# Correct test for correlation between change and baseline measurement

In the book Medical statistics, by Cambell, Machin and Walters, you can read, in the very interesting section ‘common pitfalls’ on page 280, that the test for correlation between initial value and change is not valid since there is intrinsic negative correlation between the two measures. Instead they propose that you should test correlation between mean of the baseline and follow-up and change to produce a valid test.

I get that this would not result in intrinsic negative correlation, what I don’t get is how this would be good estimate for the correlation between baseline and change. If someone could explain this to me or point me to another reference I would be very thankful!

After doing some serious googling I found that this test is referred to as “Oldham’s test”. Apparently it can be shown that the correlation between the change and the mean of two measurements is:

$\text{Corr}[x-y, (x+y)/2] = \frac{s_{x}^{2}- s_{y}^{2}}{\sqrt{ (s_{x}^{2}- s_{y}^{2})^2 -4r_{xy}^2 s_{x}^{2} s_{y}^{2}}}$

This becomes a test of the difference in variances for the repeated measurements. The logic is that if there is exists a true effect of baseline measurement on change the variances should be different for baseline and follow-up. For example, if we are looking at cholesterol levels and people with higher cholesterol levels sink faster the follow-up measurement should have a smaller variance than the baseline measurement. If there is no difference in variance between baseline and follow-up values there should be little support for the hypothesis of difference in change for different levels of the baseline measurement.

References:

Tu, Y. K. and M. S. Gilthorpe (2007). "Revisiting the relation between change and initial value: a review and evaluation." Statistics in medicine 26(2): 443-457.

What is Regression to the Mean? The Relation between Change and Initial Value Revisited Yu-KangTu

I don't know anything about medical statistics, but in econometrics this thing is commonplace, and generally handled under the heading of "panel data" or "fixed effects" (which is different from "fixed effects" in other branches of statistics). You basically specify a regression which includes your treatment variable, any other explanatory variables, and individual-level "fixed-effects" -- dummy variables for each person (or unit) for which you've got repeated measurements. When calculating your standard errors, you need to account for the autocorrelated nature of those errors. Google around for "clustered standard errors" to get an in-depth description. Stata implements it using the cluster(clustvar) syntax in reg y x, cluster(clustvar). I don't know of any R packages that handle this, though its not too hard to program it yourself, and google should probably turn up some packages or code.