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i am trying to compare two regression models with different settings:

Model 1: Y ~ X + M + C, and

Model 2: M ~ X + Y + C

The purpose is to check which model is better. I think likelihood based methods will not work as the outcome variables are not the same, am I right? Then which kind of metrics can be used to compare such two models?

All advice and comments are appreciated! Thanks!

wenjia xu
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    There isn't going to be a way to compare those two. You've just switched whether Y or M is on the left hand side of the regression equation. This amounts to determining whether you want to minimize vertical or horizontal deviations to fit the regression line (cf here). If you are wondering whether Y causes M, or M causes Y, you need to run some experiments. – gung - Reinstate Monica Jul 06 '21 at 19:41
  • Thanks for your comment @gung-ReinstateMonica. Yes, you are right that I want to test if Y causes M or M causes Y. but the data I have is observational data, could you let me know what kind of experiments I could do? Or could you perhaps share some experience on how to test this given observational data? thanks! – wenjia xu Jul 07 '21 at 08:00
  • Recruit a new set of units (patients, consumers, mice, firms, whatever), & randomly assign them to groups. Independently manipulate levels of M between the groups, & see if Y differs by group. Likewise for manipulating Y & seeing if M differs. – gung - Reinstate Monica Jul 07 '21 at 11:29
  • Thanks for your reply. Unfortunately the survey has finished and there is no way to get new data or manipulate levels of M or Y as it is observational social survey. Is there any empirical way to compare two models? – wenjia xu Jul 07 '21 at 12:11
  • No, that was the point of my first comment. – gung - Reinstate Monica Jul 07 '21 at 13:12

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