Incapacitation: crime rate, share prevented and the marginal year

An offender commits crimes at rate lam a year while free, and each crime leads to a custodial sentence of s years with probability q. Over a career the crime rate is lam / (1 + lam q s): the share of offending prevented, lam q s / (1 + lam q s), never reaches 1, longer or surer sentences lower the rate, and one more year of sentence prevents lam^2 q / ((1 + lam q s) (1 + lam q (s + 1))) crimes a year, less for each year added. A sentence uniform across offenders prevents a larger share of the offending of those with the higher rate. The model holds the rate fixed over the career and replaces nobody; see incapr() for both. See proof("incap").

Usage

incap(lam, q, s, share = NULL)

Arguments

lam

Offending rates while free (crimes a year): positive numbers, one per group, as a numeric vector, a 1-d table, a list of single numbers or a one-row data frame.

q

The chance a crime leads to a custodial sentence: numbers in the interval from 0 (excluded) to 1, recycled to the length of lam; one per group, in the forms of lam and matched to it by name when both are named (unnamed, by position: only names protect against a mismatch).

s

Sentence lengths in years: non-negative numbers, recycled; one per group, in the forms of lam, matched by name.

share

NULL, or one non-negative population share per group, summing to 1, in the forms of lam, matched by name.

Value

A data frame with one row per group: group, lam, q, s, rate (the long-run crime rate), prev (the share prevented) and marg (the crimes a year one more year of sentence prevents). With share, a last row all holds the weighted free rate in lam, the weighted rate in rate and the share of all offending prevented in prev.

References

Avi-Itzhak, B. and Shinnar, R. (1973). Quantitative models in crime control. Journal of Criminal Justice, 1(3), 185-217. doi:10.1016/0047-2352(73)90061-5

Examples

incap(lam = c(low = 2, high = 10), q = 0.1, s = 1, share = c(0.8, 0.2))
#>   group  lam   q  s     rate      prev      marg
#> 1   low  2.0 0.1  1 1.666667 0.1666667 0.2380952
#> 2  high 10.0 0.1  1 5.000000 0.5000000 1.6666667
#> 3   all  3.6  NA NA 2.333333 0.3518519        NA