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The Spearman's rank-order correlation coefficient

The test of significance for the Spearman's rank-order correlation coefficient is used to verify the hypothesis determining the lack of monotonic correlation between analysed features of the population and it is based on the Spearman's rank-order correlation coefficient calculated for the sample. The closer to 0 the value of $r_s$ coefficient is, the weaker dependence joins the analysed features.

Basic assumptions:

measurement on an ordinal scale or on an interval scale.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & R_s = 0, \\ \mathcal{H}_1: & R_s \ne 0. \end{array}$

The test statistic is defined by:

$\begin{displaymath} t=\frac{r_s}{SE}, \end{displaymath}$

where $\displaystyle SE=\sqrt{\frac{1-r_s^2}{n-2}}$ .

The value of the test statistic can not be calculated when $r_s=1$ lub $r_s=-1$ or when $n<3$ .

The test statistic has the t-Student distribution with $n-2$ degrees of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

The settings window with the Spearman's monotonic correlation can be opened in Statistics menu → NonParametric tests→monotonic correlation (r-Spearman) or in ''Wizard''.

EXAMPLE (LDL weeks.pqs file)

The effectiveness of a new therapy designed to lower cholesterol levels in the LDL fraction was studied. 88 people at different stages of the treatment were examined. We will test whether LDL cholesterol levels decrease and stabilize with the duration of the treatment (time in weeks).

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & $In the population, there is no monotonic relationship between treatment time and LDL levels,$\\ \mathcal{H}_1: & $In the population, there is a monotonic relationship between treatment time and LDL levels.$ \end{array}$

Comparing $p$ <0.0001 with a significance level $\alpha=0.05$ we find that there is a statistically significant monotonic relationship between treatment time and LDL levels. This relationship is initially decreasing and begins to stabilize after 150 weeks. The Spearman's monotonic correlation coefficient and therefore the strength of the monotonic relationship for this relationship is quite high at $r_s$ =-0.78. The graph was plotted by curve fitting through local LOWESS linear smoothing techniques.

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The Spearman's rank-order correlation coefficient

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