Category response curves of the graded response model

The probability of a response above each boundary between adjacent categories is \(P^*_k(\theta) = 1 / (1 + \exp(-D a (\theta - b_k)))\), and the probability of each category is the difference between successive boundary curves, with \(P^*_0 = 1\) and \(P^*_{m+1} = 0\). The difficulties must increase, or some probabilities would be negative.

Usage

prob_grm(theta, a = 1.5, b = c(-2, -0.75, 0.75, 2), D = 1)

Arguments

theta

Numeric vector of abilities.

a

Discrimination.

b

Increasing difficulties of the boundaries, \(b_1, \ldots, b_m\).

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.

Examples

prob_grm(c(-2, 0, 2))
#>                1          2         3          4           5
#> [1,] 0.500000000 0.36703576 0.1170578 0.01343377 0.002472623
#> [2,] 0.047425873 0.19765914 0.5098300 0.19765914 0.047425873
#> [3,] 0.002472623 0.01343377 0.1170578 0.36703576 0.500000000
prob_grm(c(-2, 0, 2), a = 2, b = c(-1, 1))
#>                1         2           3
#> [1,] 0.880797078 0.1167303 0.002472623
#> [2,] 0.119202922 0.7615942 0.119202922
#> [3,] 0.002472623 0.1167303 0.880797078