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Is the (deterministic) monotone assumption for an instrumental variable in conflict with the positivity assumption for potential outcomes?

Z is the instrument, X is the treatment, Y is the outcome and all 3 are binary

Assuming Z meets all conditions to be an instrumental variable, except for the heterogeniety condition, then the (deterministic) monotone condition says that we can divide the population into 4 types

  1. compliers (always have X=1 when Z=1 and always have X=0 when Z=0)
  2. defiers (always have X=0 when Z=1 and always have X=1 when Z=0)
  3. always takers (always have X=1 regardless of the value of Z)
  4. never takers (always have X=0 regardless of the value of Z) and either there are no compliers OR there are no defiers

But if we want to use the potential outcomes framework aren't the always takers and never takers already excluded from the population by the positivity restriction? Since for every subject, the probability that X=1 can not be 0 or 1

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  • $\begingroup$ the positiity assumption is across the population, not "for every subject" $\endgroup$
    – Ben
    Jan 4, 2023 at 21:12
  • $\begingroup$ I agree if that was the way positivity was consistently defined, but it is not. From a Rubin paper (for example) "Probability is 'probabolistic' if each unit has a positive probability of receiving either treatment". And "each unit" sure sounds to me like it is defined at the subject level. $\endgroup$ Jan 4, 2023 at 21:23
  • $\begingroup$ i don't think that's right - it's not clear what the probability is over when looking at a single unit. the unit either receives the treatment or not. If possible, please provide a reference that states the assumption mathematically (rather than in text). $\endgroup$
    – Ben
    Jan 5, 2023 at 0:51
  • $\begingroup$ This is a problem with the PO framework. Exactly how the probability is defined is unclear. Define B (baseline covariates), Y(0), Y(1) potential outcomes and these are vectors (for entire population or sample, not really clearly defined), then papers frequently define as 0<P(X_i|B,Y(0),Y(1))<1. The problem is whether or not this is a probability conditional on subject i or not is not clearly defined. And it matters. And if one is attempting to glean which definition, the language in the papers "unit-level probability" implies the former". $\endgroup$ Jan 5, 2023 at 20:08
  • $\begingroup$ Also, yes after the study is conducted each subject either receives treatment or not, so conditional on the study the only way X_i can be a random variable is unconditional on subject i. But the PO framework starts with a population where treatment assignment is random. Leads to even more confusion as there are a few different ways to define X_i as a random variable. $\endgroup$ Jan 5, 2023 at 20:12

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