A NOTE ON SAMPLE SIZE CALCULATION FOR MEAN COMPARISONS BASED ON NONCENTRAL t-STATISTICS

One-sample and two-sample t-tests are commonly used in analyzing data from clinical trials in comparing mean responses from two drug products. During the planning stage of a clinical study, a crucial step is the sample size calculation, i.e., the determination of the number of subjects (patients) needed to achieve a desired power (e.g., 80%) for detecting a clinically meaningful difference in the mean drug responses. Based on noncentral t-distributions, we derive some sample size calculation formulas for testing equality, testing therapeutic noninferiority/superiority, and testing therapeutic equivalence, under the popular one-sample design, two-sample parallel design, and two-sample crossover design. Useful tables are constructed and some examples are given for illustration.     

Group sequential design and analysis of clinical equivalence assessment for generic nonsystematic drug products

Clinical trials with therapeutical endpoints are designed with three arms to demonstrate both the efficacy and the equivalence of the test generic treatment and the reference treatment. A generic drug product is determined to be equivalent to the reference drug product if the ratio or difference between the mean responses is bounded within the pre-specified equivalence limits. Often the trials are oversized for the placebo arm. For improvement, we propose a group sequential design with hierarchical testing for the purpose of terminating the placebo arm before testing equivalence between the test and the reference treatments. The hierarchical feature of the proposal will reduce the sample size of the placebo arm and provide treatments to patients in a more efficient manner in a clinical trial setting. After dropping the placebo arm, the option of allocating the planned but unused sample size from the placebo group to the test and reference groups will increase the sample size and power of the equivalence test without inflating the type I error rate by delaying spending it.     

Noninferiority testing beyond simple two-sample comparison

In order to fulfill the requirement of a new drug application, a sponsor often need to conduct multiple clinical trials. Often these trials are of designs more complicated than a randomized two-sample single-factor study. For example, these trials could be designed with multiple centers, multiple factors, covariates, group sequential and/or adaptive scheme, etc. When an active standard treatment used as the control treatment in a two-arm clinical trial, the efficacy of the test treatment is often established by performing a noninferiority test through comparison of the test treatment and the active standard treatment. Typically, the noninferiority trials are designed with either a generalized historical control approach (i.e., noninferiority margin approach or delta-margin approach) or a cross-trial comparison approach (i.e., synthesis approach or lambda-margin approach). Many of the statistical properties of the approaches discussed in the literature were focused on testing in a simple two sample comparison form. We studied the limitations of the two approaches for the consideration of switching between superiority and noninferiority testing, feasibility to be applied with group sequential design, constancy assumption requirements, test dependency in multiple trials, analysis of homogeneity of efficacy among centers in a multi-center trial, data transformation and changing analysis method from the historical studies. Our evaluation shows that the cross-trial comparison approach is more restricted to simple two sample comparison with normal approximation test because of its poor properties with more complicated design and analysis. On the other hand, the generalized historical control comparison approach may have more flexible properties when the variability of the margin delta is indeed negligibly small.     

Sample Size Estimation Based on Event Data for a Two-Stage Survival Adaptive Trial with Different Durations

In clinical development, an adaptive design combining results from two separate studies (e.g., a seamless adaptive design with a dose finding study phase and a confirmatory study phase) is commonly considered. The purpose of an adaptive design is not only to reduce lead time between the two studies, but also to evaluate the treatment effect in a more efficient way. In this paper, the focus is on the case where the study objectives are the same but the time durations of the study periods are different in the two stages. In particular, event data are collected in both stages. Statistical procedure for combining data observed from the two different stages is discussed. Furthermore, results on hypotheses testing and sample size calculation are derived for the comparison of two treatments.     

On power and sample size calculation for QT studies with recording replicates at given time point

The problem of the impact on power and sample size calculation for routine QT studies with ECG recording replicates under a parallel-group design and a crossover design is examined. Replicate ECGs are defined as single ECG recorded within several minutes of a nominal time (PhRMA, 2003). Formulas for sample size calculations with and without adjustment for covariates such as some pharmacokinetic responses (e.g., AUC or C(max)), which are known to be correlated to the QT intervals, were derived under both the parallel-group design and the crossover design. The results indicate that the approach of replicates may require a smaller sample size for achieving the same power when the correlation coefficient between the recording replicates (or repeated measures) is close to 0 (i.e., these replicate ECGs are almost independent). On the other hand, if the correlation coefficient is close to 1, then there is not much gain regardless of whether replicate ECGs are considered. In this paper, an approach to identifying optimal allocation between the number of subjects and the number of replicates per subject is proposed for achieving the maximum power under a fixed budget constraint. The proposed approach can also be applied to minimize the cost for a given power.     

Noninferiority Testing Beyond Simple Two-Sample Comparison∗

In order to fulfill the requirement of a new drug application, a sponsor often need to conduct multiple clinical trials. Often these trials are of designs more complicated than a randomized two-sample single-factor study. For example, these trials could be designed with multiple centers, multiple factors, covariates, group sequential and/or adaptive scheme, etc. When an active standard treatment used as the control treatment in a two-arm clinical trial, the efficacy of the test treatment is often established by performing a noninferiority test through comparison of the test treatment and the active standard treatment. Typically, the noninferiority trials are designed with either a generalized historical control approach (i.e., noninferiority margin approach or δ-margin approach) or a cross-trial comparison approach (i.e., synthesis approach or λ-margin approach). Many of the statistical properties of the approaches discussed in the literature were focused on testing in a simple two sample comparison form. We studied the limitations of the two approaches for the consideration of switching between superiority and noninferiority testing, feasibility to be applied with group sequential design, constancy assumption requirements, test dependency in multiple trials, analysis of homogeneity of efficacy among centers in a multi-center trial, data transformation and changing analysis method from the historical studies. Our evaluation shows that the cross-trial comparison approach is more restricted to simple two sample comparison with normal approximation test because of its poor properties with more complicated design and analysis. On the other hand, the generalized historical control comparison approach may have more flexible properties when the variability of the margin δ is indeed negligibly small.     

非虚假设多中心配对临床试验

目的:将配对设计加以扩展,以适应非虚假设和分层。方法:将多中心临床试验的每个中心看作1层,取层样本分数为权计算平均治疗-对照差。将其期望与最小可识别差量比较建立非虚假设,按其方差构造分层配对设计基本关系式,进而推导出样本量公式,检验统计量,和功效函数。将这些用于临床试验的设计、执行和分析。以Monte Carlo方法展示观测功效。结果:这些在最小可识别差量取零时还原为对应经典统计学方法,其观测功效和期望功效吻合,所需样本量小于对应的两组设计。结论:这种临床试验直观高效,适于建立试药对于对照药的临床优效性、非劣效性或等效性,并附有实例描述用法。     

Notes on Testing Noninferiority in Ordinal Data Under the Parallel Groups Design

When testing the noninferiority of an experimental treatment to a standard (or control) treatment in a randomized clinical trial (RCT), we may come across the outcomes of patient response on an ordinal scale. We focus our discussion on testing noninferiority in ordinal data for an RCT under the parallel groups design. We develop simple test procedures based on the generalized odds ratio (GOR). We note that these test procedures not only can account for the information on the order of ordinal responses without assuming any specific parametric structural model, but also can be independent of any arbitrarily subjective scoring system. We further develop sample size determination based on the test procedure using the GOR. We apply Monte Carlo simulation to evaluate the performance of these test procedures and the accuracy of sample size calculation formula proposed here in a variety of situations. Finally, we employ the data taken from a trial comparing once-daily gatifloxican with three-times-daily co-amoxiclav in the treatment of community-acquired pneumonia to illustrate the use of these test procedures and sample size calculation formula.     

Effect of dropouts on sample size estimates for test on trends across repeated measurements

Sample size calculation is an important component at the design stage of clinical trials. We investigate the implications of dropouts for the sample size estimates in testing differences in the rates of changes produced by two treatments in a randomized parallel-groups repeated measurement design. Statistical models for calculating sample sizes for repeated measurement designs often fail to take into account the impact of dropouts correctly. In this article, we examine the impact of dropouts on sample size estimate and compare the power with the approach of Jung and Ahn [Jung, S. H., Ahn, C. (2003). Sample size estimation for GEE method for comparing slopes in repeated measurements data. Stat. Med. 22: 1305-1315] with that suggested by Patel and Rowe [Patel, H., Rowe, E. (1999). Sample size for comparing linear growth curves. J. Biopharm. Stat. 9:339-350] through a simulation study.     

Sample size adaptation in fixed-dose combination drug trial

Statistical testing in clinical trials can be complex when the statistical distribution of the test statistic involves a nuisance parameter. Some type of nuisance parameters such as standard deviation of a continuous response variable can be handled without too much difficulty. Other type of nuisance parameters, specifically associated with the main parameter under testing, can be difficult to handle. Without knowledge of the possible value of such a nuisance parameter, the maximum type I error associated with testing the main parameter may occur at an extreme value of the nuisance parameter. A well known example is the intersection-union test for comparing a combination drug with its two component drugs where the nuisance parameter is the mean difference between the two components. Knowledge of the possible range of value of this mean difference may help enhance the clinical trial design. For instance, if the interim internal data suggest that this mean difference falls into a possible range of value, then the sample size may be reallocated after the interim look to possibly improve the efficiency of statistical testing. This research sheds some light into possible power advantage from such a sample size reallocation at the interim look.     

Probability monitoring procedures for sample size determination

ABSTRACT

In clinical research, power analysis is often performed for sample size calculation. The purpose is to achieve a desired power of correctly detecting a clinically meaningful difference at a pre-specified level of significance if such a difference truly exists. However, in some situations such as (i) clinical trials with extremely low incidence rates and (ii) for rare disease drug development clinical trials, power analysis for sample size calculation may not be feasible because (i) it may require a huge sample size for detecting a relatively small difference and (ii) eligible patients may not be available for a small target patient population. In these cases, other procedures for sample size determination with certain statistical assurance are needed. In this article, an innovative method based on a probability monitoring procedure is proposed for sample size determination. The concept is to select an appropriate sample size for controlling the probability of crossing safety and/or efficacy boundaries. For rare disease clinical development, an adaptive probability monitoring procedure may be applied if a multiple-stage adaptive trial design is used.     

Statistical issues including design and sample size calculation in thorough QT/QTc studies

After several drugs were removed from the market in recent years because of death due to ventricular tachycardia resulting from drug-induced QT prolongation (Khongphatthanayothin et al., 1998; Lasser et al., 2002; Pratt et al., 1994; Wysowski et al., 2001), the ICH Regulatory agencies requested all sponsors of new drugs to conduct a clinical study, named a Thorough QT/QTc (TQT) study, to assess any possible QT prolongation due to the study drug. The final version of the ICH E14 guidance (ICH, 2005) for "The Clinical Evaluation of QT/QTc Interval Prolongation and Proarrhythmic Potential for Nonantiarrhythmic Drugs" was released in May 2005. The purpose of the ICH E14 guidance (ICH, 2005) is to provide recommendations to sponsors concerning the design, conduct, analysis, and interpretation of clinical studies to assess the potential of a drug to delay cardiac repolarization. The guideline, however, is not specific on several issues. In this paper, we try to address some statistical issues, including study design, primary statistical analysis, assay sensitivity analysis, and the calculation of the sample size for a TQT study.     

Controversies in antimicrobial therapy: critical analysis of clinical trials

Problems with design and statistical evaluation of clinical efficacy trials of antimicrobial agents are reviewed. Of the three major criteria used for evaluating antimicrobial agents (efficacy, toxicity, cost), the most important is efficacy. Clinical efficacy can be evaluated in uncontrolled or controlled clinical trials. Uncontrolled trials are often conducted to satisfy Food and Drug Administration requirements during premarketing testing; the response rate is typically high because only patients with susceptible infections may be treated and large doses are given. Controlled antibiotic trials should be randomized, blinded, parallel comparisons of an investigational agent versus the best available agent at an accepted dose. However, interpretation of these studies is frequently clouded by poor study design, small sample sizes, and heterogeneous patient populations. Controlled trials are usually centered around a null hypothesis (i.e., that no difference will be found between the agents being compared). All conclusions (to reject or not reject the null hypothesis) should be carefully evaluated by clinicians seeking to apply the available data to patient care. Researchers can incorrectly conclude that two therapies have equal efficacy because of insufficient statistical power (i.e., small sample size) or poor study design. Likewise, researchers may incorrectly conclude that there is a statistical difference between two therapies because of poor design or improper sample selection. For the clinician, clinical relevance takes precedence over statistical significance. Before the results of a study are allowed to affect drug use in an institution, strong similarities between subjects and methods in the study and patients and care in the institution should be demonstrated.     

Sample sizes based on exact unconditional tests for phase II clinical trials with historical controls

Investigated in the setting of phase II clinical trials is the two-sample binomial problem of testing H0: pe = pc H1: pe > pc, where pe and pc are the unknown target population response rates for the experimental and control groups, respectively, using the usual Z-statistic with pooled variance estimator. The cornerstones that make this paper unique are as follows. First, the emphasis is on determining the sample size given that the control group information has already been collected (historical control). Second, exact unconditional inference, rather than an asymptotic method, is utilized. Sample size tables, contrasting the exact and asymptotic methods, are provided. Although asymptotic results were usually fairly close to the exact results, some important differences were observed.     

Clinical and regulatory perspectives on general/safety pharmacology tests: Results of a pharmaceutical industry survey

General/safety pharmacology is an emerging discipline within the pharmaceutical industry in which unanticipated effects of new drug candidates on major organ function (i.e., secondary pharmacological effects) are critically assessed in a variety of animal models. A survey was conducted to obtain customer input on the role and strategies of this emerging discipline. Four surveys were distributed to each of 30 U.S. pharmaceutical companies. Two surveys went to individuals within the regulatory department, and two went to individuals within the clinical department of each company. All responses were returned anonymously to the survey authors. Sixty-six responses from at least 19 different companies were obtained. The responses from the clinical and regulatory departments were essentially the same and led to the following conclusions: (1) Major organ functions are monitored routinely in clinical trials to assess potential adverse drug effects; (2) major organ function studies are considered an essential part of non-clinical drug safety testing; (3) respiratory, gastrointestinal, autonomic, and endocrine evaluations are considered an important part of major organ function studies, whereas cardiovascular, renal and central nervous evaluations are considered essential; (4) non-clinical testing of these major organ functions should be conducted early in drug development, preferably before initiation of clinical trials; and (5) organ function investigations involving mechanisms of secondary pharmacological effects or drug interaction are considered an important part of general/safety pharmacology testing. © 1995 Wiley-Liss, Inc.     

On sample size calculation based on odds ratio in clinical trials

Sample size calculation formulas for testing equality, noninferiority, superiority, and equivalence based on odds ratio were derived under both parallel and one-arm crossover designs. An example concerning the study of odds ratio between a test compound (treatment) and a standard therapy (control) for prevention of relapse in subjects with schizophrenia and schizoaffective disorder is presented to illustrate the derived formulas for sample size calculation for various hypotheses under both a parallel design and a crossover design. Simulations were performed to assess the adequacy of the sample size calculation formulas. Simulation results were given at the end of the paper.     

Group Sequential Survival Trial Design and Monitoring Using the Log-Rank Test

For randomized group sequential survival trial designs with unbalanced treatment allocation, the widely used Schoenfeld formula is inaccurate, and the commonly used information time as the ratio of number of events at interim look to the number of events at the end of trial can be biased. In this article, a sample size formula for the two-sample log-rank test under the proportional hazards model is proposed that provides more accurate sample size calculation for unbalanced survival trial designs. Furthermore, a new information time is introduced for the sequential survival trials such that the new information time is more accurate than the traditional information time when the allocation of enrollments is unbalanced in groups. Finally, we demonstrate the monitoring process using the sequential conditional probability ratio test and compare it with two other well-known group sequential procedures. An example is given to illustrate unbalanced survival trial design using available software. Supplementary materials for this article are available online.     

SAMPLE SIZE AND OPTIMAL DESIGNS IN STRATIFIED COMPARATIVE TRIALS TO ESTABLISH THE EQUIVALENCE OF TREATMENT EFFECTS AMONG TWO ETHNIC GROUPS

When a new investigational medicine is intended to be applied to populations with different ethnic backgrounds, a stratified comparative phase III trial using ethnic groups as strata may be conducted to assess the influence of ethnic factors on clinical outcomes of this new medicine. In this paper, based on a binomial model with odds ratio as the measure of the treatment effect, we derive the score test and the associated sample size formula for establishing the equivalence/noninferiority of the treatment effects of a medicine among two ethnic groups. A simplified test together with its sample size formula are also given. Taking into account the sample size, cost, and power of testing, respectively, we derive the optimal design parameters, i.e., the allocation among treatment groups and ethnic groups, based on the simplified test.