confounds
Consider the following dependence relations:
Z / \ / \ / \ / \ v v X -------> Y
Where
is not effective for men. is 100% effective for women.- Most women take
, and most men do not take
Then, looking at
How should we adjust for this? We need to look at the men who actually took the drug and find their recovery rate. Then we will look at the women who actually took the drug and find their recovery rate. We will weight each rate by the proportion of men and women in the population:
- let's find the rate of recovery per gender (
) and then pretend that each gender had that recovery rate, for all members of the gender.
1. Correction to the above
Upon reading more, I think a better teaching example would by
If it happens that
Importantly, severity causally effects the outcome as well as the medication.
Note the subtle difference between this and the previously given example. For the "severity" example, you might imagine that the effectiveness of the drugs are the same, and keeping the drugs fixed, the outcome will vary linearly with severity. So the "severe" drug-outcome distribution is just a scaled version of the "mild" drug-outcome distribution.
In contrast, in the "gender" example, there is an entirely different distribution of outcomes per drug, dependent on gender.
2. sources
- confounds wikipedia page
3. see also
- conditional dependence – especially "explaining away"
- wikipedia page on interactions – if we don't include an interaction term in our statistical model, we may be ignoring the fact that the main effect is confounded with an interaction effect between two covariates. – TODO come back to this