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

Question

Suppose you have a sample of $n$ outcomes and for subject $i$ you observe $Y_{i1}$ and $Y_{i2}$, which are 'before' and 'after' measurements on subject $i$. You have a single continuous independent variable, $X_i$, observed for subject $i$ and would like to explain the before/after difference $D_{i} = Y_{i2}-Y_{i1}$ in terms of $X_i$ in something like a simple linear regression:

$$ D_i = \beta_{0} + \beta_{1} X_i + \varepsilon_i$$

The problem is that both $Y_{i1}$ and $Y_{i2}$ are left and right censored. In particular, both are such that if $Y_{i1} \leq a$, then $Y_{i1}$ is recorded as $a$. Similarly, if $Y_{i1} \geq b$, then $Y_{i1}$ is recorded as $b$. The same is true (with the same $a,b$) for $Y_{i2}$.

So far I've considered imputing the censored values of $Y_{i1},Y_{i2}$ by making a parametric assumption and replacing the censored values with $E(Y_{i1}|Y_{i1} \leq a)$ and $E(Y_{i1}|Y_{i1} \geq b)$ (and similarly for $Y_{i2}$) 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 $Y_{i1},Y_{i2}$ potentially depends on $X_i$.

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 $\beta_{0}, \beta_{1}$? Also, any references on this or a related problem would be appreciated.

Answer 1

I would build a bivariate response model simultaneously for $Y_{i1}$ and $Y_{i2}$, introducing an underlying propensity $Y^*_{i1}$, $Y^*_{i2}$ such that $$Y_{i1}=\min(a,Y^*_{i1}), Y_{i2} = \max(b,Y^*_{i2}).$$ If $X_i$ is a treatment variable, then a version of your model is $$Y^*_{i2} = Y^*_{i1} + \beta_0 + \beta_1 X_i + \epsilon_i.$$

I think you use Stata, so you would want to look at cmp package that fits models of this kind. There might be some caveats, as Roodman talks about "fully observed" models there, in the sense that nothing depends on the latent variables $Y^*_1, Y^*_2$. You might have to change somewhat the model to make sure it is estimable.