Approximated non parametric confidence regions for the ratio of two percentiles

In the wood industry, it is common practice to compare in terms of the ratio of the same-strength properties for lumber of two different dimensions, grades, or species. Because United States lumber standards are given in terms of population fifth percentile, and strength problems arise from the weaker fifth percentile rather than the stronger mean, so the ratio should be expressed in terms of the fifth percentiles rather than the means of two strength distributions. Percentiles are estimated by order statistics. This paper assumes small samples to derive new non parametric methods such as percentile sign test and percentile Wilcoxon signed rank test, construct confidence intervals with covergage rate 1 – αx for single percentiles, and compute confidence regions for ratio of percentiles based on confidence intervals for single percentiles. Small 1 – αx is enough to obtain good coverage rates of confidence regions most of the time.     

Tests for Symmetry Based on the One-Sample Wilcoxon Signed Rank Statistic

ABSTRACT

The one-sample Wilcoxon signed rank test was originally designed to test for a specified median, under the assumption that the distribution is symmetric, but it can also serve as a test for symmetry if the median is known. In this article we derive the Wilcoxon statistic as the first component of Pearson's X ² statistic for independence in a particularly constructed contingency table. The second and third components are new test statistics for symmetry. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. A trimming proportion is used to split the observations in the tails from those around the median. We further extend the method so that no arbitrary choice for the trimming proportion has to be made. Finally, the new tests are compared to other tests for symmetry in a simulation study. It is concluded that our tests often have substantially greater powers than most other tests.     

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.     

Nonparametric confidence intervals for location in time series data

We extend the confidence interval construction procedure for location for symmetric iid data using the one-sample Wilcoxon signed rank statistic (T⁺) to stationary time series data. We propose a normal approximation procedure when explicit knowledge of the underlying dependence structure/distribution is unknown. By conducting extensive simulations from linear and nonlinear time series models, we show that the extended procedure is a strong contender for use in the construction of confidence intervals in time series analysis. Finally we demonstrate real application implementations in two case studies.     

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.     

Tests of symmetry derived as components of pearson's phi-squared distance measure

Rank statistics which arise as estimates of the first and third components of a frequency decomposition of Pearson's phi-squared distance measure, introduced by Eubank, LaRiccia, and Rosenstein (1987), are examined for their usefulness as tests of symmetry about a known median against various asymmetric alternatives, Pitman asymptotic relative efficiencies are used to compare the efficacies of the new statistics with classical test procedures, and empirical powers of the new tests are compared via simulation for a variety of asymmetric distributions. Statistics which arise from components based on Legendre polynomials are of particular interest. The first component is the familiar Wilcoxon signed rank statistic. The third component, which is a new statistic for this problem, exhibits a high level of sensitivity to a variety of asymmetric alternatives both asymptotically and in small sample studies.     

Robust likelihood inference for public policy

Issues of public policy are typically decided by non‐specialists who are increasingly informed by statistical methods. In order to be influential, inferential techniques must be widely understood and accepted. This motivates the author to propose likelihood‐based methods that prove relatively insensitive to the choice of underlying distribution because they exploit a peculiarly stable relation between two standard errors and a 95% coverage probability. The author also notes that bootstrap and jackknife estimates of variance can sometimes be strongly biased. In fact, symbolic computations in R suggest that they are reliable only for statistics that are well approximated by averages whose distributions are roughly symmetric. The author thus proposes to transform the classical likelihood ratio into a statistic whose variance can be estimated robustly. He shows that the signed root of the log‐likelihood is well approximated by an average with a roughly symmetric distribution. This leads to Cox‐Tukey intervals for a Student‐like statistic and to simple confidence intervals for most models used in public policy.     

A comparative study of tests for paired lifetime data

In this paper, we investigate different procedures for testing the equality of two mean survival times in paired lifetime studies. We consider Owen’s M-test and Q-test, a likelihood ratio test, the paired t-test, the Wilcoxon signed rank test and a permutation test based on log-transformed survival times in the comparative study. We also consider the paired t-test, the Wilcoxon signed rank test and a permutation test based on original survival times for the sake of comparison. The size and power characteristics of these tests are studied by means of Monte Carlo simulations under a frailty Weibull model. For less skewed marginal distributions, the Wilcoxon signed rank test based on original survival times is found to be desirable. Otherwise, the M-test and the likelihood ratio test are the best choices in terms of power. In general, one can choose a test procedure based on information about the correlation between the two survival times and the skewness of the marginal survival distributions.     

Log‐rank permutation tests for trend: saddlepoint p‐values and survival rate confidence intervals

Suppose p + 1 experimental groups correspond to increasing dose levels of a treatment and all groups are subject to right censoring. In such instances, permutation tests for trend can be performed based on statistics derived from the weighted log‐rank class. This article uses saddlepoint methods to determine the mid‐P‐values for such permutation tests for any test statistic in the weighted log‐rank class. Permutation simulations are replaced by analytical saddlepoint computations which provide extremely accurate mid‐P‐values that are exact for most practical purposes and almost always more accurate than normal approximations. The speed of mid‐P‐value computation allows for the inversion of such tests to determine confidence intervals for the percentage increase in mean (or median) survival time per unit increase in dosage. The Canadian Journal of Statistics 37: 5‐16; 2009 © 2009 Statistical Society of Canada     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

A monte carlo study of the LN statistic for the multivariate nonparametric median and rank sum tests for two populations

Small sample tables are not available for the multisample multivariate rank sum test (MMRST) or the multisample multivariate median test (MMMT) L_N statistic. Consequently, the statistic usually is compared to its asymptotic Chi-square value. To investigate the appropriateness of this procedure a Monte Carlo study is used to measure both significance level and relative power for a variety of multivariate dispersion structures.     

An Affine-Invariant Generalization of the Wilcoxon Signed-Rank Test for the Bivariate Location Problem

This paper proposes an affine‐invariant test extending the univariate Wilcoxon signed‐rank test to the bivariate location problem. It gives two versions of the null distribution of the test statistic. The first version leads to a conditionally distribution‐free test which can be used with any sample size. The second version can be used for larger sample sizes and has a limiting χ₂² distribution under the null hypothesis. The paper investigates the relationship with a test proposed by Jan & Randles (1994). It shows that the Pitman efficiency of this test relative to the new test is equal to 1 for elliptical distributions but that the two tests are not necessarily equivalent for non‐elliptical distributions. These facts are also demonstrated empirically in a simulation study. The new test has the advantage of not requiring the assumption of elliptical symmetry which is needed to perform the asymptotic version of the Jan and Randles test.     

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.     

Asymptotic separability in sensitivity analysis

In an observational study in which each treated subject is matched to several untreated controls by using observed pretreatment covariates, a sensitivity analysis asks how hidden biases due to unobserved covariates might alter the conclusions. The bounds required for a sensitivity analysis are the solution to an optimization problem. In general, this optimization problem is not separable, in the sense that one cannot find the needed optimum by performing a separate optimization in each matched set and combining the results. We show, however, that this optimization problem is asymptotically separable, so that when there are many matched sets a separate optimization may be performed in each matched set and the results combined to yield the correct optimum with negligible error. This is true when the Wilcoxon rank sum test or the Hodges-Lehmann aligned rank test is applied in matching with multiple controls. Numerical calculations show that the asymptotic approximation performs well with as few as 10 matched sets. In the case of the rank sum test, a table is given containing the separable solution. With this table, only simple arithmetic is required to conduct the sensitivity analysis. The method also supplies estimates, such as the Hodges-Lehmann estimate, and confidence intervals associated with rank tests. The method is illustrated in a study of dropping out of US high schools and the effects on cognitive test scores.     

Simultaneous Small Sample Inference for Linear Combinations of Generalized Linear Model Parameters

A method is proposed to construct simultaneous confidence intervals for multiple linear combinations of generalized linear model parameters, that uses a multivariate normal- or t-distribution together with the signed likelihood root statistic. In an application to a case study simultaneous confidence bands for logistic regression are calculated. A simulation study based on the example evaluation suggests superior performance compared to the common Wald-type approaches. The proposed methods are readily implemented in the R extension package mcprofile.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

Confidence intervals based on the deviance statistic for the hyperparameters in state space models

The main objective of this work is to evaluate the performance of confidence intervals, built using the deviance statistic, for the hyperparameters of state space models. The first procedure is a marginal approximation to confidence regions, based on the likelihood test, and the second one is based on the signed root deviance profile. Those methods are computationally efficient and are not affected by problems such as intervals with limits outside the parameter space, which can be the case when the focus is on the variances of the errors. The procedures are compared to the usual approaches existing in the literature, which includes the method based on the asymptotic distribution of the maximum likelihood estimator, as well as bootstrap confidence intervals. The comparison is performed via a Monte Carlo study, in order to establish empirically the advantages and disadvantages of each method. The results show that the methods based on the deviance statistic possess a better coverage rate than the asymptotic and bootstrap procedures.     

A unified confidence interval for reliability-related quantities of two-parameter Weibull distribution

Statistical inference methods for the Weibull parameters and their functions usually depend on extensive tables, and hence are rather inconvenient for the practical applications. In this paper, we propose a general method for constructing confidence intervals for the Weibull parameters and their functions, which eliminates the need for the extensive tables. The method is applied to obtain confidence intervals for the scale parameter, the mean-time-to-failure, the percentile function, and the reliability function. Monte-Carlo simulation shows that these intervals possess excellent finite sample properties, having coverage probabilities very close to their nominal levels, irrespective of the sample size and the degree of censorship.     

Confidence intervals for dose or treatment effect in several 2 × j tables

Consider dichotomous observations taken from T strata or tables, where within each table, the effect of J>2 doses or treatments are valuated. 'Ihe dose or treatment effect may be measured by various functions of the probability of outcomes, but it is assumed that the effect is the same in each table. Previous work on finding confidence intervals is specific to a particular function of the probabilities, based on only two doses, and limited to ML estimation of the nuisance parameters. In this paper, confidence intervals are developed based on the C, test, allowing for a unification and generalization of previous work. A computational procedure is given that minimizes the number of iterations required. An extension of the procedure to the regression framework suitable when there are large numbers of sparse tables is outlined.     

Signed likelihood ratio tests in the Birnbaum-Saunders regression model

The Birnbaum-Saunders regression model is becoming increasingly popular in lifetime analyses and reliability studies. In this model, the signed likelihood ratio statistic provides the basis for testing inference and construction of confidence limits for a single parameter of interest. We focus on the small sample case, where the standard normal distribution gives a poor approximation to the true distribution of the statistic. We derive three adjusted signed likelihood ratio statistics that lead to very accurate inference even for very small samples. Two empirical applications are presented.