So I know that linear mixed models model has coefficients that are fixed and random. From what I understand, the fixed coefficients are still good since the random slopes/intercepts capture between subject heterogeneity. Which I suppose help estimate the crossectional effects better( i.e the fixed Betas)(adsbygoogle = window.adsbygoogle || []).push({});

Here is an example of LME

##Y_{i,j} \sim (\beta_{0}+b_{0,i} )+(\beta_{1}+b_{1,i})X_{i,j}+\epsilon_{i,j}##

Where ##(b_{0,i},b_{0,i})\underset{i,i,d}\sim N((0,0),G)##

##\epsilon_{i,j}\underset{i,i,d}\sim N(0,\sigma_{\epsilon})##

So I see here that the "coefficient"s consists of a random part and a fixed part

But I heard that LMEs are just special cases of Generalized Linear Mixed Effects Models(GLMMs). But isn't that a contradiction? GLMMs estimates are only for within subjects. Whereas for LMEs, I have population based estimates and within subject based ones.

Here is an example of GLMM

##link(\mu_{i,j}) \sim \beta_{0} +\beta_{1}X_{i,j}##

Where here appearantly, the betas are random.

So how can the lme be a subcase of glmm if there is no fixed component about which the individual can vary?

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# A Difference between LMEs and GLMMs?

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