Extending the Barnard’s test to non-inferiority

In 1945, George Alfred Barnard presented an unconditional exact test to compare two independent proportions. Critical regions for this test, by construction accomplish the very useful property of being Barnard convex sets. Besides, there are empirical findings suggesting that Barnard’s test is the most generally powerful. For Barnard’s test, calculation of critical regions is complicated due that they are constructed in an iterative form until is obtained a test size, as close as possible to the nominal significance level and less than or equal to it. In this article we present an extension to non-inferiority of this very leading test. This extension was contructed for any dissimilarity measure and tables were constructed for the difference between proportions. Also we calculate the critical regions for this extended test for sample sizes less or equal than 30, nominal significance level 0.01, 0.025, 0.05, and 0.10 and for non-inferiority margins 0.05, 0.10, 0.15, and 0.20. Additionally, we computed test sizes for the mentioned configurations. To do this calculations, we have written a program in the R environment.     

Win-probabilities for comparing two binary outcomes

This article considers the problem of choosing between two treatments that have binary outcomes with unknown success probabilities p₁ and p₂. The choice is based upon the information provided by two observations X₁ ～ B(n₁, p₁) and X₂ ～ B(n₂, p₂) from independent binomial distributions. Standard approaches to this problem utilize basic statistical inference methodologies such as hypothesis tests and confidence intervals for the difference p₁ ? p₂ of the success probabilities. However, in this article the analysis of win-probabilities is considered. If X*₁ represents a potential future observation from Treatment 1 while X*₂ represents a potential future observation from Treatment 2, win-probabilities are defined in terms of the comparisons of X*₁ and X*₂. These win-probabilities provide a direct assessment of the relative advantages and disadvantages of choosing either treatment for one future application, and their interpretation can be combined with other factors such as costs, side-effects, and the availabilities of the two treatments. In this article, it is shown how confidence intervals for the win-probabilities can be constructed, and examples of their use are provided. Computer code for the implementation of this new methodology is available from the authors.     

On Power Approximations and Comparison of Several Asymptotic Tests to Detect a Specified Difference between Two Proportions

A large sample test is proposed for a problem of testing for a specified difference between two binomial proportions. The test is compared to the tests by Falk and Koch (1998 Falk , R. W. , Koch , G. G. ( 1998 ). Testing a specified difference between proportions . Biometrics 54 ( 4 ): 1602 – 1614 .[Crossref] , [Google Scholar]), and Parmet and Schechtman (2007 Parmet , Y. , Schechtman , E. ( 2007 ). On a test of the difference between two binomial proportions . Communications in Statistics – Theory and Methods 36 : 887 – 895 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and is shown to dominate in terms of the Type I error rate control. Asymptotic power is derived for each test and is shown to result in values quite proximate to the simulated power values. In addition, formulas to perform sample size estimation are provided. These methods are expected to be especially valuable in the design stage when obtaining the correct power/sample size estimation is essential.     

Statistical analysis of non-inferiority via non-zero risk difference in stratified matched-pair studies

A stratified study is often designed for adjusting several independent trials in modern medical research. We consider the problem of non-inferiority tests and sample size determinations for a nonzero risk difference in stratified matched-pair studies, and develop the likelihood ratio and Wald-type weighted statistics for testing a null hypothesis of non-zero risk difference for each stratum in stratified matched-pair studies on the basis of (1) the sample-based method and (2) the constrained maximum likelihood estimation (CMLE) method. Sample size formulae for the above proposed statistics are derived, and several choices of weights for Wald-type weighted statistics are considered. We evaluate the performance of the proposed tests according to type I error rates and empirical powers via simulation studies. Empirical results show that (1) the likelihood ratio and the Wald-type CMLE test based on harmonic means of the stratum-specific sample size (SSIZE) weight (the Cochran's test) behave satisfactorily in the sense that their significance levels are much closer to the prespecified nominal level; (2) the likelihood ratio test is better than Nam's [2006. Non-inferiority of new procedure to standard procedure in stratified matched-pair design. Biometrical J. 48, 966–977] score test; (3) the sample sizes obtained by using SSIZE weight are smaller than other weighted statistics in general; (4) the Cochran's test statistic is generally much better than other weighted statistics with CMLE method. A real example from a clinical laboratory study is used to illustrate the proposed methodologies.     

Comparison of correlated proportions based on paired binary data from clustered samples

In this article, we explore hypothesis testing problems related to correlated proportions from clustered matched-pair binary data. Null hypotheses of equality in proportions, homogeneity, and non-inferiority of one to another are similar testing problems of linear contrasts of correlated proportions with suitable transformation. The covariance estimators of the test statistics are based on moment estimation under the null hypotheses. We present a general framework for testing linear contrasts of the correlated proportions from clustered matched-pair data based upon a class of unbiased estimators of the proportions. The corresponding testing procedures do not impose structure assumptions on the correlation matrix and are easy to use. Simulation results suggest that the proposed method is more likely to maintain the proper significance level and to improve power than the test proposed by Obuchowski.     

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.