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| Consider a baseline score x and a later score on the same test, y. The correlation between a baseline score and the difference score x-y (baseline score with change from baseline) is biased to be considerably greater than zero. Tu and Gilthorpe (2007) among others illustrate that for two independent ''random'' variables, x and y with the same variance the correlation between x and the x-y difference is approximately not zero as one might expect but the reciprocal of root 2 = 0.71. | Consider a baseline score x and a later score on the same test, y. The correlation, r(x,x-y), between a baseline score and the difference score x-y (baseline score with change from baseline) is biased to be considerably greater than zero. Tu and Gilthorpe (2007) among others illustrate that for two independent ''random'' variables, x and y with the same variance the correlation between x and the x-y difference is approximately not zero as one might expect but the reciprocal of root 2 = 0.71. |
Correcting r(x,x-y) for bias due to correlating two terms involving x
Consider a baseline score x and a later score on the same test, y. The correlation, r(x,x-y), between a baseline score and the difference score x-y (baseline score with change from baseline) is biased to be considerably greater than zero. Tu and Gilthorpe (2007) among others illustrate that for two independent random variables, x and y with the same variance the correlation between x and the x-y difference is approximately not zero as one might expect but the reciprocal of root 2 = 0.71.
This positive bias between x and x-y is caused by a variable, baseline x in this case, appearing in both sides of the terms being correlated. It follows that the correlation between x and y-x is biased negatively from zero which intuitively says that the smaller baseline scores tend to have larger increases.
Tu et al. (2005) therefore suggest that if it can be assumed that the variances of x and y are equal the correlation between baseline and x-y change should be compared using Fisher's test to the square root of 0.5(1-r)where r is the correlation between x and y. Both sides need to be Fisher transformed to do this comparisons. SPREADSHEET TO BE ADDED.
Other authors suggest comparing the variances' of x and y since the variances at later times would be expected to be smaller as scores 'bunch up' if there is a relationship between x and x-y. So a difference in variances in scores x and y will illustrate a relationship between baseline score and change. Myrtek and Foerster (1986) propose a t-test that compares x and y variances assuming that one variance is greater than the other. References Myrtek, M. and Foerster, F. (1986). The law of initial value: a rare exception. Tu, Y-K., Baelum, V. and Gilthorpe, M. S. (2005). The relationship between baseline value and its change: problems in categorisation and the proposal of a new method. Tu, Y-K. and Gilthorpe, M. S. (2007). [Revisiting the relation between change and initial value: A review and evaluation http://dionysus.psych.wisc.edu/lit/Topics/Statistics/RegressionToMean/tu_RegressionToTheMean_SiM2007.pdf] Statistics in Medicine