[From Bill Powers (2007.07.29.1422 MDT)]

Actually I'm still in Minneapolis waiting to come home (tomorrow) from the CSG meeting. Rick and I have talked several times about the misclassification problem, if that's the right thing to call it, and are agreed that we want to be sure we get it right before reaching any conclusions.

I'll take the "preventing fatal heart attacks" theme as the basis for constructing a thought-experiment. Here there is no question of getting multiple determinations for each individual (as Martin T. mentioned), since we die only once.

Given: a population with a certain incidence of fatal heart attacks per year, say K per capita per year. In a sample of N individuals, K*N are expected to die of a heart attack each year.

Now suppose there is a treatment that is hypothesized to have either of two effects:

1. It actually reduces the chance of a fatal heart attack by 10% for every individual in the population. That is, if the whole population is given the treatment, the incidence of heart attacks will become 0.9*K*N per year.

2. It actually reduces the incidence of heart attacks by 100% for 10% of the individuals, and has no effect on the incidence of heart attacks in the other 90% of the population. In this case, too, the incidence of heart attacks will become 0.9*K*N per year.

I claim that if the treatment is given to the whole population, these two cases are statistically indistinguishable. All we can say is that K*N of the population will die without the treatment, and that 0.9*K*N will die with it. We can't tell if this equation should be written (0.9*K)*N, 90% of the previous risk for 100% of the population, or K*(0.9*N), 100 percent of the previous risk for for 90% of the population.

If my conjecture is right, it opens the door to a continuum of statistically indistinguishable cases in which a treatment appears to provide a B% reduction of risk to individuals in a population where N% of the members are at risk. Is this the result of a B% reduction in risk for the whole population, or of a 100% reduction in risk for (100 - B)% of the population?

It is also my contention that the most likely case is that a treatment will benefit some proportion of the population for physical/physiological reasons, and will not benefit the rest at all for similar reasons. Problems are not caused by the chances of things happening, but by specific physical effects that stem from regular relationships among variables. The appearance of stochastic effects arises primarily from errors of measurement, from the use of incorrect models, and from lack of knowlege.

Best.

Bill P.

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