# Non-normal random effects in GLMM

Hi,

In a logistic model say, logit[P(Y = 1|…)] = x \beta + B_0, can one specifiy the distribution of B_0|… as t(0, scale, dof), in brms?

Özgür Asar

See https://github.com/paul-buerkner/brms/issues/231 for how to model t-distributed random effects.

For t, what parametrisation do you use? Is it \sigma Z/\sqrt{V/\nu}, where Z is standard Normal, and
V is chi-square with d.o.f. of \nu??

Hi Paul,
when I generate Stan code using {\tt brms} for binary mixed model with t-distributed random intercept:

\begin{equation} \mbox{logit}(P(Y_{ij}= 1|...)) = {\boldsymbol x}_{ij} {\boldsymbol \beta} + U_i; \ i = 1, \ldots, n, \ j = 1, \ldots, m, \end{equation}

where U_i \sim t(\phi, 0, \sigma), below is what I got. It seems you parameterise U_i as U_i = \sigma Z_i \sqrt{V * \phi}, where Z_i \sim N(0, 1), and V is scalar and V \sim {\mbox Inverse} \ \chi^2(\phi).
My question is: shouldn’t V be V_i, as use of V introduce dependency between U_i and U_i^{\prime} terms for i \neq i^{\prime}?

Thanks,

Özgür

// generated with brms 2.7.0
functions {
}
data {
int<lower=1> N;  // total number of observations
int Y[N];  // response variable
int<lower=1> K;  // number of population-level effects
matrix[N, K] X;  // population-level design matrix
// data for group-level effects of ID 1
int<lower=1> J_1[N];
int<lower=1> N_1;
int<lower=1> M_1;
vector[N] Z_1_1;
int prior_only;  // should the likelihood be ignored?
}
transformed data {
int Kc = K - 1;
matrix[N, K - 1] Xc;  // centered version of X
vector[K - 1] means_X;  // column means of X before centering
for (i in 2:K) {
means_X[i - 1] = mean(X[, i]);
Xc[, i - 1] = X[, i] - means_X[i - 1];
}
}
parameters {
vector[Kc] b;  // population-level effects
real temp_Intercept;  // temporary intercept
vector<lower=0>[M_1] sd_1;  // group-level standard deviations
vector[N_1] z_1[M_1];  // unscaled group-level effects
// parameters for student-t distributed group-level effects
real<lower=1> df_1;
real<lower=0> udf_1;
}
transformed parameters {
// group-level effects
vector[N_1] r_1_1 = sqrt(df_1 * udf_1) * sd_1 * (z_1);
}
model {
vector[N] mu = temp_Intercept + Xc * b;
for (n in 1:N) {
mu[n] += r_1_1[J_1[n]] * Z_1_1[n];
}
// priors including all constants
target += student_t_lpdf(temp_Intercept | 3, 0, 10);
target += student_t_lpdf(sd_1 | 3, 0, 10)
- 1 * student_t_lccdf(0 | 3, 0, 10);
target += normal_lpdf(z_1 | 0, 1);
target += gamma_lpdf(df_1 | 2, 0.1);
target += inv_chi_square_lpdf(udf_1 | df_1);
// likelihood including all constants
if (!prior_only) {
target += bernoulli_logit_lpmf(Y | mu);
}
}
generated quantities {
// actual population-level intercept
real b_Intercept = temp_Intercept - dot_product(means_X, b);
}


Here is the definition of (multivariate) student-t that I am using:

The definition is correct. But the thing with mixed models is that random effects belonging to different subjects are assumed to be independent. Scalar V violates this.

Based on my simulations, scalar V introduces biases to \phi and \sigma.

Özgür

Ah I get your point now, thanks! @bgoodri can you confirm that in the above Stan code, udf_1 should be a vector instead of a scalar?

It can be a vector. Whether it can be called multivariate t is a long-standing debate.

Thanks! In the GitHub version of brms, udf_* should now be a vector. @ozgurasar would you mind taking a look and confirm that the parameterization now looks correct to you?

Yes, it now expresses the model that is intended to be fit.
One observation is that fitting a mixed model for categorical data with t random effects is quite difficult, probably due to identifying heavy tailedness through categorical data.

An example is the following (sim_data.txt (58.2 KB) is a simulated data):

devtools::install_github("paul-buerkner/brms")
brms_fit <- brm(y ~ x1 + x2 + x3 + (1|gr(id, dist = "student")),
data = sim_data,
cores = 4,