In alysonvanraalte/LifeIneq: Calculate indices of lifespan variation
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.width = 6,
fig.height = 6,
fig.align = "center"
)
Introduction
What factors are driving lifespan inequalities? One way to answer this question
is to use decomposition methods. Decomposing measures of lifespan inequality
allows to attribute some of the variation in lifespans to variation between
groups in the population, such as educational groups. The remaining variation
comes from variation within the groups.
In this package, we implement several old and new decompositions of measures
of lifespan inequality. This includes decompositions of lifespan disparity
($e^{\dagger}$), the Gini coefficient, absolute inter-individual difference
(AID; also known as absolute Gini coefficient), the Theil index, mean log
variation, mean absolute deviation (MAD), and the variance.
Decomposing lifespan inequality
Additive decompositions
All decompositions covered by this package are additive and they have the
following general form:
\begin{align}
\textrm{Total inequality} = \textrm{Within-component} + \textrm{Between-component}.
\end{align}
For each measure of lifespan inequality the within-component is a
weighted sum of the measure applied to each group. That is, the
within-component captures the average lifespan inequality across groups and
thus the intra-group heterogeneity in lifespans. The between-component captures
how different lifespan inequality is between groups; i.e., the inter-group
heterogeneity in lifespan.
For several measures of lifespan inequality additive decompositions have been
published in the literature. Table 1 below provides an overview of all
decompositions covered in the package, whether new or not. $K$ is the total
number of groups and $\mathrm{Pr}(k)$ is the proportion of group $k$ in the
population, assumed to be measured at the starting age of the life table. $k$
is also used as superscript and subscript to indicate group-specific measures;
e.g., $e_0^k$ is the life expectancy at birth in group $k$. Additional details
on the notation are provided below.
library(knitr)
library(kableExtra)
df <- data.frame(Index=c("`aid`",
"`edag`",
"`eta_dag`",
"`gini` (achieved)",
"`gini` (shortfall)",
"`H` (shortfall)",
"`mad`",
"`mld` (achieved)",
"`mld` (shortfall)",
"`theil` (achieved)",
"`theil` (shortfall)",
"`var`"))
df$Symbol <- c("$\\mathrm{AID}$",
"$e^\\dagger$",
"$\\eta^\\dagger$",
"$G$",
"$G$",
"$H$",
"$\\mathrm{MAD}$",
"$T_L$",
"$T_L$",
"$T_T$",
"$T_T$",
"$\\mathrm{Var}$")
df$Within <- c("$\\sum_{k=1}^K \\mathrm{Pr}(k)\\frac{\\ell_a^k}{\\ell_a} \\mathrm{AID}_k$&",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) e^\\dagger_k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\eta^\\dagger_k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\ell_a^{2(k)}\\eta_a^k}{\\ell_a^2\\eta_a} G_k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\ell_a^{2(k)} e_a^k}{\\ell_a^2 e_a} G_k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{e^\\dagger_k}{e_a}$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\mathrm{MAD}_k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) T_L^k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) T_L^k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\eta_a^k}{\\eta_a} T_T^k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{e_a^k}{e_a} T_T^k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\mathrm{Var}_k$")
df$Between <- c("$\\frac{1}{2\\ell_a}\\sum_{k=1}^K B(k)+R(k)$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\left(e^{\\dagger*}_k - e^\\dagger_k\\right)$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\left(\\eta^{\\dagger*}_k - \\eta^\\dagger_k\\right)$",
"$\\frac{1}{2\\ell_a\\eta_a}\\sum_{k=1}^K B(k)+R(k)$",
"$\\frac{1}{2\\ell_a e_a}\\sum_{k=1}^K B(k)+R(k)$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\left(\\frac{e^{\\dagger*}_k - e^\\dagger_k}{e_a}\\right)$",
"$\\mathrm{MAD}-\\sum_{k=1}^K \\mathrm{Pr}(k) \\mathrm{MAD}_k$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\log \\frac{e_a}{e^k_a}$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\log \\frac{\\eta_a}{\\eta^k_a}$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\eta_a^k}{\\eta_0}\\log \\frac{\\eta_a^k}{\\eta_a}$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{e_a^k}{e_0}\\log \\frac{e_a^k}{e_a}$",
"$\\sum_{k=1}^K \\mathrm{Pr}(k) (e_a^k-e_a)^2$")
df %>%
kbl(escape = F, caption = "Table 1: Decomposition formulas") %>%
kable_paper("hover", full_width = T)
Existing decompositions
The variance and its decomposition are standard textbook material. The life
table formula is taken from @permanyer2019. The Theil index, often also called
Theil T index, is a special case of the Generalized Entropy Index. The formulas
and decompositions are taken from @liao2022 and @permanyer2019. The mean log
deviation is also a special case of the Generalized Entropy Index, and
sometimes called Theil L index. The formulas and decompositions are taken
from @liao2022 and @vanraalte2011. The derivation of the decomposition for
the mean absolute deviation is straightforward.
Decomposing lifespan disparity
Here, we use a notation often found in the economic and statistical literature
on inequality measurement, which makes it more easy to derive the decomposition
of lifespan disparity, $e^\dagger$. Let $N$ be the size of the population
under study; $Y$ is the variable capturing the length of life and $y_i$ is the
length of life for individual $i$. Using this notation, $e^\dagger$ can be
written and re-arranged as follows:
\begin{align}
e^\dagger & = \frac{1}{N}\sum_{i=1}^N \mathrm{M}(Y|Y\geq y_i) \
& = \frac{1}{N}\sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}(Y|Y\geq y_{ik}) \
& = \frac{1}{N}\sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}(Y|Y\geq y_{ik}) + \mathrm{M}k(Y|Y\geq y{ik}) - \mathrm{M}k(Y|Y\geq y{ik})\
&=\frac{1}{N} \sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}k(Y|Y\geq y{ik}) + \frac{1}{N} \sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}(Y|Y\geq y_{ik})-\mathrm{M}k(Y|Y\geq y{ik})\
&=\sum_{k=1}^K \mathrm{Pr}(k) e^\dagger_k + \sum_{k=1}^K \mathrm{Pr}(k) \left(e^{\dagger}_k - e^\dagger_k\right)
\end{align}
The last line uses life table notation. The within-component is the average
group-specific lifespan disparity, and the between component is the difference
between a counterfactual group-specific lifespan inequality ($e^{\dagger}_k$)
and the actual group-specific lifespan inequality. The counterfactual
group-specific lifespan inequality would arise if members of group $k$ died
according to the life table of that group, but the average remaining life
expectancy at the age of death is that of the total population. This essentially
captures how average remaining life expectancy differs between the groups.
Decomposing the Gini coefficient
@liao2022 gives the following decomposition of the Gini coefficient:
\begin{eqnarray}
W_L&=&\frac{1}{2N^2\bar{y}} \sum_{k=1}^K\sum_{i=1}^{N_k}\sum_{j=1}^{N_k} |y_{ik}-y_{jk} | \
B_L&=&\frac{1}{2N^2\bar{y}} \sum_{k=1}^K\sum_{h=1,h\neq k}^K\sum_{i=1}^{N_k}\sum_{j=1}^{N_h} |y_{ik}-y_{jh} |,
\end{eqnarray}
where $y_{ik}$ is the lifespan of individual $i$ in group $k$. This can be
written in life table notation as
\begin{eqnarray}\label{eq:W1}
W_L&=&\frac{1}{2\ell_0^2 2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z | b_x-b_z| \
B_L&=&\frac{1}{2\ell_0^2 e_0}\sum_{k=1}^K\sum_{h=1,h\neq k}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^h_z | b_x-b_z|.
\end{eqnarray}
The equation for $W$ can be rearranged:
\begin{eqnarray}
W_L&=&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z | b_x-b_z| \
&=&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \left(\sum\limits_{h=1}^K \mathrm{Pr}(h) |b^h_x-b^h_z|\right) \
&=&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \mathrm{Pr}(k) |b^k_x-b^k_z| \
&+&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \left(\sum\limits_{h=1,h\neq k}^K \mathrm{Pr}(h) |b^h_x-b^h_z|\right)
\end{eqnarray}
The first line of the last definition of $W_l$ can be re-arranged to
give
\begin{eqnarray}
W=\sum_{k=1}^K \mathrm{Pr}(k) \frac{\ell_0^{2(k)} e_0^k}{\ell_0^2 e_0} G_k,
\end{eqnarray}
as shown in the overview table. The remainder of the last definition for $W_l$
in combination with $B_L$ provides the between-component $B$,
\begin{eqnarray}
B=\frac{1}{2\ell_0 e_0}\sum_{k=1}^K B(k)+R(k)
\end{eqnarray}
with $B(k)$ and $R(k)$ defined as:
\begin{eqnarray}
B(k) &=& \sum_{h=1,h\neq k}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^h_z | b_x-b_z| \
R(k) &=& \sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \left( \sum_{h=1,h\neq k}^K \mathrm{Pr}(h) | b^h_x-b^h_z| \right)
\end{eqnarray}
The case of the Gini using the achieved age at death follows in the same way, as
does the case of the life table starting at age $a$.
Decomposing H and AID
ife table entropy, $H$, is a linear transformation of lifespan disparity
($e^\dagger/e_a$), while the absolute inter-individual difference in lifespan
(AID) is a linear transformation of the Gini coefficient ($G e_a$). This means
that the formulas for the decomposition of lifespan disparity and the Gini
coefficient only need to be slightly modified. In case of $H$, the within and
between component are given by $W_{e^\dagger}/e_a$ and $B_{e^\dagger}/e_a$,
while in case of the AID we have $W_{G}e_a$ and $B_{G}e_a$; i.e., scaling by
overall life expectancy.
Example
As an example, we will decompose the lifespan inequality in Canada using gender.
This will show us to what extent lifespan inequality in Canada is due to gender
differentials in mortality. First, we load the package and the data.
library(LifeIneq)
data(LT)
data(LTm)
LT contains the life table of Canadian women in 2016, while LTm has the same
information for men. We will pass some of this data to the function 'bw_decomp'.
To do so, we combine the relevant information in matrices:
age <- 0:110
ax <- cbind(LT$ax,LTm$ax)
dx <- cbind(LT$dx,LTm$dx)
lx <- cbind(LT$lx,LTm$lx)
ex <- cbind(LT$ex,LTm$ex)
For instance, the object 'lx' contains the life table column $\ell_x$ for both
men and women. It is a matrix where each column represents one of the groups;
the first column has $\ell_x$ of women, and the second column has $\ell_x$ of
men.
head(lx)
Generally, for all matrices the number of columns will be equal to the number
of groups.
The matrices created above are passed to the 'bw_decomp' function. In addition
to the data several other things need to be specified. The argument 'prop'
takes the proportion of each group. The argument 'method' can be used to choose
betwee the different measures of lifespan inequality. Currentl, the options
"theil", "edag","var","mld","gini","mad","aid", and "H" are supported. Finally,
for some measures the distribution type needs to be specified; i.e., whether
the measure should be calculated with remaining life expectancy ("rl") or with
the achieved age at death ("aad").
For our example, we assume that women are close to 49% of the population at the
radix. Choosing mean log deviation (MLD) as our measure and using the remaining
life expectancy, we get the following results:
bw_decomp(age = age,
ax = ax,
dx = dx,
lx = lx,
ex = ex,
prop = c(.4886,1-.4886),
method = "mld",
distribution_type = "rl")
The first component ("method") shows the method used. The second component
("group_ind") shows the value of the chosen measure for each group. In this
case, MLD is about 0.04 for women and 0.05 for men. "tot" shows
the value of MLD for the total population. The next two components, "B" and
"W", show the values of the between-component and the within-component,
respectively. Compared to the within-component, the between-component is rather
small in this case. This is also indicated by "fB" and "fW", which show the
between-component and the within-component relative to the total value. The
between-component only contributes less than 1% to the total lifespan inequality
as measured through the MLD. This means that the variation between genders is
much smaller as within both genders, at least as measured through the MLD.
As a second example, we calculate the Gini coefficient using the achieved age
at death:
bw_decomp(age = age,
ax = ax,
dx = dx,
lx = lx,
ex = ex,
prop = c(.4886,1-.4886),
method = "gini",
distribution_type = "aad")
The components of the result are as in the first example. Again, the
contribution of the between-component is rather small with slightly more than
1%. Generally, such seemingly low importance of the between-component is rather
common. For instance, @vanraalte2012 find similarly low between-components by
education in several countries. One potential reason is that there are many
factors affecting mortality, and a single grouping variable alone (such as gender
in our example) has little predictive power; i.e., after accounting for this
variable still a sizeable proportion of lifespan variability remains.
References
alysonvanraalte/LifeIneq documentation built on March 12, 2024, 1:42 p.m.
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Introduction
What factors are driving lifespan inequalities? One way to answer this question is to use decomposition methods. Decomposing measures of lifespan inequality allows to attribute some of the variation in lifespans to variation between groups in the population, such as educational groups. The remaining variation comes from variation within the groups.
In this package, we implement several old and new decompositions of measures of lifespan inequality. This includes decompositions of lifespan disparity ($e^{\dagger}$), the Gini coefficient, absolute inter-individual difference (AID; also known as absolute Gini coefficient), the Theil index, mean log variation, mean absolute deviation (MAD), and the variance.
Decomposing lifespan inequality
Additive decompositions
All decompositions covered by this package are additive and they have the following general form: \begin{align} \textrm{Total inequality} = \textrm{Within-component} + \textrm{Between-component}. \end{align} For each measure of lifespan inequality the within-component is a weighted sum of the measure applied to each group. That is, the within-component captures the average lifespan inequality across groups and thus the intra-group heterogeneity in lifespans. The between-component captures how different lifespan inequality is between groups; i.e., the inter-group heterogeneity in lifespan.
For several measures of lifespan inequality additive decompositions have been published in the literature. Table 1 below provides an overview of all decompositions covered in the package, whether new or not. $K$ is the total number of groups and $\mathrm{Pr}(k)$ is the proportion of group $k$ in the population, assumed to be measured at the starting age of the life table. $k$ is also used as superscript and subscript to indicate group-specific measures; e.g., $e_0^k$ is the life expectancy at birth in group $k$. Additional details on the notation are provided below.
library(knitr) library(kableExtra) df <- data.frame(Index=c("`aid`", "`edag`", "`eta_dag`", "`gini` (achieved)", "`gini` (shortfall)", "`H` (shortfall)", "`mad`", "`mld` (achieved)", "`mld` (shortfall)", "`theil` (achieved)", "`theil` (shortfall)", "`var`")) df$Symbol <- c("$\\mathrm{AID}$", "$e^\\dagger$", "$\\eta^\\dagger$", "$G$", "$G$", "$H$", "$\\mathrm{MAD}$", "$T_L$", "$T_L$", "$T_T$", "$T_T$", "$\\mathrm{Var}$") df$Within <- c("$\\sum_{k=1}^K \\mathrm{Pr}(k)\\frac{\\ell_a^k}{\\ell_a} \\mathrm{AID}_k$&", "$\\sum_{k=1}^K \\mathrm{Pr}(k) e^\\dagger_k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\eta^\\dagger_k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\ell_a^{2(k)}\\eta_a^k}{\\ell_a^2\\eta_a} G_k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\ell_a^{2(k)} e_a^k}{\\ell_a^2 e_a} G_k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{e^\\dagger_k}{e_a}$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\mathrm{MAD}_k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) T_L^k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) T_L^k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\eta_a^k}{\\eta_a} T_T^k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{e_a^k}{e_a} T_T^k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\mathrm{Var}_k$") df$Between <- c("$\\frac{1}{2\\ell_a}\\sum_{k=1}^K B(k)+R(k)$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\left(e^{\\dagger*}_k - e^\\dagger_k\\right)$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\left(\\eta^{\\dagger*}_k - \\eta^\\dagger_k\\right)$", "$\\frac{1}{2\\ell_a\\eta_a}\\sum_{k=1}^K B(k)+R(k)$", "$\\frac{1}{2\\ell_a e_a}\\sum_{k=1}^K B(k)+R(k)$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\left(\\frac{e^{\\dagger*}_k - e^\\dagger_k}{e_a}\\right)$", "$\\mathrm{MAD}-\\sum_{k=1}^K \\mathrm{Pr}(k) \\mathrm{MAD}_k$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\log \\frac{e_a}{e^k_a}$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\log \\frac{\\eta_a}{\\eta^k_a}$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{\\eta_a^k}{\\eta_0}\\log \\frac{\\eta_a^k}{\\eta_a}$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) \\frac{e_a^k}{e_0}\\log \\frac{e_a^k}{e_a}$", "$\\sum_{k=1}^K \\mathrm{Pr}(k) (e_a^k-e_a)^2$") df %>% kbl(escape = F, caption = "Table 1: Decomposition formulas") %>% kable_paper("hover", full_width = T)
Existing decompositions
The variance and its decomposition are standard textbook material. The life table formula is taken from @permanyer2019. The Theil index, often also called Theil T index, is a special case of the Generalized Entropy Index. The formulas and decompositions are taken from @liao2022 and @permanyer2019. The mean log deviation is also a special case of the Generalized Entropy Index, and sometimes called Theil L index. The formulas and decompositions are taken from @liao2022 and @vanraalte2011. The derivation of the decomposition for the mean absolute deviation is straightforward.
Decomposing lifespan disparity
Here, we use a notation often found in the economic and statistical literature
on inequality measurement, which makes it more easy to derive the decomposition
of lifespan disparity, $e^\dagger$. Let $N$ be the size of the population
under study; $Y$ is the variable capturing the length of life and $y_i$ is the
length of life for individual $i$. Using this notation, $e^\dagger$ can be
written and re-arranged as follows:
\begin{align}
e^\dagger & = \frac{1}{N}\sum_{i=1}^N \mathrm{M}(Y|Y\geq y_i) \
& = \frac{1}{N}\sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}(Y|Y\geq y_{ik}) \
& = \frac{1}{N}\sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}(Y|Y\geq y_{ik}) + \mathrm{M}k(Y|Y\geq y{ik}) - \mathrm{M}k(Y|Y\geq y{ik})\
&=\frac{1}{N} \sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}k(Y|Y\geq y{ik}) + \frac{1}{N} \sum_{k=1}^K\sum_{i=1}^{N_k} \mathrm{M}(Y|Y\geq y_{ik})-\mathrm{M}k(Y|Y\geq y{ik})\
&=\sum_{k=1}^K \mathrm{Pr}(k) e^\dagger_k + \sum_{k=1}^K \mathrm{Pr}(k) \left(e^{\dagger}_k - e^\dagger_k\right)
\end{align}
The last line uses life table notation. The within-component is the average
group-specific lifespan disparity, and the between component is the difference
between a counterfactual group-specific lifespan inequality ($e^{\dagger}_k$)
and the actual group-specific lifespan inequality. The counterfactual
group-specific lifespan inequality would arise if members of group $k$ died
according to the life table of that group, but the average remaining life
expectancy at the age of death is that of the total population. This essentially
captures how average remaining life expectancy differs between the groups.
Decomposing the Gini coefficient
@liao2022 gives the following decomposition of the Gini coefficient: \begin{eqnarray} W_L&=&\frac{1}{2N^2\bar{y}} \sum_{k=1}^K\sum_{i=1}^{N_k}\sum_{j=1}^{N_k} |y_{ik}-y_{jk} | \ B_L&=&\frac{1}{2N^2\bar{y}} \sum_{k=1}^K\sum_{h=1,h\neq k}^K\sum_{i=1}^{N_k}\sum_{j=1}^{N_h} |y_{ik}-y_{jh} |, \end{eqnarray} where $y_{ik}$ is the lifespan of individual $i$ in group $k$. This can be written in life table notation as \begin{eqnarray}\label{eq:W1} W_L&=&\frac{1}{2\ell_0^2 2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z | b_x-b_z| \ B_L&=&\frac{1}{2\ell_0^2 e_0}\sum_{k=1}^K\sum_{h=1,h\neq k}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^h_z | b_x-b_z|. \end{eqnarray} The equation for $W$ can be rearranged: \begin{eqnarray} W_L&=&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z | b_x-b_z| \ &=&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \left(\sum\limits_{h=1}^K \mathrm{Pr}(h) |b^h_x-b^h_z|\right) \ &=&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \mathrm{Pr}(k) |b^k_x-b^k_z| \ &+&\frac{1}{2\ell_0^2e_0}\sum_{k=1}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \left(\sum\limits_{h=1,h\neq k}^K \mathrm{Pr}(h) |b^h_x-b^h_z|\right) \end{eqnarray} The first line of the last definition of $W_l$ can be re-arranged to give \begin{eqnarray} W=\sum_{k=1}^K \mathrm{Pr}(k) \frac{\ell_0^{2(k)} e_0^k}{\ell_0^2 e_0} G_k, \end{eqnarray} as shown in the overview table. The remainder of the last definition for $W_l$ in combination with $B_L$ provides the between-component $B$, \begin{eqnarray} B=\frac{1}{2\ell_0 e_0}\sum_{k=1}^K B(k)+R(k) \end{eqnarray} with $B(k)$ and $R(k)$ defined as: \begin{eqnarray} B(k) &=& \sum_{h=1,h\neq k}^K\sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^h_z | b_x-b_z| \ R(k) &=& \sum_{x=0}^\omega \sum_{z=0}^\omega d^k_x d^k_z \left( \sum_{h=1,h\neq k}^K \mathrm{Pr}(h) | b^h_x-b^h_z| \right) \end{eqnarray} The case of the Gini using the achieved age at death follows in the same way, as does the case of the life table starting at age $a$.
Decomposing H and AID
ife table entropy, $H$, is a linear transformation of lifespan disparity ($e^\dagger/e_a$), while the absolute inter-individual difference in lifespan (AID) is a linear transformation of the Gini coefficient ($G e_a$). This means that the formulas for the decomposition of lifespan disparity and the Gini coefficient only need to be slightly modified. In case of $H$, the within and between component are given by $W_{e^\dagger}/e_a$ and $B_{e^\dagger}/e_a$, while in case of the AID we have $W_{G}e_a$ and $B_{G}e_a$; i.e., scaling by overall life expectancy.
Example
As an example, we will decompose the lifespan inequality in Canada using gender. This will show us to what extent lifespan inequality in Canada is due to gender differentials in mortality. First, we load the package and the data.
library(LifeIneq) data(LT) data(LTm)
LT contains the life table of Canadian women in 2016, while LTm has the same information for men. We will pass some of this data to the function 'bw_decomp'. To do so, we combine the relevant information in matrices:
age <- 0:110 ax <- cbind(LT$ax,LTm$ax) dx <- cbind(LT$dx,LTm$dx) lx <- cbind(LT$lx,LTm$lx) ex <- cbind(LT$ex,LTm$ex)
For instance, the object 'lx' contains the life table column $\ell_x$ for both men and women. It is a matrix where each column represents one of the groups; the first column has $\ell_x$ of women, and the second column has $\ell_x$ of men.
head(lx)
Generally, for all matrices the number of columns will be equal to the number of groups.
The matrices created above are passed to the 'bw_decomp' function. In addition to the data several other things need to be specified. The argument 'prop' takes the proportion of each group. The argument 'method' can be used to choose betwee the different measures of lifespan inequality. Currentl, the options "theil", "edag","var","mld","gini","mad","aid", and "H" are supported. Finally, for some measures the distribution type needs to be specified; i.e., whether the measure should be calculated with remaining life expectancy ("rl") or with the achieved age at death ("aad").
For our example, we assume that women are close to 49% of the population at the radix. Choosing mean log deviation (MLD) as our measure and using the remaining life expectancy, we get the following results:
bw_decomp(age = age, ax = ax, dx = dx, lx = lx, ex = ex, prop = c(.4886,1-.4886), method = "mld", distribution_type = "rl")
The first component ("method") shows the method used. The second component ("group_ind") shows the value of the chosen measure for each group. In this case, MLD is about 0.04 for women and 0.05 for men. "tot" shows the value of MLD for the total population. The next two components, "B" and "W", show the values of the between-component and the within-component, respectively. Compared to the within-component, the between-component is rather small in this case. This is also indicated by "fB" and "fW", which show the between-component and the within-component relative to the total value. The between-component only contributes less than 1% to the total lifespan inequality as measured through the MLD. This means that the variation between genders is much smaller as within both genders, at least as measured through the MLD.
As a second example, we calculate the Gini coefficient using the achieved age at death:
bw_decomp(age = age, ax = ax, dx = dx, lx = lx, ex = ex, prop = c(.4886,1-.4886), method = "gini", distribution_type = "aad")
The components of the result are as in the first example. Again, the contribution of the between-component is rather small with slightly more than 1%. Generally, such seemingly low importance of the between-component is rather common. For instance, @vanraalte2012 find similarly low between-components by education in several countries. One potential reason is that there are many factors affecting mortality, and a single grouping variable alone (such as gender in our example) has little predictive power; i.e., after accounting for this variable still a sizeable proportion of lifespan variability remains.
References
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