A modified combination test for the analysis of a clinical trial when a protocol amendment changed the inclusion criteria

Protocol amendments are often necessary in clinical trials. They can change the entry criteria and, therefore, the population. Simply analysing the pooled data is not acceptable. Instead, each phase should be analysed separately and a combination test such as Fisher's test should be applied to the resulting p-values. In this situation, an asymmetric decision rule is not appropriate. Therefore, we propose a modification of Bauer and Köhne's test. We compare this new test with the tests of Liptak, Fisher, Bauer/Köhne and Edgington. In case of differences in variance only or only small differences in mean, Liptak's Z-score approach is the best, and the new test keeps up with the rest and is in most cases slightly superior. In other situations, the new test and the Z-score approach are not preferable. But no big differences in populations are usually to be expected due to amendments. Then, the new method is a recommendable alternative.     

A more powerful unconditional exact test of homogeneity for 2 × c contingency table analysis

The classical unconditional exact p-value test can be used to compare two multinomial distributions with small samples. This general hypothesis requires parameter estimation under the null which makes the test severely conservative. Similar property has been observed for Fisher's exact test with Barnard and Boschloo providing distinct adjustments that produce more powerful testing approaches. In this study, we develop a novel adjustment for the conservativeness of the unconditional multinomial exact p-value test that produces nominal type I error rate and increased power in comparison to all alternative approaches. We used a large simulation study to empirically estimate the 5th percentiles of the distributions of the p-values of the exact test over a range of scenarios and implemented a regression model to predict the values for two-sample multinomial settings. Our results show that the new test is uniformly more powerful than Fisher's, Barnard's, and Boschloo's tests with gains in power as large as several hundred percent in certain scenarios. Lastly, we provide a real-life data example where the unadjusted unconditional exact test wrongly fails to reject the null hypothesis and the corrected unconditional exact test rejects the null appropriately.     

A combined p-value test for multiple hypothesis testing

Tests that combine p-values, such as Fisher's product test, are popular to test the global null hypothesis H₀ that each of n component null hypotheses, H₁,…,H_n, is true versus the alternative that at least one of H₁,…,H_n is false, since they are more powerful than classical multiple tests such as the Bonferroni test and the Simes tests. Recent modifications of Fisher's product test, popular in the analysis of large scale genetic studies include the truncated product method (TPM) of Zaykin et al. (2002), the rank truncated product (RTP) test of Dudbridge and Koeleman (2003) and more recently, a permutation based test—the adaptive rank truncated product (ARTP) method of Yu et al. (2009). The TPM and RTP methods require users' specification of a truncation point. The ARTP method improves the performance of the RTP method by optimizing selection of the truncation point over a set of pre-specified candidate points. In this paper we extend the ARTP by proposing to use all the possible truncation points {1,…,n} as the candidate truncation points. Furthermore, we derive the theoretical probability distribution of the test statistic under the global null hypothesis H₀. Simulations are conducted to compare the performance of the proposed test with the Bonferroni test, the Simes test, the RTP test, and Fisher's product test. The simulation results show that the proposed test has higher power than the Bonferroni test and the Simes test, as well as the RTP method. It is also significantly more powerful than Fisher's product test when the number of truly false hypotheses is small relative to the total number of hypotheses, and has comparable power to Fisher's product test otherwise.     

The power of the modified Wilcoxon rank-sum test for the one-sided alternative

When testing hypotheses in two-sample problems, the Wilcoxon rank-sum test is often used to test the location parameter, and this test has been discussed by many authors over the years. One modification of the Wilcoxon rank-sum test was proposed by Tamura [On a modification of certain rank tests. Ann Math Stat. 1963;34:1101–1103]. Deriving the exact critical value of the statistic is difficult when the sample sizes are increased. The normal approximation, the Edgeworth expansion, the saddlepoint approximation, and the permutation test were used to evaluate the upper tail probability for the modified Wilcoxon rank-sum test given finite sample sizes. The accuracy of various approximations to the probability of the modified Wilcoxon statistic was investigated. Simulations were used to investigate the power of the modified Wilcoxon rank-sum test for the one-sided alternative with various population distributions for small sample sizes. The method was illustrated by the analysis of real data.     

Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

Testing Equality of Means Under Order Restrictions Using Multiple Contrasts

The likelihood ratio test for equality of ordered means is known to have power characteristics that are generally superior to those of competing procedures. Difficulties in implementing this test have led to the development of alternative approaches, most of which are based on contrasts. While orthogonal contrasts can be chosen to simplify the distribution theory, we propose a class of tests that is easy to implement even if the contrasts used are not orthogonal. An overall measure of significance may be obtained by using Fisher's combination statistic to combine the dependent p-values arising from these contrasts. This method can be easily implemented for testing problems involving unequal sample sizes and any partial order, and has power properties that compare well with those of the likelihood ratio test and other contrast-based tests.     

An Efficient Method to Generate Data and Compute Exact P-Values in Goodness-of-Fit Testing

In this article, we use a characterization of the set of sample counts that do not match with the null hypothesis of the test of goodness of fit. Two direct applications arise: first, to instantaneously generate data sets whose corresponding asymptotic P-values belong to a certain pre-defined range; and second, to compute exact P-values for this test in an efficient way. We present both issues before illustrating them by analyzing a couple of data sets. Method's efficiency is also assessed by means of simulations. We focus on Pearson's X ² statistic but the case of likelihood-ratio statistic is also discussed.     

On the Power of Bootstrapped Specification Tests

Abstract

Decisions based on econometric model estimates may not have the expected effect if the model is misspecified. Thus, specification tests should precede any analysis. Bierens' specification test is consistent and has optimality properties against some local alternatives. A shortcoming is that the test statistic is not distribution free, even asymptotically. This makes the test unfeasible. There have been many suggestions to circumvent this problem, including the use of upper bounds for the critical values. However, these suggestions lead to tests that lose power and optimality against local alternatives. In this paper we show that bootstrap methods allow us to recover power and optimality of Bierens' original test. Bootstrap also provides reliable p-values, which have a central role in Fisher's theory of hypothesis testing. The paper also includes a discussion of the properties of the bootstrap Nonlinear Least Squares Estimator under local alternatives.     

Testing the homogeneity of several two‐parameter populations

The authors extend Fisher's method of combining two independent test statistics to test homogeneity of several two‐parameter populations. They explore two procedures combining asymptotically independent test statistics: the first pools two likelihood ratio statistics and the other, score test statistics. They then give specific results to test homogeneity of several normal, negative binomial or beta‐binomial populations. Their simulations provide evidence that in this context, Fisher's method performs generally well, even when the statistics to be combined are only asymptotically independent. They are led to recommend Fisher's test based on score statistics, since the latter have simple forms, are easy to calculate, and have uniformly good level properties.     

The Benard-Van Elteren statistic and nonparametric computation

A general computer program which generates either the Wilcoxon, Kruskal-Wallis, Friedman, or extended Friedman statistic (where the numbers of cell observations n_ij may be any positive integer or zero) can be formulated simply by using the computational algorithm for the Benard-Van Elteren statistic. It is shown that the Benard-Van Elteren statistic can be computed using matrix algebra subroutines including multiplication and inverse or g-inverse computational algorithms in the case where the rank of the matrix V of the variances and covariances of the column totals is k-1. For the case where the rank of V is less than k-1 the use of the g-inverse is shown to greatly reduce the labors of calculation. In addition, the use of the Benard-Van Elteren statistic in testing against ordered alternatives is indicated.     

The limit test statistic distribution of the maximum value test for right-censored data

ABSTRACT

In this paper, the maximum value test is proposed and considered for two-sample problem solving with lifetime data. This test is a distribution-free test under non-censoring and is a not distribution-free test under censoring. The formula of the limit distribution of the proposed maximal value test is represented in the general case. The distribution of the test statistic has been studied experimentally. Also, we propose the estimate of a p-value calculation of the maximum value test instead of the Monte-Carlo simulation. This test is useful and applicable in case of choosing among the logrank test, the Cox–Mantel test, the Q test and Generalized Wilcoxon tests, for instance, the Gehan's Generalized Wilcoxon test and the Peto and Peto's Generalized Wilcoxon test.     

New critical values for the generalized t and generalized rank-sum procedures

Student's t test as well as Wilcoxon's rank-sum test may be inefficient in situations where treatments bring about changes in both location and scale. In order to rectify this situation, O'Brien (1988, Journal of the American Statistical Association 83, 52-61) has proposed two new statistics, the generalized t and generalized rank-sum procedures, which may be much more powerful than their traditional counterparts in such situations. Recently, however, Blair and Morel (1991, Statistics in Medicine in press) have shown that referencing these new statistics to standard F tables as recommended by O'Brien results in inflations of Type I errors. This paper provides tables of critical values which do not produce such inflations. Use of these new critical values results in Type I error rates near nominal levels for the generalized t statistic and slightly conservative rates for the generalized rank-sum test. In addition to the critical values, some new power results are given for the generalized tests.     

Comparison of Nonparametric Regression Curves by Spline Smoothing

In this article, procedures are proposed to test the hypothesis of equality of two or more regression functions. Tests are proposed by p-values, first under homoscedastic regression model, which are derived using fiducial method based on cubic spline interpolation. Then, we construct a test in the heteroscedastic case based on Fisher's method of combining independent tests. We study the behaviors of the tests by simulation experiments, in which comparisons with other tests are also given. The proposed tests have good performances. Finally, an application to a data set are given to illustrate the usefulness of the proposed test in practice.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

Exact Tests for 2 × 2 Contingency Tables

Fisher's exact test, difference in proportions, log odds ratio, Pearson's chi-squared, and likelihood ratio are compared as test statistics for testing independence of two dichotomous factors when the associated p values are computed by using the conditional distribution given the marginals. The statistics listed above that can be used for a one-sided alternative give identical p values. For a two-sided alternative, many of the above statistics lead to different p values. The p values are shown to differ only by which tables in the opposite tail from the observed table are considered more extreme than the observed table.     

Combining the t test and Wilcoxon's rank-sum test

In the two-sample location-shift problem, Student's t test or Wilcoxon's rank-sum test are commonly applied. The latter test can be more powerful for non-normal data. Here, we propose to combine the two tests within a maximum test. We show that the constructed maximum test controls the type I error rate and has good power characteristics for a variety of distributions; its power is close to that of the more powerful of the two tests. Thus, irrespective of the distribution, the maximum test stabilizes the power. To carry out the maximum test is a more powerful strategy than selecting one of the single tests. The proposed test is applied to data of a clinical trial.     

Optimal sample size planning for the Wilcoxon–Mann–Whitney and van Elteren tests under cost constraints

Sampling cost is a crucial factor in sample size planning, particularly when the treatment group is more expensive than the control group. To either minimize the total cost or maximize the statistical power of the test, we used the distribution-free Wilcoxon–Mann–Whitney test for two independent samples and the van Elteren test for randomized block design, respectively. We then developed approximate sample size formulas when the distribution of data is abnormal and/or unknown. This study derived the optimal sample size allocation ratio for a given statistical power by considering the cost constraints, so that the resulting sample sizes could minimize either the total cost or the total sample size. Moreover, for a given total cost, the optimal sample size allocation is recommended to maximize the statistical power of the test. The proposed formula is not only innovative, but also quick and easy. We also applied real data from a clinical trial to illustrate how to choose the sample size for a randomized two-block design. For nonparametric methods, no existing commercial software for sample size planning has considered the cost factor, and therefore the proposed methods can provide important insights related to the impact of cost constraints.     

On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: Robustness and the effect of outliers

The distribution of the sample correlation coefficient is derived when the population is a mixture of two bivariate normal distributions with zero mean but different covariances and mixing proportions 1 - λ and λ respectively; λ will be called the proportion of contamination. The test of ρ = 0 based on Student's t, Fisher's z, arcsine, or Ruben's transformation is shown numerically to be nonrobust when λ, the proportion of contamination, lies between 0.05 and 0.50 and the contaminated population has 9 times the variance of the standard (bivariate normal) population. These tests are also sensitive to the presence of outliers.     

Book Reviews

The Levene test is a widely used test for detecting differences in dispersion. The modified Levene transformation using sample medians is considered in this article. After Levene's transformation the data are not normally distributed, hence, nonparametric tests may be useful. As the Wilcoxon rank sum test applied to the transformed data cannot control the type I error rate for asymmetric distributions, a permutation test based on reallocations of the original observations rather than the absolute deviations was investigated. Levene's transformation is then only an intermediate step to compute the test statistic. Such a Levene test, however, cannot control the type I error rate when the Wilcoxon statistic is used; with the Fisher–Pitman permutation test it can be extremely conservative. The Fisher–Pitman test based on reallocations of the transformed data seems to be the only acceptable nonparametric test. Simulation results indicate that this test is on average more powerful than applying the t test after Levene's transformation, even when the t test is improved by the deletion of structural zeros.     

On Two Forms of Fisher's Measure of Information

ABSTRACT

Fisher's information number is the second moment of the “score function” where the derivative is with respect to x rather than Θ. It is Fisher's information for a location parameter, and also called shift-invariant Fisher information. In recent years, Fisher's information number has been frequently used in several places regardless of parameters of the distribution or of their nature. Is this number a nominal, standard, and typical measure of information? The Fisher information number is examined in light of the properties of classical statistical information theory. It has some properties analogous to those of Fisher's measure, but, in general, it does not have good properties if used as a measure of information when Θ is not a location parameter. Even in the case of location parameter, the regularity conditions must be satisfied. It does not possess the two fundamental properties of the mother information, namely the monotonicity and invariance under sufficient transformations. Thus the Fisher information number should not be used as a measure of information (except when Θ a location parameter). On the other hand, Fisher's information number, as a characteristic of a distribution f(x), has other interesting properties. As a byproduct of its superadditivity property a new coefficient of association is introduced.