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I have the dataset for the health of patients along with various treatments they were given.

In a normal case, I would just use linear regression to fit a model [y ~ t1 + t2 + t3 ... +tn]. This will give me the correlation of each treatment t with the health of patient y.

What I want to do is to identify the causal effect instead of just correlation of each treatment.

Upon looking on the internet and doing some readings, I have understood that I need to match the data before feeding it to a linear model (propensity score matching etc.)

There are methods out there which are used to match data , but only work when there is 1 treatment variable. i.e y~ t1. There are other methods aswell for multiple treatments but assume that only 1 treatment is being applied at one time.

In my case several of the treatments are being applied at the same time and hence the above mentioned methods do not work for my study.

Can anyone explain to me which method I should go about using to match the data if I have multiple treatments and several of them being applied at the same time.

Thanks in advance.

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    $\begingroup$ Just to make sure: were these treatments non-randomly assigned? Did units select into receiving their treatments, or otherwise influence their own treatment status? If not, you're fine -- OLS yields an unbiased causal effect. $\endgroup$ – generic_user Apr 29 '16 at 16:03
  • $\begingroup$ Yes the treatments were non-randomly assigned. I have observational data from which I want to make causal inference. $\endgroup$ – show_stopper Apr 29 '16 at 16:56
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    $\begingroup$ Matching no more deals with causal effects than regression does. The only difference is that sometimes with propensity scores you can handle a larger number of adjustment variables. $\endgroup$ – Frank Harrell Apr 29 '16 at 18:32
  • $\begingroup$ Thanks for the input Frank. What do you think on how should I go about making causal inference from the observational data with multiple treatments and several of them being applied at the same time. $\endgroup$ – show_stopper May 5 '16 at 18:02
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One straightforward but laborious way would be to form groups based on patterns of treatment, then perform matching or regression on the patterns. For example, one pattern might be "T1, T2, but not T3," and another pattern might be "just T2." In realty, the treatment pattern is the true cause you want to examine, because in a sense there is no "average treatment effect" of any one individual treatment since multiple can be applied. Each treatment pattern can be considered its own treatment with its own treatment effect.

For matching, if after matching across the treatment pattern groups the covariates are balanced, you can do a simple ANOVA with contrasts to compare treatment pattern conditions. The twang package in R allows for "multinomial" propensity score matching and balance assessment. If you can't find a large set of individuals across all treatment pattern groups that have the same covariates, you might be able to do pairwise comparisons between groups, but your causal estimand will shift from the ATE to the ATT.

I do think a simple regression would work fine in your case as long as you are not making any extreme counterfactuals. Perhaps read the work of McCaffrey on this matter.

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