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?

Thanks in advance,

Özgür Asar

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?

Thanks in advance,

Özgür Asar

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[1] * (z_1[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[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,
control = list(adapt_delta = 0.999),
seed = 123,
family = "bernoulli")
summary(brms_fit)
```

Thanks for checking!

I have made similar observations with categorical responses or other responses not containing a lot of information. I still want to do extensive simulations with t-distributed random effects (and perhaps write a paper about it) at some point, before I “officially” support them.