UN a(x) estimates from either M(x), q(x), or both

Source: R/nAx.R

The UN a(x) formula uses Coale-Demeny for ages 0, and 1-4, values of 2.5 for ages 5-9 and 10-14, and the Greville formula for higher ages. In the original sources these are referred to as separation factors.

lt_a_un(
  nMx,
  nqx,
  lx,
  IMR = NA,
  Age,
  AgeInt,
  a0rule = "ak",
  Sex = "m",
  region = "w",
  SRB = 1.05,
  tol = .Machine$double.eps,
  maxit = 1000,
  mod = TRUE,
  extrapLaw = "kannisto",
  extrapFrom = max(Age),
  extrapFit = Age[Age >= 60],
  ...
)

Arguments

nMx

numeric. Event exposure mortality rates.
nqx

numeric. Vector of age specific death probabilities in standard abridged age groups.
lx

numeric. Vector of lifetable survivorship in standard abridged age groups.
IMR

numeric. Optional. q0, the death probability in first year of life, in case available separately.
Age

integer. A vector of ages of the lower integer bound of the age classes.
AgeInt

integer. Vector of age interval widths.
a0rule

character. Either "ak" (default) or "cd".
Sex

character. "m", "f" or "b" for male, female, or both.
region

character. "n", "e", "s" or "w" for North, East, South, or West.
SRB

numeric. The sex ratio at birth (boys/girls), default 1.05.
tol

numeric. The tolerance for the qx-based iterative method. Default .Machine$double.eps.
maxit

integer. The maximum number of iterations for the qx-based iterative method. Default 1000.
mod

logical. Whether or not to use Gerland's modification for ages 5-14. Default TRUE.
extrapLaw

character. If extrapolating, which parametric mortality law should be invoked? Options include "Kannisto", "Kannisto_Makeham", "Makeham","Gompertz", "GGompertz", "Beard", "Beard_Makeham", "Quadratic". Default "Kannisto". See details.
extrapFrom

integer. Age from which to impute extrapolated mortality.
extrapFit

integer vector. Ages to include in model fitting. Defaults to all ages > =60.
...

Other arguments to be passed on to the MortalityLaw function.

Value

nax average contribution to exposure of those dying in the interval.

Details

a(x) for age 0 and age group 1-4 are based on Coale-Demeny q0-based lookup tables. If the main input is nMx, and if IMR is not given, we first approximate q0 for the Coale-Demeny approach before applying the formula. The final a(x) value is closed out using the lt_a_closeout() method (reciprocal and Mortpak methods are deprecated). For nMx inputs this method is rather direct, but for qX or l(x) inputs it is iterative. Age groups must be standard abridged. No check on age groups are done.

References

Greville TN (1977). “Short methods of constructing abridged life tables.” In Mathematical Demography, 53--60. Springer. United Nations (1982). Model Life Tables for Developing Countries. United Nations Department of International Economic and Social Affairs. Arriaga EE, Johnson PD, Jamison E (1994). Population analysis with microcomputers, volume 1. Bureau of the Census. United Nations (1988). Mortpak- The United Nations Software Package for Mortality Measurement. United Nations Department of International Economic and Social Affairs.

Examples

# example data from UN 1982 Model Life Tables for Developing Countries.
# first Latin American model table for males (p. 34).
Mx <- c(.23669,.04672,.00982,.00511,.00697,.01036,.01169,
    .01332,.01528,.01757,.02092,.02517,.03225,.04241,.06056,
    .08574,.11840,.16226,.23745)
ax <- c(0.330,1.352,2.500,2.500,2.633,2.586,2.528,2.528,
    2.526,2.529,2.531,2.538,2.542,2.543,2.520,2.461,2.386,2.295,4.211)

AgeInt     <- inferAgeIntAbr(vec = Mx)
Age <- int2age(AgeInt)
nAx1       <- lt_a_un(nMx = Mx,
                   Age = Age,
                    AgeInt = AgeInt,
                    a0rule = "cd",
                    Sex = "m",
                  region = "w",
                    mod = FALSE)
nAx2       <- lt_a_un(nMx = Mx,
                   Age = Age,
                    AgeInt = AgeInt,
                    a0rule = "cd",
                    Sex = "m",
                  region = "w",
                    mod = TRUE)
# this is acceptable...
round(nAx2,3) - ax # only different in ages 5-9 and 10-14, and last two ages

#>  [1]  0.000  0.000 -0.506 -0.082  0.000  0.000  0.000  0.000  0.000  0.000
#> [11]  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.012 -0.566
# ignore open age, which is treated differently
N <- length(ax)
# default unit test...
stopifnot(all(round(nAx1[Age<80],3) - ax[Age<80] == 0)) # spot on

# another example:

nMx <- c(0.11621,0.02268,0.00409,0.00212,0.00295,
         0.00418,0.00509,0.00609,0.00714,0.00808,
         0.00971,0.0125,0.0175,0.02551,0.03809,
         0.05595,0.0809,0.15353,0.2557)
AgeInt <- inferAgeIntAbr(vec = nMx)
Age    <- int2age(AgeInt)
nAx1   <- lt_a_un(nMx, 
                  Age = Age,
                  AgeInt = AgeInt, 
                  a0rule = "ak",
                  mod = TRUE, 
                  closeout = TRUE)

nqx    <- lt_id_ma_q(nMx = nMx, 
                     nax = nAx1, 
                     AgeInt = AgeInt, 
                     closeout = FALSE)
nAx2 <- lt_a_un(nqx = nqx, 
                Age = Age,
                AgeInt = AgeInt, 
                a0rule = "ak", 
                mod= TRUE, 
                closeout = TRUE)

stopifnot(all(abs(nAx1[Age<75] - nAx2[Age<75]) < 1e-7))