LEdecomp (version 1.0.4)

arriaga: classic Arriaga decomposition

Description

Following the notation given in Preston et al (2000), Arriaga's decomposition method can written as: $$_{n}\Delta_{x} = \frac{l_x^1}{l_0^1}\cdot \left( \frac{_{n}L_{x}^{2}}{l_{x}^{2}} - \frac{_{n}L_{x}^{1}}{l_{x}^{1}}\right) + \frac{T^{2}_{x+n}}{l_{0}^{1}} \cdot \left( \frac{l_{x}^{1}}{l_{x}^{2}} - \frac{l_{x+n}^{1}}{l_{x+n}^{2}} \right) $$

Usage

arriaga(
  mx1,
  mx2,
  age = 0:(length(mx1) - 1),
  nx = rep(1, length(mx1)),
  sex1 = "t",
  sex2 = sex1,
  closeout = TRUE
)

Value

cc numeric vector with one element per age group, and which sums to the total difference in life expectancy between population 1 and 2.

Arguments

mx1

numeric vector of the mortality rates (central death rates) for population 1

mx2

numeric vector of the mortality rates (central death rates) for population 2

age

integer vector of the lower bound of each age group (currently only single ages supported)

nx

integer vector of age intervals, default 1.

sex1

character either the sex for population 1: Male ("m"), Female ("f"), or Total ("t")

sex2

character either the sex for population 2: Male ("m"), Female ("f"), or Total ("t") assumed same as sex1 unless otherwise specified.

closeout

logical. Default TRUE. Shall we use the HMD Method Protocol to close out the ax and qx values? See details.

Details

A little-known property of this decomposition method is that it is directional, in the sense that we are comparing a movement of mx1 to mx2, and this is not exactly symmetrical with a comparison of mx2 with mx1. Note also, if decomposing in reverse from the usual, you may need a slight adjustment to the closeout value in order to match sums properly (see examples for a demonstration).

setting closeout to TRUE will result in value of 1/mx for the final age group, of ax and a value of 1 for the closeout of qx.

References

arriaga1984measuringLEdecomp preston2000demographyLEdecomp

Examples

a <- .001
b <- .07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- arriaga(mx1, mx2, age = x)
e01 <- mx_to_e0(mx1, age = x)
e02 <- mx_to_e0(mx2, age = x)
(delta <- e02 - e01)
sum(cc)

# \donttest{
 plot(x, cc)
# }
# asymmetrical with a decomposition in the opposite direction
cc2 <- -arriaga(mx1 = mx2, mx2 = mx1, age = x)
plot(x, cc)
lines(x,cc2)
# also we lose some precision?
sum(cc);sum(cc2)
# found it!
delta-sum(cc2); cc2[length(cc2)] / 2

# But this is no problem if closeout = FALSE
-arriaga(mx1 = mx2, mx2 = mx1, age = x,closeout=FALSE) |> sum()
arriaga(mx1 = mx1, mx2 = mx2, age = x,closeout=FALSE) |> sum()