data simulation from IRT 2PL to IRT 3PL model

General
2

Hi! The 3PL IRT model is usually defined as shown here.

First, I we can define a function, irt_prob that calculates the probability of a correct response, with or without c parameters. Then we pass that function to one more apply() function. The full code is below:

irt_prob <- function(logit, c = NULL) {
  if (is.null(c)) {
    1 / ( 1 + exp( -logit ) )
  } else {
    c + ((1 - c) / (1 + exp( -logit ) ) )
  }
}

irt.sim <- function( nitem = 20, npers = 100 ) {
  
  i.loc <- rnorm( nitem )
  p.loc <- rnorm( npers )
  i.slp <- rlnorm( nitem, sdlog = .4 )
  i.gus <- rbeta( nitem, shape1 = 5, shape2 = 17)
  
  temp <- matrix( rep( p.loc, length( i.loc ) ), ncol = length( i.loc ) )
  
  logits <- t( apply( temp , 1, '-', i.loc ) )
  logits <- t( apply( logits, 1, '*', i.slp ) )
  
  # For 2PL: 
  # probabilities <- t( apply( logits, 1, irt_prob ) )
  
  # For 3PL:
  probabilities <- t( apply( logits, 1, irt_prob, c = i.gus ) )
  
  resp.prob <- matrix( probabilities, ncol = nitem)

  obs.resp <- matrix( sapply( c(resp.prob), rbinom, n = 1, size = 1), ncol = length(i.loc) )
  
  output <- list()
  output$i.loc <- i.loc
  output$i.slp <- i.slp
  output$p.loc <- p.loc
  output$resp <- obs.resp
  
  output
}

I've generated the c-parameters from a beta(5,17) distribution, which will give an average value of ~.2.