Skip to contents

Differences between groups in a risk tool's decisions

Source: R/fairdf.R
fairdf.Rd

How the share of cases a tool flags (predicts positive) differs between groups, as a difference and as a ratio against a base group, each with an interval; with the outcomes, how its true and false positive rates differ; and the bias amplification score, the largest parity gap times the Gini coefficient of the groups' shares.

Usage

fairdf(
  pred,
  grp = NULL,
  y = NULL,
  base = NULL,
  conf = 0.95,
  nboot = 0,
  seed = NULL
)

Arguments

pred

The tool's decisions: 1 (or TRUE) for a case it flags, 0 (or FALSE) otherwise, one per case. A clpre list from clprep() is taken too: its elements fill the arguments by name.

grp

Group labels: numbers, text or a factor, one per case, no missing values. Groups are ordered as the factor's levels, or sorted.

y

NULL, or the outcomes: 0 or 1 (or FALSE/TRUE), one per case.

base

The base group: NULL for the group with the largest share flagged (the first in order on ties), its label, or its position from 1.

conf

The confidence level of the intervals, strictly between 0 and 1.

nboot

The bootstrap replicates for the intervals of the Gini coefficient and the score: a whole number from 0 (none).

seed

NULL to draw the seed of the bootstrap (or the permutations) from R's generator, so that set.seed() fixes it, or one whole number from 0.

Value

A list: tab, a data frame with one row per group (in the order of grp): group, n, npos (cases flagged), ppos (their share) with lo and hi, dp with dplo and dphi, di with dilo and dihi, and with y also tpr, fpr, dtpr (dtprlo, dtprhi), dfpr (dfprlo, dfprhi), aod (aodlo, aodhi) and eo; the base group's own intervals are NaN. sum, a named vector: di (the smallest finite ratio of another group), dp (the gap of largest size, signed), gini, bas, ginilo, ginihi, baslo, bashi (NaN when nboot is 0) and with y also aod and eo (those of largest size, signed). base, the base group's label.

Details

fair() reports each group's rates and their ratios; fairdf() reports the comparisons the fairness literature names. The parity gap dp is a group's share flagged minus the base group's (demographic parity holds when every gap is 0), with Newcombe's (1998) hybrid score interval. The ratio di is the disparate impact ratio, with the log interval of Katz et al. (1978); under the four-fifths rule a ratio below 0.8 against the group with the highest selection rate is the usual sign of adverse impact, so by default that group is the base. With the outcomes y, dtpr and dfpr are the gaps in true and false positive rates (both 0 is equalized odds), aod their mean (the average odds difference) with the interval of Zou and Donner (2008) built from the two Newcombe intervals, and eo the larger of the two gaps in size. The bias amplification score of Barman and Barman (2026) is large only when a sizeable gap comes with unequal shares across all the groups; its bootstrap interval, and the Gini coefficient's, are percentile intervals from redrawing each group's decisions with replacement (Efron and Tibshirani 1993), the base group held.

These numbers describe what the tool did to each group. They do not say why: a gap in shares flagged can come from a gap in the outcomes the tool was built to predict, from how those outcomes were recorded, or from the tool; and when the groups' base rates differ, equal error rates and equal predictive values cannot all hold at once (Chouldechova 2017, see impfpr()). A ratio's interval is NaN when either share is 0, and a ratio against a base group with no case flagged is Inf (or NaN for 0 over 0).

References

Newcombe, R. G. (1998). Interval estimation for the difference between independent proportions: comparison of eleven methods. Statistics in Medicine, 17(8), 873-890. doi:10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I

Katz, D., Baptista, J., Azen, S. P. and Pike, M. C. (1978). Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics, 34(3), 469-474. doi:10.2307/2530610

Zou, G. Y. and Donner, A. (2008). Construction of confidence limits about effect measures: a general approach. Statistics in Medicine, 27(10), 1693-1702. doi:10.1002/sim.3095

Barman, P. K. and Barman, P. K. (2026). Spatial disparities in simulated patrol detection: a reproducible generative-model audit across Baltimore and Chicago. doi:10.48550/arXiv.2603.18987

Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall. doi:10.1007/978-1-4899-4541-9

Examples

set.seed(1)
g <- rep(c("a", "b"), each = 200)
y <- rbinom(400, 1, 0.3)
p <- rbinom(400, 1, ifelse(g == "a", 0.4, 0.25))
r <- fairdf(p, g, y, nboot = 199, seed = 1)
r$tab[, c("group", "ppos", "dp", "dplo", "dphi", "di", "dilo", "dihi")]
#>   group  ppos     dp       dplo        dphi        di      dilo      dihi
#> 1     a 0.435  0.000        NaN         NaN 1.0000000       NaN       NaN
#> 2     b 0.260 -0.175 -0.2640056 -0.08182849 0.5977011 0.4507581 0.7925463
r$sum
#>           di           dp         gini          bas       ginilo       ginihi 
#>  0.597701149 -0.175000000  0.125899281 -0.022032374  0.059494196  0.194227687 
#>        baslo        bashi          aod           eo 
#> -0.054404735 -0.004745647 -0.159044465 -0.202023149