Sample size calculation for testing non-inferiority and equivalence under Poisson distribution

When counting the number of chemical parts in air pollution studies or when comparing the occurrence of congenital malformations between a uranium mining town and a control population, we often assume Poisson distribution for the number of these rare events. Some discussions on sample size calculation under Poisson model appear elsewhere, but all these focus on the case of testing equality rather than testing equivalence. We discuss sample size and power calculation on the basis of exact distribution under Poisson models for testing non-inferiority and equivalence with respect to the mean incidence rate ratio. On the basis of large sample theory, we further develop an approximate sample size calculation formula using the normal approximation of a proposed test statistic for testing non-inferiority and an approximate power calculation formula for testing equivalence. We find that using these approximation formulae tends to produce an underestimate of the minimum required sample size calculated from using the exact test procedure. On the other hand, we find that the power corresponding to the approximate sample sizes can be actually accurate (with respect to Type I error and power) when we apply the asymptotic test procedure based on the normal distribution. We tabulate in a variety of situations the minimum mean incidence needed in the standard (or the control) population, that can easily be employed to calculate the minimum required sample size from each comparison group for testing non-inferiority and equivalence between two Poisson populations.     

Sample size determination for testing equality in Poisson frequency data under an AB/BA crossover trial

Assuming that the frequency of occurrence follows the Poisson distribution, we develop sample size calculation procedures for testing equality based on an exact test procedure and an asymptotic test procedure under an AB/BA crossover design. We employ Monte Carlo simulation to demonstrate the use of these sample size formulae and evaluate the accuracy of sample size calculation formula derived from the asymptotic test procedure with respect to power in a variety of situations. We note that when both the relative treatment effect of interest and the underlying intraclass correlation between frequencies within patients are large, the sample size calculation based on the asymptotic test procedure can lose accuracy. In this case, the sample size calculation procedure based on the exact test is recommended. On the other hand, if the relative treatment effect of interest is small, the minimum required number of patients per group will be large, and the asymptotic test procedure will be valid for use. In this case, we may consider use of the sample size calculation formula derived from the asymptotic test procedure to reduce the number of patients needed for the exact test procedure. We include an example regarding a double‐blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these sample size formulae. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiple-arm superiority and non-inferiority designs with various endpoints

Multiple-arm dose-response superiority trials are widely studied for continuous and binary endpoints, while non-inferiority designs have been studied recently in two-arm trials. In this paper, a unified asymptotic formulation of a sample size calculation for k-arm (k>0) trials with different endpoints (continuous, binary and survival endpoints) is derived for both superiority and non-inferiority designs. The proposed method covers the sample size calculation for single-arm and k-arm (k> or =2) designs with survival endpoints, which has not been covered in the statistic literature. A simple, closed form for power and sample size calculations is derived from a contrast test. Application examples are provided. The effect of the contrasts on the power is discussed, and a SAS program for sample size calculation is provided and ready to use.     

Sample size determination for testing equality in frequency data under an incomplete block crossover design

When there are more than two treatments under comparison, we may consider the use of the incomplete block crossover design (IBCD) to save the number of patients needed for a parallel groups design and reduce the duration of a crossover trial. We develop an asymptotic procedure for simultaneously testing equality of two treatments versus a control treatment (or placebo) in frequency data under the IBCD with two periods. We derive a sample size calculation procedure for the desired power of detecting the given treatment effects at a nominal-level and suggest a simple ad hoc adjustment procedure to improve the accuracy of the sample size determination when the resulting minimum required number of patients is not large. We employ Monte Carlo simulation to evaluate the finite-sample performance of the proposed test, the accuracy of the sample size calculation procedure, and that with the simple ad hoc adjustment suggested here. We use the data taken as a part of a crossover trial comparing the number of exacerbations between using salbutamol or salmeterol and a placebo in asthma patients to illustrate the sample size calculation procedure.     

An exact test for trend among binomial proportions based on a modified Baumgartner-Weiß-Schindler statistic

The Cochran-Armitage test is the most frequently used test for trend among binomial proportions. This test can be performed based on the asymptotic normality of its test statistic or based on an exact null distribution. As an alternative, a recently introduced modification of the Baumgartner-Weiß-Schindler statistic, a novel nonparametric statistic, can be used. Simulation results indicate that the exact test based on this modification is preferable to the Cochran-Armitage test. This exact test is less conservative and more powerful than the exact Cochran-Armitage test. The power comparison to the asymptotic Cochran-Armitage test does not show a clear winner, but the difference in power is usually small. The exact test based on the modification is recommended here because, in contrast to the asymptotic Cochran-Armitage test, it guarantees a type I error rate less than or equal to the significance level. Moreover, an exact test is often more appropriate than an asymptotic test because randomization rather than random sampling is the norm, for example in biomedical research. The methods are illustrated with an example data set.     

Assessing non-inferiority to an aggregate response with an application to development of pneumococcal conjugate vaccines

The development of a new pneumococcal conjugate vaccine involves assessing the responses of the new serotypes included in the vaccine. The World Health Organization guidance states that the response from each new serotype in the new vaccine should be compared with the aggregate response from the existing vaccine to evaluate non-inferiority. However, no details are provided on how to define and estimate the aggregate response and what methods to use for non-inferiority comparisons. We investigate several methods to estimate the aggregate response based on binary data including simple average, model-based, and lowest response methods. The response of each new serotype is then compared with the estimated aggregate response for non-inferiority. The non-inferiority test p-value and confidence interval are obtained from Miettinen and Nurminen's method, using an effective sample size. The methods are evaluated using simulations and demonstrated with a real clinical trial example.     

A nonparametric confidence interval for slope based on spearman's rho

An algorithm is presented for computing an exact nonparametric interval estimate of the slope parameter in a simple linear regression model. The confidence interval is obtained by inverting the hypothesis test for slope that uses Spearman's rho. This method is compared to an exact procedure based on Kendall's tau. The Spearman rho procedure will generally give exact levels of confidence closer to desired levels, especially in small samples. Monte carlo results comparing these two methods with the parametric procedure are given     

Designing historical control studies with survival endpoints using exact statistical inference

Historical control trials compare an experimental treatment with a previously conducted control treatment. By assigning all recruited samples to the experimental arm, historical control trials can better identify promising treatments in early phase trials compared with randomized control trials. Existing designs of historical control trials with survival endpoints are based on asymptotic normal distribution. However, it remains unclear whether the asymptotic distribution of the test statistic is close enough to the true distribution given relatively small sample sizes in early phase trials. In this article, we address this question by introducing an exact design approach for exponentially distributed survival endpoints, and compare it with an asymptotic design in both real examples and simulation examples. Simulation results show that the asymptotic test could lead to bias in the sample size estimation. We conclude the proposed exact design should be used in the design of historical control trials.     

Exact methods for testing homogeneity of proportions for multiple groups of paired binary data

We consider four exact procedures to test the homogeneity of proportions for correlated multiple clustered data. Exact procedures are compared with the asymptotic approach based on the score statistic. We use a real example from a double-blind clinical trial studying the treatment of otitis media to illustrate the various test procedures and provide extensive numerical studies to compare procedures with regards to Type I error rates and powers under the unconditional framework. The exact unconditional procedure based on estimation followed by maximization is generally more powerful than other procedures.     

A Rank-Based Test for Comparison of Multidimensional Outcomes

For comparison of multiple outcomes commonly encountered in biomedical research, Huang et al. (2005) improved O'Brien's (1984) rank-sum tests through the replacement of the ad hoc variance by the asymptotic variance of the test statistics. The improved tests control the Type I error rate at the desired level and gain power when the differences between the two comparison groups in each outcome variable fall into the same direction. However, they may lose power when the differences are in different directions (e.g., some are positive and some are negative). These tests and the popular Bonferroni correction failed to show important significant difference when applied to compare heart rates from a clinical trial to evaluate the effect of a procedure to remove the cardioprotective solution HTK. We propose an alternative test statistic, taking the maximum of the individual rank-sum statistics, which controls the type I error and maintains satisfactory power regardless of the directions of the differences. Simulation studies show the proposed test to be of higher power than other tests in certain alternative parameter space of interest. Furthermore, when used to analyze the heart rates data the proposed test yields more satisfactory results.     

Weighted Logrank Permutation Tests for Randomly Right Censored Life Science Data

In biomedical research, weighted logrank tests are frequently applied to compare two samples of randomly right censored survival times. We address the question how to combine a number of weighted logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone of alternatives, which are given by hazard rates. This leads to a new class of semiparametric projection tests that are motivated by likelihood ratio tests for an asymptotic model. We show that these tests can be carried out as permutation tests and discuss their asymptotic properties. A simulation study together with the analysis of a classical data set illustrates the advantages.     

基于平均自下而上时间的两种分类方法的比较

内容提要：诸如疾病分类系统的预后预测和分类方法,常可用于帮助进行临床管理决策。同一疾病总体常可得到多种分类方法,因此有必要比较这些方法以确定最优分类,或者寻找不逊于最优分类的替代方法。本文基于约束平均寿命引入分离度指标来度量分类方法的预后分类效率,这个指标可用来比较以生存时间为结局的两种分类方法的功效,特别是用于非劣性和等效性检验。我们给出了基于配对数据的两个分离度的估计与检验方法。模拟结果提示,检验方法在适当的样本量条件下能够控制第一类错误,两个实例表明在医学临床中的应用。     

On the Gini Mean Difference Test for Circular Data

We introduce a new test of isotropy or uniformity on the circle, based on the Gini mean difference of the sample arc-lengths and obtain both the exact and asymptotic distributions under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a “generalized” Gini mean difference test has better asymptotic efficiency than a corresponding “generalized” Rao's test in the sense of Pitman asymptotic relative efficiency.     

Testing hypotheses about the common mean of two normal populations

Two simple tests which allow for unequal sample sizes are considered for testing hypothesis for the common mean of two normal populations. The first test is an exact test of size a based on two available t-statistics based on single samples made exact through random allocation of α among the two available t-tests. The test statistic of the second test is a weighted average of two available t-statistics with random weights. It is shown that the first test is more efficient than the available two t-tests with respect to Bahadur asymptotic relative efficiency. It is also shown that the null distribution of the test statistic in the second test, which is similar to the one based on the normalized Graybill-Deal test statistic, converges to a standard normal distribution. Finally, we compare the small sample properties of these tests, those given in Zhou and Mat hew (1993), and some tests given in Cohen and Sackrowitz (1984) in a simulation study. In this study, we find that the second test performs better than the tests given in Zhou and Mathew (1993) and is comparable to the ones given in Cohen and Sackrowitz (1984) with respect to power..     

A general procedure for deriving distributions

A general procedure for deriving the exact and asymptotic distributions of a certain class of test statistics in multivariate analysis is proposed. The method is based on an asymptotic expansion of gamma ratios in terms of generalized Bernoulli polynomials. The exact and asymptotic results are obtained and the method is illustrated in the problem of testing linear hypotheses in the multinomial case. In this problem the method yields Box's (1949) expansion as a special case.     

Sample size estimation for non-inferiority trials of time-to-event data

We consider the problem of sample size calculation for non-inferiority based on the hazard ratio in time-to-event trials where overall study duration is fixed and subject enrollment is staggered with variable follow-up. An adaptation of previously developed formulae for the superiority framework is presented that specifically allows for effect reversal under the non-inferiority setting, and its consequent effect on variance. Empirical performance is assessed through a small simulation study, and an example based on an ongoing trial is presented. The formulae are straightforward to program and may prove a useful tool in planning trials of this type.     

Distribution-free test for symmetry based on Bonferroni's measure

We propose a test based on Bonferroni's measure of skewness. The test detects the asymmetry of a distribution function about an unknown median. We study the asymptotic distribution of the given test statistic and provide a consistent estimate of its variance. The asymptotic relative efficiency of the proposed test is computed along with Monte Carlo estimates of its power. This allows us to perform a comparison of the test based on Bonferroni's measure with other tests for symmetry.     

Exact solutions to the Behrens–Fisher Problem: Asymptotically optimal and finite sample efficient choice among

The problem of testing the equality of two normal means when variances are not known is called the Behrens–Fisher Problem. This problem has three known exact solutions, due, respectively, to Chapman, to Prokof’yev and Shishkin, and to Dudewicz and Ahmed. Each procedure has level alpha and power beta when the means differ by a given amount delta, both set by the experimenter. No single-sample statistical procedures can make this guarantee. The most recent of the three procedures, that of Dudewicz and Ahmed, is asymptotically optimal. We review the procedures, and then compare them with respect to both asymptotic efficiency and also (using simulation) in finite samples. Of these exact procedures, based on finite-sample comparisons the Dudewicz–Ahmed procedure is recommended for practical use.     

Methods for one-sided testing of the difference between proportions and sample size considerations related to non-inferiority clinical trials

Assessment of non-inferiority is often performed using a one-sided statistical test through an analogous one-sided confidence limit. When the focus of attention is the difference in success rates between test and active control proportions, the lower confidence limit is computed, and many methods exist in the literature to address this objective. This paper considers methods which have been shown to be popular in the literature and have surfaced in this research as having good performance with respect to controlling type I error at the specified level. Performance of these methods is assessed with respect to power and type I error through simulations. Sample size considerations are also included to aid in the planning stages of non-inferiority trials focusing on the difference in proportions. Results suggest that the appropriate method to use depends on the sample size allocation of subjects in the test and active control groups.     

Exact power computation for stratified dose-response studies

Toxicological and pharmaceutical studies often use the stratified dose-response design. This paper deals with the problem of exact power computation for such a design. Recourse to exact methods is advised in study with sparse data. We give three algorithms for exact power calculation and apply them to three data sets. The algorithms include an exact power computation approach and two alternatives to further reduce the computation time: a -precision approach and Monte Carlo approach. Four exact and two asymptotic tests are used for illustration.