Generalized partial credit model

Muraki's model exponentiates the running sum of the step logits. The v = 0 term is common to every numerator and cancels, which is why the first step parameter is arbitrary; it is kept so the printed formula is followed literally.

Usage

Gpcm(y, theta, a, b_j)

Arguments

y

Observed categories, 0-based, one per person.

theta

Person abilities, same length as y.

a

Item slope, positive.

b_j

Step parameters; their count sets the category count.

Value

List with estimate (mean probability of the observed response), p_observed, probs_first, loglik, categories, n, method.

Details

Formula: P_k = exp(sum_{v=0}^{k} a (theta - b_v)) / sum_c exp(sum_{v=0}^{c} a (theta - b_v)).

References

Muraki (1992), A generalized partial credit model, Applied Psychological Measurement 16(2):159-176. doi:10.1177/014662169201600206

Examples

Gpcm(y = c(0, 1, 2, 1), theta = c(0, 0.5, -0.5, 1), a = 1.2, b_j = c(-1, 0, 1))
#> $estimate
#> [1] 0.3502939
#> 
#> $p_observed
#> [1] 0.4345570 0.4767300 0.0553317 0.4345570
#> 
#> $probs_first
#> [1] 0.434557 0.434557 0.130886
#> 
#> $loglik
#> [1] -5.302071
#> 
#> $categories
#> [1] 3
#> 
#> $n
#> [1] 4
#> 
#> $method
#> [1] "GPCM eq. (1) of Muraki (1992)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"