5.3 Spearmans Rank Correlation

In studying the relationship between two variables, a measure of correlation assumes that neither variable is functionally dependent on the other e.g we must ask whether body weight is related to height for a 35 year old man or exam scores in music are related to exam scores in mathematics for first year college students.
If the distributions underlying the variables are far from bivariate normal, or if the data are ordinal, then non-parametric correlation techniques should be employed to test hypothesis about the relationship of the variables.

The assumptions underlying non-parametric correlation are:

(i).

the \(n\) pairs of ratio, interval or ordinal data constitute a random sample;

(ii).

the two members of each pair are measured on the same subject.

Nothing is assumed about the shape of either distribution, which is the whole point: Pearson’s coefficient requires bivariate normality before its significance test is valid, and these methods require none.

The rank correlation coefficient proposed by C. Spearman is Pearson’s coefficient computed on the ranks of the observations rather than on the observations themselves.
The Spearman rank correlation coefficient computed from a sample is written \(r_s\). Then the parameter is denoted as \(\rho _s\).
If each of the \(n\) measurements of \(X_i\) (i.e \(X_1,\dots ,X_n\)), then \(R(X_i)\) represents the rank of \(X_i\) where each \(i\in \mathbb {Z}\) each rank is a integer \(1,\dots ,n\) indicating relative magnitude.