Spearman'S Rho

Since its inception more than a century ago, the correlation coefficient has remained one of the most commonly used statistical indices in empirical research. A correlation coefficient provides an index of the relationship between two variables, or bivariate relationship. When both variables are quantitative in nature and either (a) they depart from normality or (b) they represent ordinal scales of measurement, then a researcher can use Spearman's rho (ρ), also known as the Spearman rank correlation coefficient. Charles Spearman introduced the concept of Spearman's rho in his seminal article in 1904, titled “The Proof and Measurement of Association Between Two Things,” published in the American Journal of Psychology.

Technically speaking, Spearman's rho, a nonparametric measure of association, is appropriate whenever each variable is measured on at least an ordinal scale and both variables are taken from any continuous bivariate distribution. Moreover, Spearman's rho does not require any assumption about the frequency distribution of the variables. However, when the variables are normally distributed, the procedure known as Pearson's r, or the Pearson product-moment correlation, has more statistical power than does Spearman's rho and thus is more appropriate. In fact, the asymptotic relative efficiency of Spearman's rho with respect to Pearson's r is 91.2%. In other words, Pearson's r is more efficient than Spearman's rho in that for a desired level of statistical significance (i.e., Type I error), Pearson's r has the same power for detecting statistical significance as does Spearman's rho using 91.2% of the sample size needed for Spearman's rho. Alternatively stated, if a sample size of 1,000 is needed for a relationship to be declared statistically significant for a nominal alpha level using Spearman's rho, then a smaller sample size (i.e., 912) is needed to yield the same p value when using Pearson's r. However, when the normality assumption is violated, the Type I error rate is inflated when Pearson's r is used, and thus nonparametric alternatives should be employed.

Spearman's rho is the second-most-popular bivariate correlational technique. Computation of Spearman's rho is as follows: Suppose we have n pairs of continuous data. The first step in computing Spearman's rho is to rank the X scores in the paired sample data from 1 (smallest score) to n (largest score), and independently rank the Y scores in the paired sample data from 1 (smallest score) to n (largest score). Therefore, the original (X1Y1), (X2Y2),…, (XnYn) pairs of observations will change to [Rank (X1), Rank (Y1)], [Rank (X2), Rank (Y2)],…, [Rank (Xi), Rank (Yi)]. The next step in the process is to calculate a difference d for each pair as the difference between the ranks of the corresponding X and Y variables. The sum of the differences always will equal zero. The test statistic, denoted by rs, is defined as the sum of squares of these differences3. The formula for this expression is

An examination of the extreme cases helps us see how this formula works. Suppose both X and Y scores represent identical ordered arrays from smallest to largest. In this case, each di = 0, and thus Σdi2 = 0, and substitution in Equation 1 provides an rs value of +1. Thus, rs = 1 describes a perfect positive relationship between ranks. Conversely, if the ranks of the X scores [Page 928]are the complete opposite of the ranks of the Y scores, then it can be demonstrated

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