The Rank Correlation Coefficient as an Indicator of the Product-Moment Correlation Coefficient for Small Samples

On the basis of 2200 small (10-100) samples obtained from four random number populations, it is shown that the rank correlation coefficient, p, is 88 adequate a measure of concordance 88 the Pearson product-moment correlation, T. The samples were obtained by using pseudo-random linear and normal number generators for bivariate populations with expected correlation cC'efficientB, r~, of 0.0 and 0.5. The sampling distributions for p and r appear 80 similar that one may conJlude that at least 95% of the values of Ip TI should lie within one standard error of T. It is suggested that geophysicists may benefit from the speed of calculating p when a rapid and objective measure of concordance is sought. Introduction. In meteorology, oceanography, and other geophysical disciplines, correlation methods are required in order to glean informa.tion from limited data. The Pearson productmoment correlation coefficient, r, is generally used as the indicator of concordance in a series of paired varlates. Although regression equations result from the determination of r, in a large number of geophysical problems only a rapid and objective measure of concordance is sought. For example, McDonald and Green [1960] mentioned studies of climatological homogeneity of a region where one is interested in determining the spatial pattern of correlation between one or more base stations and a large number of neighboring stations. In these problems the Spearman or rank correlation coefficient, p, becomes the preferred geophysical correlation coefficient when the sample size is small (less than 50) and a rapid method is desired. McDonald and Green examined the difference, r p, using 50-year records of ha.lf-year precipitation from 220 stations throughout the United States. They concluded that the distributions of p and r are so similar that for most applications of correlation coefficients to geophysical problems p is as adequate a statistic as r and in certain cases is preferable to r. Their work, however, does not provide information on the relationship between p and r for smaller sample sizes (l~ than SO). It is for these small samples that the rank correlation coefficient is genuinely useful for rapid hand calculation. Also, McDonald and Green provide no comparison of p and r for various parent populations. This information is required in order to strengthen the confidence of geophysicists in the reliability of the rw correlation coefficient as an indicator of the magnitude of the productmoment correlation coefficient. A series of empirical experiments has been designed to provide information on the sampling distribution of p and r for these small samples. It will be shown that p is an adequate measure of r over their entire range for samples of size 10-100 drawn from both random linear and random normal populations. If exact information on the probability of existence of p is required, one may refer to Owen [1962] (sample size, 1-11) or Old" [1938] (sample size, 11-30). These authors assume no relationship between the paired variates, and they have given significance tables for a bivariate population of zero correlation. The sampling distributions of p for nonzero bivariate populations are not known. To determine a rank correlation coefficient, the investigator replaces the precise values of the variables (x" V,) by ranks. The smallest x, is assigned the rank 1 and the next largest value the rank 2, and so forth. The V, values are also assigned ranks. If there is a tied rank, the ranks are usua.lly split between the identical variables. After assigning ranks, the difference in the ranks