What are R-structure G-structure in a glmm?

Question

I've been using the MCMCglmm package recently. I am confused by what is referred to in the documentation as R-structure and G-structure. These seem to relate to the random effects - in particular specifying the parameters for the prior distribution on them, but the discussion in the documentation seems to assume that the reader knows what these terms are. For example:

optional list of prior specifications having 3 possible elements: R (R-structure) G (G-structure) and B (fixed effects)............ The priors for the variance structures (R and G) are lists with the expected (co)variances (V) and degree of belief parameter (nu) for the inverse-Wishart

...taken from from here.

EDIT: Please note that I have re-written the rest of the question following the comments from Stephane.

Can anyone shed light on what R-structure and G-structure are, in the context of a simple variance components model where the linear predictor is $$\beta_0 + e_{0ij} + u_{0j} $$ with $e_{0ij} \sim N(0,\sigma_{0e}^2)$ and $u_{0j} \sim N(0,\sigma_{0u}^2)$

I made the following example with some data that comes with MCMCglmm

> require(MCMCglmm)
> require(lme4)
> data(PlodiaRB)
> prior1 = list(R = list(V = 1, fix=1), G = list(G1 = list(V = 1, nu = 0.002)))
> m1 <- MCMCglmm(Pupated ~1, random = ~FSfamily, family = "categorical", 
+ data = PlodiaRB, prior = prior1, verbose = FALSE)
> summary(m1)


 G-structure:  ~FSfamily

         post.mean l-95% CI u-95% CI eff.samp
FSfamily    0.8529   0.2951    1.455      160

 R-structure:  ~units

      post.mean l-95% CI u-95% CI eff.samp
units         1        1        1        0

 Location effects: Pupated ~ 1 

            post.mean l-95% CI u-95% CI eff.samp  pMCMC    
(Intercept)   -1.1630  -1.4558  -0.8119    463.1 <0.001 ***
---

> prior2 = list(R = list(V = 1, nu = 0), G = list(G1 = list(V = 1, nu = 0.002)))
> m2 <- MCMCglmm(Pupated ~1, random = ~FSfamily, family = "categorical", 
+ data = PlodiaRB, prior = prior2, verbose = FALSE)
> summary(m2)


 G-structure:  ~FSfamily

         post.mean l-95% CI u-95% CI eff.samp
FSfamily    0.8325   0.3101    1.438    79.25

 R-structure:  ~units

      post.mean l-95% CI u-95% CI eff.samp
units    0.7212  0.04808    2.427    3.125

 Location effects: Pupated ~ 1 

            post.mean l-95% CI u-95% CI eff.samp  pMCMC    
(Intercept)   -1.1042  -1.5191  -0.7078    20.99 <0.001 ***
---

> m2 <- glmer(Pupated ~ 1+ (1|FSfamily), family="binomial",data=PlodiaRB)
> summary(m2)
Generalized linear mixed model fit by the Laplace approximation 
Formula: Pupated ~ 1 + (1 | FSfamily) 
   Data: PlodiaRB 
  AIC  BIC logLik deviance
 1020 1029   -508     1016
Random effects:
 Groups   Name        Variance Std.Dev.
 FSfamily (Intercept) 0.56023  0.74849 
Number of obs: 874, groups: FSfamily, 49

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -0.9861     0.1344  -7.336  2.2e-13 ***

So based on the comments from Stephane I think the G structure is for $\sigma_{0u}^2$. But the comments also say that the R structure is for $\sigma_{0e}^2$ yet this does not seem to appear in the lme4 output.

Note that the results from lme4/glmer() are consistent with both examples from MCMC MCMCglmm.

So, is the R structure for $\sigma_{0e}^2$ and why doesn't this appear in the output for lme4/glmer() ?

Answer 1

I would prefer to post my comments below as a comment but this would not be enough. These are questions rather than an answer (simlarly to @gung I don't feel strong enough on the topic).

I am under the impression that MCMCglmm does not implement a "true" Bayesian glmm. The true Bayesian model is described in section 2 of this paper. Similarly to the frequentist model, one has $g(E(y \mid u)) = X\beta + Zu$ and there is a prior required on the dispersion parameter $\phi_1$ in addition to the fixed parameters $\beta$ and the "G" variance of the random effect $u$.

But according to this MCMCglmm vignette, the model implemented in MCMCglmm is given by $g(E(y \mid u,e)) = X\beta + Zu + e$ , and it does not involve the dispersion parameter $\phi_1$. It is not similar to the classical frequentist model.

Therefore I would be not surprised that there is no analogue of $\sigma_e$ with glmer.

Please apologize for these rough comments, I just took a quick look about that.

Answer 2

I am late to the game, but a few notes. The $\mathbf{R}$ structure is the residual structure. In your case, the "structure" only has a single element (but this need not be the case). For Gaussian response variable, the residual variance, $\sigma^{2}_{e}$ is typically estimated. For binary outcomes, it is held constant. Because of how MCMCglmm is setup, you cannot fix it at zero, but it is relatively standard to fix it at $1$ (also true for a probit model). For count data (e.g., with a poisson distribution), you do not fix it and this automatically estimates an overdispersion parameter essentially.

The $\mathbf{G}$ structure is the random effects structure. Again in your case, just a random intercept, but if you had multiple random effects, they would form a variance-covariance matrix, $\mathbf{G}$.

A final note, because the residual variance is not fixed at zero, the estimates will not match those from glmer. You need to rescale them. Here is a little example (not using random effects, but it generalizes). Note how the R structure variance is fixed at 1.

# example showing how close the match is to ML without separation
m2 <- MCMCglmm(vs ~ mpg, data = mtcars, family = "categorical",
  prior = list(
    B = list(mu = c(0, 0), V = diag(2) * 1e10),
    R = list(V = 1, fix = 1)),
  nitt = 1e6, thin = 500, burnin = 10000)
summary(m2)

Here is the rescaling constant for the binomial family:

k <- ((16*sqrt(3))/(15*pi))^2

Now divide the solution by it, and get the posterior modes

posterior.mode(m2$Sol/(sqrt(1 + k)))

Which should be fairly close to what we get from glm

summary(glm(vs ~mpg, data = mtcars, family = binomial))