Constructing tests to compare two proportions whose critical regions guarantee to be Barnard convex sets

In drug development, non inferiority tests are often employed to determine the odds ratio between two independent binomial proportions. Some traditional test statistics for non inferiority based on different statistical schools of thought are available, but they can produce markedly different behaviors. Cornfield's exact conditional test is criticized for its conservativeness, while the Wald and score tests may be applicable for large samples but tend to be liberal with small samples. Although the mid-p value method has recently gained wide acceptance for its better behavior than those of other tests, it has almost no solid theoretical foundation attached to it. In the view of the fiducial school of thought, the fiducial test based on Fisher's fiducial argument also tends to perform unsatisfactorily for small samples. In this paper, we suggest an IM-based efficient test using the randomized plausibility function based on the Inferential Model. We prove that our new method is not only valid but also efficient, i.e., it maintains a nominal type I error rate, and we show, via simulations, that the proposed test also has satisfactory power when compared with those of other competing tests. A numerical example illustrates the proposed method.

This paper uses graphical methods to illustrate and compare the coverage properties of a number of methods for calculating confidence intervals for the difference between two independent binomial proportions. We investigate both small-sample and large-sample properties of both two-sided and one-sided coverage, with an emphasis on asymptotic methods. In terms of aligning the smoothed coverage probability surface with the nominal confidence level, we find that the score-based methods on the whole have the best two-sided coverage, although they have slight deficiencies for confidence levels of 90% or lower. For an easily taught, hand-calculated method, the Brown-Li 'Jeffreys' method appears to perform reasonably well, and in most situations, it has better one-sided coverage than the widely recommended alternatives. In general, we find that the one-sided properties of many of the available methods are surprisingly poor. In fact, almost none of the existing asymptotic methods achieve equal coverage on both sides of the interval, even with large sample sizes, and consequently if used as a non-inferiority test, the type I error rate (which is equal to the one-sided non-coverage probability) can be inflated. The only exception is the Gart-Nam 'skewness-corrected' method, which we express using modified notation in order to include a bias correction for improved small-sample performance, and an optional continuity correction for those seeking more conservative coverage. Using a weighted average of two complementary methods, we also define a new hybrid method that almost matches the performance of the Gart-Nam interval.

The use of noninferiority (NI) clinical trials continues to increase. At the same time, this study design remains complex with numerous issues related to the planning, implementation, and analysis of such trials. In this systematic review, we expand previous reviews, and evaluate reporting of the statistical methods used for design and analysis of NI trials with binomial proportions. We assessed 70 NI clinical trials, published between June 2017 and May 2018. Based on these trials, we outline the gaps in statistical reporting in the published articles, and provide recommendations to minimize the gaps. The recommendations, and the awareness they raise are a helpful addition to the efforts the research community is facing today in combating reproducibility and replicability issues.

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Efficient calculation of test sizes for non-inferiority

In this article, we propose considering an approximate exact score (AES) test for noninferiority comparisons and we derive its test-based confidence interval for the difference between two independent binomial proportions. This test was published in the literature, but not its associated confidence interval. The p-value for this test is obtained by using exact binomial probabilities with the nuisance parameter being replaced by its restricted maximum likelihood estimate. Calculated type I errors revealed that the AES method has important advantages for noninferiority comparisons over popular asymptotic methods for adequately powered confirmatory clinical trials, at 80% or 90% statistical power. For unbalanced sample sizes of the compared groups, type I errors for the asymptotic score method were shown to be higher than the nominal level in a systematic pattern over a range of true proportions, but the AES method did not suffer from such a problem. On average, the true type I error of the AES method was closer to the nominal level than all considered methods in the empirical comparisons. In rare cases, type I errors of the AES test exceeded the nominal level, but only by a small amount. Presented examples showed that the AES method can be more attractive in practice than practical exact methods. In addition, p-value and confidence interval of the AES method can be obtained in <30 s of computer time for most confirmatory trials. Theoretical arguments, combined with empirical evidence and fast computation time should make the AES method attractive in statistical practice.

The relationship between two independent binomial proportions is commonly estimated and presented using the difference between proportions, the number needed to treat, the ratio of proportions or the odds ratio. Several different confidence intervals are available, but they can produce markedly different results. Some of the traditional approaches, such as the Wald interval for the difference between proportions and the Katz log interval for the ratio of proportions, do not perform well unless the sample size is large. Better intervals are available. This article describes and compares approximate and exact confidence intervals that are - with one exception - easy to calculate or available in common software packages. We illustrate the performances of the intervals and make recommendations for both small and moderate-to-large sample sizes.

The problem of testing equality of two independent binomial proportions is reexamined. A recent article by D'Agostino, Chase, and Belanger (1988) argues that the Fisher exact test and the Yates chi-squared test with continuity correction are much too conservative in small samples. The authors proposed using a studentized version of the Pearson chi-squared test, and showed that the empirical size of this test is generally close to the nominal level in repeated product-binomial sampling. Although the article is persuasive on its own terms, two central issues are completely ignored: (a) the propriety of analyzing tests based on discrete data using fixed nominal levels of significance, and (b) the question of whether the empirical size should be computed in repeated samples that fix one or both of the margins. The latter is the key issue; Yates and others (1984) argued that both margins should be held fixed for inference, even though only one margin is fixed by the product-binomial sampling design. T...

The difference of two independent binomial proportions is frequently of interest in biomedical research. The interval estimation may be an important tool for the inferential problem. Many confidence intervals have been proposed. They can be classified into the class of exact confidence intervals or the class of approximate confidence intervals. Ore may prefer exact confidence interval s in that they guarantee the minimum coverage probability greater than the nominal confidence level. However, someone, for example Agresti and Coull (1998) claims that "approximation is better than exact." It seems that when sample size is large, the approximate interval is more preferable to the exact interval. However, the choice is not clear when sample, size is small. In this note, an exact confidence and an approximate confidence interval, which were recommended by Santner et al. (2007) and Lee (2006b), respectively, are compared in terms of the coverage probability and the expected length.

A recent article in this journal by Tabak (2006) highlighted a potentially serious source of bias that can arise when multiple statistical tests of a damages theory are performed and even one test rejecting the null hypothesis is regarded as supporting the damages theory. Repeated testing will eventually produce a “false discovery,” that is, rejection of the null hypothesis in favor of the alternative hypothesis when the null hypothesis is true, which statisticians refer to as type I error. Consequently, performing multiple tests without adjusting the critical value can be problematical because it can lead to improperly accepting statistical evidence that apparently supports rejection of the null hypothesis as reliable when it is not. Tabak (2006) recommends making the Sidak (1968, 1971) multiple-comparison adjustment to the standard statistical t-test to correct for the false-discovery bias inherent in multiple-comparison testing. In particular, he recommends making this adjustment when performing 10b-5 securities fraud event studies when more than one corrective disclosure date is involved. This article clarifies the circumstances in which a multiple-comparison adjustment is appropriate and explains why the correction is normally not needed in securities fraud event-study testing. More generally, I explain why it is not required when each of several tests is performed and its results are reported separately, as for example, where the objective is simply to test the statistical significance of the abnormal stock return on each day on which a new and distinct curative disclosure occurred. I show that the Sidak multiple-comparison adjustment is nearly as stringent as the classical Bonferroni procedure (Simes, 1986), which can increase the risk of type II error. This article discusses a more powerful alternative to the Sidak adjustment due to Benjamini and Hochberg (1993), which directly corrects for the false-discovery bias in multiple-comparison testing and reduces the risk of type II error. II. Application of Multiple-Comparison Adjustments to Securities Fraud Event Studies Multiple-comparison-false-discovery bias can arise when (a) several statistical tests are performed on subsets of the same larger data set in an effort to

BackgroundClinical Trials (CTs) resulting in publication in biomedical journals is a strategy to guarantee good quality research. Clinical research promoters are obligated to publish either positive or negative results; regretfully...

A careful derivation of an approximate sample size formula for testing the difference between two independent binomial proportions with specified probabilities of type 1 and type 2 error shows why the formula developed by Casagrande, Pike, and Smith is an accurate approximation. The analysis uses the one-half continuity correction in a logically consistent manner, to approximate both the critical region of the test and the power function, thus resolving a conundrum in previous descriptions of the method. The key idea of the analysis also yields highly accurate sample size determination for a single-sample study.

It is unclear whether the reporting quality of antiretroviral (ARV) noninferiority (NI) randomized controlled trials (RCTs) has improved since the CONSORT guideline release in 2006. The primary objective of this systematic review was assessing the methodological and reporting quality of ARV NI-RCTs. We also assessed reporting quality by funding source and publication year. We searched Medline, Embase, and Cochrane Central from inception to 14 November 2022. We included NI-RCTs comparing ≥2 ARV regimens used for human immunodeficiency virus treatment or prophylaxis. We used the Cochrane Risk of Bias 2.0 tool to assess risk of bias. Screening and data extraction were performed blinded and in duplicate. Descriptive statistics were used to summarize data; statistical tests were 2 sided, with significance defined as P < .05. The systematic review was prospectively registered (PROSPERO CRD42022328586), and not funded. We included 160 articles reporting 171 trials. Of these articles, 101 (63.1%) did not justify the NI margin used, and 28 (17.5%) did not provide sufficient information for sample size calculation. Eighty-nine of 160 (55.6%) reported both intention-to-treat and per-protocol analyses, while 118 (73.8%) described missing data handling. Ten of 171 trials (5.9%) reported potentially misleading results. Pharmaceutical industry-funded trials were more likely to be double-blinded (28.1% vs 10.3%; P = .03) and to describe missing data handling (78.5% vs 59.0%; P = .02). The overall risk of bias was low in 96 of 160 studies (60.0%). ARV NI-RCTs should improve NI margin justification, reporting of intention-to-treat and per-protocol analyses, and missing data handling to increase CONSORT adherence.