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.     

Wilcoxon's signed‐rank statistic: what null hypothesis and why it matters

In statistical literature, the term ‘signed‐rank test’ (or ‘Wilcoxon signed‐rank test’) has been used to refer to two distinct tests: a test for symmetry of distribution and a test for the median of a symmetric distribution, sharing a common test statistic. To avoid potential ambiguity, we propose to refer to those two tests by different names, as ‘test for symmetry based on signed‐rank statistic’ and ‘test for median based on signed‐rank statistic’, respectively. The utility of such terminological differentiation should become evident through our discussion of how those tests connect and contrast with sign test and one‐sample t‐test. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

The split sample permutation t-tests

Without the exchangeability assumption, permutation tests for comparing two population means do not provide exact control of the probability of making a Type I error. Another drawback of permutation tests is that it cannot be used to test hypothesis about one population. In this paper, we propose a new type of permutation tests for testing the difference between two population means: the split sample permutation t-tests. We show that the split sample permutation t-tests do not require the exchangeability assumption, are asymptotically exact and can be easily extended to testing hypothesis about one population. Extensive simulations were carried out to evaluate the performance of two specific split sample permutation t-tests: the split in the middle permutation t-test and the split in the end permutation t-test. The simulation results show that the split in the middle permutation t-test has comparable performance to the permutation test if the population distributions are symmetric and satisfy the exchangeability assumption. Otherwise, the split in the end permutation t-test has significantly more accurate control of level of significance than the split in the middle permutation t-test and other existing permutation tests.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

Adjusted Wilcoxon signed rank test tables for ratio of percentiles

The ordinary Wilcoxon signed rank test table provides confidence intervals for the median of one population. Adjusted Wilcoxon signed rank test tables which can provide confidence intervals for the median and the 10th percentile of one population are created in this paper. Base-(n + 1) number system and theorems about property of symmetry of the adjusted Wilcoxon signed rank test statistic are derived for programming. Theorem 1 states that the adjusted Wilcoxon signed rank test statistic are symmetric around n(n + 1)/4. Theorem 2 states that the adjusted Wilcoxon signed rank test statistic with the same number of negative ranks m are symmetric around m(n+1)/2. 87.5% and 85% confidence intervals of the median are given in the table for n = 12, 13,…, 29 to create approximated 95% confidence intervals of the ratio of medians for two independent populations. 95% and 92.5% confidence intervals of the 10th percentile are given in the table for n = 26, 27, 28, 29 to create approximated 95% confidence regions of the ratio of the 10th percentiles for two independent populations. Finally two large datasets from wood industry will be partitioned to verify the correctness of adjusted Wilcoxon signed rank test tables for small samples.     

A new adaptive test for paired data for small to moderate sample sizes

When carrying out data analysis, a practitioner has to decide on a suitable test for hypothesis testing, and as such, would look for a test that has a high relative power. Tests for paired data tests are usually conducted using t-test, Wilcoxon signed-rank test or the sign test. Some adaptive tests have also been suggested in the literature by O'Gorman, who found that no single member of that family performed well for all sample sizes and different tail weights, and hence, he recommended that choice of a member of that family be made depending on both the sample size and the tail weight. In this paper, we propose a new adaptive test. Simulation studies for n=25 and n=50 show that it works well for nearly all tail weights ranging from the light-tailed beta and uniform distributions to t(4) distributions. More precisely, our test has both robustness of level (in keeping the empirical levels close to the nominal level) and efficiency of power. The results of our study contribute to the area of statistical inference.     

Rank Tests for Matched Survival Data

In a clinical trial with the time to an event as the outcome of interest, we may randomize a number of matched subjects, such as litters, to different treatments. The number of treatments equals the number of subjects per litter, two in the case of twins. In this case, the survival times of matched subjects could be dependent. Although the standard rank tests, such as the logrank and Wilcoxon tests, for independent samples may be used to test the equality of marginal survival distributions, their standard error should be modified to accommodate the possible dependence of survival times between matched subjects. In this paper we propose a method of calculating the standard error of the rank tests for paired two-sample survival data. The method is naturally extended to that for K-sample tests under dependence.     

An extended class of permutation techniques for matched pairs

A class of matched-pairs permutation techniques based on distances between each pair of observed signed values is considered. Although many commonly-used inference techniques for matched pairs are members of this class, some of the more appealing inference techniques among this class have received very little attention. Two new simple rank tests of this class jointly possess both intuitive properties and location-alternative power characteristics which appear more appealing than the corresponding characteristics of either the sign test or the Wllcoxon signed-ranks test. In particular, power comparisons based on slmula-tions indicate that these new rank tests are jointly as good or even vastly superior to the sign test or the Wilcoxon signed-ranks test for location alternatives involving five symmetric distributions. The five distributions selected for these com-parisons include the Laplace, logistic, normal, uniform and a U-shaped distribution     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

Power of One-Sample Location Tests Under Distributions with Equal Lévy Distance

In this article, we study the power of one-sample location tests under classical distributions and two supermodels which include the normal distribution as a special case. The distributions of the supermodels are chosen in such a way that they have equal distance to the normal as the logistic, uniform, double exponential, and the Cauchy, respectively. As a measure of distance we use the Lévy metric. The tests considered are two parametric tests, the t-test and a trimmed t-test, and two nonparametric tests, the sign test and the Wilcoxon signed-rank tests. It turns out that the power of the tests, first of all, does not depend on the Lévy distance but on the special chosen supermodel.     

Performance of confidence interval tests for the ratio of two lognormal means applied to Weibull and gamma distribution data

Five estimation approaches have been developed to compute the confidence interval (CI) for the ratio of two lognormal means: (1) T, the CI based on the t-test procedure; (2) ML, a traditional maximum likelihood-based approach; (3) BT, a bootstrap approach; (4) R, the signed log-likelihood ratio statistic; and (5) R*, the modified signed log-likelihood ratio statistic. The purpose of this study was to assess the performance of these five approaches when applied to distributions other than lognormal distribution, for which they were derived. Performance was assessed in terms of average length and coverage probability of the CIs for each estimation approaches (i.e., T, ML, BT, R, and R*) when data followed a Weibull or gamma distribution. Four models were discussed in this study. In Model 1, the sample sizes and variances were equal within the two groups. In Model 2, the sample sizes were equal but variances were different within the two groups. In Model 3, the variances were different within the two groups and the larger variance was paired with the larger sample size. In Model 4, the variances were different within the two groups and the larger variance was paired with the smaller sample size. The results showed that when the variances of the two groups were equal, the t-test performed well, no matter what the underlying distribution was and how large the variances of the two groups were. The BT approach performed better than the others when the underlying distribution was not lognormal distribution, although it was inaccurate when the variances were large. The R* test did not perform well when the underlying distribution was Weibull or gamma distributed data, but it performed best when the data followed a lognormal distribution.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Comparison of Confidence Intervals on Between Group Variance in Unbalanced Heteroscedastic One-Way Random Models

A computer algorithm for computing the alternative distributions of the Wilcoxon signed rank statistic under shift alternatives is discussed. An explicit error bound is derived for the numeric integration approximation to these distributions.

A nonparametric process control procedure in which the standard CUSUM procedure is applied to the Wilcoxon signed rank statistic is discussed. In order to implement this procedure, the distribution of the Wilcoxon statistic under shift of the underlying distribution from its point of symmetry needs to be computed. The average run length of the nonparametric and parametric CUSUM are compared.     

COMPARISON OF THREE NON-PARAMETRIC TESTS FOR BIVARIATE INTERCHANGEABILITY1

Given a random sample(X₁, Y₁), …,(X_n, Y_n) from a bivariate (BV) absolutely continuous c.d.f. H (x, y), we consider rank tests for the null hypothesis of interchangeability H₀: H(x, y). Three linear rank test statistics, Wilcoxon (W_N), sum of squared ranks (SSR_N) and Savage (S_N), are described in Section 1. In Section 2, asymptotic relative efficiency (ARE) comparisons of the three types of tests are made for Morgenstern (Plackett, 1965) and Moran (1969)BV alternatives with marginal distributions satisfying G(x) = F(x/θ) for some θ≠ 1. Both gamma and lognormal marginal distributions are used.     

Efficiency of t-Test and Hotelling's T 2-Test After Box-Cox Transformation

Early investigations of the effects of non-normality indicated that skewness has a greater effect on the distribution of t-statistic than does kurtosis. When the distribution is skewed, the actual p-values can be larger than the values calculated from the t-tables. Transformation of data to normality has shown good results in the case of univariate t-test. In order to reduce the effect of skewness of the distribution on normal-based t-test, one can transform the data and perform the t-test on the transformed scale. This method is not only a remedy for satisfying the distributional assumption, but it also turns out that one can achieve greater efficiency of the test. We investigate the efficiency of tests after a Box-Cox transformation. In particular, we consider the one sample test of location and study the gains in efficiency for one-sample t-test following a Box-Cox transformation. Under some conditions, we prove that the asymptotic relative efficiency of transformed t-test and Hotelling's T ²-test of multivariate location with respect to the same statistic based on untransformed data is at least one.     

Bayesian analysis of paired survival data using a bivariate exponential distribution

We consider a Bayesian analysis method of paired survival data using a bivariate exponential model proposed by Moran (1967, Biometrika 54:385–394). Important features of Moran’s model include that the marginal distributions are exponential and the range of the correlation coefficient is between 0 and 1. These contrast with the popular exponential model with gamma frailty. Despite these nice properties, statistical analysis with Moran’s model has been hampered by lack of a closed form likelihood function. In this paper, we introduce a latent variable to circumvent the difficulty in the Bayesian computation. We also consider a model checking procedure using the predictive Bayesian P-value.     

The performance of tests when observations have different variances

It is common to test if there is an effect due to a treatment. The commonly used tests have the assumption that the observations differ in location, and that their variances are the same over the groups. Different variances can arise if the observations being analyzed are means of different numbers of observations on individuals or slopes of growth curves with missing data. This study is concerned with cases in which the unequal variances are known, or known to a constant of proportionality. It examines the performance of the ttest, the Mann–Whitney–Wilcoxon Rank Sum test, the Median test, and the Van der Waerden test under these conditions. The t-test based on the weighted means is the likelihood ratio test under normality and has the usual optimality properties. The other tests are compared to it. One may align and scale the observations by subtracting the mean and dividing by the standard deviation of each point. This leads to other, analogous test statistics based on these adjusted observations. These statistics are also compared. Finally, the regression scores tests are compared to the other procedures.     

A combined test for differences in scale based on the interquantile range

A class of tests due to Shoemaker (Commun Stat Simul Comput 28: 189–205, 1999) for differences in scale which is valid for a variety of both skewed and symmetric distributions when location is known or unknown is considered. The class is based on the interquantile range and requires that the population variances are finite. In this paper, we firstly propose a permutation version of it that does not require the condition of finite variances and is remarkably more powerful than the original one. Secondly we solve the question of what quantile choose by proposing a combined interquantile test based on our permutation version of Shoemaker tests. Shoemaker showed that the more extreme interquantile range tests are more powerful than the less extreme ones, unless the underlying distributions are very highly skewed. Since in practice you may not know if the underlying distributions are very highly skewed or not, the question arises. The combined interquantile test solves this question, is robust and more powerful than the stand alone tests. Thirdly we conducted a much more detailed simulation study than that of Shoemaker (1999) that compared his tests to the F and the squared rank tests showing that his tests are better. Since the F and the squared rank test are not good for differences in scale, his results suffer of such a drawback, and for this reason instead of considering the squared rank test we consider, following the suggestions of several authors, tests due to Brown–Forsythe (J Am Stat Assoc 69:364–367, 1974), Pan (J Stat Comput Simul 63:59–71, 1999), O’Brien (J Am Stat Assoc 74:877–880, 1979) and Conover et al. (Technometrics 23:351–361, 1981).     

Which Measure Should be Used for Testing in a Paired Design: Simple Difference,Percent Change,or Symmetrized Percent Change?

We aimed to determine the most proper change measure among simple difference, percent, or symmetrized percent changes in simple paired designs. For this purpose, we devised a computer simulation program. Since distributions of percent and symmetrized percent change values are skewed and bimodal, paired t-test did not give good results according to Type I error and the test power. To be to able use percent change or symmetrized percent change as change measure, either the distribution of test statistics should be transformed to a known theoretical distribution by transformation methods or a new test statistic for these values should be developed.     

On the power of the t-test and some rank tests when outliers may be present

The power of some rank tests, used for testing the hypothesis of shift, is found when the underlying distributions contain outliers. The outliers are assumed to occur as the result of mixing two normal distributions with common variance. A small sample case shows how the scores for the rank tests are found and the exact power is computed for each of these rank tests. A Monte Carlo study provides an estimate of the power of the usual two sample t-test.