Positive and
negative Spearman rank correlations
The
sign of the Spearman correlation indicates the direction of association
between X (the independent variable) and Y (the
dependent variable). If Y tends to increase when X increases,
the Spearman correlation coefficient is positive. If Y tends
to decrease when X increases, the Spearman correlation coefficient
is negative. A Spearman correlation of zero indicates that there is no tendency
for Y to either increase or decrease when X increases.
The Spearman correlation increases in magnitude as X and Y become
closer to being perfect monotone functions of each other. When X and Y are
perfectly monotonically related, the Spearman correlation coefficient becomes
1. A perfect monotone increasing relationship implies that for any two pairs of
data values Xi, Yi and Xj, Yj,
that Xi − Xj and Yi − Yj always
have the same sign. A perfect monotone decreasing relationship implies that
these differences always have opposite signs.
The Spearman correlation coefficient
is often described as being "nonparametric". This can have two
meanings: First, a perfect Spearman correlation results when X and Y are
related by any monotonic function. Contrast this with the Pearson correlation, which
only gives a perfect value when X and Y are
related by a linear function. The other sense in which the
Spearman correlation is nonparametric in that its exact sampling distribution
can be obtained without requiring knowledge (i.e., knowing the
parameters) of the joint probability distribution of X and Y.
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