In alysonvanraalte/LifeIneq: Calculate indices of lifespan variation
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.width = 6,
fig.height = 6,
fig.align = "center",
tidy.opts = list(width.cutoff = 60), tidy = TRUE
)
Introduction
How unequally distributed are the ages at death within a population? There are many indices around to answer this question, all generally going under the umbrella terms lifespan inequality, lifespan variation, age-at-death variation, compression of mortality, or rectangularization of the survival curve.
In this R package we have implemented code to calculate such indices from life tables. In what follows, we will go through how to use the package with example Canadian and Hungarian life table data from the Human Mortality Database [@barbieri2015; @HMD2023].
For a good overview of the mathematical properties of the indices, as well as comparisons of how they perform when applied to empirical data, we suggest consulting the following [@wilmoth1999; @shkolnikov2003; @vanraalte2013; @wrycza2015].
Background
Choosing a measure of lifespan inequality
Variation is a broad concept. Since it is often used to make judgements on the inequality of a distribution, different indices have been developed which differ in their mathematical properties and sensitivities to different parts of the distribution. As Paul @allison1978 (p. 865) noted,"The choice of an inequality measure is properly regarded as a choice among alternative definitions of inequality rather than a choice among alternative ways of measuring a single theoretical construct." Here we outline some of the more important considerations.
Absolute versus relative inequality
Indices of lifespan variation measure absolute or relative inequality in ages at death. Absolute measures are scale invariant, meaning that their values do not change when the same life years are added to everyone's lifespan (i.e. the distribution is shifted to higher ages). Relative measures are translation invariant, meaning that they do not change when adding the same proportion of life to everyone's lifespan. If an age-at-death distribution shifts to the right with increasing life expectancy, absolute inequality stays unchanged, but relative inequality declines.
Absolute measures are generally measured in years, making them easier to interpret, but relative measures have their own advantages. Relative measures are preferred by biologists to compare species with very different life histories and life expectancies [@eakin1995; @baudisch2011; @colchero2016]. This is less of an issue in comparing contemporary human populations, either across or within countries. In the end, we feel that the choice between measuring absolute versus relative inequality is a normative one. If there is no inherent reason for preferring absolute or relative inequality, it is good practice to present both.
Attainment versus shortfall distribution
Relative measures of lifespan inequality have an additional specification: whether the inequality is relative to an average achieved age at death (e.g. from an attainment distribution) or to a remaining life expectancy (e.g. from a shortfall distribution). When following a cohort from birth, the means of the two distributions are the same---average age at death equals remaining life expectancy at birth.
Often though, we want to measure the lifespan inequality of populations or groups based on a conditional age-at-death distribution---in other words, a distribution with a starting age greater than zero, where the lifespan inequality is calculated conditional on surviving to that starting age. As an example, lifespan inequality by level of completed education can only be defined from early adulthood.
In such cases, should inequality be made relative to the average age at death, conditional upon survival to age $x$, or to the average remaining life expectancy at age $x$? So far, few researchers have paid attention to this difference. There are plenty of studies based on conditional distributions, including our own, but no reflection on how the choice of denominator (the distribution type) could impact the result.
Indeed, a quick survey of the literature to date hints that the choice of conditional distribution type seems to mostly depend on the measure used rather than any theoretical justification. The coefficient of variation, the Theil Index, and the Mean Log Deviation are generally calculated from the achieved age at death distribution [@hakkert1987; @vanraalte2012; @permanyer2019; @shkolnikov2010; @nemeth2017; @permanyer2019; @seaman2020]. The Keyfitz-Leser life table entropy measure is instead calculated from shortfall distributions [@bramajo2022]. The Gini coefficient, when calculated from the Kendall and Stuart definition, has used attained ages [@permanyer2019], while when calculated based on the Hanada formulation has used remaining life expectancy [@shkolnikov2003].
Is there any reason to prefer one concept over the other? We ourselves cannot decide, and prefer to leave this debate to play out in the literature. All of the relative measures of lifespan variation included in the LifeIneq package can be calculated in both ways, and we have added an argument to relative measures to make this choice explicit. In one of the empirical examples of this vignette, we demonstrate how important this distinction is.
Built-in indices of variation
Currently, the LifeIneq package contains the following indices of lifespan inequality (listed alphabetically):
aid the Absolute Inter-individual Difference, also known as the absolute Gini Coefficientcov the Coefficient of Variation of survival or remaining lifecp Kannisto's C measures---the shortest distance between two ages containing $p$ percent of the life table cohort's deaths.edag the life disparity $e^{\dagger}$ measure, also called the average life years lost due to deatheta_dag the mean age at death lost at death $\eta^{\dagger}$ (newly derived for this package--equivalent to edag but calculated by weighting the distribution of ages at death instead of remaining life by the death density)drewnowski the Drewnowski index, which is the complement of the Gini coefficientgini the Gini coefficient of survival or remaining life H the Leser-Keyfitz life table index, also called the life table entropy, which is more properly described as the elasticity of life expectancy to proportional change in mortality. It is also the relative inequality index of $e^{\dagger}$ [@mitra1978, @vaupel1986, @goldman1986].iqr the interquartile range of the age-at-death distribution, e.g. the difference between the 75th and 25th percentile of survival ages in the life table cohortmad the mean absolute deviation in lifetable ages at death. This may be with respect to either the mean or median age at deathmld the mean log deviation of survival or remaining lifequantile and quantile_lower are quantiles of survivorship (the former is a vector made conditional upon survival to each age, the latter is a scalar). These are not inequality measures by themselves, generally, but are included for users to design own quantile-based measures beyond the iqr and cp measures.sd the standard deviation in ages at deaththeil the Theil index of survival or remaining lifevar the variance in ages at death
Table 1 shows the formula we implemented to calculate the measure from starting age $a$.
We have dropped the 'n' subscript typically used to denote the width of the age interval since we prefer to calculate indices from single-year of age life tables. However all formulas could be calculated using age-intervals $[x,x+n)$ with some loss of accuracy.
The notation is as follows.
- $x$ is age
- $m_x$ is the death rate in the interval $[x,x+1)$
- $q_x$ is the probability of dying in the interval $[x,x+1)$
- $a_x$ is the number of person-years lived in the interval $[x,x+1)$
- $\ell_x$ is the number of survivors at exact age $x$
- $d_x$ is the number of deaths in the interval $[x,x+1)$
- $L_x$ is the number of person-years in the interval $[x,x+1)$
- $T_x$ is the number of person-years above age $x$
- $e_x$ is the remaining life expectancy above age $x$
- $\eta_x$ is the average age at death, conditional on survival to age $x$ (equivalent to $e_x$ + $x$)
- $\alpha_x$ is the average age at death in the interval $[x,x+1)$ (equivalent to $a_x$ + $x$)
- $p$ is the proportion of life table deaths, used for calculating Kannisto's compression indices
- $q_1$ and $q_3$ are the first and third quantiles of survivorship, defaulting to the age achieved by the 25th and 75th percentiles of survivors.
In order to calculate some indices of lifespan inequality, two columns were added to a standard life table: $\alpha_{x}$ and $\eta_{x}$ defined above. Two indices, the iqr and the cp measures, use spline interpolation of the survival curve to estimate the age (to several decimals) at which a given percentile or quantile of the life table cohort survive to.
The formulas are given in the discrete form which are easier to work with from life tables. Since mortality is a continuous process, different assumptions made on how to discretize survival have resulted in several different formulations, not all of which are completely equivalent (although they are close).
The column Type refers to whether the index measures absolute or relative inequality in ages at death. For relative indices, we indicate in brackets after the index name whether the index was calculated from an achieved (achieved age at death) or shortfall (remaining life expectancy) distribution.
library(knitr)
library(kableExtra)
df <- data.frame(Index=c("`aid`",
"`cov` (achieved)",
"`cov` (shortfall)",
"`cp`",
"`edag`",
"`eta_dag`",
"`drewnowski` (achieved)",
"`drewnowski` (shortfall)",
"`gini` (achieved)",
"`gini` (shortfall)",
"`H` (shortfall)$^1$",
"`iqr`",
"`mad`$^2$",
"`mld` (achieved)",
"`mld` (shortfall)",
"`sd`",
"`theil` (achieved)",
"`theil` (shortfall)",
"`var`"))
df$Formula <- c("$\\frac{1}{2\\ell_{a}^2 } \\sum_{x=a}^{\\omega}\\sum_{y=a}^{\\omega}d_{x}d_{y}\\left | \\alpha_{x} - \\alpha_{y} \\right |$",
"$\\frac{S_{a}}{{\\eta}_{x}}$",
"$\\frac{S_{a}}{e_{x}}$",
"$min(x_{2}-x_{1} | x_{2} \\gt x_{1}) \\ell(x_{1}) - \\ell(x_{2}) = p$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left [ e_{x} + \\frac{a_x}{n}\\left ( e_{x+1} - e_{x} \\right ) \\right ]$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left [ \\eta_{x} + \\frac{a_x}{n}\\left ( \\eta_{x+1} - \\eta_{x} \\right ) \\right ]$",
"1 - `G`(achieved)",
"1 - `G`(shortfall)",
"$\\frac{1}{2\\ell_{a}^2 \\eta_{a}} \\sum_{x=a}^{\\omega} \\sum_{y=a}^{\\omega} d_{x}d_{y} \\left | \\alpha_{x} - \\alpha_{y} \\right |$",
"$\\frac{1}{2\\ell_{a}^2 e_{a}} \\sum_{x=a}^{\\omega} \\sum_{y=a}^{\\omega} d_{x}d_{y} \\left | \\alpha_{x} - \\alpha_{y} \\right |$",
"$e^{\\dagger} / e_{a}$",
"$q_{3} - q_{1}$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left | \\alpha_{x} - \\eta_{x} \\right |$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\log \\left (\\frac{\\eta_{x}}{\\alpha_{x}} \\right )$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\log \\left (\\frac{e_{x}}{a_{x}} \\right )$",
"$\\sqrt{\\frac{1}{\\ell_{a}} \\sum_{x=a}^{\\omega} d_{x} \\left ( \\alpha_{x}-\\eta_{x} \\right )^2}$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left (\\frac{\\alpha_{x}}{\\eta_{x}} \\right ) \\log \\left (\\frac{\\alpha_{x}}{\\eta_{x}} \\right )$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left (\\frac{a_{x}}{e_{x}} \\right ) \\log \\left (\\frac{a_{x}}{e_{x}} \\right )$",
"$\\frac{1}{\\ell_{a}} \\sum_{x=a}^{\\omega} d_{x} \\left ( \\alpha_{x}-\\eta_{x} \\right )^2$")
#\\frac{1}{\\ell_{a}} \\sum_{x=a}^{\\omega} d_{x} \\textup{log} \\left (\\frac{\\eta_{x}}{\\alpha_{x}} \\right )
df$Type <- c("Absolute",
"Relative",
"Relative",
"Absolute",
"Absolute",
"Absolute",
"Relative",
"Relative",
"Relative",
"Relative",
"Relative",
"Absolute",
"Absolute",
"Relative",
"Relative",
"Absolute",
"Relative",
"Relative",
"Absolute")
df %>%
kbl(escape = F, caption = "Table 1: LifeIneq Formulas") %>%
kable_paper("hover", full_width = F) %>%
footnote(number = c("achieved age is also covered in the package as a distribution type, but we're not so sure of the demographic interpretation because remaining life expectancy is in the numerator and achieved age at death is in the denominator",
"the `mad` can also be calculated by using the distance to the median - see help file"))
# column_spec(1, bold = T)
# column_spec(2, border_right = T)
Installing and loading the package
Before using the package, you will need to install it, either from CRAN, like so:
install.packages("LifeIneq")
or directly from GitHub like so:
remotes::install_github("alysonvanraalte/LifeIneq")
On subsequent uses, you can start from here, where we first load the package into our workspace.
library(LifeIneq)
# some other packages used in this tutorial:
library(ggplot2)
library(scales)
library(dplyr)
library(tidyr)
We recommend to use data in a single-age format, although the package can handle abridged-data formats as well. We also recommend closing out a life table at an age with few deaths. Lifespan variation is far more sensitive to mortality at higher ages than life expectancy [@vanraalte2013], and assumptions on the mortality hazard in the open-ended age interval can make a considerable difference when this age is close to the modal age at death.
If your data have a lower than desired upper age bound, consider extrapolation methods, for instance a parametric Kannisto model (implemented in MortalityLaws::MortalityLaw). If your data are abridged, consider first smoothing over age, and calculating a life table by single year of age (for instance by smoothing with a pclm model in the package ungroup [@Pascariu2019] or with a penalized B-spline approach in the package MortalitySmooth [@Camarda2012]).
Minor issues to be aware of
Some of the indices will return NaNs at the very oldest ages, beyond around the last one-tenth of a percent of deaths. This can happen when $\ell_x$ counts are zero.
Empirical examples
Example 1: Using LifeIneq to calculate lifespan inequality
This first example calculates various measures of lifespan inequality from a lifetable.
Installed in the package are the Canadian female 1x1 LT and 5x1 LTabr 2016 period life tables, taken from the HMD. The HMD data is freely available after registration. The 1x1 lifetable presents data by single year of age and single calendar year, and the 5x1 lifetable is the classic age-abridged format for a single calendar year. Both lifetables have an open-ended age interval of 110+.
Here are the first few lines of the life tables. To make the data easier to handle, the Age column is an integer referring to the starting age. The column OpenInterval is an indicator column which is FALSE for all single or abridged age intervals, but TRUE for the open-ended age interval 110+. This handy life table structure comes from reading in the HMD data using the HMDHFDPlus R package [@Riffe2023].
head(LifeIneq::LT)
head(LifeIneq::LTabr)
The ineq function calculates lifespan inequality. Life table columns are taken as standard arguments, specifically age, dx, lx, ex, and ax, all defined earlier. The method argument asks you to choose the index of variation. As of 2024, the indices listed in the previous section Built-in indices of variation are implemented. An up-to-date list can be accessed by typing args(ineq).
As a first example, we can calculate the standard deviation from the 1x1 table.
# calculating the standard deviation in 2016
sd <- ineq(age=LT$Age,
dx=LT$dx,
lx=LT$lx,
ex=LT$ex,
ax=LT$ax,
method='sd')
head(sd)
The function returns a vector (i.e., a life table column) of standard deviations, conditional upon survival to each age, e.g. sd[1] = r sd[1] is the standard deviation in survival age at birth (age 0); sd[11] = r sd[11] is the standard deviation conditional upon survival to age 10.
Not all measures require all input arguments. In the example above, the $\ell_x$ column was superfluous input for the standard deviation calculations, and running the code above returned the warning message following arguments not used: lx. Thus, the lx argument could be (but does not need to be) deleted from the code in this case.
Changing the index can be as simple as changing one line of code. Here we ask for $e^\dagger$, the life disparity or average life lost at death (@vaupel1986 @goldman1986).
# calculating the life disparity
edag <- ineq(age=LT$Age,
dx=LT$dx,
lx=LT$lx,
ex=LT$ex,
ax=LT$ax,
method='edag')
head(edag)
No warning messages were returned because the edag method uses all input arguments.
Additional arguments
For some measures, additional arguments are possible. For example, Kannisto's compression measures allow the user to determine whether they are interested in finding the smallest age interval containing any given proportion of life table deaths. The measure defaults onto $C50$.
# calculating Kannisto's compression measures, e.g. C50
C50 <- ineq(age=LT$Age,
dx=LT$dx,
lx=LT$lx,
ex=LT$ex,
ax=LT$ax,
method='cp')
C50
If another proportion is desired, this can be changed. Any additional arguments for the measures (and their defaults) can be seen from the help file for the measure, e.g. ?LifeIneq::ineq_cp. In this case, we are told that in addition to age, and lx, we require p "numeric. What proportion of the life table cohort do you want captured in the C measure? The default is .5".
Let's now change this to the smallest age range containing 20 percent of the life table deaths.
# calculating Kannisto's compression measures, C20
C20 <- ineq(age=LT$Age,
dx=LT$dx,
lx=LT$lx,
ex=LT$ex,
ax=LT$ax,
method='cp',
p=0.2)
C20
Abridged versus single-age lifetables
Calculating measures from abridged life tables does not require any additional input compared to the single age tables--just be sure that your age vector is numeric, and contains the lower age bounds (e.g. Age = c(0,1,seq(5,110,5)). Here we compare results for the case of the IQR.
# calculating the interquartile range in 2016
iqr <- ineq(age = LT$Age,
lx = LT$lx,
method = 'iqr')
iqr_abr <- ineq(age = LTabr$Age,
lx = LTabr$lx,
method = 'iqr')
# reasonably close
head(iqr_abr)
head(iqr)
Example 2: Variation in achieved ages at death compared to remaining life expectancy
Relative measures of variation all require an additional argument distribution_type.
This example compares relative measures of lifespan variation using Hungarian period male life table data from the HMD. Hungary was explicitly chosen because it experienced mortality trends which differed in magnitude and sometimes even direction across the age span.
First we read in a time series of Hungarian male life tables, downloaded from the HMD.
LTs <- read.csv("HUNmltper_1x1.csv", header = TRUE)
head(LTs)
Next we calculate relative measures of lifespan variation for each year. While in the first example we used the generic ineq function, specifying the index with the measure argument, here we show that you can also use functions that are specific to each measure (e.g. ineq_cov = ineq(..., measure=cov)).
Setting distribution_type = "aad" calculates the relative measure from a distribution of ages at death. Setting distribution_type = "rl" calculates the relative measure from a distribution of remaining life expectancies.
ineq_compare <- LTs |>
group_by(Year) |>
mutate(
COV_AAD = ineq_cov(age=Age,dx,ex,ax,distribution_type = "aad"),
COV_RL = ineq_cov(age=Age,dx,ex,ax,distribution_type = "rl"),
H_AAD = ineq_H(age=Age,dx,lx,ex,ax,distribution_type = "aad"),
H_RL = ineq_H(age=Age,dx,lx,ex,ax,distribution_type = "rl"),
T_AAD = ineq_theil(age=Age,dx,ex,ax,distribution_type = "aad"),
T_RL = ineq_theil(age=Age,dx,ex,ax,distribution_type = "rl"),
GINI_AAD = ineq_gini(age=Age,dx,ex,ax,distribution_type = "aad"),
GINI_RL = ineq_gini(age=Age,dx,ex,ax,distribution_type = "rl"),
RL = ex,
AAD = ex + Age) |>
# keeping results for conditional survival to ages 30 or 65
filter(Age==30 | Age==65) |>
select(Year, Age,
RL, AAD,
COV_AAD, COV_RL,
H_AAD, H_RL,
T_AAD, T_RL,
GINI_AAD, GINI_RL)
head(ineq_compare)
We can then plot these.
To use ggplot2 it is handy to first manipulate data into long format.
# putting our results into a 'long format' and adding new column
# adding a couple of columns to describe the measure more fully
New_cols <- bind_cols(
Measure = c("AAD","RL",
"COV_AAD", "COV_RL",
"H_AAD", "H_RL",
"T_AAD", "T_RL",
"GINI_AAD", "GINI_RL"),
Dist_type = rep(c("Age at Death","Remaining Life"),5),
Family = c("*Life Expectancy", "*Life Expectancy",
"Coef. of variation", "Coef. of variation",
"Keyfitz-Leser H","Keyfitz-Leser H",
"Theil Index", "Theil Index",
"Gini Coefficient", "Gini Coefficient"))
# long format
ineq_long <- ineq_compare %>%
pivot_longer(cols = RL:GINI_RL,
names_to = "Measure",
values_to = "Value") %>%
left_join(New_cols, by = join_by(Measure))
To put these measures on the same scale, we will look at relative change in the measure since 1950.
ineq_change <- ineq_long %>%
group_by(Age, Measure) %>%
mutate(Prop_change = Value / Value[1])
tail(ineq_change)
The top rows show trends in the relative measures, while the bottom rows show trends in the average (remaining life expectancy or average age at death).
# comparison conditional upon survival to ages 30 and 65
ggplot(data=ineq_change,
aes(x=Year, y=Prop_change, colour=Dist_type)) +
geom_line(size=1) +
facet_grid(Age ~ Family) +
ylab("Proportional change") +
scale_colour_manual(values=c("darkred", "darkorange")) +
labs(title = "Proportional change in measure since 1950",
subtitle = "Conditional upon survival to ages 30 (top row) and 65 (bottom row)",
caption = "*Life expectancy is not a lifespan inequality measure, but is shown for interpretation") +
coord_trans(y="log2") +
theme_minimal() +
theme(strip.text.x = element_text(size=8),
legend.position="bottom",
legend.title=element_blank(),
axis.text.x = element_text(angle=45, vjust=0.5, size=10))
So as this example makes clear, what goes in the denominator for conditional measures of lifespan variation can make an enormous difference in interpreting the variability. This is especially important at older ages, when relative differences over time (or between populations) in remaining life will be much larger than relative differences in ages at death.
References
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Introduction
How unequally distributed are the ages at death within a population? There are many indices around to answer this question, all generally going under the umbrella terms lifespan inequality, lifespan variation, age-at-death variation, compression of mortality, or rectangularization of the survival curve.
In this R package we have implemented code to calculate such indices from life tables. In what follows, we will go through how to use the package with example Canadian and Hungarian life table data from the Human Mortality Database [@barbieri2015; @HMD2023].
For a good overview of the mathematical properties of the indices, as well as comparisons of how they perform when applied to empirical data, we suggest consulting the following [@wilmoth1999; @shkolnikov2003; @vanraalte2013; @wrycza2015].
Background
Choosing a measure of lifespan inequality
Variation is a broad concept. Since it is often used to make judgements on the inequality of a distribution, different indices have been developed which differ in their mathematical properties and sensitivities to different parts of the distribution. As Paul @allison1978 (p. 865) noted,"The choice of an inequality measure is properly regarded as a choice among alternative definitions of inequality rather than a choice among alternative ways of measuring a single theoretical construct." Here we outline some of the more important considerations.
Absolute versus relative inequality
Indices of lifespan variation measure absolute or relative inequality in ages at death. Absolute measures are scale invariant, meaning that their values do not change when the same life years are added to everyone's lifespan (i.e. the distribution is shifted to higher ages). Relative measures are translation invariant, meaning that they do not change when adding the same proportion of life to everyone's lifespan. If an age-at-death distribution shifts to the right with increasing life expectancy, absolute inequality stays unchanged, but relative inequality declines.
Absolute measures are generally measured in years, making them easier to interpret, but relative measures have their own advantages. Relative measures are preferred by biologists to compare species with very different life histories and life expectancies [@eakin1995; @baudisch2011; @colchero2016]. This is less of an issue in comparing contemporary human populations, either across or within countries. In the end, we feel that the choice between measuring absolute versus relative inequality is a normative one. If there is no inherent reason for preferring absolute or relative inequality, it is good practice to present both.
Attainment versus shortfall distribution
Relative measures of lifespan inequality have an additional specification: whether the inequality is relative to an average achieved age at death (e.g. from an attainment distribution) or to a remaining life expectancy (e.g. from a shortfall distribution). When following a cohort from birth, the means of the two distributions are the same---average age at death equals remaining life expectancy at birth.
Often though, we want to measure the lifespan inequality of populations or groups based on a conditional age-at-death distribution---in other words, a distribution with a starting age greater than zero, where the lifespan inequality is calculated conditional on surviving to that starting age. As an example, lifespan inequality by level of completed education can only be defined from early adulthood.
In such cases, should inequality be made relative to the average age at death, conditional upon survival to age $x$, or to the average remaining life expectancy at age $x$? So far, few researchers have paid attention to this difference. There are plenty of studies based on conditional distributions, including our own, but no reflection on how the choice of denominator (the distribution type) could impact the result.
Indeed, a quick survey of the literature to date hints that the choice of conditional distribution type seems to mostly depend on the measure used rather than any theoretical justification. The coefficient of variation, the Theil Index, and the Mean Log Deviation are generally calculated from the achieved age at death distribution [@hakkert1987; @vanraalte2012; @permanyer2019; @shkolnikov2010; @nemeth2017; @permanyer2019; @seaman2020]. The Keyfitz-Leser life table entropy measure is instead calculated from shortfall distributions [@bramajo2022]. The Gini coefficient, when calculated from the Kendall and Stuart definition, has used attained ages [@permanyer2019], while when calculated based on the Hanada formulation has used remaining life expectancy [@shkolnikov2003].
Is there any reason to prefer one concept over the other? We ourselves cannot decide, and prefer to leave this debate to play out in the literature. All of the relative measures of lifespan variation included in the LifeIneq package can be calculated in both ways, and we have added an argument to relative measures to make this choice explicit. In one of the empirical examples of this vignette, we demonstrate how important this distinction is.
Built-in indices of variation
Currently, the LifeIneq package contains the following indices of lifespan inequality (listed alphabetically):
aidthe Absolute Inter-individual Difference, also known as the absolute Gini Coefficientcovthe Coefficient of Variation of survival or remaining lifecpKannisto's C measures---the shortest distance between two ages containing $p$ percent of the life table cohort's deaths.edagthe life disparity $e^{\dagger}$ measure, also called the average life years lost due to deatheta_dagthe mean age at death lost at death $\eta^{\dagger}$ (newly derived for this package--equivalent toedagbut calculated by weighting the distribution of ages at death instead of remaining life by the death density)drewnowskithe Drewnowski index, which is the complement of the Gini coefficientginithe Gini coefficient of survival or remaining lifeHthe Leser-Keyfitz life table index, also called the life table entropy, which is more properly described as the elasticity of life expectancy to proportional change in mortality. It is also the relative inequality index of $e^{\dagger}$ [@mitra1978, @vaupel1986, @goldman1986].iqrthe interquartile range of the age-at-death distribution, e.g. the difference between the 75th and 25th percentile of survival ages in the life table cohortmadthe mean absolute deviation in lifetable ages at death. This may be with respect to either the mean or median age at deathmldthe mean log deviation of survival or remaining lifequantileandquantile_lowerare quantiles of survivorship (the former is a vector made conditional upon survival to each age, the latter is a scalar). These are not inequality measures by themselves, generally, but are included for users to design own quantile-based measures beyond theiqrandcpmeasures.sdthe standard deviation in ages at deaththeilthe Theil index of survival or remaining lifevarthe variance in ages at death
Table 1 shows the formula we implemented to calculate the measure from starting age $a$.
We have dropped the 'n' subscript typically used to denote the width of the age interval since we prefer to calculate indices from single-year of age life tables. However all formulas could be calculated using age-intervals $[x,x+n)$ with some loss of accuracy.
The notation is as follows.
- $x$ is age
- $m_x$ is the death rate in the interval $[x,x+1)$
- $q_x$ is the probability of dying in the interval $[x,x+1)$
- $a_x$ is the number of person-years lived in the interval $[x,x+1)$
- $\ell_x$ is the number of survivors at exact age $x$
- $d_x$ is the number of deaths in the interval $[x,x+1)$
- $L_x$ is the number of person-years in the interval $[x,x+1)$
- $T_x$ is the number of person-years above age $x$
- $e_x$ is the remaining life expectancy above age $x$
- $\eta_x$ is the average age at death, conditional on survival to age $x$ (equivalent to $e_x$ + $x$)
- $\alpha_x$ is the average age at death in the interval $[x,x+1)$ (equivalent to $a_x$ + $x$)
- $p$ is the proportion of life table deaths, used for calculating Kannisto's compression indices
- $q_1$ and $q_3$ are the first and third quantiles of survivorship, defaulting to the age achieved by the 25th and 75th percentiles of survivors.
In order to calculate some indices of lifespan inequality, two columns were added to a standard life table: $\alpha_{x}$ and $\eta_{x}$ defined above. Two indices, the iqr and the cp measures, use spline interpolation of the survival curve to estimate the age (to several decimals) at which a given percentile or quantile of the life table cohort survive to.
The formulas are given in the discrete form which are easier to work with from life tables. Since mortality is a continuous process, different assumptions made on how to discretize survival have resulted in several different formulations, not all of which are completely equivalent (although they are close).
The column Type refers to whether the index measures absolute or relative inequality in ages at death. For relative indices, we indicate in brackets after the index name whether the index was calculated from an achieved (achieved age at death) or shortfall (remaining life expectancy) distribution.
library(knitr) library(kableExtra) df <- data.frame(Index=c("`aid`", "`cov` (achieved)", "`cov` (shortfall)", "`cp`", "`edag`", "`eta_dag`", "`drewnowski` (achieved)", "`drewnowski` (shortfall)", "`gini` (achieved)", "`gini` (shortfall)", "`H` (shortfall)$^1$", "`iqr`", "`mad`$^2$", "`mld` (achieved)", "`mld` (shortfall)", "`sd`", "`theil` (achieved)", "`theil` (shortfall)", "`var`")) df$Formula <- c("$\\frac{1}{2\\ell_{a}^2 } \\sum_{x=a}^{\\omega}\\sum_{y=a}^{\\omega}d_{x}d_{y}\\left | \\alpha_{x} - \\alpha_{y} \\right |$", "$\\frac{S_{a}}{{\\eta}_{x}}$", "$\\frac{S_{a}}{e_{x}}$", "$min(x_{2}-x_{1} | x_{2} \\gt x_{1}) \\ell(x_{1}) - \\ell(x_{2}) = p$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left [ e_{x} + \\frac{a_x}{n}\\left ( e_{x+1} - e_{x} \\right ) \\right ]$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left [ \\eta_{x} + \\frac{a_x}{n}\\left ( \\eta_{x+1} - \\eta_{x} \\right ) \\right ]$", "1 - `G`(achieved)", "1 - `G`(shortfall)", "$\\frac{1}{2\\ell_{a}^2 \\eta_{a}} \\sum_{x=a}^{\\omega} \\sum_{y=a}^{\\omega} d_{x}d_{y} \\left | \\alpha_{x} - \\alpha_{y} \\right |$", "$\\frac{1}{2\\ell_{a}^2 e_{a}} \\sum_{x=a}^{\\omega} \\sum_{y=a}^{\\omega} d_{x}d_{y} \\left | \\alpha_{x} - \\alpha_{y} \\right |$", "$e^{\\dagger} / e_{a}$", "$q_{3} - q_{1}$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left | \\alpha_{x} - \\eta_{x} \\right |$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\log \\left (\\frac{\\eta_{x}}{\\alpha_{x}} \\right )$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\log \\left (\\frac{e_{x}}{a_{x}} \\right )$", "$\\sqrt{\\frac{1}{\\ell_{a}} \\sum_{x=a}^{\\omega} d_{x} \\left ( \\alpha_{x}-\\eta_{x} \\right )^2}$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left (\\frac{\\alpha_{x}}{\\eta_{x}} \\right ) \\log \\left (\\frac{\\alpha_{x}}{\\eta_{x}} \\right )$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a} ^{\\omega} d_{x} \\left (\\frac{a_{x}}{e_{x}} \\right ) \\log \\left (\\frac{a_{x}}{e_{x}} \\right )$", "$\\frac{1}{\\ell_{a}} \\sum_{x=a}^{\\omega} d_{x} \\left ( \\alpha_{x}-\\eta_{x} \\right )^2$") #\\frac{1}{\\ell_{a}} \\sum_{x=a}^{\\omega} d_{x} \\textup{log} \\left (\\frac{\\eta_{x}}{\\alpha_{x}} \\right ) df$Type <- c("Absolute", "Relative", "Relative", "Absolute", "Absolute", "Absolute", "Relative", "Relative", "Relative", "Relative", "Relative", "Absolute", "Absolute", "Relative", "Relative", "Absolute", "Relative", "Relative", "Absolute") df %>% kbl(escape = F, caption = "Table 1: LifeIneq Formulas") %>% kable_paper("hover", full_width = F) %>% footnote(number = c("achieved age is also covered in the package as a distribution type, but we're not so sure of the demographic interpretation because remaining life expectancy is in the numerator and achieved age at death is in the denominator", "the `mad` can also be calculated by using the distance to the median - see help file")) # column_spec(1, bold = T) # column_spec(2, border_right = T)
Installing and loading the package
Before using the package, you will need to install it, either from CRAN, like so:
install.packages("LifeIneq")
or directly from GitHub like so:
remotes::install_github("alysonvanraalte/LifeIneq")
On subsequent uses, you can start from here, where we first load the package into our workspace.
library(LifeIneq) # some other packages used in this tutorial: library(ggplot2) library(scales) library(dplyr) library(tidyr)
We recommend to use data in a single-age format, although the package can handle abridged-data formats as well. We also recommend closing out a life table at an age with few deaths. Lifespan variation is far more sensitive to mortality at higher ages than life expectancy [@vanraalte2013], and assumptions on the mortality hazard in the open-ended age interval can make a considerable difference when this age is close to the modal age at death.
If your data have a lower than desired upper age bound, consider extrapolation methods, for instance a parametric Kannisto model (implemented in MortalityLaws::MortalityLaw). If your data are abridged, consider first smoothing over age, and calculating a life table by single year of age (for instance by smoothing with a pclm model in the package ungroup [@Pascariu2019] or with a penalized B-spline approach in the package MortalitySmooth [@Camarda2012]).
Minor issues to be aware of
Some of the indices will return NaNs at the very oldest ages, beyond around the last one-tenth of a percent of deaths. This can happen when $\ell_x$ counts are zero.
Empirical examples
Example 1: Using LifeIneq to calculate lifespan inequality
This first example calculates various measures of lifespan inequality from a lifetable.
Installed in the package are the Canadian female 1x1 LT and 5x1 LTabr 2016 period life tables, taken from the HMD. The HMD data is freely available after registration. The 1x1 lifetable presents data by single year of age and single calendar year, and the 5x1 lifetable is the classic age-abridged format for a single calendar year. Both lifetables have an open-ended age interval of 110+.
Here are the first few lines of the life tables. To make the data easier to handle, the Age column is an integer referring to the starting age. The column OpenInterval is an indicator column which is FALSE for all single or abridged age intervals, but TRUE for the open-ended age interval 110+. This handy life table structure comes from reading in the HMD data using the HMDHFDPlus R package [@Riffe2023].
head(LifeIneq::LT) head(LifeIneq::LTabr)
The ineq function calculates lifespan inequality. Life table columns are taken as standard arguments, specifically age, dx, lx, ex, and ax, all defined earlier. The method argument asks you to choose the index of variation. As of 2024, the indices listed in the previous section Built-in indices of variation are implemented. An up-to-date list can be accessed by typing args(ineq).
As a first example, we can calculate the standard deviation from the 1x1 table.
# calculating the standard deviation in 2016 sd <- ineq(age=LT$Age, dx=LT$dx, lx=LT$lx, ex=LT$ex, ax=LT$ax, method='sd') head(sd)
The function returns a vector (i.e., a life table column) of standard deviations, conditional upon survival to each age, e.g. sd[1] = r sd[1] is the standard deviation in survival age at birth (age 0); sd[11] = r sd[11] is the standard deviation conditional upon survival to age 10.
Not all measures require all input arguments. In the example above, the $\ell_x$ column was superfluous input for the standard deviation calculations, and running the code above returned the warning message following arguments not used: lx. Thus, the lx argument could be (but does not need to be) deleted from the code in this case.
Changing the index can be as simple as changing one line of code. Here we ask for $e^\dagger$, the life disparity or average life lost at death (@vaupel1986 @goldman1986).
# calculating the life disparity edag <- ineq(age=LT$Age, dx=LT$dx, lx=LT$lx, ex=LT$ex, ax=LT$ax, method='edag') head(edag)
No warning messages were returned because the edag method uses all input arguments.
Additional arguments
For some measures, additional arguments are possible. For example, Kannisto's compression measures allow the user to determine whether they are interested in finding the smallest age interval containing any given proportion of life table deaths. The measure defaults onto $C50$.
# calculating Kannisto's compression measures, e.g. C50 C50 <- ineq(age=LT$Age, dx=LT$dx, lx=LT$lx, ex=LT$ex, ax=LT$ax, method='cp') C50
If another proportion is desired, this can be changed. Any additional arguments for the measures (and their defaults) can be seen from the help file for the measure, e.g. ?LifeIneq::ineq_cp. In this case, we are told that in addition to age, and lx, we require p "numeric. What proportion of the life table cohort do you want captured in the C measure? The default is .5".
Let's now change this to the smallest age range containing 20 percent of the life table deaths.
# calculating Kannisto's compression measures, C20 C20 <- ineq(age=LT$Age, dx=LT$dx, lx=LT$lx, ex=LT$ex, ax=LT$ax, method='cp', p=0.2) C20
Abridged versus single-age lifetables
Calculating measures from abridged life tables does not require any additional input compared to the single age tables--just be sure that your age vector is numeric, and contains the lower age bounds (e.g. Age = c(0,1,seq(5,110,5)). Here we compare results for the case of the IQR.
# calculating the interquartile range in 2016 iqr <- ineq(age = LT$Age, lx = LT$lx, method = 'iqr') iqr_abr <- ineq(age = LTabr$Age, lx = LTabr$lx, method = 'iqr') # reasonably close head(iqr_abr) head(iqr)
Example 2: Variation in achieved ages at death compared to remaining life expectancy
Relative measures of variation all require an additional argument distribution_type.
This example compares relative measures of lifespan variation using Hungarian period male life table data from the HMD. Hungary was explicitly chosen because it experienced mortality trends which differed in magnitude and sometimes even direction across the age span.
First we read in a time series of Hungarian male life tables, downloaded from the HMD.
LTs <- read.csv("HUNmltper_1x1.csv", header = TRUE) head(LTs)
Next we calculate relative measures of lifespan variation for each year. While in the first example we used the generic ineq function, specifying the index with the measure argument, here we show that you can also use functions that are specific to each measure (e.g. ineq_cov = ineq(..., measure=cov)).
Setting distribution_type = "aad" calculates the relative measure from a distribution of ages at death. Setting distribution_type = "rl" calculates the relative measure from a distribution of remaining life expectancies.
ineq_compare <- LTs |> group_by(Year) |> mutate( COV_AAD = ineq_cov(age=Age,dx,ex,ax,distribution_type = "aad"), COV_RL = ineq_cov(age=Age,dx,ex,ax,distribution_type = "rl"), H_AAD = ineq_H(age=Age,dx,lx,ex,ax,distribution_type = "aad"), H_RL = ineq_H(age=Age,dx,lx,ex,ax,distribution_type = "rl"), T_AAD = ineq_theil(age=Age,dx,ex,ax,distribution_type = "aad"), T_RL = ineq_theil(age=Age,dx,ex,ax,distribution_type = "rl"), GINI_AAD = ineq_gini(age=Age,dx,ex,ax,distribution_type = "aad"), GINI_RL = ineq_gini(age=Age,dx,ex,ax,distribution_type = "rl"), RL = ex, AAD = ex + Age) |> # keeping results for conditional survival to ages 30 or 65 filter(Age==30 | Age==65) |> select(Year, Age, RL, AAD, COV_AAD, COV_RL, H_AAD, H_RL, T_AAD, T_RL, GINI_AAD, GINI_RL) head(ineq_compare)
We can then plot these.
To use ggplot2 it is handy to first manipulate data into long format.
# putting our results into a 'long format' and adding new column # adding a couple of columns to describe the measure more fully New_cols <- bind_cols( Measure = c("AAD","RL", "COV_AAD", "COV_RL", "H_AAD", "H_RL", "T_AAD", "T_RL", "GINI_AAD", "GINI_RL"), Dist_type = rep(c("Age at Death","Remaining Life"),5), Family = c("*Life Expectancy", "*Life Expectancy", "Coef. of variation", "Coef. of variation", "Keyfitz-Leser H","Keyfitz-Leser H", "Theil Index", "Theil Index", "Gini Coefficient", "Gini Coefficient")) # long format ineq_long <- ineq_compare %>% pivot_longer(cols = RL:GINI_RL, names_to = "Measure", values_to = "Value") %>% left_join(New_cols, by = join_by(Measure))
To put these measures on the same scale, we will look at relative change in the measure since 1950.
ineq_change <- ineq_long %>% group_by(Age, Measure) %>% mutate(Prop_change = Value / Value[1]) tail(ineq_change)
The top rows show trends in the relative measures, while the bottom rows show trends in the average (remaining life expectancy or average age at death).
# comparison conditional upon survival to ages 30 and 65 ggplot(data=ineq_change, aes(x=Year, y=Prop_change, colour=Dist_type)) + geom_line(size=1) + facet_grid(Age ~ Family) + ylab("Proportional change") + scale_colour_manual(values=c("darkred", "darkorange")) + labs(title = "Proportional change in measure since 1950", subtitle = "Conditional upon survival to ages 30 (top row) and 65 (bottom row)", caption = "*Life expectancy is not a lifespan inequality measure, but is shown for interpretation") + coord_trans(y="log2") + theme_minimal() + theme(strip.text.x = element_text(size=8), legend.position="bottom", legend.title=element_blank(), axis.text.x = element_text(angle=45, vjust=0.5, size=10))
So as this example makes clear, what goes in the denominator for conditional measures of lifespan variation can make an enormous difference in interpreting the variability. This is especially important at older ages, when relative differences over time (or between populations) in remaining life will be much larger than relative differences in ages at death.
References
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