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18.4: Randomization Association

Last updated

Apr 23, 2022
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- 18.3: Randomization Tests - Two or More Conditions
- 18.5: Fisher's Exact Test

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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.

Table $\PageIndex{1}$ : Example data
X	1.0	2.4	3.8	4.0	11.0
Y	1.0	2.0	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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