In the time-varying models, living kin are stocks: they refer to the expected number of relatives alive at the beginning of the period, by the Focal’s age, the kin’s age, and calendar year. Deaths are flows over the following year of risk. In the output, those deaths are located at the year and Focal age at which the risk interval begins.
This means that, for a given kin type and kin age, the ratio between
deaths during the year and living kin at the start of the year should
reproduce the period death probability, 1 - p. The small
example below uses the one-sex time-varying model and checks this
relationship for mothers.
years_use <- c("1900", "1901", "1902")
# model
kin_mothers <- kin_time_variant(
p = swe_px[, years_use],
f = swe_asfr[, years_use],
output_period = as.integer(years_use[1:2]),
output_kin = "m"
)
#> Length of birth_female lower than years of risk. Completed with last value...
#> Running kinship model...
#> Preparing output...
# pick a kin to check
check_mothers <- subset(
kin_mothers,
year == 1900 &
kin == "m" &
age_focal == 0 &
living > 0
)
# check probabilities
check_mothers$death_over_living <- check_mothers$dead / check_mothers$living
check_mothers$period_death_prob <- 1 - swe_px[check_mothers$age_kin + 1, "1900"]
head(
check_mothers[
check_mothers$age_kin >= 15,
c("year", "age_focal", "kin", "age_kin",
"death_over_living", "period_death_prob")
],
8
)
#> year age_focal kin age_kin death_over_living period_death_prob
#> <int> <int> <char> <int> <num> <num>
#> 1: 1900 0 m 15 0.00519 0.00519
#> 2: 1900 0 m 16 0.00534 0.00534
#> 3: 1900 0 m 17 0.00604 0.00604
#> 4: 1900 0 m 18 0.00590 0.00590
#> 5: 1900 0 m 19 0.00598 0.00598
#> 6: 1900 0 m 20 0.00597 0.00597
#> 7: 1900 0 m 21 0.00634 0.00634
#> 8: 1900 0 m 22 0.00569 0.00569The columns death_over_living and
period_death_prob coincide because they describe the same
interval: deaths of mothers during 1900 divided by mothers alive at the
start of 1900. Living kin are therefore interpreted as a stock, while
deaths are interpreted as a period flow.
Because deaths are flows over the interval after a stock is observed, the final stock in a time series has no following interval inside the model. Therefore, deaths for the last input year are not available unless the model is projected one additional period.
For the case of time-invariant models, all previous is also valid but only considering age of Focal, time is not included.