The false positive rate the other rates imply

Chouldechova's (2017) identity: in any confusion table, the false positive rate is p / (1 - p) (1 - ppv) / ppv (1 - fnr), from the base rate p, the positive predictive value ppv and the false negative rate fnr. So two groups whose scores are equally calibrated (the same ppv) and miss equally (the same fnr) but whose base rates differ must have different false positive rates: no score, however it is built, can equalise all three. The identity is arithmetic, true of any tool on any data; it says which equalities cannot hold together, not which one should give way. See proof("impfpr").

Usage

impfpr(p, ppv, fnr)

Arguments

p

Base rates, strictly between 0 and 1.

ppv

Positive predictive values, above 0 and at most 1.

fnr

False negative rates, from 0 and below 1. The three have one length (or length 1); named ones are matched by name.

Value

The false positive rates, one per triple.

References

Chouldechova, A. (2017). Fair prediction with disparate impact: a study of bias in recidivism prediction instruments. Big Data, 5(2), 153-163. doi:10.1089/big.2016.0047

Examples

# the same ppv and fnr, different base rates: different fpr
impfpr(p = c(a = 0.3, b = 0.5), ppv = 0.6, fnr = 0.25)
#>         a         b 
#> 0.2142857 0.5000000