Item Characteristic Curves
The three-parameter logistic (3PL) model defines the probability of a correct response as:
\[P(\theta) = c + \frac{1 - c}{1 + e^{-a(\theta - b)}}\]where $a$ is the discrimination parameter (slope), $b$ is the difficulty (location), and $c$ is the pseudo-guessing lower asymptote. Setting $c = 0$ yields the 2PL; additionally fixing $a = 1$ yields the 1PL (Rasch-family).
| Parameter | Symbol | Interpretation |
|---|---|---|
| Discrimination | $a$ | Steepness of the ICC at $b$ |
| Difficulty | $b$ | $\theta$ at which $P = (1+c)/2$ |
| Guessing | $c$ | Lower asymptote ($0 \le c < 1$) |
Item and Test Information
The item information function quantifies how precisely an item measures ability:
\[I_i(\theta) = \frac{[P'_i(\theta)]^2}{P_i(\theta)\,Q_i(\theta)}\]where $Q_i = 1 - P_i$. For the 2PL this simplifies to $I_i(\theta) = a_i^2 P_i Q_i$. Test information is additive across items:
\[I(\theta) = \sum_{i=1}^{k} I_i(\theta)\]The standard error of measurement at a given ability level is $\text{SE}(\theta) = 1/\sqrt{I(\theta)}$.
Parameter Estimation
Item parameters are estimated via marginal maximum likelihood (MML), integrating over the latent ability distribution. Person parameters are recovered by maximum likelihood or expected a posteriori (EAP) estimation given calibrated item parameters. The likelihood for person $n$ is:
\[L(\theta_n) = \prod_{i=1}^{k} P_i(\theta_n)^{u_{ni}}\,[1 - P_i(\theta_n)]^{1 - u_{ni}}\]Fit is assessed using $\chi^2$-based outfit and infit statistics.