Category response curves of the nominal response model

The probabilities of the unordered categories of an item, with the first category as the reference, whose exponent is 0, and each of the others with its own discrimination and difficulty (the parameterization of Baker, 1992): $$P(X = k \mid \theta) = \frac{\exp(z_k)}{\sum_h \exp(z_h)}, \quad z_1 = 0, \quad z_k = D a_k (\theta - b_k).$$

Usage

prob_nrm(theta, a = c(-4, -2, 2, 4), b = c(-4, -2, 2, 4), D = 1)

Arguments

theta

Numeric vector of abilities.

a

Discriminations of the categories after the first.

b

Difficulties of the categories after the first.

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 a, b, c, ...

References

Baker, F. B. (1992). Item Response Theory: Parameter Estimation Techniques. New York: Marcel Dekker.

Examples

prob_nrm(c(-2, 0, 2))
#>              a            b            c            d            e
#> [1,] 0.4998323 1.676751e-04 0.4998323249 0.0001676751 1.886934e-11
#> [2,] 0.9646629 1.085585e-07 0.0176684182 0.0176684182 1.085585e-07
#> [3,] 0.4998323 1.886934e-11 0.0001676751 0.4998323249 1.676751e-04