Hi all,

I am trying to construct some Cormack-Jolly_Seber (CJS) models in Stan, but am running into difficulties incorporating covariates. I am following code associated with the book “Bayesian Population Analysis” and the Stan code version of it supplied here. https://github.com/stan-dev/example-models/tree/master/BPA. I am also tyring to look at the code supplied here https://mc-stan.org/docs/2_18/stan-users-guide/mark-recapture-models.html, that appears to mirror the former link. The Stan users guide says that in the code, “All individuals are modeled as having the same capture probability, but this model could be easily generalized to use a logistic regression here based on individual-level inputs to be used as predictors.” In section 7.5.3 of the BPA book, it states that the individual mode "provides the base for modeling survival as a function of an individual covariate Xi (e.g. size of individual). I am interested in incorporating a flexible design matrix into the CJS model that allows for individual level covariates that are predictors of survivorship, but just not sure where to start. I am guessing it would be in the constraints secion of the code, but just not sure how to do it.

here is (I think) the relevant code from the BPA book.

```
// This models is derived from section 12.3 of "Stan Modeling Language
// User's Guide and Reference Manual"
functions {
int first_capture(int[] y_i) {
for (k in 1:size(y_i))
if (y_i[k])
return k;
return 0;
}
int last_capture(int[] y_i) {
for (k_rev in 0:(size(y_i) - 1)) {
// Compoud declaration was enabled in Stan 2.13
int k = size(y_i) - k_rev;
// int k;
// k = size(y_i) - k_rev;
if (y_i[k])
return k;
}
return 0;
}
matrix prob_uncaptured(int nind, int n_occasions,
matrix p, matrix phi) {
matrix[nind, n_occasions] chi;
for (i in 1:nind) {
chi[i, n_occasions] = 1.0;
for (t in 1:(n_occasions - 1)) {
// Compoud declaration was enabled in Stan 2.13
int t_curr = n_occasions - t;
int t_next = t_curr + 1;
/*
int t_curr;
int t_next;
t_curr = n_occasions - t;
t_next = t_curr + 1;
*/
t_curr = n_occasions - t;
t_next = t_curr + 1;
chi[i, t_curr] = (1 - phi[i, t_curr])
+ phi[i, t_curr] * (1 - p[i, t_next - 1]) * chi[i, t_next];
}
}
return chi;
}
}
data {
int<lower=0> nind; // Number of individuals
int<lower=2> n_occasions; // Number of capture occasions
int<lower=0,upper=1> y[nind, n_occasions]; // Capture-history
}
transformed data {
int n_occ_minus_1 = n_occasions - 1;
// int n_occ_minus_1;
int<lower=0,upper=n_occasions> first[nind];
int<lower=0,upper=n_occasions> last[nind];
// n_occ_minus_1 = n_occasions - 1;
for (i in 1:nind)
first[i] = first_capture(y[i]);
for (i in 1:nind)
last[i] = last_capture(y[i]);
}
parameters {
real<lower=0,upper=1> mean_phi; // Mean survival
real<lower=0,upper=1> mean_p; // Mean recapture
vector[nind] epsilon;
real<lower=0,upper=5> sigma;
// In case a weakly informative prior is used
// real<lower=0> sigma;
}
transformed parameters {
matrix<lower=0,upper=1>[nind, n_occ_minus_1] phi;
matrix<lower=0,upper=1>[nind, n_occ_minus_1] p;
matrix<lower=0,upper=1>[nind, n_occasions] chi;
real mu;
// Constraints
mu = logit(mean_phi);
for (i in 1:nind) {
for (t in 1:(first[i] - 1)) {
phi[i, t] = 0;
p[i, t] = 0;
}
for (t in first[i]:n_occ_minus_1) {
phi[i, t] = inv_logit(mu + epsilon[i]);
p[i, t] = mean_p;
}
}
chi = prob_uncaptured(nind, n_occasions, p, phi);
}
model {
// Priors
// Uniform priors are implicitly defined.
// mean_phi ~ uniform(0, 1);
// mean_p ~ uniform(0, 1);
// sigma ~ uniform(0, 5);
// In case a weaily informative prior is used
// sigma ~ normal(2.5, 1.25);
epsilon ~ normal(0, sigma);
// Likelihood
for (i in 1:nind) {
if (first[i] > 0) {
for (t in (first[i] + 1):last[i]) {
1 ~ bernoulli(phi[i, t - 1]);
y[i, t] ~ bernoulli(p[i, t - 1]);
}
1 ~ bernoulli(chi[i, last[i]]);
}
}
}
generated quantities {
real<lower=0> sigma2;
sigma2 = square(sigma);
}
```

I am guessing that the design matrix could replace mu, but just not sure how to do it.

Any help would be appreciated.