Hi all,

I have an odd finding when it comes to fixed population effects when using different random effect distributions. Thanks to @paul.buerkner, brms has an option of a Student t distribution for the random effect distribution. However, when I compare the values of the fixed-effect parameters between the Gaussian and t distributions, they are significantly different. I confirmed that the models converged in both cases and also reran the data with non-informative priors of normal(0,1000). I also reviewed the Stan code generated by brms and found no clear errors.

Below is the code used for the two models, followed by the respective output.

Gaussian model:

```
nd_full = brm(
md ~ time_sap + (1+ time_sap|eyeid),
data = final_sap,
family = gaussian(),
set_prior("normal(0,1000)", class='b'),
warmup = 1000, iter = 8000, chains = 4,
control=list(adapt_delta=0.9, max_treedepth = 15))
```

Output:

summary(nd_full)

Family: gaussian

Links: mu = identity; sigma = identity

Formula: md ~ time_sap + (1 + time_sap | eyeid)

Data: final_sap (Number of observations: 10081)

Samples: 4 chains, each with iter = 8000; warmup = 1000; thin = 1;

total post-warmup samples = 28000

Group-Level Effects:

~eyeid (Number of levels: 1242)

Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS

sd(Intercept) 5.20 0.11 4.99 5.42 1.00 5000

sd(time_sap) 0.35 0.01 0.33 0.38 1.00 13125

cor(Intercept,time_sap) 0.13 0.04 0.06 0.20 1.00 15928

Tail_ESS

sd(Intercept) 9456

sd(time_sap) 19009

cor(Intercept,time_sap) 19562

Population-Level Effects:

Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS

Intercept -4.02 0.15 -4.31 -3.73 1.00 2652 5896

time_sap -0.15 0.01 -0.17 -0.12 1.00 16717 19839

Family Specific Parameters:

Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS

sigma 1.62 0.01 1.59 1.65 1.00 19701 19799

Student t model:

```
t_full = brm(
md ~ time_sap + (1+ time_sap|gr(eyeid, dist = "student")),
data = final_sap,
family = gaussian(),
set_prior("normal(0,1000)", class='b'),
warmup = 1000, iter = 8000, chains = 4,
control=list(adapt_delta=0.9, max_treedepth = 15))
```

Output:

summary(t_full)

Family: gaussian

Links: mu = identity; sigma = identity

Formula: md ~ time_sap + (1 + time_sap | gr(eyeid, dist = “student”))

Data: final_sap (Number of observations: 10081)

Samples: 4 chains, each with iter = 8000; warmup = 1000; thin = 1;

total post-warmup samples = 28000

Group-Level Effects:

~eyeid (Number of levels: 1242)

Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS

df 1.46 0.10 1.28 1.65 1.00 1888

sd(Intercept) 2.19 0.11 1.98 2.41 1.00 2082

sd(time_sap) 0.14 0.01 0.12 0.15 1.00 2467

cor(Intercept,time_sap) 0.22 0.05 0.13 0.32 1.00 6798

Tail_ESS

df 4714

sd(Intercept) 5625

sd(time_sap) 6468

cor(Intercept,time_sap) 12640

Population-Level Effects:

Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS

Intercept -1.93 0.10 -2.12 -1.74 1.00 3656 8260

time_sap -0.07 0.01 -0.08 -0.05 1.00 12615 18970

Family Specific Parameters:

Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS

sigma 1.60 0.01 1.58 1.63 1.00 26854 21498

You will notice a significant difference in the estimates of the Population-level Effects between the Gaussian and t models. Why would altering the random effect distribution affect the fixed effect values? Many thanks in advance!

- Operating System: Windows
- brms Version: 2.13.3