Information and targeting of a paired-comparison design

Calculates the Fisher information supplied by the observed comparison design. For location difference \(d=\beta_a-\beta_b\), one dichotomous comparison contributes $$I(d)=P(a\succ b)\{1-P(a\succ b)\}.$$ For an ordered comparison, the contribution is the variance of the response score. Information is summed over the comparisons involving each object, including replication counts. Observed-design information retains the fitted position and dependence effects; it is conditional on the recorded comparison history.

Usage

btl_information(fit)

Arguments

fit

A paired-comparison fit from btl.

Value

A list of class "rasch_btl_info": objects (per object: location, the fit's se, n_comparisons, the design information, and se_naive); pairs (per observed pair: n, the mean location gap, and the pair's information); comparisons (per comparison: the signed gap, weight, and the single-comparison information); the scalar total information; m; the clustered flag; and notes.

Details

se_naive = 1/sqrt(information) treats each object's comparisons in isolation. It is a description of the design, not the fitted standard error or a bound on it. The fitted standard error also reflects joint estimation, the identifying constraint, and judge clustering. The fitted model must have converged.

References

Pollitt, A. (2012). The method of adaptive comparative judgement. Assessment in Education, 19(3), 281-300.

See also

plot_btl_targeting, btl_next_pairs

Examples

set.seed(1)
beta <- c(A = -1, B = -0.3, C = 0.4, D = 0.9)
pr <- t(combn(names(beta), 2))
d <- data.frame(a = rep(pr[, 1], each = 30), b = rep(pr[, 2], each = 30))
d$win <- ifelse(runif(nrow(d)) < plogis(beta[d$a] - beta[d$b]), d$a, d$b)
btl_information(btl(d, "a", "b", "win"))
#> Paired-comparison design information: 4 objects, total 30.04
#> One-comparison Fisher information about the location gap (dichotomous: P(1 - P))
#>  object location    se n_comparisons information se_naive
#>       A   -1.238 0.214            90      12.487    0.283
#>       B   -0.354 0.186            90      17.100    0.242
#>       C    0.448 0.180            90      17.030    0.242
#>       D    1.144 0.209            90      13.463    0.273
#> Note: se is the Godambe sandwich standard error; se_naive = 1/sqrt(information) is a design-only yardstick (the object's comparisons treated in isolation), not a bound -- the fitted se can sit below or above it