# How to estimate the covariance matrix of a mirt model parameter in rstan

**URL:** <https://discourse.mc-stan.org/t/how-to-estimate-the-covariance-matrix-of-a-mirt-model-parameter-in-rstan/9956>\
**Category:** Modeling\
**Tags:** techniques\
**Created:** [July 30, 2019, 3:11am UTC](https://discourse.mc-stan.org/t/how-to-estimate-the-covariance-matrix-of-a-mirt-model-parameter-in-rstan/9956 "2019-07-30T03:11:02Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![Lucius](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/lucius/32/4302_2.png) [@Lucius](https://discourse.mc-stan.org/u/Lucius)\
**Post date:** [July 30, 2019, 3:11am UTC](https://discourse.mc-stan.org/t/how-to-estimate-the-covariance-matrix-of-a-mirt-model-parameter-in-rstan/9956/1 "2019-07-30T03:11:02Z")

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Dear stan users,  
I am a fresh stan learner, and I want to know that how could I use the stan to estimate **the covariance matrix** of model parameter, before I meet this question I have read a paper which use the software **jags** to estimated the covariance matrix of a IRT model parameter.  
So I was wondering if I could do the same work in rstan, and here is my stan code for a MIRT(Multidensional item response theory).

* * *

```stan
// M3PL-parameter estimated with stan

data {
	int<lower=1> n_stu; // the number of students
	int<lower=1> n_itm; // the number of items
	int<lower=0,upper=1> Y[n_stu, n_itm]; // the response score
	int<lower=1> D; // the number of dimensions
}

transformed data {
	row_vector [D] mu_theta = rep_row_vector(0, D);	
	cov_matrix [D] Sigma;
	Sigma = rep_matrix(0, D, D); 
	for(i in 1:D){
		Sigma[i,i] = 1;
	} 
}

parameters {
	vector<lower=-3,upper=3>[D] theta [n_stu];
	matrix<lower=0.5,upper=2.5>[n_itm,D] alpha;
	vector<lower=-3,upper=3>[n_itm] beta;
	vector<lower=0,upper=0.25>[n_itm] gamma;
}

model {
	for(i in 1:n_stu){
		theta[i] ~ multi_normal(mu_theta, Sigma); 
	}
	beta ~ normal(0,1);
	to_vector(alpha) ~ lognormal(0, 0.5);
	gamma ~ uniform(0, 0.25);

	for(j in 1:n_itm){
		for(i in 1:n_stu){
				real p;
				p = inv_logit(1.7*(row(alpha,j)*theta[i]+beta[j]));
				Y[i,j] ~ bernoulli(gamma[j]+(1-gamma[j])*p); 
		}
	}
}

generated quantities { // computate the modelbased log-likelihood for loo and WAIC
	vector[n_itm] log_lik[n_stu];
	vector[n_itm] pY[n_stu];
	for(j in 1:n_itm){
		for(i in 1:n_stu){
			pY[i,j] = gamma[j]+(1-gamma[j])*inv_logit(1.7*(row(alpha,j)*theta[i]+beta[j]));
			log_lik[i,j] = bernoulli_lpmf(Y[i,j] | pY[i,j]); 
		}
	}
}

```

* * *

I would appreciate your help very much.  
Young Luis.

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<div class="post-metadata">

**Author:** ![Lucius](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/lucius/32/4302_2.png) [@Lucius](https://discourse.mc-stan.org/u/Lucius)\
**Post date:** [October 13, 2019, 8:41am UTC](https://discourse.mc-stan.org/t/how-to-estimate-the-covariance-matrix-of-a-mirt-model-parameter-in-rstan/9956/2 "2019-10-13T08:41:01Z")

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This problem was solved by myself !

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**Author:** ![martinmodrak](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/martinmodrak/32/133_2.png) [@martinmodrak](https://discourse.mc-stan.org/u/martinmodrak)\
**Post date:** [October 16, 2019, 8:11am UTC](https://discourse.mc-stan.org/t/how-to-estimate-the-covariance-matrix-of-a-mirt-model-parameter-in-rstan/9956/3 "2019-10-16T08:11:19Z")

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Sorry we weren’t able to help you before. If you have the time, could you share an outline of what halped you and possibly the final Stan code so that others may learn from it! (but it is OK if you don’t :-) )

Thanks!