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I would greatly appreciate some advise on a statistical problem that haunts me.

Suppose you wish to estimate the effect of $x$ on $y$, but the probability to observe $\{y_i, x_i\}$ also depends on the effect of $x$. Say, if $x_i > z_i$, then you don't observe $\{y_i, x_i\}$. If $x_i < z_i$, then you do.

If this was a standard textbook example of sample selection models, we would observe $z$, and could adjust, but in my case I don't. I have no information on how many individuals did not make it into my sample. To be concrete: I don't know they ever existed. That is, they died before the sample was collected, and thus do not appear in my sample. My sample consists of survivors, so to say.

Following the comments below, I augment my question to make it a bit more precise. Suppose a group of $m$ people was exposed to a disaster, say. 20 years later I want to see how much this experience shines through today, but those that were most severely affected died along the way. I nevertheless collect data on all remaining $n$ survivors. Here, $n < m$. Unfortunately, I have no information whatsoever about the exposed population and I do not observe their characteristics $\{x_i, y_i\}$. I do not know $m$, and I do not know how many of the initial population survived. All I have is my group of survivors.

If I want to estimate the effect of the disaster on some outcome, that estimate will be an biased because the whole sample is conditioned on survival. Does anyone know any solution to this problem? Are there any methods out there to, say, compute bounds maybe?

Any hints are highly appreciated.

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  • $\begingroup$ Not sure I've understood right, but regression models for the response $y$ are conditional on observed values of the predictor $x$, so there's not a problem except of, perhaps, extrapolation. $\endgroup$ Mar 18 '14 at 0:16
  • $\begingroup$ The problem is not observing x, but observing y (or strictly speaking the observational unit). Say you want to see if some treatment x has negative effects on your health, but you only observe the entire sample 20 years after treatment has occured. Hence, those people where treatment had a particularly bad impact already died. The whole sample is thus conditional on survival, i.e. conditional on the negative impact of x being not as strong as to kill you. This is a sample selection problem I am not sure how to address. $\endgroup$ Mar 18 '14 at 10:00
  • $\begingroup$ You've got, say, people who eat 1 hamburger a month with normal blood cholesterol, people who eat 5 hamburgers a day with high blood cholesterol; but no info. on all the people who ate 5 a day & dropped dead of heart attacks when their cholesterol levels went through the roof: & you're worried that you're therefore under-estimating the effect of hamburger-eating on cholesterol? If that's the problem, it's that the probability of inclusion in the sample depends on $y$ (if it depended only on $x$ there'd be no problem). If you agree amend the question. $\endgroup$ Mar 18 '14 at 15:52
  • $\begingroup$ Yes, you are right. I have augmented my question to make it somewhat more precise. $\endgroup$ Mar 18 '14 at 17:07
  • $\begingroup$ That's better, although the 2nd sentence of the 2nd paragraph still seems to talk about selection based on $x$. If I randomly shoot all of the five-hamburger-a-day people & half of the four-hamburger-a-day people, I can still regress blood cholesterol on hamburger consumption without fear of bias. But if the fatter ones are slower, bigger targets, or whatever, I may inadvertently & indirectly be selecting on cholesterol level, which is a concern. $\endgroup$ Mar 18 '14 at 17:32
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You could try some form of truncated regression if you know threshold point which truncates observations.

Here is link for implementation in SAS:

http://www.ats.ucla.edu/stat/sas/dae/truncreg.htm

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