I'm trying to figure out, if the F test in ANOVA is the Wald's test or LRT?
I learned, that the LRT compare nested models and "assess" the reduction in residual variance. This would justify the name "analysis of variance" actually.
Then I learned, that Wald's test approaches the LRT and effectively does the same - compares model, but by testing model coefficients. That's why it doesn't need two models and doesn't compare Likelihoods, so can be used in case like the GEE estimation, where there's no likelihood, but we can still use Wald's to jointly test model coefficients to obtain the main effects.
In case of the GEE "analysis of variance" makes no sense, as there is no assessment in reduction of variance.
But in the classic general linear model, a likelihood based model, there is.
But... when I look at the formula of F I can see a ratio of two variances, one for empty model and one with certain effect.
And now I'm totally confused. Is F in ANOVA a LRT test or Wald's test? And if Wald's, which doesn't assess the reduction of variance, but rather tests coefficients of the mode, why is the method names analysis of variance?
Could someone explain to me how the two methods are "equivalent" in the classic ANOVA?
Clearly Wald's is more general if we can use it with non-likelihood methods. Then how does it relate to "reduction in variance" or "deviance" (like in GLM) if there is no deviance in GEE, for instance?
Does it mean that F and LRT (Chi2) are equivalent only in the generalized linear model, so BOTH test the reduction of variance in this case, but only here?
How is testing model coefficients equivalent to testing the reduction in residual variance?
EDIT: OK, I found answers to all my questions. F can be BOTH Wald's or LRT, similarly to Chi2 - it's just a matter of approach to the degrees of freedom, coming from whether we estimate the denominator's variance or not. In the ANOVA, which bases on the general linear model, it's the LRT. And the LRT in case of general linear model is EQUIVALENT to testint appropriate coefficients via Wald's approach.
When the likelihood cannot be used, we leave with the Wald's approach as still valid, but then it's not "analysis of variance", only analysis of main effects (equivalent to ANOVA in general linear models).
