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

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

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() ?

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.