# Interpreting multinomial logit coefficients

**URL:** <https://discourse.mc-stan.org/t/interpreting-multinomial-logit-coefficients/27119>\
**Category:** Modeling\
**Created:** [April 11, 2022, 2:37pm UTC](https://discourse.mc-stan.org/t/interpreting-multinomial-logit-coefficients/27119 "2022-04-11T14:37:53Z")\
**Posts on this page:** 1\
**Showing post:** 2

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**Author:** ![Bob\_Carpenter](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bob_carpenter/32/9230_2.png) [@Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter)\
**Post date:** [April 11, 2022, 7:51pm UTC](https://discourse.mc-stan.org/t/interpreting-multinomial-logit-coefficients/27119/2 "2022-04-11T19:51:23Z")

</div>

First, I’d think about rearranging `x` and `beta` so you don’t have to transpose in the density. It’d be even better to collect an array of the `x_beta[i]` and of `DOWELL` and then vectorize the whole thing. But that’s just efficiency, not interpretability.

It sounds like you have discrete predictors, but you seem to be using something like a continuous encoding of them in `x`. The usual way to code your predictors for regression would be this way:

```stan
data {
  int<lower=0> N; // observations
  // predictors
  array[N] int<lower=0, upper=1> male; // assuming 1 if male, 0 if female
  array[N] int<lower= 1, upper=5> books;
  vector[N] motivation;

```

and then the parameters look like this:

```stan
parameters {
  vector[4] alpha;
  vector[4] beta_sex;
  array[5] vector[4] beta_books;
  vector[4] beta_motivation;
  array[N] int<lower=1, upper=4> dowell;

```

and then the likelihood looks like this:

```stan
for (n in 1:N)
  dowell[n] ~ categorical_logit(alpha + beta_sex * sex[n] + beta_books[books[n]] + beta_motivation * x[n]);

```

There are lots of choices here for parameterizations when you get into multiple outcomes. For more efficiency, you can pack the first three terms into a matrix indexed by sex and books.

Now the `alpha` is an intercept, `beta_sex` the male effect (female rolled into intercept), and `beta_books[j]` the factors for having j units of books, and so on.

The problem is that `categorical_logit` is only identified up to an additive constant. You can handle this by pinning one of the categories to 0 and having everything else be relative, or you can let the priors sort it out. Or you can set the coefficients for the random effects so that they sum to 0 and everything else gets pushed into the intercept.

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