2 Tools for data analysts
2.1 Labels
2.1.1 Age
Producers of demographic data follow a wide variety of styles for labeling age groups. poputils contains tools for parsing and manipulating age group labels.
Age label functions in poputils require that age labels belong to one of three types:
"single". Single years of age, possibly including an open age group, eg"0",“81”,“17”,“100+”`."five". Five-year age groups, possibly including an open age group, eg"0-4","80-84","15-19","100+"."lt". Life table age groups. Like"five", but with the"0-4"age group split into"0"and"1-4".
Age labels created by poputils functions such as age_labels() follow a standard set of rules. Many age labels created using other rules can, however, be parsed by poputils functions,
library(poputils)
library(dplyr, warn.conflicts = FALSE)
tibble(original = c("5 to 9", "5_9", "05-09"),
reformated = reformat_age(original))
#> # A tibble: 3 × 2
#> original reformated
#> <chr> <fct>
#> 1 5 to 9 5-9
#> 2 5_9 5-9
#> 3 05-09 5-9Functions age_lower(), age_upper(), and age_mid() extract information about lower limits, upper limits, and centers of age groups. This can be useful for ordering data
df <- data.frame(age = c("5-9", "0-4", "15-19", "10-14"),
population = c(3, 7, 2, 4))
df
#> age population
#> 1 5-9 3
#> 2 0-4 7
#> 3 15-19 2
#> 4 10-14 4
df |>
arrange(age_lower(age))
#> age population
#> 1 0-4 7
#> 2 5-9 3
#> 3 10-14 4
#> 4 15-19 2and plotting
among other things.
Functions combine_age() and set_age_open() can be used to collapse age groups,
tibble(age = age_labels("lt", max = 30),
age_5 = combine_age(age, to = "five"),
age_25plus = set_age_open(age, lower = 20))
#> # A tibble: 8 × 3
#> age age_5 age_25plus
#> <chr> <chr> <chr>
#> 1 0 0-4 0
#> 2 1-4 0-4 1-4
#> 3 5-9 5-9 5-9
#> 4 10-14 10-14 10-14
#> 5 15-19 15-19 15-19
#> 6 20-24 20-24 20+
#> 7 25-29 25-29 20+
#> 8 30+ 30+ 20+The aim is that users should be able to with age group labels throughout the analysis.
2.2 Life tables and life expectancy
A life table a way of summarizing mortality conditions. It consists of quantities calculated from age-specific mortality rates. The most widely-used life table quantity is life expectancy at birth.
2.2.1 Basic functionality
Life tables can be calculated from age-specific mortality rates using function lifetab().
nzmort |>
filter(year == 2022,
gender == "Female") |>
lifetab(mx = mx)
#> # A tibble: 21 × 10
#> year gender age deaths popn qx lx dx Lx ex
#> <int> <chr> <fct> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2022 Female Infant 84 29680 0.00283 100000 283. 99859. 83.4
#> 2 2022 Female 1-4 years 18 118420 0.000608 99717. 60.6 398748. 82.6
#> 3 2022 Female 5-9 years 12 156820 0.000383 99657. 38.1 498189. 78.7
#> 4 2022 Female 10-14 years 15 164830 0.000455 99619. 45.3 497980. 73.7
#> 5 2022 Female 15-19 years 42 154150 0.00136 99573. 136. 497528. 68.8
#> 6 2022 Female 20-24 years 63 156860 0.00201 99438. 199. 496690. 63.8
#> 7 2022 Female 25-29 years 72 172770 0.00208 99238. 207. 495675. 59.0
#> 8 2022 Female 30-34 years 78 194570 0.00200 99032. 198. 494663. 54.1
#> 9 2022 Female 35-39 years 111 175050 0.00317 98833. 313. 493385. 49.2
#> 10 2022 Female 40-44 years 147 160070 0.00458 98521. 451. 491474. 44.3
#> # ℹ 11 more rowslifetab() and lifeexp() both have a by argument. Separate results are calculated for each combination of the by variables,
nzmort |>
lifeexp(mx = mx,
by = c(gender, year))
#> # A tibble: 4 × 3
#> gender year ex
#> <chr> <int> <dbl>
#> 1 Female 2021 84.0
#> 2 Male 2021 80.5
#> 3 Female 2022 83.4
#> 4 Male 2022 80.0The same effect can be obtained using dplyr::group_by(),
nzmort |>
group_by(gender, year) |>
lifeexp(mx = mx)
#> # A tibble: 4 × 3
#> gender year ex
#> <chr> <int> <dbl>
#> 1 Female 2021 84.0
#> 2 Male 2021 80.5
#> 3 Female 2022 83.4
#> 4 Male 2022 80.0The input data for life tables and life expectancies can be probabilities of dying (qx), rather than mortality rates (mx)
west_lifetab |>
group_by(level, sex) |>
lifeexp(qx = qx)
#> # A tibble: 50 × 3
#> level sex ex
#> <int> <chr> <dbl>
#> 1 1 Female 20.1
#> 2 1 Male 18.1
#> 3 2 Female 22.5
#> 4 2 Male 20.5
#> 5 3 Female 25.0
#> 6 3 Male 22.9
#> 7 4 Female 27.5
#> 8 4 Male 25.3
#> 9 5 Female 30.0
#> 10 5 Male 27.7
#> # ℹ 40 more rowsBy default, lifeexp() calculates life expectancy at age zero. It can, however, be used to calculate life expectancy at other ages.
2.2.2 Calculation methods
Alternative methods for calculating life tables differ mainly in their assumptions variation within age groups (Preston, Heuveline, and Guillot 2001; Keyfitz and Caswell 2005). It turns out that, for the purposes of constructing life tables, all the relevant information about the way that mortality varies by age within each age group can be captured by a single number: the average length of time lived in an interval by people who die in that interval (Preston, Heuveline, and Guillot 2001, 43). This number is denoted \(_na_x\), where \(x\) is exact age at the start of the internal, and \(n\) is the length of the interval. The quantity \(_5a_{20}\), for instance, refers to the average number of years lived after their 20th birthday by people who die between their 20th and 25th birthdays. When \(n=1\), the \(n\) subscript is typically omitted.
Functions lifetab() and lifeexp() have four arguments for specifying calculation methods:
infant, which specifies how \(a_0\) is calculated,child, which specifies how \(_4a_1\) is calculated,closed, which specifies how \(_na_x\) for all other closed intervals are calculated, andopen, which specifies how the final interval, \(_{\infty}a_x\) is calculated.
Different choices of method are available for each argument. In some cases, different formulas are used for females and males. The formulas can also differ depending on whether the input data is of mortality rates or probabilities of dying.
argument |
sex |
method |
input | formula |
|---|---|---|---|---|
infant |
<any> | "constant" |
mx |
\[a_0 = \frac{1 - (m_0 + 1) e^{-m_0}}{m_0 (1 - e^{-m_0})}\] |
infant |
<any> | "constant" |
qx |
\[a_0 = \frac{(1 - \log(1 - q_0) (1 - q_0)) - 1}{\log(1 - q_0) q_0}\] |
infant |
<any> | "linear" |
mx |
\[a_0 = 0.5\] |
infant |
<any> | "linear" |
qx |
\[a_0 = 0.5\] |
infant |
Female | "CD" |
mx |
\[a_0 = \begin{cases} 0.053 + 2.8 m_0 & 0 \le m_0 < 0.107 \\ 0.35 & m_0 \ge 0.107 \end{cases}\] |
infant |
Female | "CD" |
qx |
\[a_0 = \begin{cases} 0.05 + 3 q_0 & 0 \le m_0 < 0.1 \\ 0.35 & q_0 \ge 0.1 \end{cases}\] |
infant |
Male | "CD" |
mx |
\[a_0 = \begin{cases} 0.045 + 2.684 m_0 & 0 \le m_0 < 0.107 \\ b0.33 & m_0 \ge 0.107 \end{cases}\] |
infant |
Male | "CD" |
qx |
\[a_0 = \begin{cases} 0.0425 + 2.875 q_0 & 0 \le q_0 < 0.1 \\ 0.33 & q_0 \ge 0.1 \end{cases}\] |
infant |
Female | "AK" |
mx |
\[a_0 = \begin{cases} 0.14903 - 2.05527 m_0 & 0 \le m_0 < 0.01724 \\ 0.04667 + 3.88089 m_0 & 0.01724 \le m_0 < 0.06891 \\ 0.31411 & m_0 \ge 0.06891 \end{cases}\] |
infant |
Female | "AK" |
qx |
\[a_0 = \begin{cases} 0.149 - 2.0867 q_0 & 0 \le q_0 < 0.017 \\ 0.0438 + 4.1075 q_0 & 0.017 \le q_0 < 0.0658 \\ 0.3141 & q_0 \ge 0.0658 \end{cases}\] |
infant |
Male | "AK" |
mx |
\[a_0 = \begin{cases} 0.14929 - 1.99545 m_0 & 0 \le m_0 < 0.023 \\ 0.02832 + 3.26021 m_0 & 0.023 \le m_0 < 0.08307 \\ 0.29915 & m_0 \ge 0.08307 \end{cases}\] |
infant |
Male | "AK" |
qx |
\[a_0 = \begin{cases} 0.1493 - 2.0367 q_0 & 0 \le q_0 < 0.0226 \\ 0.0244 + 3.4994 q_0 & 0.0226 \le q_0 < 0.0785 \\ 0.2991 & q_0 \ge 0.0785 \end{cases}\] |
child |
<any> | "constant" |
mx |
\[_4a_1 = \frac{1 - (4 \times {_4}m_1 + 1) e^{-4 \times {_4}m_1}}{_4m_1 (1 - e^{-4 \times {_4}m_1})}\] |
child |
<any> | "constant" |
qx |
\[_4a_1 = \frac{4((1 - \log(1-{_4}q_1)) (1 - {_4}m_1) - 1)}{\log(1 - {_4q_1}) {_4}q_1}\] |
child |
<any> | "linear" |
mx |
\[_4a_1 = 2\] |
child |
<any> | "linear" |
qx |
\[_4a_1 = 2\] |
child |
Female | "CD" |
mx |
\[_4a_1 = \begin{cases} 1.522 - 1.518 m_0 & 0 \le m_0 < 0.107 \\ 1.361 & m_0 \ge 0.107 \end{cases}\] |
child |
Female | "CD" |
qx |
\[_4a_1 = \begin{cases} 1.542 - 1.625 q_0 & 0 \le q_0 < 0.1 \\ 1.361 & q_0 \ge 0.1 \end{cases}\] |
child |
Male | "CD" |
mx |
\[_4a_1 = \begin{cases} 1.651 - 2.816 m_0 & 0 \le m_0 < 0.107 \\ 1.352 & m_0 \ge 0.107 \end{cases}\] |
child |
Male | "CD" |
qx |
\[_4a_1 = \begin{cases} 1.653 - 3.013 q_0 & 0 \le q_0 < 0.1 \\ 1.352 & q_0 \ge 0.1 \end{cases}\] |
closed |
<any> | "constant" |
mx |
\[_na_x = \frac{1 - (n \times {_n}m_x + 1) e^{-n \times {_n}m_x}}{_nm_x (1 - e^{-n \times {_n}m_x})}\] |
closed |
<any> | "constant" |
qx |
\[_na_x = \frac{n((1 - \log(1 - {_n}q_x))(1 - {_nq_x}) - 1)}{\log(1 - {_nq_x}) {_n}q_x}\] |
closed |
<any> | "linear" |
mx |
\[_na_x = 0.5 n\] |
closed |
<any> | "linear" |
qx |
\[_na_x = 0.5 n\] |
open |
<any> | "constant" |
mx |
\[_{\infty}a_{\omega} = \frac{1}{_{\infty}m_{\omega}}\] |
open |
<any> | "constant" |
qx |
\[_{\infty}a_{\omega} = \frac{1}{_{n}m_{\omega-n}}\] |
In the table above, the values for "CD" are from Coale, Demeny, and Vaughan (1983), p20, and Preston, Heuveline, and Guillot (2001), p48; the values for "AK" are from Andreev and Kingkade (2015), p376, and Wilmoth et al. (2021), p37; and the values for "constant" are expected values for an exponential distribution that has been right-truncated at \(n\).
When the inputs data are \(_nq_x\), the value of \(_na_x\) for the last age group is based in mortality rates in the second-to-last age group. This is an expedient to deal with the fact that \(_nq_x\) is always 1 in the last age group, and therefore provides no information about mortality conditions in that age group.
Once the \(_na_x\) have been determined, the life table is fully specified, and the required calculations can be carried out with no further input from the user.
The probability of dying within each interval is
\[_nq_x = \frac{n \times {_n}m_x}{1 + (n - {_n}a_x) \times {_nm_x}},\]
with \(_{\infty}q_{\omega} = 1\). Quantity \(l_x\) is the number of people surviving to exact age \(x\). In lifetab(), by default, \(l_0 = 100,000\). Remaining values are calculated using
\[l_{x+n} = (1 - {_nq_x}) \times l_x.\] Quantity \(_nd_x\) is the number of people who die between exact ages \(x\) and \(x+n\),
\[_nd_x = l_x - l_{x+n}.\]
Quantity \(_nL_x\) is the number of person-years lived between exact ages \(x\) and \(x+n\). It consists of person-years lived by people who survive the interval, plus person-years lived by people who die within the interval,
\[_nL_x = l_{x+n} \times n + {_nd_x} \times {_na_x}.\] Finally, \(e_x\), the number of years of life remaining to a person aged exactly \(x\), is \[e_x = {_nL_x} + {_nL_{x+n}} + \cdots + {_{\infty}L_{\omega}}\].
Although the results for lifetab() and lifeexp() do vary with difference choices for infant, child, or closed, the differences are often small,
lin <- nzmort |>
lifeexp(mx = mx,
by = c(gender, year),
infant = "linear",
suffix = "lin")
ak <- nzmort |>
lifeexp(mx = mx,
sex = gender,
by = year,
infant = "AK",
suffix = "ak")
inner_join(lin, ak, by = c("year", "gender")) |>
mutate(diff = ex.lin - ex.ak)
#> # A tibble: 4 × 5
#> gender year ex.lin ex.ak diff
#> <chr> <int> <dbl> <dbl> <dbl>
#> 1 Female 2021 84.0 84.0 0.000906
#> 2 Male 2021 80.5 80.5 0.00110
#> 3 Female 2022 83.4 83.4 0.000771
#> 4 Male 2022 80.0 80.0 0.0009652.2.3 Uncertainty
The examples of life tables and life expectancy so far have all been based on a deterministic input, mx column of data frame nzmort,
nzmort
#> # A tibble: 84 × 6
#> year gender age deaths popn mx
#> <int> <chr> <fct> <int> <int> <dbl>
#> 1 2021 Female Infant 108 29570 0.00365
#> 2 2021 Female 1-4 years 30 118950 0.000252
#> 3 2021 Female 5-9 years 12 158350 0.0000758
#> 4 2021 Female 10-14 years 21 163800 0.000128
#> 5 2021 Female 15-19 years 51 152960 0.000333
#> 6 2021 Female 20-24 years 54 160250 0.000337
#> 7 2021 Female 25-29 years 60 180290 0.000333
#> 8 2021 Female 30-34 years 63 192070 0.000328
#> 9 2021 Female 35-39 years 105 171970 0.000611
#> 10 2021 Female 40-44 years 153 157740 0.000970
#> # ℹ 74 more rowsThe data frame nzmort_rvec instead uses a rvec to represent mortality rates,
library(rvec)
#>
#> Attaching package: 'rvec'
#> The following objects are masked from 'package:stats':
#>
#> sd, var
#> The following object is masked from 'package:base':
#>
#> rank
nzmort_rvec
#> # A tibble: 84 × 4
#> year gender age mx
#> <int> <chr> <fct> <rdbl<1000>>
#> 1 2021 Female Infant 0.0032 (0.0028, 0.0037)
#> 2 2021 Female 1-4 years 0.00018 (0.00014, 0.00022)
#> 3 2021 Female 5-9 years 9.2e-05 (7.3e-05, 0.00012)
#> 4 2021 Female 10-14 years 0.00012 (9.9e-05, 0.00014)
#> 5 2021 Female 15-19 years 0.00027 (0.00023, 0.00031)
#> 6 2021 Female 20-24 years 0.00032 (0.00028, 0.00036)
#> 7 2021 Female 25-29 years 0.00034 (0.00031, 0.00038)
#> 8 2021 Female 30-34 years 0.00042 (0.00038, 0.00046)
#> 9 2021 Female 35-39 years 6e-04 (0.00054, 0.00065)
#> 10 2021 Female 40-44 years 0.00091 (0.00084, 0.00098)
#> # ℹ 74 more rowsThe mx rvec holds 1000 draws from the posterior distribution from a Bayesian model of mortality. The posterior distribution for infant mortality for females in 2021, for instance, has a posterior median of 0.0032, and a 95% credible interval of (0.0028, 0.0037).
If the input to lifetab() or lifeexp() is an rvec, then the output will be too. Uncertainty about mortality rates is propagated through to quantities derived from these rates.
library(rvec)
nzmort_rvec |>
filter(year == 2022,
gender == "Female") |>
lifetab(mx = mx) |>
select(age, qx, lx)
#> # A tibble: 21 × 3
#> age qx lx
#> <fct> <rdbl<1000>> <rdbl<1000>>
#> 1 Infant 0.0034 (0.0029, 0.0039) 1e+05 (1e+05, 1e+05)
#> 2 1-4 years 0.00075 (6e-04, 0.00095) 99661 (99613, 99708)
#> 3 5-9 years 5e-04 (0.00039, 0.00063) 99587 (99534, 99638)
#> 4 10-14 years 0.00064 (0.00053, 0.00076) 99538 (99479, 99594)
#> 5 15-19 years 0.0014 (0.0012, 0.0016) 99474 (99411, 99532)
#> 6 20-24 years 0.0017 (0.0015, 0.0019) 99332 (99265, 99393)
#> 7 25-29 years 0.0018 (0.0017, 0.0021) 99163 (99093, 99231)
#> 8 30-34 years 0.0023 (0.0021, 0.0025) 98979 (98901, 99053)
#> 9 35-39 years 0.0032 (0.0029, 0.0035) 98753 (98666, 98837)
#> 10 40-44 years 0.0048 (0.0044, 0.0052) 98438 (98339, 98530)
#> # ℹ 11 more rows