MIRT a to factor-loading conversion (and inverse)

lambda_il = a_il / sqrt(1 + a_i P a_i') (normal-ogive metric; reduces to the inverse of Reckase Eq. 2.28 for m = 1), thresholds tau_i = -d_i / sqrt(1 + a_i P a_i') per Eq. 6.12; the inverse map is a_il = lambda_il / sqrt(1 - lambda_i R lambda_i').

Usage

morie_mfird(a, d = NULL, P = NULL, inverse = FALSE)

Arguments

a: Matrix (items x m) of discriminations (or loadings when inverse).
d: Optional intercept vector (enables thresholds).
P: Optional latent covariance (default identity).
inverse: Convert loadings back to discriminations.

Value

A list with elements loadings (or discriminations), norming, thresholds, communalities, inverse, method.

References

Reckase, M. D. (2009). Multidimensional Item Response Theory. Springer.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
morie_mfird(V)
#> $norming
#> [1] 1.414214 2.236068 3.162278 4.123106 5.099020 6.082763 7.071068 8.062258
#> 
#> $thresholds
#> NULL
#> 
#> $communalities
#> [1] 0.5000000 0.8000000 0.9000000 0.9411765 0.9615385 0.9729730 0.9800000
#> [8] 0.9846154
#> 
#> $inverse
#> [1] FALSE
#> 
#> $method
#> [1] "MIRT a <-> lambda (Reckase Eq. 2.28 / Eqs. 6.11-6.12)"
#> 
#> $loadings
#>           [,1]
#> [1,] 0.7071068
#> [2,] 0.8944272
#> [3,] 0.9486833
#> [4,] 0.9701425
#> [5,] 0.9805807
#> [6,] 0.9863939
#> [7,] 0.9899495
#> [8,] 0.9922779
#>