Lab: Life Tables (1948-2016)

Published

April 3, 2026

M1 MIDS/MFA/LOGOS

Université Paris Cité
Année 2025

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Code
params <- list(
  year_e=2017, 
  year_p=1948
)
Code
datafile <- '../../resources/full_life_table.Rds'
fpath <- datafile #str_c("./DATA/", datafile) # here::here('DATA', datafile)   # check getwd() if problem 

if (! file.exists(fpath)) {
  download.file("https://stephane-v-boucheron.fr/data/tamed_life_table.Rds", 
                fpath,
                mode="wb")
}

life_table <- readr::read_rds(fpath)
Code
life_table <- life_table |>
  mutate(Country = as_factor(Country)) |>
  mutate(Country = fct_relevel(Country, 
                               "Spain", "Italy", "France", "England & Wales", "Netherlands", "Sweden", "USA")) |>
  mutate(Gender = as_factor(Gender)) |>
  mutate(Area = fct_collapse(Country, 
    SE = c("Spain", "Italy", "France"), 
    NE = c("England & Wales", "Netherlands", "Sweden"), 
    USA = "USA"))

Definitions

Check on http://www.mortality.org the meaning of the different columns.

See: Demography: Measuring and Modeling Population Processes by SH Preston, P Heuveline, and M Guillot. Blackwell. Oxford. 2001.

Document Tables de mortalité françaises pour les XIXe et XXe siècles et projections pour le XXIe siècle contains detailed information on the construction of Life Tables for France.

Period tables versus cohort tables

Two kinds of Life Tables can be distinguished: Period tables (Table du moment) which contain for each period (here a period is a calendar year), the mortality risks at different age ranges (here, we have one year ranges) for that very period; and Tables de génération which contain for a given birthyear, the mortality risks at which an individual born during that year has been exposed.

The life tables investigated in this lab are Table du moment. According to the document by Vallin and Meslé, building the life tables required ,decisions and doctoring.

Lexis diagrams

Lexis diagrams provide a graphical device that summarizes the construction of mortality quotients (and other rates in demography).

Code
birth_dates <- as_date("1999-01-01") + duration(sample(2*365, size=20, replace=T),units="day")
death_dates <- as_date("2009-07-01") + duration(sample(3*365, size=20, replace=T),units="day")

b_period <- as_date("2010-01-01")
b_frame <-  as_date(b_period - duration(1, units = "year"))
b_age <- 10L

tb_ld <- tibble(birth=birth_dates, death=death_dates)

tb_ld |> 
  ggplot() +
  geom_segment(aes(x=b_frame,
                   xend=death,
                   y=interval(birth, b_frame)/years(1),
                   yend=interval(birth, death)/years(1))
               ) +
  annotate(geom="rect",
           xmin=b_period,
           xmax=as_date(b_period + duration(1, units = "year")),
           ymin=b_age,
           ymax=b_age + 1L,
           fill="grey",
          alpha=.5) +
  ylab("Age") +
  xlab("Time") +
  coord_cartesian(xlim=c(as_date(b_period - duration(6, units = "months")), 
                         as_date(b_period + duration(18, units = "months"))),
                  ylim=c(b_age - .5, b_age+1.5)) +
  labs(
    title="A Lexis diagram",
    subtitle = "for mortality quotient at Age 10 during Year 2010-11"
  )

Each line represents the life line of an individual born during years 1999 and 2000 and deceased beetween mid 2009 and mid 2012. In order to compute the mortality quotient at age 10 for year 2010, we have to compute the relevant number of occurrences, that is the number of segments ending in the grey rectangle, and the sum of exposure times, which is proportional to the sum of the lengths of the segments crossing the grey rectangle.

Have a look at Lexis diagram or at Preston et al.

Definitions can be obtained from www.lifeexpectancy.org. We translate it into mathematical (rather than demographic) language.

The mortality quotients define a probability distribution over \(\mathbb{N}\). This probability distribution is a construction that reflects the health situation in a population at a given time. This probability distribution does not describe the sequence of sanitary situations experienced by a cohort (people born during a specific year).

One works with a period, or current, life table (table du moment). This summarizes the mortality experience of persons across all ages in a short period, typically one year or three years. More precisely, the death probabilities \(q_x\) for every age \(x\) are computed for that short period, often using census information gathered at regular intervals. These \(q_x\)’s are then applied to a hypothetical cohort of \(100 000\) people over their life span to produce a life table.

Code
small_tb <- life_table |> 
  filter(Country=='France', Year== 2010, Gender=='Female', Age < 10 | between(Age, 80, 89)) |> 
  select(Age, qx, mx, lx, dx, Lx, Tx, ex)

small_tb
# A tibble: 20 × 8
     Age      qx      mx     lx    dx    Lx      Tx    ex
   <int>   <dbl>   <dbl>  <int> <int> <int>   <int> <dbl>
 1     0 0.00324 0.00325 100000   324 99722 8465207 84.6 
 2     1 0.00032 0.00032  99676    32 99660 8365484 83.9 
 3     2 0.00015 0.00015  99645    15 99637 8265824 83.0 
 4     3 0.00011 0.00011  99630    11 99624 8166187 82.0 
 5     4 0.00008 0.00008  99619     8 99615 8066563 81.0 
 6     5 0.00005 0.00005  99611     5 99608 7966948 80.0 
 7     6 0.00008 0.00008  99606     8 99602 7867339 79.0 
 8     7 0.00008 0.00008  99598     8 99594 7767737 78.0 
 9     8 0.00008 0.00008  99590     8 99586 7668143 77   
10     9 0.00007 0.00007  99582     7 99578 7568557 76   
11    80 0.0298  0.0302   75619  2252 74493  802295 10.6 
12    81 0.0346  0.0352   73367  2535 72099  727802  9.92
13    82 0.0398  0.0406   70832  2818 69423  655702  9.26
14    83 0.0464  0.0475   68014  3158 66435  586280  8.62
15    84 0.0539  0.0554   64856  3493 63109  519845  8.02
16    85 0.0610  0.0630   61362  3745 59490  456736  7.44
17    86 0.0699  0.0725   57617  4029 55603  397246  6.89
18    87 0.0793  0.0826   53588  4249 51464  341643  6.38
19    88 0.0922  0.0966   49339  4547 47066  290180  5.88
20    89 0.105   0.111    44792  4706 42440  243114  5.43
NoteQuestion

The table above is not as readable as it should. Use package gt to get a more tunable outpout.

Reorder and filter the columns so that Age comes first (they identify rows), then qx, mx up to ex. You can use select or relocate, or both to do this. Note that Gender, Country, Year are constant in this tibble and need to be reported in the table header, but nowhere else.

Columns qx and mx (for mortality quotient and central death rate) should be dsplayed in scientific notation so that the fact that their range extends over several orders of magnitude shows up.

Columns lx, dx, Lx, Tx contain integer values.

Column ex (residual life expectancy) is a (fictional) decimal number of years

Code
small_tb |> 
  gt() |> 
  tab_header(
    title = "Life table (extract)",
    subtitle = "France, Women, 2010"
  ) |> 
  fmt_integer(columns=c(lx, dx, Lx, Tx)) |> 
  fmt_engineering(columns = c(qx,mx), exp_style = "e", drop_trailing_zeros = T ) |> 
  tab_source_note(source_note = "From https://mortality.org")
Life table (extract)
France, Women, 2010
Age qx mx lx dx Lx Tx ex
0 3.24e−03 3.25e−03 100,000 324 99,722 8,465,207 84.65
1 320e−06 320e−06 99,676 32 99,660 8,365,484 83.93
2 150e−06 150e−06 99,645 15 99,637 8,265,824 82.95
3 110e−06 110e−06 99,630 11 99,624 8,166,187 81.97
4 80e−06 80e−06 99,619 8 99,615 8,066,563 80.97
5 50e−06 50e−06 99,611 5 99,608 7,966,948 79.98
6 80e−06 80e−06 99,606 8 99,602 7,867,339 78.98
7 80e−06 80e−06 99,598 8 99,594 7,767,737 77.99
8 80e−06 80e−06 99,590 8 99,586 7,668,143 77.00
9 70e−06 70e−06 99,582 7 99,578 7,568,557 76.00
80 29.78e−03 30.24e−03 75,619 2,252 74,493 802,295 10.61
81 34.55e−03 35.16e−03 73,367 2,535 72,099 727,802 9.92
82 39.78e−03 40.59e−03 70,832 2,818 69,423 655,702 9.26
83 46.44e−03 47.54e−03 68,014 3,158 66,435 586,280 8.62
84 53.86e−03 55.36e−03 64,856 3,493 63,109 519,845 8.02
85 61.03e−03 62.95e−03 61,362 3,745 59,490 456,736 7.44
86 69.93e−03 72.46e−03 57,617 4,029 55,603 397,246 6.89
87 79.29e−03 82.56e−03 53,588 4,249 51,464 341,643 6.38
88 92.15e−03 96.6e−03 49,339 4,547 47,066 290,180 5.88
89 105.05e−03 110.88e−03 44,792 4,706 42,440 243,114 5.43
From https://mortality.org

Understanding the columns of the life table

In the sequel, we denote by \(F_{t}\) the cumulative distribution function for year \(t\). We agree on \(\overline{F}_t = 1 - F_t\) and \(F_t(-1)=0\). Henceforth, \(\overline{F}\) is called the survival function.

qx
(age-specific) risk of death at age \(x\), or mortality quotient at given age \(x\) for given year \(t\).
NoteAbout the definition of q_{t,x}

Defining and computing q_{t,x} does not boil down to knowing the number of people at age \(x\) at the beginning of ear \(t\) and knowing how many of them died during year \(t\). If we want to be rigorous, we need to know all life lines in the Lexis diagram, or equivalently, how many people at Age \(x\) were alive on each day of Year \(t\).

CautionMortality quotients define a probability distribution

For a given year \(t\), the sequence of mortality quotients define a survival function \(\overline{F}_t\) using the following recursion:

\[q_{t,x} = \frac{\overline{F}_t(x) - \overline{F}_t(x+1)}{\overline{F}_t(x)}\] with boundary condition \(\overline{F}_t(-1) =1\).

This recursion can also be read as:

\[\overline{F}_{t}(x+1) = \overline{F}_{t}(x) \times (1-q_{t,x+1})\, .\]

This artificial probability distribution is used to define and compute life expectancies.

Note

\(q_{t,x}\) is the hazard rate of \(\overline{F}_t\) at age \(x\).

mx
central death rate at age \(x\) during year \(t\). This is connected with \(q_{t,x}\) by \[m_{t,x} = -\log(1- q_{t,x}) \,,\]

or equivalently \[q_{t,x} = 1 - \exp(-m_{t,x})\]

NoteAbout central death rate

If we want to define a continuous probability distribution \(G\) over \([0,\infty)\) so that \(G\) and \(F\) coincide over integers and \(G\) has piecewise constant hazard rate, we can pick \(m_{t,x}\) as the piecewise constant hazard rate.

lx
the so-called survival function: the scaled proportion of persons alive at age \(x\). These values are computed recursively from the \(q_{t,x}\) values using the formula

\[l_t(x+1) = l_t(x) \times (1-q_{t,x}) \, ,\] with \(l_{t,0}\), the radix of the table, (arbitrarily) set to \(100000\). In the table lx is rounded to the next integer

Function \(l_{t,\cdot}\) and \(\overline{F}_t\) are connected by

\[l_{t,x + 1} = l_{t,0} \times \overline{F}_t(x)\,.\]

dx
\(d_{t,x} = q_{t,x} \times l_{t,x}\). The fictitious number of deaths occurring at age \(x\) during year \(t\). Again this is a rounded quantity.
Tx
Total number of person-years lived by the cohort from age \(x\) to \(x+1\). This is the sum of the years lived by the \(l_{t, x+1}\) persons who survive the interval, and the \(d_{t,x}\) persons who die during the interval. The former contribute exactly \(1\) year each, while the latter contribute, on average, approximately half a year, so that \(L_{t,x} = l_{t,x+1} + 0.5 \times d_{t,x}\). This approximation assumes that deaths occur, on average, half way in the age interval x to x+1. Such is satisfactory except at age 0 and the oldest age, where other approximations are often used.

Compare with the denominator in the definition of qx and its description using the Lexis diagram.

We will stick to a simplified vision \(L_{t,x}= l_{t,x+1}\)

ex:
Residual Life Expectancy at age \(x\) and year \(t\)

This is the expectation of \(X -x\) for a random variable \(X\) distributed according to \(\overline{F}_t\) conditionnally on the event \(\{ X \geq x \}\). That is \(e_{t,x}\) is the expectation of the probability distribution defined by \(\overline{F}_t(\cdot + x-1)/\overline{F}_t(x-1)\).

NoteQuestion

Check dependencies between columns

Code
life_table |> 
  filter( Year>=1948, Age < 90, Gender != "Both") |> 
  group_by(Country, Year, Gender) |> 
  summarise(m1 =max(abs(lx -dx -lead(lx))/lx, na.rm = T), 
            m2 =max(abs(lx * qx -dx)/dx, na.rm=T),
            m3 =max(abs(Lx -lx * (1 + qx * (ax-1)))/Lx, na.rm=T),
            m4 =max(abs(1-exp(-mx)-qx)/qx, na.rm=T),
            .groups = "drop") |> 
  select(Year, Country, Gender, m1, m2, m3, m4) |>  
  rename(lx=m1, dx=m2, Lx=m3, qx=m4) |> 
  group_by(Country, Gender) |> 
  slice_max(order_by = desc(qx), n = 1)  |> 
  ungroup() |> 
  gt() |> 
  tab_header(
    title = "Life table (relative discrepancies)",
    subtitle = ""
  ) |> 
  fmt_engineering(columns = ends_with("x"), 
                  decimals=2,
                  drop_trailing_zeros = T ) |> 
  tab_source_note(source_note = "From https://mortality.org")
Life table (relative discrepancies)
Year Country Gender lx dx Lx qx
2013 England & Wales Female 20.33 × 10−6 77.95 × 10−3 10.96 × 10−6 1.51 × 10−3
2002 England & Wales Male 55.97 × 10−6 41.69 × 10−3 31.12 × 10−6 3.15 × 10−3
2005 France Female 10.7 × 10−6 31.67 × 10−3 12.56 × 10−6 1.93 × 10−3
2007 France Male 34.2 × 10−6 105.17 × 10−3 28.96 × 10−6 2.22 × 10−3
2010 Italy Female 19.78 × 10−6 49.96 × 10−3 16.84 × 10−6 1.5 × 10−3
2012 Italy Male 31.08 × 10−6 95.38 × 10−3 27.53 × 10−6 2.31 × 10−3
2008 Netherlands Female 21.37 × 10−6 85.69 × 10−3 14.31 × 10−6 2.08 × 10−3
2009 Netherlands Male 25.74 × 10−6 65.96 × 10−3 27.43 × 10−6 2.74 × 10−3
2015 Spain Female 17 × 10−6 10.9 × 10−3 11.22 × 10−6 2.16 × 10−3
2016 Spain Male 31.74 × 10−6 33.98 × 10−3 21.2 × 10−6 2.14 × 10−3
2004 Sweden Female 19.22 × 10−6 25.87 × 10−3 19.81 × 10−6 1.87 × 10−3
2005 Sweden Male 28.5 × 10−6 32.16 × 10−3 26.8 × 10−6 2.88 × 10−3
2012 USA Female 25.87 × 10−6 17.3 × 10−3 14.73 × 10−6 1.98 × 10−3
2010 USA Male 40.17 × 10−6 62 × 10−3 36.57 × 10−6 2.53 × 10−3
From https://mortality.org

Western countries in 1948

Several pictures share a common canvas:

NoteQuestion

Plot mortality quotients (qx) against age using a logarithmic scale on the \(y\) axis. Countries are identified by aesthetics (shape, color, linetype).

  1. Use facetting to plot qx of all countries at all ages for years 1950, 1960, …, 2010.
  2. Use plotly to build an animated plot using Year for the frame aesthetics.

Abiding to the DRY principle, define a prototype ggplot (alternatively plotly) object.

The prototype will then be fed with different datasets and decorated and arranged for the different figures.

Code
dummy_data <- filter(life_table, FALSE)

proto_plot <- dummy_data |>
  ggplot() +
    aes(x=Age,
        y=qx,
        col=Country,
        linetype=Country,
        shape=Area) +
    scale_y_log10() +
    scale_x_continuous(breaks = c(seq(0, 100, 10), 109)) +
    ylab("Mortality quotients") +
    labs(linetype="Country") +
    theme_minimal()
Code
p_1 |> 
    plotly::ggplotly()
Code
proto_plt2 <-
  ggplot() +
  aes(x=Age, y=qx, colour=Area, frame=Year, linetype=Country) +
  geom_point(size=.1) +
  geom_line(size=.1) +
  scale_y_log10() +
  labs(linetype=c("Country")) +
  scale_x_continuous(breaks = c(seq(0, 100, 10), 109)) +
  xlab("Age") +
  ylab("Mortality quotients") +
  facet_grid(cols=vars(Gender)) +
  theme_minimal()
Code
p_2 <- with(params,
  (proto_plt2 %+%
    (life_table |>
      filter(between(Year, year_p, year_e), 
             Year %% 10 == 0, 
             Gender != 'Both', 
             Age < 90))  +
    ggtitle(glue("Mortality quotient {{year_p}}-{{year_e}}: Europe catches up")))
)
Code
p_2 |>
  plotly::ggplotly()

In 1948, NE and the USA exhibit comparable mortality quotients at all ages for the two genders, the USA looking like a more dangerous place for young adults. Spain lags behind, Italy and France showing up at intermediate positions.

By year 1962, SE has almost caught up the USA. Italy and Spain still have higher infant mortality while mortality quotients in the USA and France are almost identical at all ages for both genders. Mortality quotients attain a minimum around 10-12 for both genders. In Spain the minium central death rate has been divided by almost ten between 1948 and 1962.

If we dig further we observe that the shape of the male mortality quotients curve changes over time. In 1962, in the USA and France, mortality quotients exhibit a sharp increase between years 12 and 18, then remain almost constant between 20 and 30 and afterwards increase again. This pattern shows up in other countries but in a less spectacular way.

Twenty years afterwards, during years 1980-1985, death rates at age 0 have decreased at around \(1\%\) in all countries while it was \(7\%\) in Spain in 1948. The male central death curve exhibits a plateau between ages 20 and 30. Mortality quotients at this age look higher in France and the USA.

By year 2000, France is back amongst European countries (at least with respect to mortality quotients). Young adult mortality rates are higher in the USA than in Europe. This phenomenon became more pregnant during the last decade.

NoteQuestion

Plot ratios between mortality quotients (qx) in European countries and mortality quotients in the USA in 1948.

Code
p <- with(params,
  (filter(eur_us_table, Year %% 10==0)  |>
    ggplot() +
    aes(x=Age,
        y=Ratio,
        frame=Year) +
    aes(linetype=Country, show.legend = FALSE) +
    aes(color=Area, show.legend = FALSE) +
    scale_y_log10() +
    scale_x_continuous(breaks = c(seq(0, 100, 10), 109)) +
    geom_point(size=.1) +
    geom_smooth(method="loess",
                formula= 'y~ x',
                se=FALSE,
                span=.1,
                size=.1) +
    ylab("Ratio of mortality quotients with respect to US") +
    ggtitle(label = glue("European countries with respect to US,{year_p}-{year_e}"), 
            subtitle = "Sweden consistently ahead") +
   facet_grid(rows = vars(Gender))
  )) 

gp <- p |>
  ggplotly()

gp
Code
for (i in seq_along(gp$x$data)) {
  gp$x$data[[i]]$showlegend <- FALSE
}

gp
Code
(p %+% with(params,
  filter(eur_us_table, Year %% 10==0, Age <= 10)) +
  geom_point(size=.5) ) |>
  ggplotly()
Code
(p + with(params,
  filter(eur_us_table, Year %% 10==0, between(Age, 15, 25))) ) |>
  ggplotly()
NoteComment

This animation reveals less than the preceding one since we just have ratios with respect to the USA. But the patterns followed by European societies emerge in a more transparent way. The divide between northern and southern Europe at the onset of the period is even more visible. The ratios are important across the continent: there is a factor of 10 between spanish and swedish infant mortality rates. But the ratios at ages 50 and above tend to be similar. By the early 60s, the gap between southern and northern Europe has shrinked. By now, the ratios between mortality quotients tend to be within a factor of 2 across all ages, and even less at ages 50 and above.

Death rates evolution since WW II

Question

Plot mortality quotients (column qx) for both genders as a function of Age for years 1946, 1956, ... up to 2016 . Use aesthetics to distinguish years. You will need to categorize the Year column (forcats:: may be helpful).

  1. Facet by Gender and Country
  2. Pay attention to axes labels, to legends. Assess logarithmic scales.
Code
p_3 <- (p  %+%
  filter(life_table,
    Year %in% post_ww_II, 
    Gender!="Both",
    Age < 90,
    Country %in% c('Spain', 'USA')) +
  labs(title="Mortality quotient per Age",
    subtitle = "Post WW II")
)
Code
p_3 |> ggplotly()
NoteQuestion

Write a function ratio_mortality_rates with signature function(df, reference_year=1946, target_years=seq(1946, 2016, 10)) that takes as input:

  • a dataframe with the same schema as life_table,
  • a reference year ref_year and
  • a sequence of years target_years

and that returns a dataframe with schema:

Column Name Column Type
Year integer
Age integer
mx double
mx.ref_year double
Country factor
Gender factor

where (Country, Year, Age, Gender) serves as a primary key, mx denotes the central death rate at Age for Year and Gender in Country whereas mx_ref_year denotes central death rate at Age for argument reference_year in Country for Gender.

Code
ratio_mortality_rates <- function(df,
                                  reference_year=1946,
                                  target_years=seq(1946, 2016, 10)){
  
  jbe <- join_by(Age, Gender, Country)
  
  right_df <- df |>
    filter(Year==reference_year) |>
    select(Age, Gender, Country, qx)
  
  df |> 
    filter(Year %in% target_years, Age <90) |>
    select(Age, Area, Gender, Country, qx, Year) |>
    inner_join(right_df, by = jbe)
    
}
NoteQuestion

Draw plots displaying the ratio \(q_{x,t}/q_{x, 1946}\) for ages \(x \in 1, \ldots, 90\) and year \(t\) for \(t \in 1946, \ldots, 2013\) where \(q_{x,t}\) is the mortality quotient at age \(x\) during year \(t\).

  1. Handle both genders and all countries
  2. One properly facetted plot is enough.
Code
geom_smooth_line <- geom_smooth(method="loess",
  formula = y ~ x,
  se= FALSE,
  size =.2,
  span= .1
)
Code
q <- df_ratios |>
  ggplot() +
  aes(x=Age,
      y=qx.x/qx.y) +
  geom_smooth_line +
  scale_y_log10() 

q1 <- q + 
  aes(linetype=as_factor(Year),
      col=as_factor(Year)) +
  ylab("Ratio of mortality rates, reference Year 1946") +
  labs(linetype="Year", col="Year") +
  scale_colour_brewer() +
  theme(legend.position = "none") +
  facet_grid(
    rows = vars(Country), 
    cols =vars(Gender),
    scales = "free_y"
  )

q1

Code
q1 + (
  df_ratios |>
    filter(Age <= 20)
)