0<P(A=a|L)<1 for all a in A and L
If you want to compare two types of treatment, then you have to have people in your data who are able to & sometimes will receive all relevant treatment options!
You can’t compare apples to oranges, if you’ve only ever seen apples!
There are two types of violations which can happen:
Imagine the exposure in the scenario above was “is on a 2019 50-under-50 list”. The 52 year olds *cant* be exposed: there is structural non-positivity.
The solution is to exclude them from our data & inference entirely!
Eg to learn about 50-under-50 lists, only enroll people under 50!
In an #RCT, randomization performs 2 important functions. The first, and most commonly discussed, is that it removes confounding. The second is that it ensures positivity: everyone has a chance of being assigned either treatment.
But we *don’t* have positivity if we assign everyone to treatment (ie “single arm” trials aren’t really trials, don’t @ me!)
When we have sustained treatments, like medication use, we can get post-randomization positivity violations!!
People with contraindications are excluded before randomization and can’t be in either the treatment or control arm.
In the intention-to-treat analysis, that’s fine, because ITT is the effect of *assignment*
But if we want to estimate the statins effect, we need to build in rules for how to handle these people.
And so, we couldn’t estimate a per-protocol effect for “continuous statins” vs “no statins ever” even if we can control for confounding.
We probably *do* have positivity for that!
Unlike an RCT, in an obs study we don’t necessarily have positivity at baseline, and we need to worry about positivity over follow-up.
(1) when groups can’t be or will always be exposed at baseline, we should exclude them from our study & target pops.
(2) when people enter these groups over follow-up, we should excuse them or specify a rule in our exposure definition.
Anyone who would always or never get the exposure should not be included in our study or our target pop.