EAP ability estimate with its posterior sd and Fisher information

The posterior sd shrinks towards the prior and stays finite for an all-correct pattern; 1/sqrt(I(theta)) is the frequentist error of the ML estimate and diverges there. Both are returned. The quadrature grid is FIXED (trapezoid), never adaptive.

Usage

Eapinfo(
  items,
  x,
  D = 1,
  prior_mean = 0,
  prior_sd = 1,
  lower = -4,
  upper = 4,
  nqp = 33
)

Arguments

items

Item parameters (a, b, c, d), one row per item.

x

Responses in {0, 1}.

D

Scaling constant.

prior_mean, prior_sd

Normal prior on theta.

lower, upper

Quadrature range.

nqp

Number of quadrature points.

Value

List with estimate, se, information, se_ml, item_information, prob, J, nqp.

Details

Formula: e = exp(D a (theta - b)); P = c + (d - c) e/(1 + e); dP = D a e (d - c)/(1 + e)^2; I_j = dP^2/(P(1 - P)); EAP = int theta pi L dtheta / int pi L dtheta

References

Verified against the reference implementation in the CRAN package catR 3.17 (Magis & Raiche), functions Pi, Ii and eapEst. catR implements the procedures of van der Linden & Pashley, Item selection and ability estimation in adaptive testing, in Elements of Adaptive Testing (2010), which this row cites; that chapter was NOT obtainable, so the package source is used as the reference.

Examples

items <- matrix(c(1, 0, 0.2, 0.95, 1.2, -0.5, 0.1, 0.98, 0.8, 1, 0.15, 0.9),
                3, 4, byrow = TRUE)
Eapinfo(items, x = c(1, 0, 1))
#> $estimate
#> [1] -0.05527055
#> 
#> $se
#> [1] 0.8858893
#> 
#> $information
#> [1] 0.4784865
#> 
#> $se_ml
#> [1] 1.445657
#> 
#> $item_information
#> [1] 0.14279697 0.26782008 0.06786943
#> 
#> $prob
#> [1] 0.5646394 0.6546991 0.3754856
#> 
#> $J
#> [1] 3
#> 
#> $nqp
#> [1] 33
#> 
#> $method
#> [1] "EAP with posterior sd and test information"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"