Masters' partial credit model

The Rasch member of the partial-credit family: the slope is fixed at one, so the raw score is a sufficient statistic for theta, which is what distinguishes it from Muraki's GPCM. The category probabilities are therefore taken from the GPCM kernel evaluated at a = 1 with the step vector prefixed by a zero, rather than by writing a second softmax.

Usage

Irtprc(y, theta, delta_j)

Arguments

y

Observed categories, 0-based (0 .. m).

theta

Person abilities, same length as y.

delta_j

Step difficulties delta_1 .. delta_m.

Value

List with estimate, p_observed, probs_first, loglik, categories, n, method.

Details

Formula: P(X = k) = exp(sum_{v=1}^{k} (theta - delta_v)) / sum_{c=0}^{m} exp(sum_{v=1}^{c} (theta - delta_v)), empty sum zero.

References

Masters (1982), A Rasch model for partial credit scoring, Psychometrika 47(2):149-174. doi:10.1007/BF02296272

Examples

Irtprc(y = 5L, theta = 0.5, delta_j = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] 2.125491e-06
#> 
#> $p_observed
#> [1] 2.125491e-06
#> 
#> $probs_first
#> [1] 5.703484e-01 3.459338e-01 7.718827e-02 6.335999e-03 1.913306e-04
#> [6] 2.125491e-06 8.686395e-09 1.305947e-11 7.222987e-15
#> 
#> $loglik
#> [1] -13.06151
#> 
#> $categories
#> [1] 9
#> 
#> $n
#> [1] 1
#> 
#> $method
#> [1] "P(X=k) = exp(sum_{v<=k}(theta - delta_v)) / normaliser, Masters (1982) eq. (7)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"