The Spearman Rank Correlation Test (Numerical Example)

Consider the following data for the illustration of the detection of heteroscedasticity using the Spearman Rank correlation test. The Data file is available to download.

YX2X3
11208.1
16188.4
11228.5
14218.5
13278.8
17269
14258.9
15279.4
12309.5
18289.5

The estimated multiple linear regression model is:

$$Y_i = -34.936 -0.75X_{2i} + 7.611X_{3i}$$

The Residuals with the data table are:

YX2X3Residuals
11208.1-0.63302
16188.40.575564
11228.5-2.16954
14218.50.076455
13278.81.317102
172693.040825
14258.90.047951
15279.4-1.2497
12309.5-2.74881
18289.51.743171

We need to find the rank of absolute values of $u_i$ and the expected heteroscedastic variable $X_2$.

$Y$$X_2$$X_3$ResidualsRank of |$u_i$|Rank of $X_2$$d$$d^2$
11208.1-0.633 4224
16188.40.576 3124
11228.5-2.170 84416
14218.50.076 23-11
13278.81.317 67.5-1.52.25
172693.041 106416
14258.90.048 15-416
15279.4-1.250 57.5-2.56.25
12309.5-2.749 910-11
18289.51.743 79-24
       Total =070.5

Calculating the Spearman Rank correlation

\begin{align}
r_s&=1-\frac{6\sum d^2}{n(n-1)}\\
&=1-\frac{6\times 70.5)}{10(100-1)}=0.5727
\end{align}

Let us perform the statistical significance of $r_s$ by t-test

\begin{align}
t&=\frac{r_s \sqrt{n}}{\sqrt{1-r_s^2}}\\
&=\frac{0.5727\sqrt{8}}{\sqrt{1-(0.573)^2}}=1.977
\end{align}

The value of $t$ from the table at a 5% level of significance at 8 degrees of freedom is 2.306.

Since $t_{cal} \ngtr t_{tab}$, there is no evidence of the systematic relationship between the explanatory variables, $X_2$ and the absolute value of the residuals ($|u_i|$) and hence there is no evidence of heteroscedasticity.

Since there is more than one regressor (the example is from the multiple regression model), therefore, Spearman’s Rank Correlation test should be repeated for each of the explanatory variables.

Spearman Rank Correlation

As an assignment perform the Spearman Rank Correlation between |$u_i$| and $X_3$  for the data above. Test the statistical significance of the coefficient in the above manner to explore evidence about heteroscedasticity.

Read about Pearson’s Correlation Coefficient

R Language Interview Questions

2 thoughts on “The Spearman Rank Correlation Test (Numerical Example)”

  1. Bilal Data Analyst Ahmad

    n= 10 but this one n= 100 so this is mistake please update this value.
    Thank you.

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