The probability of a correct (or keyed) response to a dichotomous item,
$$P(X = 1 \mid \theta) = c + \frac{d - c}{1 + \exp(-D a (\theta - b))},$$
which is the Rasch model when \(a = 1\), \(c = 0\) and \(d = 1\), the
2PL when \(c = 0\) and \(d = 1\), and the 3PL when \(d = 1\).
Usage
prob_4pl(theta, a = 1, b = 0, c = 0, d = 1, D = 1)
Arguments
- theta
Numeric vector of abilities.
- a
Discrimination.
- b
Difficulty (location).
- c
Lower asymptote (guessing).
- d
Upper asymptote (inattentiveness).
- D
Scaling constant: 1 keeps the logistic metric, and 1.702 makes the
curve almost coincide with the normal ogive.
Value
A numeric vector of probabilities, one for each value of
theta.
Examples
prob_4pl(c(-2, 0, 2))
#> [1] 0.1192029 0.5000000 0.8807971
prob_4pl(c(-2, 0, 2), a = 1.5, b = 0.5, c = 0.2, D = 1.702)
#> [1] 0.2013504 0.3745173 0.9829930