Sammy at Angry Bear has a fascinating post on a (warning pdf) paper by Ohsfeldt and Schneider
Read the post.
Fascinating. I am especially impressed by the corrected life expectancy. The health care spending regressions seem to me to be of the sort that could benefit from more work. They provide strong evidence of Baumol's disease which suggests that, at least, a linear correction for per capita GDP is needed.
However, I do not find the non linear (semilog) estimate convincing. The problem is that the USA is a double outlier with high per capita GDP and high health care spending. This means that it is an extremely influential observation. I would want to see the fitted values from a regression from which the USA is dropped. In particular of countries 2-10 in per capita GDP the regression line (curve on the graph) lies above 7 passes through one (as far as I can see) and passes below only 1. This is strong evidence that the USA is taking control of the regression forcing the USA residual to be near zero. I have, of course, picked the nuimber 10 to make the case as extreme as possible.
Also the poorest 4 countries lie below the regression line (curve on the graph). There is very strong (I'd guess statistically significant) evidence that the log linear specification forces excessive convexity on the estimated relationship (and vice versa but les so for the linear specification which suggests that US spending is much higher than expected given per capita GDP)