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Calibration audit of a predictive-policing ranking

Source: R/calaud.R
calaud.Rd

Does the order in which an algorithm ranks areas by predicted risk match the order of what was later recorded there, and does the disagreement follow the areas' demographic make-up? calaud() ranks the areas both ways, measures the agreement by Spearman's correlation, and gives each group the mean gap between its areas' two ranks, with a permutation test. It generalises the district-level analysis of the SciencesPo predictive-policing study of Chicago (Lacherade et al. 2021) to any place: the five columns of the audit are area, risk, outcome, group and population.

Usage

calaud(
  area,
  risk = NULL,
  outcome = NULL,
  group = NULL,
  population = NULL,
  nperm = 999,
  seed = NULL
)

Arguments

area

The area of each record: labels; or a data frame with columns area, risk, outcome, group and optionally population, the other arguments then left out. A clpre list from clprep() is taken too: its elements fill the arguments by name.

risk

The risk the algorithm predicts for each record: finite numbers.

outcome

The recorded outcome of each record (a 0 or 1, or a count): finite numbers.

group

The group of each record (or area): labels.

population

NULL, or the population of each area: positive, one per record (the same within an area) or one per area named by area.

nperm

The number of permutations: 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: areas, a data frame with one row per area (sorted by label): area, n (records), risk, rate, group, riskrk, raterk and gap; groups, one row per group: group, n (areas), gap (mean) and p (NaN without areas or permutations); spear, a named vector: rho, s (Spearman's statistic, the sum of squared rank differences), pex, t and pt; ties (TRUE when either ranking has ties); between, ss and p; and nperm.

Details

Records are rolled up to areas first: an area's risk is the mean over its records, its rate the mean outcome (or, with population, the summed outcome per head), and its group the most frequent label among its records (the first in order on ties). One record per area is the per-area audit itself. Rank 1 is the highest; tied values share the mean of their ranks. An area's gap is its outcome rank minus its risk rank: positive when the algorithm ranks it as riskier than its recorded outcome warrants, negative when less risky.

Spearman's rho is Pearson's correlation of the two rankings (Spearman 1904). pex is its two-sided p-value by algorithm AS 89 (Best and Roberts 1975): exact for nine areas or fewer, from the Edgeworth series up to 1289, and NA when either ranking has ties; pt is the large-sample one from the t distribution, as stats::cor.test() gives them. A group's mean gap is tested against shuffling the groups across the areas, nperm times; between tests all groups at once by the between-group sum of squares of the gaps (the gaps sum to 0).

The audit compares the algorithm with the records, not with crime: a recorded outcome follows where the police looked as well as what happened, so a group's areas can look over-predicted because their incidents were recorded less, or under-predicted because they were recorded more (see fairsim()). A gap says the two orders disagree for that group's areas; it does not say which order is right or why.

References

Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15(1), 72-101. doi:10.2307/1412159

Best, D. J. and Roberts, D. E. (1975). Algorithm AS 89: the upper tail probabilities of Spearman's rho. Journal of the Royal Statistical Society, Series C, 24(3), 377-379. doi:10.2307/2347111

Lacherade, P., Szabo, B., Krikava, J. and Aeby, M. (2021). Predictive policing in the City of Chicago. Group project, Decoding Biases in AI, Sciences Po. https://github.com/marquito95/Predictive-policing-Chicago

Examples

set.seed(3)
d <- fairsim(4000, nareas = 24, bias = 0.5, seed = 1)
# the recorded outcome is `det`: the incidents that were detected
r <- calaud(d$area, d$risk, d$det, d$group, nperm = 499, seed = 1)
r$groups
#>   group  n gap     p
#> 1     A 12 -12 0.002
#> 2     B 12  12 0.002
r$spear
#>           rho             s           pex             t            pt 
#> -7.400000e-01  4.002000e+03  5.640811e-05 -5.160381e+00  3.575934e-05