Samejima graded response model

The model is stated through the cumulative operating characteristics; the category probability is a difference of two of them, so the thresholds must strictly increase or a probability goes negative. The constraint is checked rather than assumed.

Usage

Grmsam(y, theta, a, b_k)

Arguments

y

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

theta

Person abilities, same length as y.

a

Item slope, positive.

b_k

The m strictly increasing thresholds.

Value

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

Details

Formula: P*_k = 1/(1 + exp(-a(theta - b_k))), P*_0 = 1, P*_{m+1} = 0, P_k = P*_k - P*_{k+1}.

References

Samejima (1969), Estimation of latent ability using a response pattern of graded scores, Psychometrika Monograph Supplement 34(4, Pt. 2). doi:10.1007/BF03372160

Examples

Grmsam(y = c(0, 1, 2, 1), theta = c(0, 0.5, -0.5, 1), a = 1.2, b_k = c(-1, 0, 1))
#> $estimate
#> [1] 0.2011907
#> 
#> $p_observed
#> [1] 0.2314752 0.2124926 0.2124926 0.1483025
#> 
#> $probs_first
#> [1] 0.2314752 0.2685248 0.2685248 0.2314752
#> 
#> $loglik
#> [1] -6.469479
#> 
#> $categories
#> [1] 4
#> 
#> $n
#> [1] 4
#> 
#> $method
#> [1] "P_k = P*_k - P*_{k+1} with logistic P*, Samejima (1969)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"