The chance that a person experiences an event (a first arrest, a first conviction, a first jail stay, a first child protection placement) by each age, from first events and the population by age, as a synthetic cohort; with mortality as a competing risk when deaths are given.
Arguments
- age
The ages, in order.
- fireve
First events at each age.
- pop
The population at each age.
- atrisk
Optional: the persons at risk at each age, when known, in place of the synthetic-cohort assumption.
- dthrate
Optional death rates by age (probability of dying within the age).
- conflvl
Confidence level of the cumulative risk limits (default 0.95).
Details
With \(E_a\) first events at age \(a\) and \(N_a\) persons at risk (those who have not had the event), the hazard is \(h_a = E_a/N_a\). When only the population \(P_a\) is known, the persons at risk are taken as \(P_a(1 - CR_{a-1})\) (the synthetic-cohort assumption). Without mortality, \(CR_a = 1 - \prod_{x \le a}(1 - h_x)\). With death rates \(q_x\), the multiple-decrement table carries survivors \(l_{x+1} = l_x(1 - h_x - q_x)\) and \(CR_a = \sum_{x \le a} l_x h_x / l_0\), the risk of the event before death. The interval (without mortality) is Greenwood's on the log of the survival: \(\mathrm{Var}(\log S_a) = \sum_{x \le a} h_x/(N_x(1 - h_x))\).
Counting first events needs records that link a person's events over time; an agency's counts of all events overstate first events. Events of people who do not live in the area, counted against its population, bias the risk upward.
References
Bonczar TP (2003). Prevalence of imprisonment in the U.S. population, 1974-2001. Bureau of Justice Statistics, NCJ 197976.
Applied to arrests, convictions and jail in New York City: Hepburn P, Kohler-Hausmann I, Zorro Medina A (2019). Cumulative risks of multiple criminal justice outcomes in New York City. Demography 56(3):1161-1171. doi:10.1007/s13524-019-00781-7
Examples
age <- 16:25
first <- c(900, 1400, 1700, 1600, 1400, 1100, 900, 700, 600, 500)
pop <- rep(100000, 10)
cmlrisk(age, first, pop)
#> ╭── Cumulative risk of a first event ─────────────────────────────────╮
#> │ Ages 16 to 25 │
#> │ Risk by the last age 10.8% [10.6, 11.0] │
#> │ │
#> │ age hazard atrisk cumrisk interval │
#> │ --- ------- ------- ------- ------------------ │
#> │ 16 0.00900 100,000 0.00900 [0.00841, 0.00959] │
#> │ 17 0.01413 99,100 0.02300 [0.02207, 0.02393] │
#> │ 18 0.01740 97,700 0.04000 [0.03878, 0.04121] │
#> │ 19 0.01667 96,000 0.05600 [0.05457, 0.05742] │
#> │ 20 0.01483 94,400 0.07000 [0.06842, 0.07158] │
#> │ 21 0.01183 93,000 0.08100 [0.07931, 0.08269] │
#> │ 22 0.00979 91,900 0.09000 [0.08822, 0.09177] │
#> │ 23 0.00769 91,000 0.09700 [0.09516, 0.09883] │
#> │ 24 0.00664 90,300 0.10300 [0.10111, 0.10488] │
#> │ 25 0.00557 89,700 0.10800 [0.10607, 0.10992] │
#> │ │
#> │ By age 25 an estimated 10.8% of the population have had a first │
#> │ event, from the age-specific first-event hazards of a life table. │
#> ╰─────────────────────────────────────────────────────────────────────╯