LEdecomp (version 1.0.4)

chandrasekaran_III: Chandrasekaran III decomposition

Description

Implements the Chandrasekaran III decomposition as described in Ponnapalli (2005), which combines multiple directional effects into a symmetric average. The method constructs a decomposition of the difference in life expectancy into four parts: the main effect, the operative effect, their average (exclusive effect), and a non-linear interaction term. These are calculated based on life table values. Let \(e_x^i\) denote remaining life expectancy at age \(x\) for population \(i\), and \(l_x^i\) the number of survivors to age \(x\). Then:

• Main effect: $$ \frac{l_x^1}{l_x^2} \left[ l_x^2 (e_x^2 - e_x^1) - l_{x+n}^2 (e_{x+n}^2 - e_{x+n}^1) \right] $$

• Operative effect: $$ \frac{l_x^2}{l_x^1} \left[ l_x^1 (e_x^2 - e_x^1) - l_{x+n}^1 (e_{x+n}^2 - e_{x+n}^1) \right] $$

• Exclusive effect: $$ \frac{\text{Main effect} + \text{Operative effect}}{2} $$

• Interaction effect: $$ (e_{x+n}^2 - e_{x+n}^1) \cdot \frac{1}{2} \left[ \frac{l_x^1 \cdot l_{x+n}^2}{l_x^2} + \frac{l_x^2 \cdot l_{x+n}^1}{l_x^1} - (l_{x+n}^1 + l_{x+n}^2) \right] $$

The final contribution by age group is the sum of exclusive and interaction effects.

Usage

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

Value

Numeric vector of contributions by age group that sum to the total difference in life expectancy between the two mortality schedules.

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

This decomposition gives numerically identical results to arriaga_sym(), lopez_ruzicka_sym(), and chandrasekaran_II(), despite conceptual differences in their derivation. Included here for methodological completeness.

References

Ponnapalli2005LEdecomp

See Also

chandrasekaran_II, arriaga_sym, lopez_ruzicka_sym

Examples

a <- 0.001
b <- 0.07
x <- 0:100
mx1 <- a * exp(x * b)
mx2 <- a/2 * exp(x * b)
cc <- chandrasekaran_III(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, type = "l")
# }