In jschoeley/lifequal: Calculate Lifespan Equality from Life-tables

How to install?

You can install lifequal by running:

install.packages("devtools")
devtools::install_github("jschoeley/lifequal")

What does it do?

lifequal lets you calculate three measures of lifespan equality from a life-table:

1) ExDagger(x, ex, ax) Life expectancy lost by those who die in age interval [x, x+w) 2) EDagger(dx, exdagger, radix) Total life expectancy lost due to death 3) KeyfzEntro(edagger, e0) Keyfitz's entropy

They are defined as follows,

---

Measure Definition

---

Start of age interval $x$

Width of age interval starting at $x$ $w_x$

Start of last age interval $\omega$

Average time spent in age interval $a_x$ $[x, x+wx)$ when dying in that interval

Deaths in age interval $[x, x+w_x)$ $d_x$

Life-expectancy at age $x$ $e_x$

Life expectancy lost due to death in age $$e_x^\dagger = \begin{cases} \frac {a_x} {w_x}e_{x+w_x} + (1-\frac {a_x} {w_x}) e_x & x \neq \omega \ \frac{e_\omega + 1.4}{2} & x = \omega \ \end{cases}$$ interval $[x, x+w_x)$

Total life expectancy lost due to death $e^\dagger = \sum_{x=0}^{\omega} d_x e_x^\dagger$

Keyfitz's Entropy $\frac{e^\dagger} {e_0}$

Life expectancy versus lifespan equality for 1x1 Swedish life-tables

library(lifequal)
library(dplyr)
library(ggplot2)

The analysis starts with a demographic life-table. We want 1) age intervals ordered from low to high, 2) no gaps between subsequent age intervals, 3) an open last age interval (e.g. 110+). Something like this:

# Swedish 1x1 period life-tables by period and sex
sweden1x1

First, we use ExDagger() on each single life-table (separate by period and sex) to calculate the life expectancy lost in each age. We then summarise each life-table into a set of 3 numbers: Life expectancy at birth, total life years lost due to death (EDagger()) and lifespan equality (KeyfzEntro()). Note that we transform Keyfitz's Entropy by taking the negative log.

sweden1x1 %>%
  # ...for each single life-table...
  group_by(period, sex) %>%
  #...we calculate the life years lost in age x...
  mutate(exdagger = ExDagger(x, ex)) %>%
  # ...and then summarise each life-table into a set of 3 numbers:
  # e0:         Life-expectancy at birth
  # edagger:    Total life years lost due to death
  # keyfzentro: Lifespan equality
  summarise(
    e0         = ex[x == 0],
    edagger    = EDagger(dx, exdagger, radix = 100000),
    keyfzentro = -log(KeyfzEntro(edagger, e0))
  ) -> sweden1x1summary

The summarised life-tables look like this:

sweden1x1summary

For each life-table we plot the life expectancy at birth versus the lifespan equality.

plot_lifequal <-
  ggplot(sweden1x1summary, aes(x = e0, y = keyfzentro, color = sex)) +
  geom_point(size = 0.6, alpha = 0.5) +
  theme(aspect.ratio = 1)
plot_lifequal

Life expectancy versus lifespan equality for 5x5 Swedish life-tables

The same exercise as before, only now we deal with life-tables aggregated over multiple year period and age intervals.

# Swedish 5x5 period life-tables by period and sex
sweden5x5

The width of each age interval, $w_x$, is calculated by the ExDagger() function on the basis of the differences between the age interval starting points x.

sweden5x5 %>%
  # ...for each single life-table...
  group_by(period, sex) %>%
  #...we calculate the life years lost in age interval [x, x+wx)...
  mutate(exdagger = ExDagger(x, ex, ax)) %>%
  # ...and then summarise each life-table into a set of 3 numbers:
  # e0:         Life-expectancy at birth
  # edagger:    Total life years lost due to death
  # keyfzentro: Lifespan equality
  summarise(
    e0         = ex[x == 0],
    edagger    = EDagger(dx, exdagger, radix = 100000),
    keyfzentro = -log(KeyfzEntro(edagger, e0))
  ) -> sweden5x5summary

The summarised life-tables look like this:

sweden5x5summary

The aggregated life-tables follow the same trend as the single year life-tables.

plot_lifequal +
  geom_point(shape = 22, fill = "white",
             data = sweden5x5summary)

jschoeley/lifequal documentation built on May 20, 2019, 2:07 a.m.