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### The function:

This is a **very simple** data simulator for a 2PL Model. This is just to get you started, from here is easy to add function parameters for indicating item locations and slopes or person distribution characteristics.

- The function accepts only two parameters:
- The number of items
- The number of persons

- The function creates a list containing four objects:
- A vector of item locations
- A vector of item slopes
- A vector of person locations
- A matrix of simulated responses

### The code:

```
twopl.sim <- function( nitem = 20, npers = 100 ) {
i.loc <- rnorm( nitem )
p.loc <- rnorm( npers )
i.slp <- rlnorm( nitem, sdlog = .4 )
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) )
probabilities <- 1 / ( 1 + exp( -logits ) )
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
}
```

### Example:

This is a simple example that uses the IRToys (irtoys) package to estimate the model parameters after simulating the data. Do try this at home too!

```
#install.packages('irtoys')
# Note: 'irtoys' might not load right away in OSX. You can try the following sequence in that case:
# install.packages('mvtnorm')
# install.packages('msm')
# install.packages('ltm',type='source')
# install.packages('irtoys',type='source')
library(irtoys)
###### Running Simulation ######
data2pl <- twopl.sim(nitem=20,npers=10000)
sim.loc.param <- data2pl$i.loc
sim.slp.param <- data2pl$i.slp
###### Estimation Using irtoys ######
analysis.2pl <- est(data2pl$resp, model = '2PL', engine = 'ltm')
est.slp.param <- analysis.2pl[,1]
est.loc.param <- analysis.2pl[,2]
layout(matrix(c(1,2),nrow=1))
plot(est.slp.param,sim.slp.param)
plot(est.loc.param,sim.loc.param)
```

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