irf: Item response function

Description

Returns the item response function of the 3PL (1PL, 2PL) model, the i.e. the probabilities defined by $$P(U_{ij}=1|\theta_i,a_j,b_j,c_j)=c_j+(1-c_j)\frac{\displaystyle\exp(Da_j(\theta_i-b_j))}{1+\displaystyle\exp(Da_j(\theta_i-b_j))}$$ where $U_{ij}$ is a binary response given by person $i$ to item $j$, $\theta_i$ 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 $j$, and $D$ is a constant usually set to either 1.7 or 1. Some authors call the IRF "the item characteristic curve".

Usage

irf(ip, x = NULL)

Arguments

Item parameters: a matrix with one row per item, and three columns: [,1] item discrimination $a$, [,2] item difficulty $b$, and [,3] asymptote $c$.

The values of the latent variable ($\theta$ 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.

Value

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

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.

Examples

Run this code

p.2pl <- est(Scored, model = "2PL", engine = "ltm")
plot(irf(p.2pl[1, ]))