficacy of a test therapy to an active control on a clinical adverse event in the target patient population with cardiovascular disease is presented to illustrate the proposed method. Simulation studies were also conducted to assess the type I error rate and the power.     

A bootstrap test for comparing two variances: simulation of size and power in small samples

An F statistic was proposed by Good and Chernick ( 1993 ) in an unpublished paper, to test the hypothesis of the equality of variances from two independent groups using the bootstrap; see Hall and Padmanabhan ( 1997 ), for a published reference where Good and Chernick ( 1993 ) is discussed. We look at various forms of bootstrap tests that use the F statistic to see whether any or all of them maintain the nominal size of the test over a variety of population distributions when the sample size is small. Chernick and LaBudde ( 2010 ) and Schenker ( 1985 ) showed that bootstrap confidence intervals for variances tend to provide considerably less coverage than their theoretical asymptotic coverage for skewed population distributions such as a chi-squared with 10 degrees of freedom or less or a log-normal distribution. The same difficulties may be also be expected when looking at the ratio of two variances. Since bootstrap tests are related to constructing confidence intervals for the ratio of variances, we simulated the performance of these tests when the population distributions are gamma(2,3), uniform(0,1), Student's t distribution with 10 degrees of freedom (df), normal(0,1), and log-normal(0,1) similar to those used in Chernick and LaBudde ( 2010 ). We find, surprisingly, that the results for the size of the tests are valid (reasonably close to the asymptotic value) for all the various bootstrap tests. Hence we also conducted a power comparison, and we find that bootstrap tests appear to have reasonable power for testing equivalence of variances.     

Exact and Approximate Power and Sample Size Calculations for Analysis of Covariance in Randomized Clinical Trials With or Without Stratification

ABSTRACT

Analysis of covariance (ANCOVA) is commonly used in the analysis of randomized clinical trials to adjust for baseline covariates and improve the precision of the treatment effect estimate. We derive the exact power formulas for testing a homogeneous treatment effect in superiority, noninferiority, and equivalence trials under both unstratified and stratified randomizations, and for testing the overall treatment effect and treatment × stratum interaction in the presence of heterogeneous treatment effects when the covariates excluding the intercept, treatment, and prestratification factors are normally distributed. These formulas also work very well for nonnormal covariates. The sample size methods based on the normal approximation or the asymptotic variance generally underestimate the required size. We adapt the recently developed noniterative and two-step sample size procedures to the above tests. Both methods take into account the nonnormality of the t statistic, and the lower order variance term commonly ignored in the sample size estimation. Numerical examples demonstrate the excellent performance of the proposed methods particularly in small samples. We revisit the topic on the prestratification versus post-stratification by comparing their relative efficiency and power. Supplementary materials for this article are available online.