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Let's assume there is a general linear mixed model with categorical covariates with several levels, so I want to test the main effects. Let's assume I want to test only the fixed part of the model.

I can use the LRT, which will result in comparing nested models. One model will contain the term of interest, and one will not. I don't know how are the degrees of freedom calculated in this case.

I can use the Wald's joint test of model coefficients. I guess this will use the infinite degrees of freedom.

And finally, I learned that I can use the "conditional F tests" with appropriately guessed degrees. In case of small samples, I can use the Kenward-Roger or Satthertwaite correction to them. And that these "conditional F tests" are preferred over the LRT and Wald's tests.

Now my question is:

What does it mean "conditional"? Conditional to what? To the effect next to which it is displayed in the output table?

How they are calculated? By comparing models? But this is done by the LRT. What is the relationship between F, LRT and Wald's?

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