Resolution of a ranking built from noisy risk scores

Two people with true scores \(t_1 < t_2\) are ranked by estimates that noise can move by up to \(\delta\) each. Whenever the true gap is below \(2\delta\) an admissible noise pair reverses the order (Research.P5.rank_reversal_exists); whenever it exceeds \(2\delta\) no admissible noise can (rank_stable_of_gap). The resolution of a ranking is therefore twice the noise half-width; on a probability scale with interval half-widths near one half no pair has an identified order (identified_scores_le). Given point estimates and their half-widths, the function reports which pairs are identified and the share of pairs that are not (the Baldus proportionality-review situation in Weisburd & Britt ch. 1, where most intervals covered nearly all of [0, 1]).

Usage

morie_ranking_resolution(estimate, half_width)

Arguments

estimate

Estimated scores (probabilities).

half_width

Half-width of the interval around each estimate (one number, recycled, or one per estimate).

Value

A list with n_pairs, identified_pairs (a logical matrix: order identified if the estimates differ by more than the sum of the two half-widths), share_unidentified, resolution (twice the largest half-width) and theorems.

Examples

morie_ranking_resolution(c(0.2, 0.35, 0.8), half_width = c(0.1, 0.1, 0.05))
#> $n_pairs
#> [1] 3
#> 
#> $identified_pairs
#>       [,1]  [,2] [,3]
#> [1,]    NA FALSE TRUE
#> [2,] FALSE    NA TRUE
#> [3,]  TRUE  TRUE   NA
#> 
#> $share_unidentified
#> [1] 0.3333333
#> 
#> $resolution
#> [1] 0.2
#> 
#> $theorems
#> [1] "Research.P5.rank_reversal_exists" "Research.P5.rank_stable_of_gap"  
#> [3] "Research.P5.identified_scores_le"
#>