A Comparison of Correlation Coefficients via a Three-Step Bootstrap Approach

Tahani Maturi,Anga Elsayigh

doi:10.5539/jmr.v2n2p3

Abstract

In this article we compare ten correlation coefficients using a three-step bootstrap approach (TSB). A three-step bootstrap is applied to determine the optimal repetitions, $B$, to estimate the standard error of the statistic with certain degree of accuracy. The coefficients in question are Pearson product moment ($r$), Spearman's rho ($\rho$), Kendall's tau ($\tau$) , Spearman's Footrule ($F_t$), Symmetric Footrule ($C$), the Greatest deviation ($R_g$), the Top - Down ($r_T$), Weighted Kendall's tau ($\tau_w$), Blest ($\nu$), and Symmetric Blest's coefficient ($\nu^*$). We consider a standard error criterion for our comparisons. However, since the rank correlation coefficients suffer from the tied problem that results from the bootstrap technique, we use existing modified formulas for some rank correlation coefficients, otherwise, the randomization tied-treatment is applied.

Highlights

One may be interested in the relationship between two factors or two variables and would wish to represent this relationship by a number or even using a statistical technique to make an inference
Pearson coefficient considers the linearity of the relationship whereas Spearman and Kendall study the monotonicity of this relationship
The most important property is that the randomization methods do not affect the null distribution of the rank correlation coefficient, so we do not need to adapt the null distribution for these coefficients

Summary

Introduction

One may be interested in the relationship between two factors or two variables and would wish to represent this relationship by a number or even using a statistical technique to make an inference. Others may use a nonparametric rank correlation coefficient such as Spearman rho or Kendall tau for the same purpose. Borkowf (2000) presented a new nonparametric method for estimating the variance of Spearman’s rho by calculating ρ from a two-way contingency table with categories defined by the bivariate ranks. His method is a computer method depending on the data at hand like the bootstrap and jackknife methods. The common issue is how many replications, B, should run to get the most accurate results required One such approach is introduced by Andrews and Buchinsky (2000; 2001; 2002) which is called a three-step approach.

Overview of the correlation coefficients

Bootstrap Motivation and Notation

Findings

Conclusion