World Population Prospects 2024 provides estimated mortality for a reference period. This example uses the published survivor counts for Colombia in 2023, measured per 100,000 initial lives. The five-year survival probability from age 65 uses the ratio of the published counts at ages 70 and 65.
col <- subset(mortality_regions, location_code == 170L & sex == "both",
select = c(x, lx))
t_px(col, x = 65, t = 5)## [1] 0.933984The rates in the source are in base 1. Rounding can cause slight discrepancies between the published probabilities and survivor counts; using only x and lx makes the valuation basis explicit. The original 100+ interval is open: its qx of 1 is not a one-year probability and does not justify a whole-life extrapolation. See the dataset help for the published aggregate dx discrepancy at 100+.
MASS::Melanoma contains 205 patients from Andersen, Borgan, Gill and Keiding (1993). Status 1 indicates death from melanoma, 2 indicates alive at last follow-up (right-censored), and 3 indicates death from other causes. The observations are read from MASS when installed; no copy is included.
At each death time, divide deaths by cause by the number at risk just before that time. Multiply those increments by the probability of remaining alive before the time, then accumulate by cause. Patients censored at that time are in its risk set and are removed afterwards.
d <- MASS::Melanoma
times <- sort(unique(d$time[d$status != 2L]))
risk <- vapply(times, function(t) sum(d$time >= t), 0L)
death <- sapply(c(melanoma = 1L, other = 3L), function(cause) {
vapply(times, function(t) sum(d$time == t & d$status == cause), 0L)
})
increment <- death / risk
survival <- cumprod(1 - rowSums(increment))
before <- c(1, head(survival, -1))
incidence <- apply(increment * before, 2, cumsum)
tail(data.frame(days = times, survival, incidence), 1)## days survival melanoma other
## 70 3458 0.5553354 0.3387175 0.1059471These are cumulative incidences on follow-up time, with censoring accounted for. They are not annual age-specific probabilities for md_table(). For an example of that input, use the package’s synthetic multiple_decrement_sample with its three cause columns and close = FALSE.