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  1. Introduction
  2. Graphing Distributions
  3. Summarizing Distributions
  4. Describing Bivariate Data
  5. Probability
  6. Research Design
  7. Normal Distribution
  8. Advanced Graphs
  9. Sampling Distributions
  10. Estimation
  11. Logic of Hypothesis Testing
  12. Tests of Means
  13. Power
  14. Regression
  15. Analysis of Variance
  16. Transformations
  17. Chi Square

  18. Distribution Free Tests
    1. Contents
      Standard
    2. Benefits
      Standard         Video
    3. Randomization Tests: Two Conditions
      Standard         Video
    4. Randomization Tests: Two or More Conditions
      Standard         Video
    5. Randomization Association
      Standard  
    6. Fisher Exact Test
      Standard         Video
    7. Rank Randomization Two Conditions
      Standard         Video
    8. Rank Randomization Two or More Conditions
      Standard         Video
    9. Rank Randomization for Association
      Standard         Video
    10. Statistical Literacy
      Standard
    11. Exercises
      Standard

  19. Effect Size
  20. Case Studies
  21. Calculators
  22. Glossary
 

Randomization Tests: Association (Pearson's r)

Author(s)

David M. Lane

Prerequisites

Inferential Statistics for b and r

Learning Objectives
  1. 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 1. Example data.

X Y
1.0 1.0
2.4 2.0
3.8 2.3
4.0 3.7
11.0 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 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 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 2. 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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