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I want to estimate the average effect of a treatment that was given with a selection bias. To do this, I'd like to use a matching method. Basically, this method involves finding, for each treated person, their equivalent untreated person, and then measuring the difference in outcomes between the two. This should help correct the selection bias.

However, I'm in a special case where the outcome for untreated individuals is constant, specifically 0, because the treatment is the only way to get a result. For example, if I want to measure the hair regrowth rate after a hair transplant, the treatment is the transplant, and the outcome is the regrowth rate. It turns out that my study population has experienced selection bias (because those who can afford the transplant likely have better living conditions, etc.). On the other hand, my control population hasn't had a transplant, so their regrowth rate is 0 by default.

My question is, does the matching method make sense in a case where the outcome variance is 0 in the control population? Intuitively, this bothers me, but I can't prove it statistically. If someone could provide an explanation and recommend specific methods to handle this specific case (when control outcome is constant), I'd appreciate it. I've searched the literature, but haven't found anything yet.

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The point of matching is to estimate the potential outcome under control for each treated unit. But you already know the potential outcome under control for each treated unit; it is 0. So you don't need to match at all and you don't need a control group at all. You can simply do a 1-sample t-test with your treated group comparing their mean outcome to a null value of 0.

This also invites the question of why you collected data on a control group in the first place. If you knew their outcomes would be exactly equal to 0 by definition, why did you bother collecting data on them? There is a difference between all control units being forced to have an outcome of 0 and all control units happening to have an outcome of 0; in the latter case, you can just use a regular outcome analysis (although one that takes into account excess 0s) after matching. But if the control group must have outcomes of 0 by definition, then there was no point in measuring their outcomes.

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  • $\begingroup$ We know for sure that the treatment has a significant effect - the question I am trying to answer is "how significant is it, considering that there's a strong selection bias in my population?". Keep imagine a hair transplant scenario; the aim is to gauge the regrowth rate, knowing that the treated patients are not representative of the total population. There is a sharp disparity between those who proceed and the whole population suffering from alopecia. A stratification method is not feasible due to many variables and insufficient data... $\endgroup$
    – HnbBarca
    Feb 17 at 18:10
  • $\begingroup$ ...Hence, the idea of using matching to measure the ATE and adjust for this selection bias, between those undergoing the procedure (treated) and the rest of the population with alopecia (control - we have all the national data). I've seen multiple institutes employing matching methods for similar tasks, but tbh, I'm uncertain regarding the logic behind it. That's why I am trying to understand the how matching could help in this kind of situation, and if you think that there is any other method that may be more adapted for this situation, I am happy to hear ! $\endgroup$
    – HnbBarca
    Feb 17 at 18:14
  • $\begingroup$ Ah, it sounds like you want to re-weight your treated sample to resemble the full population to simulate what would have happened had the full population received the procedure. In that case, you shouldn't use matching, which is used to impute the potential outcomes under control for the treated units. Instead, you want to impute the potential outcomes under treatment for the whole population. Use inverse probability weighting instead. $\endgroup$
    – Noah
    Feb 17 at 21:47

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