Category response curves of the generalized partial credit model

The probabilities of the categories \(1, \ldots, m + 1\) of an item with \(m\) step parameters, $$P(X = k \mid \theta) = \frac{\exp\left(\sum_{v=1}^{k-1} D a (\theta - \delta_v)\right)}{\sum_{h=1}^{m+1} \exp\left(\sum_{v=1}^{h-1} D a (\theta - \delta_v)\right)},$$ where an empty sum is 0. With \(a = 1\) this is the partial credit model (Masters, 1982), otherwise the generalized partial credit model (Muraki, 1992). The curves of categories \(k\) and \(k + 1\) cross at \(\theta = \delta_k\).

Usage

prob_gpcm(theta, a = 1, steps = c(-2, -0.75, 0.75, 2), D = 1)

Arguments

theta

Numeric vector of abilities.

a

Discrimination.

steps

Step parameters \(\delta_1, \ldots, \delta_m\).

D

Scaling constant: 1 keeps the logistic metric, and 1.702 makes the curve almost coincide with the normal ogive.

Value

A matrix with one row for each value of theta and one column for each category, named 1 to m + 1.

References

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Examples

prob_gpcm(c(-2, 0, 2))
#>                 1          2         3          4            5
#> [1,] 0.4338101387 0.43381014 0.1242887 0.00794551 0.0001455271
#> [2,] 0.0308444494 0.22791137 0.4824884 0.22791137 0.0308444494
#> [3,] 0.0001455271 0.00794551 0.1242887 0.43381014 0.4338101387
prob_gpcm(c(-2, 0, 2), a = 1.5, steps = c(-1, 0, 1))
#>                 1           2           3            4
#> [1,] 8.101347e-01 0.180765495 0.008999784 9.997857e-05
#> [2,] 9.121276e-02 0.408787238 0.408787238 9.121276e-02
#> [3,] 9.997857e-05 0.008999784 0.180765495 8.101347e-01