A nonparametric confidence interval for slope based on spearman's rho

An algorithm is presented for computing an exact nonparametric interval estimate of the slope parameter in a simple linear regression model. The confidence interval is obtained by inverting the hypothesis test for slope that uses Spearman's rho. This method is compared to an exact procedure based on Kendall's tau. The Spearman rho procedure will generally give exact levels of confidence closer to desired levels, especially in small samples. Monte carlo results comparing these two methods with the parametric procedure are given     

Kendall's tau for serial dependence

The authors show how Kendall's tau can be adapted to test against serial dependence in a univariate time series context. They provide formulas for the mean and variance of circular and noncircular versions of this statistic, and they prove its asymptotic normality under the hypothesis of independence. They present also a Monte Carlo study comparing the power and size of a test based on Kendall's tau with the power and size of competing procedures based on alternative parametric and nonparametric measures of serial dependence. In particular, their simulations indicate that Kendall's tau outperforms Spearman's rho in detecting first‐order autoregressive dependence, despite the fact that these two statistics are asymptotically equivalent under the null hypothesis, as well as under local alternatives.     

A test for correlation based on Kendall's tau

A consistent estimator for the variance of Kendall's tau is proposed which allows for testing the hypothesis of no correlation in a bivariate distribution. The null distribution of the test statistic is tabulated under independence, and the properties of the test are discussed.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

The Ranges of Limiting Values of Some Partial Correlations under Conditional Independence

Pearson's partial correlation, Kendall's partial tau, and a partial correlation based on Spearman's rho need not be consistent estimators of zero under conditional independence. The ranges of possible limiting values of these correlations are computed under multivariate normality and lognormality. Students should exercise caution when interpreting these partial correlations as a measure of conditional independence.     

An approximated distribution of the gini's rank associatio coefficient

An approximate distribution is proposed for the Gini's rank association coefficient g which is, like Kendall's and Spearman's rank correlation coefficient, a statistic to test independence between two random variables. The purposed distribution can be simply transformed into a Student's T distribution; so, hypothesis testing is made much easier.     

Taux de résistance des tests de rang d'indépendance

The resistance of tests to acceptance and rejection of null hypotheses was denned and studied by Ylvisaker in the context of one-sample problems. This notion provides a measure of a test's resistance to outliers. In this paper, we propose an extension of this notion to rank-based tests of independence for bivariate random variables. We show, among other things, that Kendall's test of independence is more resistant than Spearman's test.     

On estimating the variance of a U-statistic

We compare jackknifing and bootstrapping as methods for estimating the variance of a U-statistic. The use of these estimates in calculating asymptotic confidence intervals is discussed, and the results of a numerical study involving Kendall's tau are reported. For the special case of this statistic, the bootstrap is the estimate of choice.     

ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS

The estimation of a real‐valued dependence parameter in a multivariate copula model is considered. Rank‐based procedures are often used in this context to guard against possible misspecification of the marginal distributions. A standard approach consists of maximizing the pseudo‐likelihood. Here, we investigate alternative estimators based on the inversion of two multivariate extensions of Kendall's tau developed by Kendall and Babington Smith, and by Joe. The former, which amounts to the average value of tau over all pairs of variables, is often referred to as the coefficient of agreement. Existing results concerning the finite‐ and large‐sample properties of this coefficient are summarized, and new, parallel findings are provided for the multivariate version of tau due to Joe, along with illustrations. The performance of the estimators resulting from the inversion of these two versions of Kendall's tau is compared in the context of copula models through simulations.     

A “Reverse Order” Elementary Statistics Course

A family of coefficients for measuring monotone association is presented. These include measures of association of ordinal or interval variables such as gamma of Goodman and Kruskal, Somers's d_yx , Kendall's tau, or Spearman's rho as special cases. The article shows how a large number of measures of association can be put into a single general form. These coefficients are used as a basis for defining a variety of data analysis techniques.     

Two-sample inference procedures under nonproportional hazards

We introduce a new two-sample inference procedure to assess the relative performance of two groups over time. Our model-free method does not assume proportional hazards, making it suitable for scenarios where nonproportional hazards may exist. Our procedure includes a diagnostic tau plot to identify changes in hazard timing and a formal inference procedure. The tau-based measures we develop are clinically meaningful and provide interpretable estimands to summarize the treatment effect over time. Our proposed statistic is a U-statistic and exhibits a martingale structure, allowing us to construct confidence intervals and perform hypothesis testing. Our approach is robust with respect to the censoring distribution. We also demonstrate how our method can be applied for sensitivity analysis in scenarios with missing tail information due to insufficient follow-up. Without censoring, Kendall's tau estimator we propose reduces to the Wilcoxon-Mann–Whitney statistic. We evaluate our method using simulations to compare its performance with the restricted mean survival time and log-rank statistics. We also apply our approach to data from several published oncology clinical trials where nonproportional hazards may exist.     

Remarks on multivariate analogues of kendall's tau

The aim of the paper is to discuss a decision theoretical interpretation of multivariate analogues of Kendall's tau.     

KENDALL'S TAU AND CONTINGENCY TABLES

Kendall's tau is a coefficient of concordance between two rankings of n objects. Its definition and large sample normal approximation are easily extended to the case where one of the rankings contains ties. In this paper, definition and normal approximation are extended further to the case where both rankings contain ties. The results are applied to give a fully distribution-free test for two-way contingency tables with ordered categories.     

Nonparametric measures of intraclass correlation

Three nonparametric measures of intraclass correlation based on the notion of concordance are considered. Their unbiased estimators and nonparametric tests based on the estimators are studied and it is shown that an analogue of the Kendall's tau provides small variance estimator and relatively powerful test. Furthermore, the approximate variance of the estimator is given when the correlation is small in the normal model.     

BOUNDS ON BIVARIATE DISTRIBUTION FUNCTIONS WITH GIVEN MARGINS AND MEASURES OF ASSOCIATION

We find pointwise best-possible bounds on the bivariate distribution function of continuous random variables with given margins and a given value of the population version of a nonparametric measure of association such as Kendall's tau or Spearman's rho.     

Copulas Constructed from Horizontal Sections

In analogy with the study of copulas whose diagonal sections have been fixed, we study the set _h of copulas for which a horizontal section h has been given. We first show that this set is not empty, by explicitly writing one such copula, which we call horizontal copula. Then we find the copulas that bound both below and above the set _h. Finally, we determine the expressions for Kendall's tau and Spearman's rho for the horizontal and the bounding copulas.     

DISTRIBUTION-FREE TESTS BASED ON SUB-SAMPLE EXTREMA FOR TESTING AGAINST POSITIVE DEPENDENCE

A class of distribution-free tests based on U-statistics has been proposed for testing the null hypothesis of independence against positive quadrant dependence. The tests are based on U-statistics and the Kendall's-tau test belongs to this class.     

Bias reduction for nonparametric correlation coefficients under the bivariate normal copula assumption with known detection limits

The authors derive the asymptotic mean and bias of Kendall's tau and Spearman's rho in the presence of left censoring in the bivariate Gaussian copula model. They show that tie corrections for left‐censoring brings the value of these coefficients closer to zero. They also present a bias reduction method and illustrate it through two applications.     

Large Differences between Partial Correlation Coefficients

A plot of each ranking of N objects in N-dimensional space is shown to provide geometric interpretations of Kendall's tau and Spearman's rho and also of the relationship of rho to a sum of inversion weights. The computation of rho from a sum of inversion weights is shown to allow sequential calculation of rho.     

A new extension of the FGM copula for negative association

In this paper we introduced a single parameter, absolutely continuous and radially symmetric bivariate extension of the Farlie-Gumbel-Morgenstern (FGM) family of copulas. Specifically, this extension measures the higher negative dependencies than most FGM extensions available in literature. Closed-form formulas for distribution, quantile, density, conditional distribution, regression, Spearman's rho, Kendall's tau, and Gini's gamma are obtained. In addition, a formula for random variate generations is presented in closed-form to facilitate simulation studies. We conduct both paired and multiple comparisons with Frank, Gaussian, and Plackett copulas to investigate the performance based on Vuong's test. Furthermore, the new copula is compared with Frank, Gaussian, and Plackett copulas using both Kolmogorov-Smirnov and Cramér-von Mises type test statistics. Finally, a bivariate dataset is analyzed to compare and illustrate the flexibility of the new copula for negative dependence.