I am trying to figure our the most robust way to combine two different sets and run a regression. The first dataset gives me an outcome value for each of several categorical treatment variables, each measured on a single person:

Treatment | Value

    A        1.2
    B        0.9
    A        1.3
    B        2.2
    C        1.7
    D        3.2
    A        0.2
    .         .
    .         .
    .         .

The other data set, taken on a completely different group of people, has the same treatment variable, and then several continuous covariates:

Treatment |   X1   |   X2  |   X3

    D        17.1    -2.1     6.1
    C         3.2     1.7     1.1
    C        -7.3     6.2     1.2
    A        -2.8    -3.9     7.3
    B         4.1    -1.0     6.4
    A        12.2     1.1     2.1
    D         3.1    -2.7     4.5
    .         .        .       .
    .         .        .       . 
    .         .        .       .

what I would like to do is fit a regression of the form:

value ~ X1 + X2 + X3

Since these are taken on different groups of people, I can't just merge them together directly. The easiest way to analyze this is to use the first dataset to get a mean (or median) value for each treatment, and then impute this as the outcome in the second dataset. I'd then run a regression of the form

mean_value ~ X1 + X2 + X3

This certainly works, but it ignores a lot of the variation in the data and should underestimate the uncertainty. It also will likely lead to some odd residuals since all treatments will have the same outcome value.

My other though is to do a bootstrap regression, where I randomly sample the treatment values from Dataset 1, and assign them to Dataset 2, matched by Treatment. I then do run a regression on the merged data, and I repeat for 1000 or so boostrap samples, each time resampling from Dataset 1. Intuitively I like this approach a bit better, but it's not really a standard bootstrap regression because I'm not randomly sampling the X's as well, I'm only sampling the outcome.

My last thought is to use a Bayesian approach and use Dataset 1 to define a prior on the outcome Value (mean, sd for each Treatment), but this is likewise a bit nonstandard, as it would simultaneously estimate the most probable outcome value along with the regression coefficients.

Are there any straightforward and well established ways to analyze this sort of data? Or are there some improvements or alternatives to the above approaches?


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