# 12.4: Randomization Association


skills to develop

• Compute a randomization test for Pearson's $$r$$

A significance test for Pearson's $$r$$ is described in the section inferential statistics for $$b$$ and $$r$$. The significance test described in that section assumes normality. This section describes a method for testing the significance of $$r$$ that makes no distributional assumptions.

 X Y 1 2.4 3.8 4 11 1 2 2.3 3.7 2.5

The approach is to consider the $$X$$ variable fixed and compare the correlation obtained in the actual data to the correlations that could be obtained by rearranging the $$Y$$ variable. For the data shown in Table $$\PageIndex{1}$$, the correlation between $$X$$ and $$Y$$ is $$0.385$$. There is only one arrangement of $$Y$$ that would produce a higher correlation. This arrangement is shown in Table $$\PageIndex{2}$$ and the $$r$$ is $$0.945$$. Therefore, there are two arrangements of $$Y$$ that lead to correlations as high or higher than the actual data.

Table $$\PageIndex{1}$$: The example data arranged to give the highest $$r$$
X Y
1.0 1.0
2.4 2.0
3.8 2.3
4.0 2.5
11.0 3.7

The next step is to calculate the number of possible arrangements of $$Y$$. The number is simply $$N!$$, where $$N$$ is the number of pairs of scores. Here, the number of arrangements is $$5! = 120$$. Therefore, the probability value is $$2/120 = 0.017$$. Note that this is a one-tailed probability since it is the proportion of arrangements that give an $$r$$ as large or larger. For the two-tailed probability, you would also count arrangements for which the value of $$r$$ were less than or equal to $$-0.385$$. In randomization tests, the two-tailed probability is not necessarily double the one-tailed probability.

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