# irf: Item response function In irtoys: A Collection of Functions Related to Item Response Theory (IRT)

## Description

Returns the item response function of the 3PL (1PL, 2PL) model, the i.e. the probabilities defined by

P(U_{ij}=1|θ_i,a_j,b_j,c_j)=c_j+(1-c_j)\frac{\displaystyle\exp(a_j(θ_i-b_j))}{1+\displaystyle\exp(a_j(θ_i-b_j))}

where U_{ij} is a binary response given by person i to item j, θ_i is the value of the latent variable ("ability") for person i, a_j is the discrimination parameter for item j, b_j is the difficulty parameter for item j, c_j is the asymptote for item j. Some authors call the IRF "the item characteristic curve".

## Usage

 1 irf(ip, items = NULL, x = NULL) 

## Arguments

 ip Item parameters: the output of est, or a 3-column matrix corresponding to its first element, est. items The item(s) for which irf is computed. If NULL (the default), irf for all items will be returned x The values of the latent variable (θ in the equation above), at which the IRF will be evaluated. If not given, 99 values spaced evenly between -4 and +4 will be used, handy for plotting.

## Details

In the 2PL model (model="2PL"), all asymptotes c_j are 0. In the 1PL model (model="1PL"), all asymptotes c_j are 0 and the discriminations a_j are equal for all items (and sometimes to 1).

A common use of this function would be to obtain a plot of the IRF.

## Value

A list of:

 x A copy of the argument x f A matrix containing the IRF values: persons (values of (x) as rows and items as columns

## Author(s)

Ivailo Partchev

plot.irf
 1 plot(irf(Scored2pl, item=1))