Then, a hypothetical statistical model that controlled for age when deaths in the cohort were observed (even in a simple linear hazard model framework) would soak up the effect

I don't think so, specifically the effect at each age is in part determined by the integral of the effect at every prior age, since alcohol consumption is time-varying and so are the risks of various *other* causes of death, this general effect is time-varying, and you've got a definitely nonlinear response. The "correct" model in my toy problem is the solution to the differential equation. No linear regression as a function of age for example is going to do the right thing, though it might improve things. If you can specify a family of hazard curves you might be able to fit the hazard by fitting the parameters of the hazard curves, but you'll have a hard time doing a causal model there because it will be entirely confounded with the "accidents of history" such as the price of corn syrup which maybe caused changes in cost of certain foods and maybe changed diabetes risk, and the alcohol drinkers are exposed to more of that risk because they don't die of heart attacks, but it looks like alcohol "causes" diabetes if you just fit hazard curves... Etc

]]>Super long lag in response here ðŸ™‚ I was reminded of this conversation by reading through some of the comments on Gelman's blog vis a vis Pearl's new book. I agree that a better causal model would be optimal. However, I am thinking that, if

"By construction, alcohol doesn't change your cancer risk in this model... So why do more alcohol drinkers die of cancer? The answer is that they live longer on average because they don't die of heart disease as much... So eventually they die of cancer, or "other causes"."

Then, a hypothetical statistical model that controlled for age when deaths in the cohort were observed (even in a simple linear hazard model framework) would soak up the effect, and the estimate of alcohol effect would not be biased by this phenomenon. Shouldn't be too hard to test this idea with some sort of simulation here... ]]>

Basically, age is treated as if it's a pre-treatment predictor variable, but age itself is a post-treatment outcome variable: you live longer/shorter due to the consumption. What's needed to do a good job is a causal model through time.

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