Warm's weighted likelihood estimates by raw score

Computes the weighted likelihood estimate (WLE) of person location for every possible raw score on a set of items, with standard errors. WLE estimates are finite at the extreme (zero and maximum) scores, unlike the maximum likelihood estimate.

Usage

person_wle(tau_list, disc = 1)

Arguments

tau_list: List of per-item threshold vectors.
disc: Common discrimination (frame unit) of the items; with a constant discrimination the raw score remains sufficient.

Value

A list with theta and se, each named by raw score.

Details

For raw score $R$, let $E(\theta)$, $V(\theta)$, and $\mu_3(\theta)$ be the sums of the item expected scores, variances, and third central moments. The estimate solves Warm's weighted score equation $$R-E(\theta)+\frac{\mu_3(\theta)}{2V(\theta)}=0.$$ When the equation has several solutions, competing maxima are compared using log likelihood plus one half log information. Equal maxima use the lower location; their average need not maximize the weighted likelihood. With common discrimination $d$, its explicit multiplier cancels from this equation, although the moments are evaluated under $d$. The reported standard error is $$\operatorname{SE}(\hat{\theta})= \{d^2V(\hat{\theta})\}^{-1/2}.$$

References

Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427–450.

Zhang, J. (2005). Bias correction for the maximum likelihood estimate of ability. ETS Research Report RR-05-15.

Examples

person_wle(list(c(-1, 0), c(-0.5, 0.5), c(0, 1)))
#> $theta
#>          0          1          2          3          4          5          6 
#> -2.4444062 -1.2371268 -0.5611339  0.0000000  0.5611339  1.2371268  2.4444062 
#> 
#> $se
#>         0         1         2         3         4         5         6 
#> 1.5455221 0.9731988 0.8347598 0.8003924 0.8347598 0.9731988 1.5455221 
#>