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I believe my question is similar with this one:

Likelihood-ratio test for three models?

But I do still not understand and my problem is slightly different (maybe). I want to check which model is best, by performing a likelihood ratio test with three variables. For example:

Let say model M1, M2, M3, M4, M5, M6, M7 correspond to variable A, B, C, A+B, A+C, B+C, A+B+C, respectively. First, we select the best fit model using AIC. And after that, we perform a likelihood ratio test to check if the selected model improving the simple one or not.

When we get the M4 (or M5 or M6) as the best fit model, we need to test it with M1(A) and M2(B) to check if M4(A+B) improving M1 or M2. When M4 improve M1 and M2, we select M4 as the best fit model. When M4 improve M1 but not M2, we select M2 as the best fit model and vice versa.

But I do not know if the best fit model is M7(A+B+C). How to test M7 model?

Thank you

ani jaya
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    If you select which models to test before seeing how well they perform, using an LR test is fine. However, if you select which models to test based on AIC, then I think the LR test statistic will not have its usual distribution under the null. You will suffer from a pretest bias and will have some trouble with post-selection inference. – Richard Hardy Oct 18 '21 at 12:52
  • Thank you very much for your comment Prof. Hardy. But from the references that I read, the step to select the best fit model is first select the model with lowest AIC and second is to test that model to the simple model with LRT. For example: M0: y~x M1: y~x+x1 M2: y~x+x2 M3: y~x+x1+x2 If M0 (basic model, null hypothesis) show lowest AIC, we choose M0 as our model. If M1 or M2 show lowest AIC, we test M1 or M2 prior to M0 using LRT. If M3, we test M3 to M1 and M2 prior to M0. And I completely lost when we have model M4: y~x+x1+x2+x3 show the lowest AIC. – ani jaya Oct 19 '21 at 01:07
  • I find your references suspect. – Richard Hardy Oct 19 '21 at 05:17

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