Sample sizes for clinical trials with normal data

This article gives an overview of sample size calculations for parallel group and cross-over studies with Normal data. Sample size derivation is given for trials where the objective is to demonstrate: superiority, equivalence, non-inferiority, bioequivalence and estimation to a given precision, for different types I and II errors. It is demonstrated how the different trial objectives influence the null and alternative hypotheses of the trials and how these hypotheses influence the calculations. Sample size tables for the different types of trials and worked examples are given.     

A hybrid Bayesian-frequentist approach to evaluate clinical trial designs for tests of superiority and non-inferiority

Specification of the study objective of superiority or non-inferiority at the design stage of a phase III clinical trial can sometimes be very difficult due to the uncertainty that surrounds the efficacy level of the experimental treatment. This uncertainty makes it tempting for investigators to design a trial that would allow testing of both superiority and non-inferiority hypotheses. However, when a conventional single-stage design is used to test both hypotheses, the sample size is based on the chosen primary objective of either superiority or non-inferiority. In this situation, the power of the test for the secondary objective can be low, which may lead to a large loss of resources. Potentially low reproducibility is another major concern for the single-stage design in phase III trials, because significant findings of confirmatory trials are required to be reproducible. In this paper, we propose a hybrid Bayesian-frequentist approach to evaluate reproducibility and power in single-stage designs for phase III trials to test both superiority and non-inferiority. The essence of the proposed approach is to express the uncertainty that surrounds the efficacy of the experimental treatment as a probability distribution. Then one can use Bayes formula with simple graphical techniques to evaluate reproducibility and power adequacy.     

The use of putative placebo in active control trials: two applications in a regulatory setting

For life-threatening diseases, ethical considerations preclude the inclusion of an untreated control group in the investigation of a new therapeutic agent when a standard therapy exists. In these cases, active controlled studies are conducted, and may be planned to demonstrate either superiority or equivalence/non-inferiority of the new drug over the standard therapy (active control). In the non-inferiority study, an important aspect is the ability to detect an inferior drug (assay sensitivity). It has been suggested that assay sensitivity for a non-inferiority study should be deduced from historical data, specifically placebo controlled studies with the standard therapy. The assessment of assay sensitivity may also be important in a superiority trial that fails to demonstrate a statistically significant difference between treatments, and the sponsor attempts to determine whether there is lack of inferiority as an alternative hypothesis for regulatory approval. This paper describes two methods of putative placebo analysis for assessing assay sensitivity in active controlled trials. One approach imputes a point estimate for the odds ratio (95 per cent confidence interval) for a new drug (T) compared to a placebo control (P). A Bayesian approach calculates the posterior probability that T is superior to P, or, that T is at least k per cent as good as the active control (A) and A is more effective than P. These methods are applied in two clinical/regulatory settings: a phase III trial comparing docetaxel (Taxotere) to doxorubicin in metastatic breast cancer patients, and a phase III programme with two trials comparing enoxaparin (Lovenox) plus aspirin to unfractionated heparin plus aspirin in patients with unstable angina or non-Q-wave myocardial infarction. The methodologies presented in this paper were used in securing regulatory approval for docetaxel in the treatment of locally advanced or metastatic breast cancer after failure of prior chemotherapy, and for enoxaparin in the treatment of acute coronary syndrome.     

A combined superiority and non-inferiority approach to multiple endpoints in clinical trials

Treatment comparisons in clinical trials often involve multiple endpoints. By making use of bootstrap tests, we develop a new non-parametric approach to multiple-endpoint testing that can be used to demonstrate non-inferiority of a new treatment for all endpoints and superiority for some endpoint when it is compared to an active control. It is shown that this approach does not incur a large multiplicity cost in sample size to achieve reasonable power and that it can incorporate complex dependencies in the multivariate distributions of all outcome variables for the two treatments via bootstrap resampling.     

PASS软件实现临床试验中非劣效、等效和优效性检验的样本量估算

用PASS 11软件对非劣效、等效和优效性设计的临床试验进行样本量估算,并与SAS软件运行结果进行比较,探讨PASS 11软件在临床科研中计算样本量的实用性和准确性,为科研工作者在临床试验设计阶段进行科学的样本量估算提供帮助。     

Two-stage adaptive strategy for superiority and non-inferiority hypotheses in active controlled clinical trials

In active controlled trials without a placebo arm, there are usually two study objectives: to test a superiority hypothesis that the experimental treatment is more effective than the active control therapy, and to test a non-inferiority hypothesis that the experimental treatment is therapeutically no worse than the active control within a defined margin. For a two-stage adaptive design, it is not necessary to give a fixed sample size calculation at the planning stage of the study when treatment effect information is often insufficient. Instead, decision and estimation of the design specifications can be made more reliably after the first stage when interim results are available. We propose the use of conditional power approach to determine the sample size and critical values for testing the superiority and non-inferiority hypotheses for the second stage based on the observed result of the first stage. The proposed adaptive procedure preserves the overall type I error rate for both superiority and non-inferiority, and has the flexibility of early termination of the study (for futility or efficacy) or extending the study by appropriate sample size.     

The importance of equivalence trials in showing the usefulness of treatments

There are different kinds of randomised controlled trials: trials in which the superiority of a treatment can be demonstrated (superiority trials) and trials in which the equal efficacy of two treatments can be shown (equivalence trials). The main reason for performing an equivalence trial is that for many diseases and disorders an effective treatment already exists. Equivalence trials are appropriate when a new treatment offers some advantages over an existing treatment (less cost, greater safety, improved convenience or freedom of choice for the patient), in addition to the expected equal therapeutic effectiveness. The design of equivalence trials is to a large extent comparable to that of superiority trials, but there are some methodological differences. In equivalence trials, the null hypothesis and alternative hypothesis are interchanged, compared to superiority trials. In equivalence trials, an equivalence margin is defined for the different treatments. Clinical professionals decide on the equivalence margin beforehand on the basis of the clinical relevance. To demonstrate equivalence, the confidence interval of the difference between two treatments must lie completely within the equivalence margin. In equivalence trials, there are usually more patients needed: the smaller the equivalence margin, the more patients are needed. In equivalence trials, both per-protocol analyses and intention-to-treat analyses should be used to prove the equal therapeutic effectiveness of the treatments under study.     

Blinded sample size reassessment in non-inferiority and equivalence trials

Even in situations where the design and conduct of clinical trials is highly standardized, there may be a considerable between-study variation in the observed variability of the primary outcome variable. As a consequence, performing a study in a fixed sample size design implies a considerable risk of resulting in a too high or too low sample size. This difficulty can be alleviated by applying a design with internal pilot study. After a provisional sample size calculation in the planning stage, a portion of the planned sample is recruited and the sample size is recalculated on the basis of the observed variability. To comply with the requirement of some regulatory guidelines only blinded data should be used for the reassessment procedure. Furthermore, the effect on the type I error rate should be quantified. The current literature presents analytical results on the actual level in the t-test situation only for superiority trials. In these situations, blinded sample size recalculation does not lead to an inflation of the type I error rate. We extended the methodology to non-inferiority and equivalence trials with normally distributed outcome variable and hypotheses formulated in terms of the ratio and difference of means. Surprisingly, in contrast to the case of testing superiority, we observed actual type I error rates above the nominal level. The extent of inflation depends on the required sample size, the sample size of the internal pilot study, and the standardized equivalence or non-inferiority margin. It turned out that the elevation of the significance level is negligible for most practical situations. Nevertheless, the consequences of sample size reassessment have to be discussed case by case and regulatory concerns with respect to the actual size of the procedure cannot generally be refuted by referring to the fact that only blinded data were used.     

Group sequential test strategies for superiority and non-inferiority hypotheses in active controlled clinical trials.

In a group sequential active controlled clinical trial, the study hypothesis may be a superiority hypothesis that an experimental treatment is more effective than the active control therapy or a non-inferiority hypothesis that the treatment is no worse than the active control within some non-inferiority range. When it is necessary to plan for testing the superiority and the non-inferiority hypotheses, we propose an adaptive group sequential closed test strategy by which the sample size is planned for testing superiority and is to be increased for showing non-inferiority given that it is deemed more plausible than superiority based on the observed sample path during the course of the trial. The proposed adaptive test strategy is valid in terms of having the type I error probability maintained at the targeted alpha level for both superiority and non-inferiority. It has power advantage or sample size saving over the traditional group sequential test designed for testing either superiority only or non-inferiority only.     

Modified Haybittle-Peto group sequential designs for testing superiority and non-inferiority hypotheses in clinical trials

In designing an active controlled clinical trial, one sometimes has to choose between a superiority objective (to demonstrate that a new treatment is more effective than an active control therapy) and a non-inferiority objective (to demonstrate that it is no worse than the active control within some pre-specified non-inferiority margin). It is often difficult to decide which study objective should be undertaken at the planning stage when one does not have actual data on the comparative advantage of the new treatment. By making use of recent advances in the theory of efficient group sequential tests, we show how this difficulty can be resolved by a flexible group sequential design that can adaptively choose between the superiority and non-inferiority objectives during interim analyses. While maintaining the type I error probability at a pre-specified level, the proposed test is shown to have power advantage and/or sample size saving over fixed sample size tests for either only superiority or non-inferiority, and over other group sequential designs in the literature.     

Non-inferiority trials: the 'at least as good as' criterion with dichotomous data

The 'at least as good as' criterion, introduced by Laster and Johnson for a continuous response variate, is developed here for applications with dichotomous data. This approach is adaptive in nature, as the margin of non-inferiority is not taken as a fixed difference; it varies as a function of the positive control response. When the non-inferiority margin is referenced as a high fraction of the positive control response, the procedure is seen to be uniformly more efficient than the fixed margin approach, yielding smaller sample sizes when sizing non-inferiority trials under identically specified conditions. Extending this method to proportions is straightforward, but highlights special considerations in the design of non-inferiority trials versus superiority trials, including potential trade-offs in statistical efficiency and interpretability.     

A noniterative sample size procedure for tests based on t distributions

A noniterative sample size procedure is proposed for a general hypothesis test based on the t distribution by modifying and extending Guenther's 6 approach for the one sample and two sample t tests. The generalized procedure is employed to determine the sample size for treatment comparisons using the analysis of covariance (ANCOVA) and the mixed effects model for repeated measures in randomized clinical trials. The sample size is calculated by adding a few simple correction terms to the sample size from the normal approximation to account for the nonnormality of the t statistic and lower order variance terms, which are functions of the covariates in the model. But it does not require specifying the covariate distribution. The noniterative procedure is suitable for superiority tests, noninferiority tests, and a special case of the tests for equivalence or bioequivalence and generally yields the exact or nearly exact sample size estimate after rounding to an integer. The method for calculating the exact power of the two sample t test with unequal variance in superiority trials is extended to equivalence trials. We also derive accurate power formulae for ANCOVA and mixed effects model for repeated measures, and the formula for ANCOVA is exact for normally distributed covariates. Numerical examples demonstrate the accuracy of the proposed methods particularly in small samples.     

Assessing non-inferiority: a combination approach

Non-inferiority designs are growing in importance as a strategy for comparing new drugs with established therapies. Because it is not possible to show that a new drug and the established therapy have identical efficacy profiles, non-inferiority trials are designed to demonstrate that the new drug is not inferior to an established drug (the 'control') relative to a prespecified 'non-inferiority margin'. No objective principle guides the choice of the non-inferiority margin, and controversies about the margin have, in some cases, had important consequences for drug development.We argue that some of these controversies have arisen because non-inferiority trials must achieve two objectives. They must demonstrate not only that the new drug is not inferior to the control drug by the non-inferiority margin, but also that the new drug is superior to placebo. When the second objective is not considered explicitly, it can distort the choice of the non-inferiority margin. Some methods designed to address both objectives through the choice of the non-inferiority margin lead to overly stringent non-inferiority criteria.We describe an approach to non-inferiority analysis that combines two tests, a traditional test for non-inferiority and a test for superiority based on a synthetic estimate of the effect of the new treatment relative to placebo. The synthetic estimate may be 'discounted' to address concerns about assay inconstancy. We discuss power and sample size considerations for the proposed procedure.     

Therapeutic equivalence: fallacies and falsification

The number of studies designed specifically to demonstrate therapeutic equivalence or alternatively non-inferiority of pharmaceutical treatments has increased dramatically in recent years, during which time awareness of the methodological issues has increased. Regulatory authorities have been quick to recognize the need for specific support and have either published or initiated the creation of relevant guidance. Common misconceptions prevail however regarding sample size estimation and the choice of the most appropriate patient population to analyse while other areas such as equivalence margin specification and covariate adjustment have been neglected. This paper challenges some of the regulatory advice and the interpretation that others have made of this guidance with the aim of stimulating further debate.     

A note on the conservativeness of the confidence interval approach for the selection of non-inferiority margin in the two-arm active-control trial

Compared with placebo-control clinical trials, the interpretation of efficacy results from active-control trials requires more caution. This is because efficacy results from such trials cannot be reliably interpreted without a thorough understanding of the efficacy evidence that formed the basis for the approval of the active control, especially when such drug efficacy is to be established on the basis of clinical evidence from the traditional two-arm active-control clinical equivalence studies as opposed to the multi-arm active control. This is because in addition to over-reliance on the quantification of a clinically irrelevant acceptable margin of inferiority from historical data, such interpretation also depends on cross-trial inference for demonstration of experimental drug effect. We provide a brief overview of some design issues with the traditional two-arm active-control clinical trial and discuss regulators' concern regarding Type I error rate control (with the two most popular methods for the quantification of the non-inferiority margin) in cross-trial demonstration of experimental drug effect. Simulation results are presented to show that the point estimate method provides adequate control of the Type I error rate with > or =75 per cent retention of known active-control effect and that the confidence interval approach is uniformly ultra-conservative. We also report (via a numerical example from real clinical trial data) a couple of potentially less stringent alternative approaches for establishing the non-inferiority of a test drug over a control, which have been used in the past to provide additional efficacy evidence in NDA submission.     

Tests for equivalence or non-inferiority for paired binary data.

Assessment of therapeutic equivalence or non-inferiority between two medical diagnostic procedures often involves comparisons of the response rates between paired binary endpoints. The commonly used and accepted approach to assessing equivalence is by comparing the asymptotic confidence interval on the difference of two response rates with some clinical meaningful equivalence limits. This paper investigates two asymptotic test statistics, a Wald-type (sample-based) test statistic and a restricted maximum likelihood estimation (RMLE-based) test statistic, to assess equivalence or non-inferiority based on paired binary endpoints. The sample size and power functions of the two tests are derived. The actual type I error and power of the two tests are computed by enumerating the exact probabilities in the rejection region. The results show that the RMLE-based test controls type I error better than the sample-based test. To establish an equivalence between two treatments with a symmetric equivalence limit of 0.15, a minimal sample size of 120 is needed. The RMLE-based test without the continuity correction performs well at the boundary point 0. A numerical example illustrates the proposed procedures.     

Test non-inferiority (and equivalence) based on the odds ratio under a simple crossover trial

For testing the non-inferiority (or equivalence) of a generic drug to a standard drug, the odds ratio (OR) of patient response rates has been recommended to measure the relative treatment efficacy. On the basis of a random effects logistic regression model, we develop asymptotic test procedures for testing non-inferiority and equivalence with respect to the OR of patient response rates under a simple crossover design. We further derive exact test procedures, which are especially useful for the situations in which the number of patients in a crossover trial is small. We address sample size calculation for testing non-inferiority and equivalence based on the asymptotic test procedures proposed here. We also discuss estimation of the OR of patient response rates for both the treatment and period effects. Finally, we include two examples, one comparing two solution aerosols in treating asthma, and the other one studying two inhalation devices for asthmatics, to illustrate the use of the proposed test procedures and estimators.     

Tests of equivalence and non-inferiority for diagnostic accuracy based on the paired areas under ROC curves

Assessment of equivalence or non-inferiority in accuracy between two diagnostic procedures often involves comparisons of paired areas under the receiver operating characteristic (ROC) curves. With some pre-specified clinically meaningful limits, the current approach to evaluating equivalence is to perform the two one-sided tests (TOST) based on the difference in paired areas under ROC curves estimated by the non-parametric method. We propose to use the standardized difference for assessing equivalence or non-inferiority in diagnostic accuracy based on paired areas under ROC curves between two diagnostic procedures. The bootstrap technique is also suggested for both non-parametric method and the standardized difference approach. A simulation study was conducted empirically to investigate the size and power of the four methods for various combinations of distributions, data types, sample sizes, and different correlations. Simulation results demonstrate that the bootstrap procedure of the standardized difference approach not only can adequately control the type I error rate at the nominal level but also provides equivalent power under both symmetrical and skewed distributions. A numerical example using published data illustrates the proposed methods.     

Choosing the analysis population in non-inferiority studies: per protocol or intent-to-treat

For superiority trials, the intent-to-treat population (ITT) is considered the primary analysis population because it tends to avoid the over-optimistic estimates of efficacy that results from a per-protocol (PP) population. However, the roles of the ITT population and PP population in non-inferiority studies are not clearly defined as in superiority trials. In this paper, a simulation study is conducted to systematically investigate the impact of different types of missingness and protocol violations on the conservatism or anticonservatism of analyses based on the ITT and the PP population in non-inferiority trials. We find that conservatism or anticonservatism of the PP or ITT analysis depends on many factors, including the type of protocol deviation and missingness, the treatment trajectory (for longitudinal study) and the method of handling missing data in ITT population. The requirement that non-inferiority be shown for both PP and ITT populations does not necessarily guarantee the validity of a non-inferiority conclusion and a sufficiently powered PP analysis is not necessarily powered for ITT analysis. It is important to assess the potential types and rates of protocol deviation and missingness that might occur in a non-inferiority trial and to obtain some prior knowledge regarding the treatment trajectory of the test treatment versus the active control at the design stage so that a proper analysis plan and appropriate power estimation can be carried out. In general, for the types of protocol violations and missingness considered, we find that hybrid ITT/PP analysis, which excludes non-compliant patients as in the PP analysis and properly addresses the impact of non-trivial missing data as in the MLE-based ITT analysis, is more promising by way of providing reliable non-inferiority tests.     

Assessing noninferiority in a three-arm trial using the Bayesian approach

Non-inferiority trials, which aim to demonstrate that a test product is not worse than a competitor by more than a pre-specified small amount, are of great importance to the pharmaceutical community. As a result, methodology for designing and analyzing such trials is required, and developing new methods for such analysis is an important area of statistical research. The three-arm trial consists of a placebo, a reference and an experimental treatment, and simultaneously tests the superiority of the reference over the placebo along with comparing this reference to an experimental treatment. In this paper, we consider the analysis of non-inferiority trials using Bayesian methods which incorporate both parametric as well as semi-parametric models. The resulting testing approach is both flexible and robust. The benefit of the proposed Bayesian methods is assessed via simulation, based on a study examining home-based blood pressure interventions.