Difference between LMEs and GLMMs?

[FallenApple]

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)

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?

 

[andrewkirk]

My understanding was that GLMMs are like GLMs except that they contain random as well as fixed effects. If so then, strictly speaking, that last one would not be a GLMM because it contains no fixed effects. But I suspect that practical use may be that GLMMs are understood to include all models that, via a link transformation, are linear combinations of random and/or fixed effects, in which case we would include the second example as a somewhat-degenerate GLMM.

I would have thought that, in the above, the first model can be expressed as an instance of the second by mapping $(\beta_0+b_{0,i})+\epsilon_{i,j}$ and $(\beta_1+b_{1,i})$ in the first model to $\beta_0$ and $\beta_1$ in the second.

Thus the remaining difference in structure between the two is that the second has a potentially non-linear link function, making the first a special case of the second, with identity link function.

Although I am very far from expert on this, I did a bit of modelling last year using function glmer from R package lme4, which implements GLMMs. The function is general enough that it can have all random effects, all mixed effects or a mixture of both, and can have a range of link functions including identity (giving a linear model).

If you want input from the real experts, a good place to get answers to statistical practice questions like this is Cross-Validated. This is distinct from Stack Overflow, where one asks questions about how to make specific programs like SAS, Stata or R do certain things.