Output from multinomial logistic regression in Stan - stan

Question

I'm running a multinomial logistic regression. The outcome has four categories and there are two predictors (Male =1, a measure of the number of books in the home as a 5 point scale, and measure of motivation to read which is continuous. Here is the essential aspects of the code. I'm following "K" parameterization in https://mc-stan.org/docs/stan-users-guide/multi-logit.html. Thanks.
DOWELL <- as.factor(DOWELL)
Canadareg2 <- data.frame(Canadareg2)
n <- nrow(Canadareg2)
f <- as.formula("DOWELL ~ Male + booksHome + motivRead")
m <- model.matrix(f,Canadareg2)
data.list <- list(n=nrow(Canadareg2),
k=length(unique(Canadareg2[,1])),
d=ncol(m),x=m, Male=Male, booksHome=booksHome,
motivRead=motivRead,DOWELL=as.numeric(Canadareg2[,1]))
ReadMultiNom <- "
data {
int<lower = 2> k; // The variable has at least two categories
int<lower = 1> d; // number of predictors
int<lower = 0> n;
vector[n] Male;
vector[n] booksHome;
vector[n] motivRead;
int <lower=1, upper=k> DOWELL[n];
matrix[n, d] x;
}
parameters {
matrix[d, k] beta;
}
transformed parameters {
matrix[n, k] x_beta= x * beta;
}
model {
to_vector(beta) ~ normal(0,2); // vectorizes beta and assigns same prior
for (i in 1:n) {
DOWELL[i] ~ categorical_logit(x_beta[i]');
}
}
generated quantities {
int <lower=1, upper=k> DOWELL_rep[n];
vector[n] log_lik;
for (i in 1:n) {
DOWELL_rep[i] = categorical_logit_rng(x_beta[i]');
log_lik[i] = categorical_logit_lpmf(DOWELL[i] |x_beta[i]');
}
}
"
nChains = 4
nIter= 10000
thinSteps = 10
burnInSteps = floor(nIter/2)
DOWELL = data.list$DOWELL
MultiNomRegFit = stan(data=data.list,model_code=ReadMultiNom,
chains=nChains,control = list(adapt_delta = 0.99),
iter=nIter,warmup=burnInSteps,thin=thinSteps)
Everything runs beautifully and all convergence criterion are met. However, I am struggling to interpret the betas. I'm not sure where the Male effects are located. That is, it seems to be only for the two other predictors, but even then, one of them is a 5 point scale. It would seem to me that each beta would have three elements, e.g. beta(1,1,1), beta(1,2,1), etc. Here is the output. I'm just unsure how to interpret the betas.
Inference for Stan model: 7be33c603bd35d82ad7f6b200ccee16f.
## 4 chains, each with iter=10000; warmup=5000; thin=10;
## post-warmup draws per chain=500, total post-warmup draws=2000.
##
## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
## beta[1,1] -2.07 0.03 1.05 -4.16 -2.80 -2.08 -1.36 0.00 1705 1
## beta[1,2] 1.16 0.03 1.04 -0.97 0.47 1.18 1.84 3.25 1652 1
## beta[1,3] 1.06 0.03 1.07 -1.03 0.36 1.06 1.76 3.15 1703 1
## beta[1,4] -0.01 0.03 1.07 -2.17 -0.70 -0.01 0.68 2.06 1657 1
## beta[2,1] 0.51 0.03 1.04 -1.52 -0.18 0.52 1.20 2.51 1618 1
## beta[2,2] -0.01 0.03 1.04 -2.08 -0.72 -0.02 0.67 2.07 1636 1
## beta[2,3] -0.31 0.03 1.03 -2.31 -1.00 -0.31 0.38 1.70 1617 1
## beta[2,4] -0.26 0.03 1.04 -2.28 -0.94 -0.26 0.44 1.72 1648 1
## beta[3,1] 0.26 0.03 1.01 -1.73 -0.40 0.27 0.95 2.17 1525 1
## beta[3,2] 0.03 0.03 1.01 -2.01 -0.65 0.05 0.70 1.93 1525 1
## beta[3,3] 0.02 0.03 1.01 -2.04 -0.64 0.04 0.70 1.95 1522 1
## beta[3,4] -0.30 0.03 1.01 -2.36 -0.99 -0.29 0.38 1.59 1528 1
## beta[4,1] 0.18 0.02 0.98 -1.70 -0.49 0.14 0.85 2.17 1561 1
## beta[4,2] -0.09 0.02 0.98 -1.98 -0.77 -0.14 0.58 1.89 1568 1
## beta[4,3] -0.12 0.02 0.98 -2.01 -0.80 -0.15 0.56 1.85 1566 1
## beta[4,4] 0.11 0.02 0.99 -1.79 -0.56 0.07 0.79 2.10 1559 1
##
## Samples were drawn using NUTS(diag_e) at Tue Oct 18 09:58:48 2022.
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at
## convergence, Rhat=1).
I'm not quite sure what is going on, so any advice would be appreciated.