Linear Rank Tests Research Articles

Saddlepoint p-values for a class of location-scale tests

ABSTRACT The main idea of this paper is to approximate the exact p-value of a class of non-parametric, two-sample location-scale tests. In this paper, the most famous non-parametric two-sample location-scale tests are formulated in a class of linear rank tests. The permutation distribution of this class is derived from a random allocation design. This allows us to approximate the exact p-value of the non-parametric two-sample location-scale tests of the considered class using the saddlepoint approximation method. The proposed method shows high accuracy in approximating the exact p-value compared to the normal approximation method. Moreover, the proposed method only requires a few calculations and time, as in the case of the simulated method. The procedures of the proposed method are clarified through four sets of real data that represent applications for a number of different fields. In addition, a simulation study compares the proposed method with the traditional methods to approximate the exact p-value of the specified class of the non-parametric two-sample location-scale tests.

Nonparametric tests for combined location-scale and Lehmann alternatives using adaptive approach and max-type metric

The paper deals with the classical two-sample problem for the combined location-scale and Lehmann alternatives, known as the versatile alternative. Recently, a combination of the square of the standardized Wilcoxon, the standardized Ansari–Bradley and the standardized Anti-Savage statistics based on the Euclidean distance has been proposed. The Anti-Savage test is the locally most powerful rank test for the right-skewed Gumbel distribution. Furthermore, the Savage test is the locally most powerful linear rank test for the left-skewed Gumbel distribution. Then, a test statistic combining the Wilcoxon, the Ansari–Bradley, and Savage statistics is proposed. The limiting distribution of the proposed statistic is derived under the null and the alternative hypotheses. In addition, the asymptotic power of the suggested statistic is investigated. Moreover, an adaptive test is proposed based on a selection rule. We compare the power performance against various fixed alternatives using Monte Carlo. The proposed test statistic displays outstanding performance in certain situations. An illustration of the proposed test statistic is presented to explain a biomedical experiment. Finally, we offer some concluding remarks.

Linear rank tests for left-truncated data using randomized block design: saddlepoint p-values and confidence intervals

Survival analysis is most typically characterized by left truncation; left-truncated data are composed of two recorded times, the time of occurrence of an event before the event of interest, and the time of the interested event. In this study, p-values of the permutation distribution of the linear rank tests are computed using saddlepoint approximation method for left-truncated data using a randomized block design. Simulation studies demonstrate the saddlepoint approximation method’s accuracy in approximating p-value of the linear rank tests. Due to the efficiency of the proposed method over the normal approximation, 95% confidence intervals for the treatment impact are performed.

P-values and confidence intervals of linear rank tests for left-truncated data under truncated binomial design

ABSTRACT The paper provides computations comparing the accuracy of the saddlepoint approximation approach and the normal approximation method in approximating the mid-p-value of Wilcoxon and log-rank tests for the left-truncated data using a truncated binomial design. The paper uses real data examples to apply the comparison, along with some simulated studies. Confidence intervals are provided by the inversion of the tests under consideration.

Sample size analysis for two-sample linear rank tests

Sample size analysis is a key part of the planning phase of any research. So far, however, hardly any literature focuses on sample size analysis methods for two-sample linear rank tests, although these methods have optimal properties for different distributions. This article provides a new sample size analysis method for linear rank tests for location shift alternatives based on score-generating functions. Results show a slightly anti-conservative behavior, no severe risk of an occurring circular argument at small to moderate variances of the population’s distribution, and good performance compared to alternate sample size analysis methods for the most well-known linear rank test, the Wilcoxon-Mann-Whitney test.

Using Cox regression to develop linear rank tests with zero-inflated clustered data.

Zero-inflated data arise in many fields of study. When comparing zero-inflated data between two groups with independent subjects, a two degree-of-freedom test has been developed, which is the sum of a 1 degree-of-freedom Pearson chi-square test for the 2×2 table of group vs dichotomized outcome (0,> 0) and a 1 degree-of-freedom Wilcoxon rank-sum test for the values of the outcome > 0. Here, we extend this 2 degree-of-freedom test to clustered data settings. We first propose using an estimating equations score statistic from a time-varying weighted Cox regression model under naive independence, with a robust sandwich variance estimator to account for clustering. Since our proposed test statistics can be put in the framework of a Cox model, to gain efficiency over naive independence, we apply a generalized estimating equations (GEE) Cox model with a non-independence 'working correlation' between observations in a cluster. The proposed methods are applied to a General Social Survey study of days with mental health problems in a month, in which 52.3% of subjects report they have no days with problems, a zero-inflated outcome. A simulation study is used to compare our proposed test statistics to previously proposed zero-inflated test statistics.

Maximal Length Projections in Group Algebras with Applications to Linear Rank Tests of Uniformity

Let \(G\) be a finite group, let \(\mathbb{C}G\) be the complex group algebra of \(G\), and let \(p \in \mathbb{C}G\). In this paper, we show how to construct submodules\(S\) of \(\mathbb{C}G\) of a fixed dimension with the property that the orthogonal projection of \(p\) onto \(S\) has maximal length. We then provide an example of how such submodules for the symmetric group \(S_n\) can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of \(n\) items.

Linear-rank testing of a non-binary, responder-analysis, efficacy score to evaluate pharmacotherapies for substance use disorders.

The design of pharmacological trials for management of substance use disorders is shifting toward outcomes of successful individual-level behavior (abstinence or no heavy use). While binary success/failure analyses are common, McCann and Li (CNS Neurosci Ther 2012; 18: 414-418) introduced "number of beyond-threshold weeks of success" (NOBWOS) scores to avoid dichotomized outcomes. NOBWOS scoring employs an efficacy "hurdle" with values reflecting duration of success. Here, we evaluate NOBWOS scores rigorously. Formal analysis of mathematical structure of NOBWOS scores is followed by simulation studies spanning diverse conditions to assess operating characteristics of five linear-rank tests on NOBWOS scores. Simulations include assessment of Fisher's exact test applied to hurdle component. On average, statistical power was approximately equal for five linear-rank tests. Under none of conditions examined did Fisher's exact test exhibit greater statistical power than any of the linear-rank tests. These linear-rank tests provide good Type I and Type II error control for comparing distributions of NOBWOS scores between groups (e.g. active vs. placebo). All methods were applied to re-analyses of data from four clinical trials of differing lengths and substances of abuse. These linear-rank tests agreed across all trials in rejecting (or not) their null (equality of distributions) at ≤ 0.05.

Adrenal cortex expression quantitative trait loci in a German Holstein × Charolais cross.

BackgroundThe importance of the adrenal gland in regard to lactation and reproduction in cattle has been recognized early. Caused by interest in animal welfare and the impact of stress on economically important traits in farm animals the adrenal gland and its function within the stress response is of increasing interest. However, the molecular mechanisms and pathways involved in stress-related effects on economically important traits in farm animals are not fully understood. Gene expression is an important mechanism underlying complex traits, and genetic variants affecting the transcript abundance are thought to influence the manifestation of an expressed phenotype. We therefore investigated the genetic background of adrenocortical gene expression by applying an adaptive linear rank test to identify genome-wide expression quantitative trait loci (eQTL) for adrenal cortex transcripts in cattle.ResultsA total of 10,986 adrenal cortex transcripts and 37,204 single nucleotide polymorphisms (SNPs) were analysed in 145 F2 cows of a Charolais × German Holstein cross. We identified 505 SNPs that were associated with the abundance of 129 transcripts, comprising 482 cis effects and 17 trans effects. These SNPs were located on all chromosomes but X, 16, 24 and 28. Associated genes are mainly involved in molecular and cellular functions comprising free radical scavenging, cellular compromise, cell morphology and lipid metabolism, including genes such as CYP27A1 and LHCGR that have been shown to affect economically important traits in cattle.ConclusionsIn this study we showed that adrenocortical eQTL affect the expression of genes known to contribute to the phenotypic manifestation in cattle. Furthermore, some of the identified genes and related molecular pathways were previously shown to contribute to the phenotypic variation of behaviour, temperament and growth at the onset of puberty in the same population investigated here. We conclude that eQTL analysis appears to be a useful approach providing insight into the molecular and genetic background of complex traits in cattle and will help to understand molecular networks involved.Electronic supplementary materialThe online version of this article (doi:10.1186/s12863-016-0442-x) contains supplementary material, which is available to authorized users.

Cumulative Paired φ-Entropy

A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general “entropy generating function” φ, the same way Burbea et al. (IEEE Trans. Inf. Theory 1982, 28, 489–495) and Liese et al. (Convex Statistical Distances; Teubner-Verlag, 1987) did in the context of φ-divergences. Combining the ideas of φ-entropy and cumulative entropy leads to the new “cumulative paired φ-entropy” ( C P E φ ). This new entropy has already been discussed in at least four scientific disciplines, be it with certain modifications or simplifications. In the fuzzy set theory, for example, cumulative paired φ-entropies were defined for membership functions, whereas in uncertainty and reliability theories some variations of C P E φ were recently considered as measures of information. With a single exception, the discussions in the scientific disciplines appear to be held independently of each other. We consider C P E φ for continuous cdfs and show that C P E φ is rather a measure of dispersion than a measure of information. In the first place, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction of a fixed variance. Next, this paper specifically shows that C P E φ satisfies the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator, containing all its known asymptotic properties. C P E φ is the basis for several related concepts like mutual φ-information, φ-correlation, and φ-regression, which generalize Gini correlation and Gini regression. In addition, linear rank tests for scale that are based on the new entropy have been developed. We show that almost all known linear rank tests are special cases, and we introduce certain new tests. Moreover, formulas for different distributions and entropy calculations are presented for C P E φ if the cdf is available in a closed form.

Comparing the empirical powers of several independence tests in generalized FGM family

The powers of some tests for independence hypothesis against positive (negative) quadrant dependence in generalized Farlie-Gumbel-Morgenstern distribution are compared graphically by simulation. Some of these tests are usual linear rank tests of independence. Two other possible rank tests of independence are locally most powerful rank test and a powerful nonparametric test based on the Mises statistic. We also evaluate the empirical power of the class of distribution-free tests proposed by Kochar and Gupta (1987) based on the asymptotic distribution of a U-statistic and the test statistic proposed by and Kotz (2008) in generalized Farlie-Gumbel-Morgenstern distribution. Tests of independence are also compared for sample sizes n

A moment generating function of a combination of linear rank tests and its asymptotic efficiency

When testing hypotheses in two-sample problems, the Lepage test has often been used to jointly test the location and scale parameters, and has been discussed by many authors over the years. The Lepage test was a combination of the Wilcoxon statistic and the Ansari–Bradley statistic. Various Lepage-type tests were proposed with discussions of an asymptotic relative efficiency (Duran et al., Biometrika 63:173–176, 1976; Goria, Stat Neerl 36:3–13, 1982), a robustness and a power comparison (Neuhauser, Commun Stat Theory Methods 29:67–78, 2000; Buning, J Appl Stat 29:907–924, 2002) and an adaptive test (Buning and Thadewald, J Stat Comput Sim 65:287–310, 2000). We derive an expression for the moment generating function of a linear combination of two linear rank statistics. As a suggested Lepage-type test, we use a combination of the generalized Wilcoxon statistic and the generalized Mood statistic. Deriving the exact critical value of the statistic can be difficult when the sample sizes are increasing. In this situation, an approximation method to the distribution function of the test statistic can be useful with a higher order moment. We use a moment-based approximation with an adjusted gamma polynomial to evaluate the upper tail probability of a Lepage-type test for a finite sample size. We determine the asymptotic efficiencies of the Lepage and Lepage-type tests for various distributions.

Saddlepoint p-values for a class of tests for comparing competing risks with censored data

One of the general problems in clinical trials and mortality rates is the comparison of competing risks. Most of the test statistics used for independent and dependent risks with censored data belong to the class of weighted linear rank tests in its multivariate version. In this paper, we introduce the saddlepoint approximations as accurate and fast approximations for the exact p-values of this class of tests instead of the asymptotic and permutation simulated calculations. Real data examples and extensive simulation studies showed the accuracy and stability performance of the saddlepoint approximations over different scenarios of lifetime distributions, sample sizes and censoring.

Saddlepoint $$p$$ p -values and confidence intervals for the class of linear rank tests for censored data under generalized randomized block design

One of the commonly used classes of tests for testing treatment effects for censored data is the linear rank class. The underlying distribution of this class is determined by the randomization design used to collect the data. Many randomization designs are used in clinical trials. The randomized block design is an important design that reduces selection bias and accidental bias. In this paper, a double saddlepoint approximation for the exact underlying randomization distribution for the linear rank class under generalized randomized block design is presented. Extensive simulation studies are used to assess the performance of the saddlepoint approximation. This approximation shows a great improvement in accuracy over the asymptotic normal approximation. This accuracy enables us to calculate almost exact confidence intervals for the treatment effect.

Asymptotic efficiency and small sample power of a locally most powerful linear rank test for the log-logistic distribution

Assume that two independent random samples are distributed according to a log-logistic distribution (LLD). In this study, the score functions for the locally most powerful rank test were derived for the location and scale parameters. The Wilcoxon rank-sum test was shown to be locally most powerful rank test for the LLD. The asymptotic efficiency of the Wilcoxon rank-sum test was derived and compared with that of the modified Wilcoxon rank-sum test for the LLD.

P-Value for two-Sample Linear Rank Tests Using Permutation Simulations and the Normal Approximation Method

Test statistics from the class of two-sample linear rank tests are commonly used to compare a treatment group with a control group. Two independent random samples of sizes m and n are drawn from two populations. As a result, N = m + n observations in total are obtained. The aim is to test the null hypothesis of identical distributions. The alternative hypothesis is that the populations are of the same form but with a different measure of central tendency. This article examines mid p-values from the null permutation distributions of tests based on the class of two-sample linear rank statistics. The results obtained indicate that normal approximation-based computations are very close to the permutation simulations, and they provide p-values that are close to the exact mid p-values for all practical purposes.

A Simple GMM Estimator for the Semiparametric Mixed Proportional Hazard Model

Ridder and Woutersen (Ridder, G., and T. Woutersen. 2003. “The Singularity of the Efficiency Bound of the Mixed Proportional Hazard Model.” Econometrica 71: 1579–1589) have shown that under a weak condition on the baseline hazard, there exist root-N consistent estimators of the parameters in a semiparametric Mixed Proportional Hazard model with a parametric baseline hazard and unspecified distribution of the unobserved heterogeneity. We extend the linear rank estimator (LRE) of Tsiatis (Tsiatis, A. A. 1990. “Estimating Regression Parameters using Linear Rank Tests for Censored Data.” Annals of Statistics 18: 354–372) and Robins and Tsiatis (Robins, J. M., and A. A. Tsiatis. 1992. “Semiparametric Estimation of an Accelerated Failure Time Model with Time-Dependent Covariates.” Biometrika 79: 311–319) to this class of models. The optimal LRE is a two-step estimator. We propose a simple one-step estimator that is close to optimal if there is no unobserved heterogeneity. The efficiency gain associated with the optimal LRE increases with the degree of unobserved heterogeneity. Keywords: counting process; linear rank estimation; mixed proportional hazard

Adaptive linear rank tests for eQTL studies

Expression quantitative trait loci (eQTL) studies are performed to identify single-nucleotide polymorphisms that modify average expression values of genes, proteins, or metabolites, depending on the genotype. As expression values are often not normally distributed, statistical methods for eQTL studies should be valid and powerful in these situations. Adaptive tests are promising alternatives to standard approaches, such as the analysis of variance or the Kruskal-Wallis test. In a two-stage procedure, skewness and tail length of the distributions are estimated and used to select one of several linear rank tests. In this study, we compare two adaptive tests that were proposed in the literature using extensive Monte Carlo simulations of a wide range of different symmetric and skewed distributions. We derive a new adaptive test that combines the advantages of both literature-based approaches. The new test does not require the user to specify a distribution. It is slightly less powerful than the locally most powerful rank test for the correct distribution and at least as powerful as the maximin efficiency robust rank test. We illustrate the application of all tests using two examples from different eQTL studies.

Linear rank tests of uniformity: understanding inconsistent outcomes and the construction of new tests

Several nonparametric tests exist to test for differences among alternatives when using ranked data. Testing for differences among alternatives amounts to testing for uniformity over the set of possible permutations of the alternatives. Well-known tests of uniformity, such as the Friedman test or the Anderson test, are based on the impact of the usual limiting theorems (e.g. central limit theorem) and the results of different summary statistics (e.g. mean ranks, marginals, and pairwise ranks). Inconsistencies can occur among statistical tests’ outcomes – different statistical tests can yield different outcomes when applied to the same ranked data. In this paper, we describe a conceptual framework that naturally decomposes the underlying ranked data space. Using the framework, we explain why test results can differ and how their differences are related. In practice, one may choose a test based on the power or the structure of the ranked data. We discuss the implications of these choices and illustrate that for data meeting certain conditions, no existing test is effective in detecting nonuniformity. Finally, using a real data example, we illustrate how to construct new linear rank tests of uniformity.

Testing the equality among distribution functions from independent and right censored samples via Cramér–von Mises criterion

The traditional Cramér–von Mises criterion is used in order to develop a test to compare the equality of the underlying lifetime distributions in the presence of independent censoring times. Its asymptotic distribution is proved and a resampling plan, which is valid for unbalanced data situations, is proposed. Its statistical power is studied and compared with commonly used linear rank tests by Monte Carlo simulations and a real data analysis is also considered. It is observed that the new test is clearly more powerful than the traditional ones when there exists no uniform dominance among involved distributions and in the presence of late differences. Its statistical power is also good in the other considered scenarios.