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Regression when your dependent variable is the difference between two doubly censored variables

Suppose you have a sample of n outcomes and for subject i you observe Yi1 and Yi2, which are 'before' and 'after' measurements on subject i. You have a single continuous independent variable, Xi, observed for subject i and would like to explain the before/after difference Di=Yi2Yi1 in terms of Xi in something like a simple linear regression:

Di=β0+β1Xi+εi

The problem is that both Yi1 and Yi2 are left and right censored. In particular, both are such that if Yi1a, then Yi1 is recorded as a. Similarly, if Yi1b, then Yi1 is recorded as b. The same is true (with the same a,b) for Yi2.

So far I've considered imputing the censored values of Yi1,Yi2 by making a parametric assumption and replacing the censored values with E(Yi1|Yi1a) and E(Yi1|Yi1b) (and similarly for Yi2) but this doesn't really help because you're just replacing one constant with another. I've also considered making a parametric assumption and simulating data, conditional on being outside of (a,b), to impute the data. I'm not crazy about either of these ideas because neither takes into account the fact that the mean of Yi1,Yi2 potentially depends on Xi.

If I can't find a good reference I'm probably going to just end up binning the data into ordinal categories and simply modeling the probabilities of switching categories after treatment (although I'm not crazy about this idea either).

Is there some relatively simple way for me to get estimates of β0,β1? Also, any references on this or a related problem would be appreciated.

Macro
  • 45.8k
  • 12
  • 158
  • 158
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