See sample correlation coefficient for additional details.
2.
The sample correlation coefficient is written:
3.
Where is the sample correlation coefficient between and; and and are the sample standard deviation of and.
4.
Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written ( calculation is under expectation, assumes Gaussian statistics)
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Examples of variance-stabilizing transformations are the Fisher transformation for the sample correlation coefficient, the square root transformation or Anscombe transform for type-II error.
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In other words, if linear regression is the appropriate model for a set of data points whose sample correlation coefficient is not perfect, then there is regression toward the mean.
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