SpearmanRho

Spearman's rho, denoted ρs or rs, is a nonparametric measure of rank correlation that assesses the strength and direction of a monotonic relationship between two variables. It is defined as the Pearson correlation coefficient between the ranked variables. Given paired observations (x_i, y_i), assign ranks R(x_i) and R(y_i) (ties are typically assigned the average of the ranks that would have been assigned). The coefficient can be computed as rs = cov(R_x, R_y) / (sd(R_x) sd(R_y)). An explicit formula for the case with no ties is rs = 1 - 6 ∑ d_i^2 / [n(n^2 - 1)], where d_i = R(x_i) - R(y_i). In the presence of ties, rs is computed via the correlation of the ranks, or a tie-adjusted formula.

Interpretation: rs values range from -1 to 1. Values near 1 indicate a strong positive monotonic association; values near -1 indicate a strong negative monotonic association; values near 0 suggest little or no monotonic relationship. Spearman's rho is robust to outliers relative to Pearson's r and does not assume a linear relationship or normally distributed data; it is appropriate for ordinal data or for continuous data that violate parametric assumptions. It is important to note that rho measures monotonic, not necessarily linear, relationships and can be influenced by restricted ranges or heterogeneity in the data.

Computation and usage: Most statistics packages provide a function to compute Spearman's rho and an associated p-value for testing independence. Related measures include Kendall's tau, another nonparametric rank correlation coefficient.