Four Correlation Coefficients with a Third Blocking Variable: Their Efficacy,Relative Efficiency,and Test Statistics

Abstract

The efficacy and the asymptotic relative efficiency (ARE) of a weighted sum of Kendall's taus, a weighted sum of Spearman's rhos, a weighted sum of Pearson's r's, and a weighted sum of z-transformation of the Fisher–Yates correlation coefficients, in the presence of a blocking variable, are discussed. The method of selecting the weighting constants that maximize the efficacy of these four correlation coefficients is proposed. The estimate, test statistics and confidence interval of the four correlation coefficients with weights are also developed. To compare the small-sample properties of the four tests, a simulation study is performed. The theoretical and simulated results all prefer the weighted sum of the Pearson correlation coefficients with the optimal weights, as well as the weighted sum of z-transformation of the Fisher–Yates correlation coefficients with the optimal weights.     

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.     

Pravastatin improves the impaired nitric oxide-mediated neurogenic and endothelium-dependent relaxation of corpus cavernosum in aged rats

Abstract

Aim: The aim of this study was to investigate the effect of pravastatin treatment on diminished corpus cavernosum (CC) function associated with aging.

Methods: Male rats were divided into three groups as adult rats (12–14 weeks old), aged rats (72–80 weeks old) and aged rats given 10?mg/kg/d pravastatin in drinking water for six weeks. Blood pressure was measured by tail-cuff method. Total cholesterol, low-density lipoprotein-cholesterol, high-density lipoprotein-cholesterol, triglycerides and testosterone levels were estimated in blood. Changes in expression levels of endothelial nitric oxide synthase (eNOS), phosphorylated eNOS (p-eNOS) (Ser-1177), neuronal nitric oxide synthase (nNOS), NADPH oxidase subunit gp91^phox, Rho A and Rho kinase (ROCK2) in CC were assessed by immunohistochemistry. Nitric oxide (NO)-mediated endothelium-dependent and neurogenic CC relaxation were evaluated by acetylcholine (ACh, 0.1?nM–100?µM) and electrical field stimulation (EFS; 30?V, 5?ms, 2–32?Hz), respectively.

Results: In aged rats, NO-mediated, both endothelium-dependent and neurogenic CC relaxation, were significantly impaired as compared to adult rats. Besides, eNOS, p-eNOS and nNOS expressions decreased significantly in CC from aged rats, while gp91^phox, RhoA and ROCK2 expressions increased significantly. The diminished relaxation in response to ACh or EFS as well as the changes in expression of these proteins in aged rats were significantly improved by pravastatin treatment.

Conclusion: Pravastatin improves NO-mediated CC relaxations of aged rats probably by inhibiting NADPH oxidase/Rho kinase pathways, and this effect does not seem to be associated with lipid lowering effect of this drug.     

Nonparametric tests of hypotheses for umbrella alternatives

The author proposes a general method for constructing nonparametric tests of hypotheses for umbrella alternatives. Such alternatives are relevant when the treatment effect changes in direction after reaching a peak. The author's class of tests is based on the ranks of the observations. His general approach consists of defining two sets of rankings: the first is induced by the alternative and the other by the data itself. His test statistic measures the distance between the two sets. The author determines the asymptotic distribution for some special cases of distances under both the null and the alternative hypothesis when the location of the peak is known or unknown. He shows the good power of his tests through a limited simulation study     

Tests of serial independence based on Kendall's process

The authors propose new rank statistics for testing the white noise hypothesis in a time series. These statistics are Cramér‐von Mises and Kolmogorov‐Smirnov functionals of an empirical distribution function whose mean is related to a serial version of Kendall's tau through a linear transform. The authors determine the asymptotic behaviour of the underlying serial process and the large‐sample distribution of the proposed statistics under the null hypothesis of white noise. They also present simulation results showing the power of their tests.     

Gluing Copulas

We present a new way of constructing n-copulas, by scaling and gluing finitely many n-copulas. Gluing for bivariate copulas produces a copula that coincides with the independence copula on some grid of horizontal and vertical sections. Examples illustrate how gluing can be applied to build complicated copulas from simple ones. Finally, we investigate the analytical as well as statistical properties of the copulas obtained by gluing, in particular, the behavior of Spearman's ρ and Kendall's τ.     

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.     

Correlation,Regression Lines,and Moments of Inertia

The R language, a freely available environment for statistical computing and graphics is widely used in many fields. This “expert-friendly” system has a powerful command language and programming environment, combined with an active user community. We discuss how R is ideal as a platform to support experimentation in mathematical statistics, both at the undergraduate and graduate levels. Using a series of case studies and activities, we describe how R can be used in a mathematical statistics course as a toolbox for experimentation. Examples include the calculation of a running average, maximization of a nonlinear function, resampling of a statistic, simple Bayesian modeling, sampling from multivariate normal, and estimation of power. These activities, often requiring only a few dozen lines of code, offer students the opportunity to explore statistical concepts and experiment. In addition, they provide an introduction to the framework and idioms available in this rich environment.     

On the relationship between Spearman's rho and Kendall's tau for pairs of continuous random variables

It has long been known that for many joint distributions exhibiting weak dependence, the sample value of Spearman's rho is about 50% larger than the sample value of Kendall's tau. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho to tau approaches 3/2$3/2$ as the joint distribution approaches that of two independent random variables. We also find sufficient conditions for determining the direction of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely continuous.     

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.