# Data frame(s) of crude estimates of quarterly (and annual) central rates of mortality estimated using shortcut 3. Computes for a general/insured population crude estimates of quarterly (and annual) central rates of mortality by age (in a set of integer ages) for each combination of age and calendar quarter, using quarterly age aggregated data of population stocks and age-season quarterly counts of deaths, immigrants/new policies (entries) and emigrants/lapses/expirations (exits) by employing shortcut 3, based on equation (2.9), proposed in Pavia and Lledo (2023). ```r crude_mx_sh3( pop.start, pop.end, events.death, events.out, events.in, annual = FALSE ) ``` ## Arguments - `pop.start`: A data frame, corresponding to the target population/portfolio, containing by age the stock of population/portfolio by quarter collected at the beginning of the year. This data frame must have three columns. The first refers to integer ages (e.g., 20, 21, 22,...), the second to the quarter (with values 1, 2, 3 and 4) and the third to the number of people in the population/portfolio at the beginning of the year corresponding to each integer age and quarter. - `pop.end`: A data frame, corresponding to the target population/portfolio, containing by age the stock of population/portfolio by quarter collected at the end of the year. This data frame must have three columns. The first refers to integer ages (e.g., 20, 21, 22,...), the second to the quarter (with values 1, 2, 3 and 4) and the third to the number of people in the population/portfolio at the end of the year corresponding to each integer age and quarter. - `events.death`: A data frame with the number of deaths recorded in the population/portfolio during the target year in a set of integer ages for each combination of age and season/calendar quarter. Typically, this is an output of the function `count_events_quarter` or a data frame with the same structure. If the range of ages in `events.death` does not cover the range of ages in the intersection of ages defined by `pop.start` and `pop.end`, zeros are imputed for the missing ages. - `events.out`: A data frame with the number of exits (emigrants/lapses/expirations) recorded in the population/portfolio during the target year in a set of integer ages for each combination of age and season/calendar quarter. Typically, this is an output of the function `count_events_quarter` or a data frame with the same structure. If the range of ages in `events.out` does not cover the range of ages in the intersection of ages defined by `pop.start` and `pop.end`, zeros are imputed for the missing ages. - `events.in`: A data frame with the number of entries (immigrants/new policies) recorded in the population/portfolio during the target year in a set of integer ages for each combination of age and season/calendar quarter. Typically, this is an output of the function `count_events_quarter` or a data frame with the same structure. If the range of ages in `events.in` does not cover the range of ages in the intersection of ages defined by `pop.start` and `pop.end`, zeros are imputed for the missing ages. - `annual`: A character string informing whether the annual crude central rates of mortality should also be computed. Default, `FALSE`. ## Returns When `annual = FALSE` a data frame with estimated crude central rates of mortality for each combination of age and calendar quarter in the set of integer ages determined by the intersection of ages in `pop.start` and `pop.end`. The data frame has the following components: - **age**: Integer age to which the crude central rate of mortality corresponds. - **quarter.age**: Age quarter to which the crude central rate of mortality corresponds. - **quarter.calendar**: Calendar (time, season) quarter to which the crude central rate of mortality corresponds. - **exposed**: Total exposure-at-risk times in the target population, calulated according to shortcut 2, for each combination of `age`, `quarter.age` and `quarter.calendar`. - **deaths**: Number of deaths in the target population for each combination of `age`, `quarter.age` and `quarter.calendar`. - **mx**: Estimated crude central rate of mortality corresponding to the combination of `age`, `quarter.age` and `quarter.calendar`. When `annual = TRUE` the output is a list with two data frames `mx.quarterly` and `mx.annual`. `mx.quarterly` is a data frame with the estimated quarterly crude central rates of mortality as just described and `mx.annual` is the corresponding data frame with the estimated annual crude central rates of mortality. ## Note It is the responsibility of the user to assure that both stocks of population (which determine the ages for which estimates are computed) and number of events correspond to the same year. ## Examples ```r # This can take a while pop <- 2005 - as.numeric(substr(pop_2006$date.birth, 1 , 4)) pop <- as.data.frame(table(pop)) pop[, 1] <- as.numeric(as.character(pop[, 1])) pop[, 2] <- pop[, 2]/4 pop <- cbind(apply(pop, 2, rep, each = 4), quarter = rep(1:4, nrow(pop)))[, c(1, 3, 2)] pop <- as.data.frame(pop) temp <- quarterly_variables(death_2006$date.birth, death_2006$date.death) e.death <- count_events_quarter(temp) temp <- quarterly_variables(emi_2006$date.birth, emi_2006$date.emi) e.emi <- count_events_quarter(temp) temp <- quarterly_variables(immi_2006$date.birth, immi_2006$date.immi) e.immi <- count_events_quarter(temp) out <- crude_mx_sh3(pop, pop, e.death, e.emi, e.immi) # Fast example pop.1 <- data.frame(age = c(rep(40, 4), rep(41, 4)), quarter = rep(1:4, 2), people = c(4134, 4353)) pop.2 <- data.frame(age = c(rep(40, 4), rep(41, 4)), quarter = rep(1:4, 2), people = c(4250, 4213)) dates.b <- c("1980-04-12") dates.d <- c("2020-08-23") x <- quarterly_variables(dates.b, dates.d) e.death <- count_events_quarter(x) dates.b <- c("1980-05-12") dates.e <- c("2020-06-23") x <- quarterly_variables(dates.b, dates.e) e.emi <- count_events_quarter(x) dates.b <- c("1980-07-12") dates.i <- c("2020-12-10") x <- quarterly_variables(dates.b, dates.i) e.immi <- count_events_quarter(x) out <- crude_mx_sh3(pop.1, pop.2, e.death, e.emi, e.immi) ``` ## References Pavia, JM and Lledo, J (2023). Shortcuts for the construction of sub-annual life tables. *ASTIN Bulletin: The Journal of the International Actuarial Association*, 53(2), 332-350. tools:::Rd_expr_doi("10.1017/asb.2023.16") ## See Also `crude_mx`, `crude_mx_sh2`. ## Author(s) Jose M. Pavia pavia@uv.es Josep Lledo josep.lledo@uv.es

crude_mx_sh3 function