Exchangeable correlations and constant variance in MCMCglmm? - r

Question

I am attempting to fit a very simple model with MCMCglmm, but am getting quite stuck.
Imagine a class (30 students) get grades for two papers throughout the semester where the paper assignments are exactly the same (we don't want to model a difference in average scores between the papers, there are no "learning effects", and we can assume that the variance in grades are the same.)
Let $i = 1...30$ index the student, $y_{i1}$ and $y_{i2}$ index the scores for that student's first and second papers.
One way to model this data is using random intercepts for student scores to account for correlation between each students scores. Let $\mu_i$ be the student intercept, $sigma$ be the residual sd, and $\sigma_{\mu}$ be the sd of the intercepts. Then we write (in shorthand) our random intercept model at $f(y_{ij}|\mu_i) = Normal(\mu_i, \sigma)$ and $f(\mu_i) = Normal(\mu, \sigma_{\mu)$.
An alternative way to write this model would be to model the residual correlation structure more explicitly. That is, we would write that ${y_{i1}, y_{i2}}$ have a multivariate normal distribution with mean ${\mu, \mu}$ variance $\tau = \sigma^2 + \sigma_{\mu}^2$ and correlation $\rho = \sigma_{\mu}^2 / (\sigma^2 + \sigma_{\mu}^2)$.
To be clear, these models are mathematically equivalent, but statistical software will often have a specific implementation for each. For example we can fit the two approaches separately with nlme:
library(nlme)
library(tidyverse)
library(MCMCglmm)
df <-
tibble(id = factor(rep(1:100, each = 20))) %>%
mutate(paper = 1:n()) %>%
group_by(id) %>%
mutate(mu = rnorm(1),
y = mu + rnorm(n(), 0, 3))
gls(data = df,
model = y~1,
correlation = corCompSymm(form = ~ 1 | id))
lme(data = df, fixed = y ~ 1, random = ~1|id)
It seems MCMCglmm can fit the first parameterization (random intercepts) of the model just fine.
MCMCglmm(data = df,
fixed = y ~ 1,
random = ~id,
nitt = 1000, burnin = 0, thin = 1)
However, I am not seeing a way to implement the second approach. My best attempt involves "widening" the data frame and fitting a multiple response model.
df.wide <- df %>% select(- paper) %>%
pivot_wider(values_from = "y",
names_from = "obs", names_prefix = "paper") %>%
as.data.frame
MCMCglmm(fixed = cbind(paper1, paper2) ~ 1,
rcov = ~us(trait):units,
data = df.wide)
However, (1) I am not sure that I am fitting this model correctly, (2) I am not sure how to interpret the fitted values (especially since my posterior mean covariances seem much too small) and (3) there doesn't seem to be a way to get a constant variance across traits.
p.s. I would appreciate not being told to just fit the random intercept model. I am writing some course materials, and would like students to be able to more directly compare the exchangeable correlation model with other types of correlation structures that we might use when we have more than two observations (i.e. AR, Toeplitz, etc.), and I would like my students to be able to do the comparison of the two parameterizations themselves, as I would do when I used nlme.
FOLLOW-UP: I am currently trying to fit the model with BRMS, though would still be open to any "hacks" in MCMCglmm.
model1 <- brms::brm(data = df,
formula = y ~ 1 + cosy(gr = id, time = obs),
family = "gaussian",
chains = 4, thin = 1, iter = 5000, warmup = 100)

Answer 1

Is exchangeability + equal variances the same as what I would call compound symmetry? (I guess so, since you're using corCompSymm() in nlme) ...
As far as I can tell this isn't possible (I can't rule out that there's some way to hack it with the available variance structures, but it's far from obvious ...) From ?MCMCglmm:
Currently, the only ‘variance.functions’ available are ‘idv’,
‘idh’, ‘us’, ‘cor[]’ and ‘ante[]’. ‘idv’ fits a constant
variance across all components in ‘formula’. Both ‘idh’ and
‘us’ fit different variances across each component in
‘formula’, but ‘us’ will also fit the covariances. ‘corg’
fixes the variances along the diagonal to one and ‘corgh’
fixes the variances along the diagonal to those specified in
the prior. ‘cors’ allows correlation submatrices. ‘ante[]’
fits ante-dependence structures of different order (e.g
ante1, ante2), and the number can be prefixed by a ‘c’ to
hold all regression coefficients of the same order equal. The
number can also be suffixed by a ‘v’ to hold all innovation
variances equal (e.g ‘antec2v’ has 3 parameters).
By using the us() (unstructured, what nlme would call pdSymm for "positive-definite symmetric") structure, I believe you're not constraining the correlation parameters to be all the same (i.e., violating exchangeability).
For what it's worth, one reason (other than pedagogy) to want to specify a compound-symmetric correlation matrix explicitly rather than by composing the sum of group-level and individual-level random effects would be if you wanted to model negative compound symmetry (the sum-of-random-effects approach can only model rho>0).
My guess is that you're also restricted to answers using MCMCglmm, but if "some Bayesian MCMC approach" is good enough, then you could do this via brms or (somewhat more obscurely, sort of) glmmTMB + tmbstan (although this combination does not currently use informative priors!)