# Using rstan to estimate Response Times IRT Model parameters

Dear All Stan users:
I’m a student and Stan’s beginner.
Recently, I try to use stan to estimate the Response Times IRT Model (RT-IRTM), and the formula for this model is as follows: As we can see, the tij in this model is response time, so it is a given fixed variable that obtained during test, just like the response data.
I I wrote some code to estimate the item parameters for the model, Here is my Stan code:

data {
int<lower=0> n_participant;
int<lower=0> n_item;
int<lower=0, upper=1> Y[n_participant,n_item]; // the response of each student to answer each item
real RT[n_participant,n_item]; // the response times of each student to answer each item
}
parameters {
vector<lower=-3,upper=3> [n_participant] theta;
vector<lower=0,upper=2> [n_participant] rho;
vector<lower=0,upper=2.5> [n_item] alpha;
vector<lower=-3,upper=3>[n_item] beta;
vector<lower=0,upper=0.2> [n_item] gamma;
vector<lower=0,upper=10> [n_item] delta;
}
model {
theta ~ normal(0,1);
rho ~ uniform(0,2);
beta ~ normal(0,1);
alpha ~ lognormal(0,1);
gamma ~ uniform(0,0.2);
delta ~ uniform(0,10);
for(i in 1:n_participant){
for(j in 1:n_item){
real p;
p=inv_logit(-1.7alpha[j](theta[i]-((rho[i]*delta[j])/RT[i,j])-beta[j]));
Y[i,j] ~ bernoulli(gamma[j]+(1-gamma[j])*p);
}}}

However, I get some errors when I try to run the model:
SAMPLING FOR MODEL ‘4PL’ NOW (CHAIN 1).

Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can’t start sampling from this initial value.

And the R code are as follows:

library(“rstan”)
rstan_options(auto_write=TRUE)
options(mc.cores=parallel::detectCores())
#data
I<-nrow(data)
J<-ncol(data)
data_4pl<-list(n_participant=I,n_item=J,RT=RT,Y=data)

irt_4pl<-stan(file=“4PL.stan”,data=data_4pl,iter=1000,chains=1)

I would be very grateful if you could take the time to give me some guidance.

4/25
Today, I try to change the RT[i,j] term to a constant 10, of course I deleted the code related to response time, and I found the model could be fit. So I think it might be a code problem with the response time data.

p=inv_logit(-1.7alpha[j](theta[i]-((rho[i]delta[j])/RT[i,j])-beta[j]));
p=inv_logit(-1.7
alpha[j]*(theta[i]-((rho[i]*delta[j])/10)-beta[j]));

But, I still can not solve this problem. so upset… : (

Not quite sure why your model fails exactly but there are a few issues with your specification. In general it is discouraged to set hard constraints on parameters when they are not needed. For example the limits on `theta` and `beta` are not needed. Also, the uniform priors are not necessary, because they are implicitly assumed if no prior is specified.

You could try to minimize the hard constraints on your parameters (e.g. `vector[n_participant] theta`, `vector<lower=0,upper=1>[n_item] gamma`, …) and use some generic priors (e.g. Half-Cauchy/Half-Normal for positively contrained parameters, Beta for `gamma`, …) and see if the model can be fit.

If that does not work it is always a good idea to build your model from the most simple model. In your case I would start with the 1PL and then add response time parameters, item discrimination, and pseudo-guessing parameter successively. This makes debugging your model much easier.

1 Like

I really appreciate your advice and I will try it as you say.