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I'm planning a replication study, and am exploring various ways of analyzing the data. In a single study, I will have two outcome measures, which are expressed on completely different scales - one is a 5-point Likert-type, and the other is an amount of money. I have reason to believe that these two measures should be correlated with each other.

Therefore, it seems to me that it would be sensible to, for example, conduct some sort of multivariate regression, and assume that the outcomes come from a multivariate normal distribution.

However, that exact approach seems to me to be plain wrong, since the two outcomes could/should have different, non-normal distributions. Also, I imagine that something like a probit-model would be better suited for the Likert-type outcome, for example.

Do I just fit the two models (e.g. a probit model for the Likert-type, and lognormal for the amount of money) simultaneously, and use each outcome as a predictor of the other one? But I have the feeling that this approach doesn't completely capture the covariance between the outcomes (can't put my finger on the exact problem).

I'm not sure where to even start reading up on this.

Some additional info:

  • the authors of the original study used two t-test for the two outcomes, but since there's reason to believe that the outcomes are correlated, I believe that's not a good strategy
  • planning to do a Bayesian analysis.

I'd appreciate any pointers, and would definitely prefer some that aren't overly mathematical. But I'll take what I can get.

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  • $\begingroup$ The most general approach is to used a mixed distribution copula. Be prepared for complexity though. But the Bayesian machine handles this elegantly once you get past the likelihood specifications. $\endgroup$ Oct 14, 2020 at 12:18
  • $\begingroup$ Thank you, professor Harrell! Would you by any chance know a good resource to get started? $\endgroup$
    – Potato
    Oct 14, 2020 at 12:21
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    $\begingroup$ onlinelibrary.wiley.com/doi/abs/10.1002/sim.7985 $\endgroup$ Oct 14, 2020 at 12:32
  • $\begingroup$ Thank you, I'll check it out :) $\endgroup$
    – Potato
    Oct 14, 2020 at 12:44

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