# Constructing Life Tables with R

I have been using the package dplyr to handle with data for a while, and I thought I can use it with ease until I was stuck with my homework on contructing a life table. I found spreadsheets (either Excel or Google Spreadsheets) easy for handling this task, but had a hard time dealing with it in R. I think it was due to my unfamiliarity with the built-in functions and insufficient practice in R. So, I wrote this post as a review and practice of my data-wrangling skills in R.

I will illustrate how I constructed a life table with R, and you’ll find out how easy it is (and wonder how could I stumble on it).

I used these packages to construct a Life table.

library(readr)
library(dplyr)
library(knitr)


Load the csv file to life_table. The raw data contains 3 columns: Age, Survivorship at Age x ($l_x$), and Fecundity at Age x ($m_x$). I added options(scipen=999) to disable scientific notations.

options(scipen=999)  # Disable Scientific Notation
life_table

# A tibble: 10 x 3
Age        lx    mx
<int>     <dbl> <int>
1     0 1.0000000     0
2     1 0.0000620  4600
3     2 0.0000340  8700
4     3 0.0000200 11600
5     4 0.0000155 12700
6     5 0.0000110 12700
7     6 0.0000065 12700
8     7 0.0000020 12700
9     8 0.0000020 12700
10     9 0.0000000     0


## Variables to Construct

Here are the variables that need to be calculated.

Statistic Notation Calculation Formula
$l_x m_x$
$x l_x m_x$
$l_x m_x e^{-rx}$
Average survivorship
(age class)
$L_x$ $L_x = (l_x + l_{x+1})/2$
Life expectancy $e_x$ $e_x = (L_x + L_{x+1} + … + L_{max})/l_x$
Reproductive value $V_x$ $\displaystyle \frac{\sum_{y=x}^{max\hspace{0.3mm}x} e^{-ry} l_y m_y}{e^{-rx} l_x}$
Net reproductive rate $R_0$ $\sum_{all \hspace{0.3mm} x} l_x m_x$
Generation time $G$ $\frac{\sum_{all \hspace{0.3mm} x} x l_x m_x}{R_0}$
Intrinsic rate of increase Approximate $r$ $r \approx \frac{ln(R_0)}{G}$
Intrinsic rate of increase (True) $r$ $\displaystyle \sum_{all \hspace{0.3mm} x} e^{-rx}l_x m_x = 1$

$l_xm_xe^{-rx}$ and $V_x$ will be calculated twice for the approximate and the true $r$.

## Constructing Variables

### mutate: Creating new columns

Using the pipe %>% and the function mutate in package dplyr, I first constructed 7 new variables. Note the dependencies of the variables, so that I couldn’t construct the life tables at once.

life_table <- life_table %>%
mutate("lx*mx"=lx*mx,
"x*lx*mx"=Age*lx*mx,
"Lx"=replace(Lx, 10, 0),
"ex"=rev(cumsum(rev(Lx)))/lx,
"R0"=sum(lx*mx),
"G"=sum(Age*lx*mx)/R0,
"approx.r"=log(R0)/G
)


Two things worth noting in the mutate function:

1. The code "Lx"=(lx+lead(lx))/2, "Lx"=replace(Lx, 10, 0)
• lead(lx) shifts the whole column of $l_x$ to its next value, i.e. the column $l_x$ becomes $l_{(x+1)}$.
• Due to lead(lx), the last entry of the new column $L_x$ must be a NA, so I have to assign 0 to it (otherwise all calculations based on it will become NAs).
2. The code "ex"=rev(cumsum(rev(Lx)))/lx (This is where I was stuck)
1. The numerator of ex is calculated by summing over $L_x$ to $L_{max}$, the maximum age of $L_x$. This is not so intuitive when working with R.
2. rev(Lx) reverse the order of $L_x$, and cumsum() is for cummulative sum. cumsum(rev(Lx)) then is equivalent to summing $L_x$ backwards (i.e. from $L_{max}$ to $L_x$).
3. But since $L_x$ is reversed in the first place, cumsum(rev(Lx)) is also in reverse order. Reversing cumsum(rev(Lx)) with rev() then gives what I want.

### Calculating $r$ by while loop

By the Eular-Lotka equation, I can calculate $r$.

Using while loop and Approximate $r$ calculated earlier as the starting value for r, $r$ is calculated as below.

df <- as.data.frame(life_table)
r <- 0.0812198
x <- sum(exp(-r*df$Age)*df$lx*mx)
while (abs(x-1) >= 0.000001) {
if (x-1>0){
r <- r+0.00000001
}
else{
r <- r-0.00000001
}
x <- sum(exp(-r*df$Age)*df$lx*mx)
r
}


The rest is simple, and the logic applied is the same.

life_table <- life_table %>%
mutate(
"approx.r"=log(R0)/G,
"r"=r,
"Vx"=rev(cumsum(rev(exp(-r*Age)*lx*mx)))/exp(-r*Age)*lx,
"lx*mx*e^-rx"= lx*mx*exp(-r*Age),
"approx.Vx"=rev(cumsum(rev(exp(-approx.r*Age)*lx*mx)))/exp(-approx.r*Age)*lx,
"approx.lx*mx*e^-rx"= lx*mx*exp(-approx.r*Age)
)


Now, I’m done constructing a life table.

## Printing pretty Life Table

kable(life_table, format="markdown", align="c")

$Age$ $l_x$ $m_x$ $l_x m_x$ $x l_x m_x$ $L_x$ $e_x$ $R_0$ $G$ $Approximate$
$r$
$r$ $V_x$ $l_x m_x e^{-rx}$ $Approximate$
$V_x$
$Approximate$
$l_x m_x e^{-rx}$
0 1.0000000 0 0.00000 0.0000 0.5000310 0.500153 1.2829 3.06727 0.0812198 0.0847117 1.0000010 0.0000000 1.0099103 0.0000000
1 0.0000620 4600 0.28520 0.2852 0.0000480 1.967742 1.2829 3.06727 0.0812198 0.0847117 0.0000675 0.2620352 0.0000679 0.2629518
2 0.0000340 8700 0.29580 0.5916 0.0000270 2.176471 1.2829 3.06727 0.0812198 0.0847117 0.0000297 0.2497000 0.0000299 0.2514499
3 0.0000200 11600 0.23200 0.6960 0.0000178 2.350000 1.2829 3.06727 0.0812198 0.0847117 0.0000126 0.1799362 0.0000126 0.1818310
4 0.0000155 12700 0.19685 0.7874 0.0000132 1.887097 1.2829 3.06727 0.0812198 0.0847117 0.0000067 0.1402737 0.0000067 0.1422467
5 0.0000110 12700 0.13970 0.6985 0.0000087 1.454546 1.2829 3.06727 0.0812198 0.0847117 0.0000028 0.0914634 0.0000028 0.0930743
6 0.0000065 12700 0.08255 0.4953 0.0000042 1.115385 1.2829 3.06727 0.0812198 0.0847117 0.0000008 0.0496567 0.0000008 0.0507081
7 0.0000020 12700 0.02540 0.1778 0.0000020 1.500000 1.2829 3.06727 0.0812198 0.0847117 0.0000001 0.0140380 0.0000001 0.0143853
8 0.0000020 12700 0.02540 0.2032 0.0000010 0.500000 1.2829 3.06727 0.0812198 0.0847117 0.0000001 0.0128978 0.0000001 0.0132632
9 0.0000000 0 0.00000 0.0000 0.0000000 NaN 1.2829 3.06727 0.0812198 0.0847117 0.0000000 0.0000000 0.0000000 0.0000000

Note that I edited the column names of the table in a text editor to make it display in LaTeX style. There is no simple knitr function (at least I don’t know) that print out pretty displayed style text in a table generated from a data frame