Built-in models:3PL

Model

Let \(y_{ij}, i=1,\ldots,N, j=1,\ldots,T\) denote the response for person \(i\) on item \(j\) that takes value 0 or 1. The 2PL model can be written as

\[ y_{ij} \sim B(p_{ij}) \] \[ p_{ij} = c_j + \frac{1-c_j}{1+\exp[-\alpha_j(\theta_i - \beta_j)]} \] \[ \theta_i \sim N(0,1) \]

where \(\theta_i\) is the latent trait for person \(i\), \(\beta_j\) is the item difficulty parameter, \(\alpha_j\) is the item discrimination parameter and \(c_j\) is a guessing parameter for item \(j\).

BUGS code

model{
  for (i in 1:N){
    for (j in 1:T){
      p[i,j]<-c[j]+(1-c[j])/(1+exp(-alpha[j]*(theta[i]-beta[j])))
      y[i,j]~dbern(p[i,j])
    }
    theta[i]~dnorm(0,1)
  }
  for (j in 1:T){
    alpha[j]~dnorm(0,.0001)
    beta[j]~dnorm(0, .0001)
    c[j]~dnorm(0,.001)
  }
}