Item Response Theory
TODO: what is it
Glossary
| Term | Definition |
|---|---|
| $i$ | subject $i$ |
| $j$ | item $j$ |
| $a_j$ | (item) discrimination parameter |
| $b_j$ | (item) difficulty parameter |
| $c_j$ | (item) guessing parameter; the chance of a correct response for a very low $\theta$ |
| $d_j$ | probability of endorsing an (item) incorrect answer in spite of high ability |
| $D=1.7$ | scaling constant |
| $\theta_i$ | (subject) ability. Formally cited as $\theta \in (-\infty, \infty)$, it more often is the case $\theta \in [-3, +3]$. An estimated ability of 1.2 can be interpreted as 1.2 standard deviations above the average ability in the population. |
| $\mathbf{r}_{ij}$ | response from subject $i$ to item $j$ |
Assumptions
Models
Unidimensional, Dichotomous
| Parameter | Formula |
|---|---|
| 1PL | $\mathbf{P}[\mathbf{r}_{ij}=1 | \theta_i, b_j] = \frac{\exp(D(\theta-b_j))}{1+\exp(Da(\theta-b_j))} = \frac{1}{1+\exp[-Da(\theta_i-b_j)]}$ |
| 2PL | $\mathbf{P}[\mathbf{r}_{ij}=1 | \theta_i, a_j, b_j] = \frac{\exp(\theta_i-b_j)}{1+\exp[a_j(\theta_i-b_j)]} = \frac{1}{1+\exp[-Da_j(\theta_i-b_j)]}$ |
| 3PL | $\mathbf{P}[\mathbf{r}_{ij}=1 | \theta_i, a_j, b_j, c_j] = c_j + (1-c_j)\frac{1}{1+\exp[-Da_j(\theta_i-b_j)]}$ |
| 4PL | $\mathbf{P}[\mathbf{r}_{ij}=1 | \theta_i, a_j, b_j, c_j, d_j] = c_j + (d_j-c_j)\frac{1}{1+\exp[-Da_j(\theta_i-b_j)]}$ |