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When there are more treated than control, is it possible to do 1-to-1 matching (without replacement)?

It makes sense to do 1-to-1 when there are less treated than control but I am not sure if one can do that if there are more treated. It looks like a good way to introduce bias and doesn't make sense to selectively throw out treated.

I guess full matching is a good possibility when there are less control than treated but what if I want to do 1-to-1 in the same case?

The only way I can think of that may be valid is to randomly sample smaller number of treated than control and do 1-to-1, and do this over many times...

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  • $\begingroup$ David, can you explain to me please what's the difference between full matching and 1-to-1 matching? $\endgroup$ – ttnphns Jul 14 '12 at 6:40
  • $\begingroup$ sure, 1 to 1 matching without replacement as name suggests pairs one control to 1 treatment. full matching algorithm matches 1 treated to many control or many treated to 1 control but usually uses the whole sample - of course if you do something called caliper matching, then it throws out those samples who doesn't have close enough (previously specified distance) matches. one can even give partial weights to controls and treated. there are so many variations - you can check out Observational Studies by Paul Rosenbaum :) $\endgroup$ – David Lee Jul 14 '12 at 21:04
  • $\begingroup$ Well, this is not a real solution to your problem, but case-control studies don't have to be matched. Let's say you wanted to match by age and sex: just keep all your subjects and don't forget to include age and sex as covariates in your models (probably logistic regression?). $\endgroup$ – boscovich Jul 14 '12 at 21:21
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I would match by propensity score. For each control find the treatment with tuhgthe highest propensity score to match. Start with the best matches. If there are two or more controls have the same treatment with the same propensity score randomly break the tie. If you got ties then when you are finished there may be controls left that tied initially. Find among the remaining treatments the one with the highest propensity score. Keep doing this until all controls are matched to a treatment. Unfortunately some treatment cases will not be matched but in a sense you the best matches for the controls.

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  • $\begingroup$ i think that way is one way to do 1-to-1 matching when there are more controls than treated. but throwing out treated seems wrong as i mentioned. Doesn't it potentially introduce bias because those treated units not in support may systematically be different. Throwing out controls when there more controls seem acceptable because it's not ignoring any treated and it's just trying to come up with the best matched population sample of controls. But in case of treated>control? i am not sure how to interpret throwing out treated samples. this doesn't seem kosher because depending on control samples $\endgroup$ – David Lee Jul 14 '12 at 5:08
  • $\begingroup$ the treatment effect result from different 1-to-1 matching may even give different signs... $\endgroup$ – David Lee Jul 14 '12 at 5:09
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    $\begingroup$ No the whole point of matching is to reduce the bias that results from lack of randomization. Throwing out treatment cases loses information but so does throwing out controls in the opposite situation. It could be that the set of treatment cases that you wind up with can differ from the original sample. But that is done deliberately to make the treatment and control cases balanced in the characteristic that could confound the treatment effect. Reduced bias and confounding due to matching will probably more than compensate for bias that comes from the lost treatment cases. $\endgroup$ – Michael Chernick Jul 14 '12 at 11:14
  • $\begingroup$ Nothing ever compensates for the lost information from throwing out cases. $\endgroup$ – Michael Chernick Jul 14 '12 at 11:15

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