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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.

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  • \begingroup (+1) For a well-posed and interesting question. (Did you mean to call X_i a dependent or independent variable?) \endgroup
    – cardinal
    Commented Feb 14, 2012 at 19:29

1 Answer 1

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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.

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  • \begingroup X_i is not the treatment variable, it is thought to be modify the treatment effect. The treatment takes place between observation of Y_{i1} and Y_{i2}. How does that change your answer? \endgroup
    – Macro
    Commented Feb 14, 2012 at 19:59
  • \begingroup I guess the biggest question here, in terms of model specification, is whether X_i enters the equation for Y_{i2} only, or both Y_{i1} and Y_{i2}. The former is of course easier to handle, and it looks like this is the case if X_i is somehow associated with the treatment. \endgroup
    – StasK
    Commented Feb 15, 2012 at 5:04
  • \begingroup I think in the application it may be reasonable to say X_i only affects Y_{i2} which, of course, depends on Y_{i1}. I like the idea of thinking of it in terms of a latent variable data generating mechanism but I'm still not sure exactly how you specify this model. Also, are the \min and \max reversed in your first equation? Otherwise, I don't get why you specify it that way. \endgroup
    – Macro
    Commented Feb 15, 2012 at 5:24
  • \begingroup Right, I mixed the min and max up. Check out Roodman's paper, it explains the concept and implementation details. \endgroup
    – StasK
    Commented Feb 15, 2012 at 15:39

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