# Nominal Regression in STAN

I was talking to a colleague about modeling nominal outcomes in STAN, and wrote up this example. Just put it here in case it’s helpful for anyone (probably myself in the future). This is based on an example I made for a course, where you can find the brms code for nominal regression. Please also check out the Multi-logit regression session on the Stan User’s guide.

library(cmdstanr)
library(dplyr)
library(ggplot2)

Check out this paper: https://journals.sagepub.com/doi/full/10.1177/2515245918823199

stemcell <- read.csv("https://osf.io/vxw73/download")
stemcell |>
ggplot(aes(x = rating)) +
geom_bar() +
facet_wrap(~ gender) https://www.thearda.com/archive/files/Codebooks/GSS2006_CB.asp

The outcome is attitude towards stem cells research, and the predictor is gender.

Recently, there has been controversy over whether the government should provide any funds at all for scientific research that uses stem cells taken from human embryos. Would you say the government . . .

• 1 = Definitely, should fund such research
• 2 = Probably should fund such research
• 3 = Probably should not fund such research
• 4 = Definitely should not fund such research

## Nominal Logistic Regression

Ordinal regression is a special case of nominal regression with the proportional odds assumption.

## Model

\begin{align} \text{rating}_i & \sim \mathrm{Categorical}(\pi^1_{i}, \pi^2_{i}, \pi^3_{i}, \pi^4_{i}) \\ \pi^1_{i} & = \frac{1}{\exp(\eta^2_{i}) + \exp(\eta^3_{i}) + \exp(\eta^4_{i}) + 1} \\ \pi^2_{i} & = \frac{\exp(\eta^2_{i})}{\exp(\eta^2_{i}) + \exp(\eta^3_{i}) + \exp(\eta^4_{i}) + 1} \\ \pi^3_{i} & = \frac{\exp(\eta^3_{i})}{\exp(\eta^2_{i}) + \exp(\eta^3_{i}) + \exp(\eta^4_{i}) + 1} \\ \pi^4_{i} & = \frac{\exp(\eta^4_{i})}{\exp(\eta^2_{i}) + \exp(\eta^3_{i}) + \exp(\eta^4_{i}) + 1} \\ \eta^2_{i} & = \beta^2_{0} + \beta^2_{1} \text{male}_{i} \\ \eta^3_{i} & = \beta^3_{0} + \beta^3_{1} \text{male}_{i} \\ \eta^4_{i} & = \beta^4_{0} + \beta^4_{1} \text{male}_{i} \\ \end{align}

mod <- cmdstan_model("nominal_reg.stan")
mod
## //
## // This Stan program defines a nominal regression model.
## //
## // It is based on
## //   https://mc-stan.org/docs/stan-users-guide/multi-logit.html
## //
##
## // The input data is a vector 'y' of length 'N'.
## data {
##   int<lower=0> K;  // number of response categories
##   int<lower=0> N;  // number of observations (data rows)
##   int<lower=0> D;  // number of predictors
##   array[N] int<lower=1, upper=K> y;  // response vector
##   matrix[N, D] x;  // predictor matrix
## }
##
## transformed data {
##   vector[D] zeros = rep_vector(0, D);
## }
##
## // The parameters accepted by the model.
## parameters {
##   vector[K - 1] b0_raw;  // intercept for second to last categories
##   matrix[D, K - 1] beta_raw;
## }
##
## // The model to be estimated.
## model {
##   // Add zeros for reference category
##   vector[K] b0 = append_row(0, b0_raw);
##   matrix[D, K] beta = append_col(zeros, beta_raw);
##   to_vector(beta_raw) ~ normal(0, 5);
##   y ~ categorical_logit_glm(x, b0, beta);
## }
stan_dat <- with(stemcell,
list(K = n_distinct(rating),
N = length(rating),
D = 1,
y = rating,
x = matrix(as.integer(gender == "male")))
)
# Draw samples
fit <- mod$sample(data = stan_dat, seed = 123, chains = 4, parallel_chains = 2, refresh = 500, iter_sampling = 2000, iter_warmup = 2000) fit$summary() |>
knitr::kable()
variable mean median sd mad q5 q95 rhat ess_bulk ess_tail
lp__ -1035.3518437 -1035.0200000 1.7612849 1.6160340 -1038.6900000 -1033.1300000 1.000639 3809.196 4742.492
b0_raw 0.4703141 0.4697820 0.1118201 0.1128904 0.2877985 0.6527007 1.001230 3667.338 5286.149
b0_raw -0.6762280 -0.6757360 0.1507270 0.1497626 -0.9251864 -0.4307565 1.000439 4465.972 5274.345
b0_raw -0.9675208 -0.9653220 0.1672082 0.1658889 -1.2447755 -0.6988120 1.001632 4428.634 5072.438
beta_raw[1,1] -0.1760178 -0.1774040 0.1680554 0.1646985 -0.4533714 0.1036652 1.000391 3881.178 5200.826
beta_raw[1,2] -0.0121064 -0.0135833 0.2209685 0.2195530 -0.3762467 0.3532682 1.000601 4376.717 5480.247
beta_raw[1,3] -0.1751981 -0.1765975 0.2540991 0.2564196 -0.5939876 0.2370495 1.001283 4382.617 5279.069

Compare to brms:

library(brms)
brm(rating ~ gender, data = stemcell, family = categorical(link = "logit"),
file = "mlogit")
##  Family: categorical
##   Links: mu2 = logit; mu3 = logit; mu4 = logit
## Formula: rating ~ gender
##    Data: stemcell (Number of observations: 829)
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
##
## Population-Level Effects:
##                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## mu2_Intercept      0.47      0.11     0.26     0.69 1.00     3990     3108
## mu3_Intercept     -0.67      0.15    -0.97    -0.39 1.00     3809     3130
## mu4_Intercept     -0.96      0.17    -1.29    -0.63 1.00     4027     3031
## mu2_gendermale    -0.18      0.17    -0.51     0.15 1.00     4272     2829
## mu3_gendermale    -0.01      0.22    -0.43     0.41 1.00     4141     2902
## mu4_gendermale    -0.17      0.25    -0.67     0.34 1.00     3922     2937
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

The estimates are pretty much the same. ##### Hok Chio (Mark) Lai 黎學昭
###### Assistant Professor of Psychology (Quantitative Methods)

My research interests include statistics, multilevel and latent variable models, and psychometrics.