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I have two sets of variables in the same dataset. Say DATA_free and DATA_exp. DATA_exp, however, consists of variables which are very expensive/difficult to obtain whereas DATA_free are always available easily.

I was wondering if it was possible to use Canonical Correlation Analysis between these two sets of variables so that I can find a linear combination of variables in DATA_free that best explains what DATA_exp does, in hopes that, if the correlation between the two linear combinations is high enough I could stop relying on DATA_exp to compute my results.

Is this reasonable? Any advice/example you could give me? Is this theoretically correct?

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A linear combination of variables in DATA_free can explain, at best, as much as the matching linear combination of variables in DATA_exp. That is, if the first canonical correlation is 1, you would obtain as much information from the first canonical variable of DATA_free as you would from the first canonical variable of DATA_exp.

However: 1) The largest canonical correlation will rarely be 1 (and all the others will be smaller), and 2) If DATA_free is of dimension $p$ and DATA_exp is of dimension $q$ with $p < q$, the very best you can hope for is to replicate a $p$-dimensional projection of DATA_exp.

So what you are able to do is dependent on the size of the canonical correlations and the relative dimensions of DATA_free and DATA_exp.

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  • $\begingroup$ I agree, however, can we agree that replacing DATA_exp with the first canonical variable of DATA_free is better than deleting DATA_exp completely? DATA_exp is indeed very difficult to obtain (in the future) so I'm looking for ways to go around it with the information at hand. Can you suggest another multivariate technique for this issue? $\endgroup$
    – user010203
    Jan 3, 2018 at 18:13
  • $\begingroup$ Well, in general "something" is better than "nothing", although a one-dimensional reconstruction of DATA_exp might badly distort it so as to render it useless. Why limit yourself to one canonical variate? You might also consider multivariate regression (i.e., multivariate response regression, not scalar response multivariable regression). $\endgroup$
    – F. Tusell
    Jan 3, 2018 at 18:56
  • $\begingroup$ If I decided to use a multivariate response regression model (which I'm doing as I type this), what would be the difference in the interpretation between its results and the ones I get from separate univariate regression models for each one of the variables in DATA_exp? (perhaps I should select a few variables before I do it, DATA_exp consists of 258 variables and it's becoming computationally costly to run it) $\endgroup$
    – user010203
    Jan 3, 2018 at 21:15
  • $\begingroup$ 258!!! Then, go ahead with scalar regressions. Considering vector responses might get you some advantage in terms of efficiency (you benefit from considering cross correlations in the error terms if there are some). But I would think this is not worthwile with your problem dimension. $\endgroup$
    – F. Tusell
    Jan 3, 2018 at 21:35
  • $\begingroup$ Better to use scalar regression or perhaps non- or partially parametric alternatives, of which trees are probably the easiest and most promising --have a look at rpart if you are using R. $\endgroup$
    – F. Tusell
    Jan 3, 2018 at 21:36

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