Troels Ring writes:

I have measured total phosphorus, TP, on a number of dialysis patients, and also measured conventional phosphate, Pi. Now P is exchanged with the environment as Pi, so in principle a correlation between TP and Pi could perhaps be expected. I’m really most interested in the fraction of TP which is not Pi, that is TP-Pi. I would also expect that to be positively correlated with Pi. However, looking at the data using a mixed model an insignificant negative correlation is obtained. Then I thought, that since TP-Pi is bound to be small if Pi is large a negative correlation is almost dictated by the math even if the biology would have it otherwise in so far as the the TP-Pi, likely organic P, must someday have been Pi. Hence I thought about correcting the slight negative correlation between TP-Pi and Pi for the expected large negative correlation due to the math – to eventually recover what I came from: a positive correlation. People seems to agree that this thinking is nonsense. They say I can just keep to the analysis and forget about RTM. I cannot help thinking that if I could measure TP-Pi by a method not requiring me to subtract Pi, I would get at least a cleaner result.

My reply: I’m getting confused on the details here, but, yes, it is typical that if you have two variables A and B measured on a common scale, that A-B has a negative correlation with B. This comes up, for example, in adjusting for pretest scores in education. People often have the intuition that they should be analyzing posttest – pretest, but it typically makes more sense to look at posttest – 0.8*pretest or something like that. Jennifer and I discuss this in our book in the context of the Electric Company example. Ultimately I suppose the solution is to go beyond correlations and to have a generative model for the joint distribution of TP and Pi.