(Related to Over-ensapsulation and Subtext is not invariant under linear transformation)

Between 2004 and 2007, Goran Bjelakovic et al. published 3 famous meta-analysis of vitamin supplements, concluding that vitamins don't help people but instead kill people. This is now the accepted dogma; and if you ask your doctor about vitamins, she's likely to tell you not to take them, based on reading either one of these articles, or one of the many summaries of these articles made in secondary sources like The Mayo Clinic Journal.

The 2007 study claims that beta-carotene and vitamins A and E are positively correlated with death - the more you take, the more likely you are to die. Therefore, vitamins kill. The conclusion on E requires a little explanation, but the data on beta-carotene and A is simple and specific:

Univariate meta-regression analyses revealed significant influencesof dose of beta carotene (Relative Risk (RR), 1.004; 95% CI, 1.001-1.007; P = .012),dose of vitamin A (RR, 1.000006; 95% CI, 1.000002-1.000009;P = .003), ... on mortality.

This appears to mean that, for each mg of beta carotene that you take, your risk of death increases by a factor (RR) of 1.004; for each IU of vitamin A that you take, by a factor of 1.000006. "95% CI, 1.001-1.007" means that the standard deviation of the sample indicates a 95% probability that the true RR lies somewhere between 1.001 and 1.007. "P = .012" means that there's only a 1.2% chance that you would be so unlucky as to get a sample giving that result, if in fact the true RR were 1.

A risk factor of 1.000006 doesn't sound like much; but I'm taking 2,500 IU of vitamin A per day. That gives a 1.5% increase in my chance of death! (Per 3.3 years.) And look at those P-values: .012, .003!

So why do I still take vitamins?

What all of these articles do, in excruciating detail with regard to sample selection (though not so much with regard to the math), is to run a linear regression on a lot of data from studies of patients taking vitamins. A linear regression takes a set of data where each datapoint looks like this:

Y = a₁X₁ + c

and a multiple linear regression takes a set of data where each datapoint usually looks like this:

Y = a₁X₁ + a₂X₂ + ... a_nX_n + c

where Y and all the X_i's are known. In this case, Y is a ...

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