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Suppose that I have two models, A and B (A nested in B), which are tailored to explain data from a single participant in an experiment. Example: I am modeling response times in a single participant. However, I have 20 participants.

I can fit A and B using MLE to each participant independently, and then obtain likelihood ratios for each participant and test the significance with a chi-square test. But how can I make inferences over the population?

I read in one (rather obscure and not totally trusted) source that I can simply sum the $\chi^2$ values for each participant. The resulting value will be $\chi^2$ with df=20 in this case.

  1. Is this a valid procedure?
  2. Can you recommend any sources that go into detail about this sort of method of aggregating model fits?

Bonus: What if my models aren't nested, but all of the above is still true?

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    $\begingroup$ "how can I make inferences over the population?" - if I follow what you're saying, it sounds like you want mixed effects models, where participant has a random effect. $\endgroup$
    – Glen_b
    Commented Jun 13, 2013 at 0:53

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I agree with @Glen_b that you seem to be looking for mixed models. You can use a variant of Akaike's Information Criterion for model selection here. Bonus: AIC works for non-nested models. Here is a reference.

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  • $\begingroup$ I see, so I should assume individual-level parameters are drawn from overarching distributions; the fitting step is now to estimate the parameters of those distributions. Then I'd compare based on AIC... $\endgroup$
    – Questioneering
    Commented Jun 13, 2013 at 20:04

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