Tuesday, April 7, 2009

When non-linear relationships attack

In a lot of cases, in epidemiology, it is conventional to assume that relationships are linear. In a lot of cases this assumption seems to be pretty reasonable. Occasionally you get a relationship where very large values have biological plausibility (like income or albumin excretion rate) and a base 2 logarithmic transformation is a more logical way to proceed.

So far, so good.

Now look at the rate of rate of change of bone mineral density by age for women as reported in this article from the Canadian Medical Association Journal. Between ages 25 and 40 the rate of change of bone mineral density is positive. Then, between ages 40 and 60 the rate of change is negative. Suddenly, at age 60+, the rate of change becomes positive again. Yes, the first derivative of the relationship between age and bone mineral density is quadratic! That would rather imply that the actual relationship is cubic!!

Without seeing this sort of high quality descriptive data, I would have screamed "overfit" if I saw a cubic age term in a statistical model. As it is, I am having pessimistic thoughts about just have big the sample size needs to be in order to estimate the polynomial association between age and bone mineral density.

Now imagine that you are trying to remove confounding by age from another association; just how are you going to be sure that you don't have residual confounding?

Wow, juSt wow!

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