irf: Item response function

Description

Returns the item response function of the 3PL (1PL, 2PL) model, the i.e. the probabilities defined by $$P(U_{ij}=1|\theta_i,a_j,b_j,c_j)=c_j+(1-c_j)\frac{\displaystyle\exp(a_j(\theta_i-b_j))}{1+\displaystyle\exp(a_j(\theta_i-b_j))}$$ 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 $j$. Some authors call the IRF "the item characteristic curve".

Usage

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

Arguments

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

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:

A copy of the argument x

A 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

# NOT RUN {
plot(irf(Scored2pl, item=1))

# }