---
title: "Inequalities in Disease, Lifespan and Mortality"
author: 36-313, Fall 2022
date: 6 October 2022 (Lecture 12)
header-includes:
  - \DeclareMathOperator*{\argmin}{argmin}
output:
  pdf_document:
    toc: true
    toc_depth: 4
    number_sections: true
bibliography: locusts.bib
---

```{r, include=FALSE}
# Set up global options for displaying code and so forth
library(knitr)
opts_chunk$set(size="small", 
               tidy=TRUE, tidy.opts=list(comment=FALSE),
               cache=TRUE, cache.comments=FALSE, autodep=TRUE,
               warning=FALSE, message=FALSE, error=FALSE,
               echo=FALSE)
```

\newcommand{\Prob}[1]{\mathbb{P}\left( #1 \right) }
\newcommand{\Expect}[1]{\mathbb{E}\left[ #1 \right] }
\newcommand{\Var}[1]{\mathrm{Var}\left[ #1 \right] }
\newcommand{\Cov}[1]{\mathrm{Cov}\left[ #1 \right] }
\newcommand{\myexp}[1]{\exp{\left\{ #1 \right\}}}
\newcommand{\xmin}{x_{\min}}
\newcommand{\SurvivalCurve}{\overline{F}}
\newcommand{\SurvivalFunction}{\SurvivalCurve}

# Highlights of the previous lecture

We have just looked at how to measure differences in rates, proportions or
probabilities of outcomes.  The general setting was that there was some
binary variable $Y_i$ for each individual, with $Y_i=1$ representing some
kind of "success" (admission to a school, hiring, graduation, promotion,
getting a loan, etc.), and $Y_i=0$ representing the corresponding failure.
The main goal was to estimate the rate or probability of success for
every group defined by a variable $X$, perhaps adjusting for or controlling
for covariates $Z, W, \ldots$.

If everyone in the sample has the same probability $p=\Prob{Y=1}$ of success, and successes
are independent (or at least uncorrelated), then the natural or direct estimator
of $p$ is the sample or empirical proportion,
\[
\hat{p} = \frac{1}{n}\sum_{i=1}^{n}{Y_i}
\]
The reason $\hat{p}$ is a sensible estimator is that (i) it's unbiased,
\[
\Expect{\hat{p}} = \Expect{Y} = \Prob{Y=1}=p
\]
and (ii) its variance shrinks towards zero quite fast, like $1/n$:
\[
\Var{\hat{p}} = \frac{p(1-p)}{n}
\]
Thus as $n\rightarrow\infty$, $\hat{p} \rightarrow p$.

It is not necessary to have individual-level data on successes or failures;
it's enough to know the number of successes $N$, since $N =
\sum_{i=1}^{n}{Y_i}$, and hence $\hat{p}=N/n$.

If we are dealing with multiple groups defined by $X$ and/or covariates, the
most straightforward thing to do is calculate a separate sample proportion for
each group.

As an alternative, we can lean yet more heavily on the facts that
\begin{eqnarray}
\Prob{Y=1} & = & \Expect{Y}\\
\Prob{Y=1|X} & = & \Expect{Y|X}\\
\Prob{Y=1|X, Z} & = & \Expect{Y|X,Z}
\end{eqnarray}
etc., etc., to turn the problem of estimating the conditional probability of
success into the problem of regressing $Y$ on covariates.  This is essential
when some of the covariates are continuous numbers, and often _useful_ when
there are a very large number of discrete covariates.  The reason it's useful
in the many-discrete-covariates situation is that then the groups defined by
the intersection of covariates tend to have very small sample sizes, but groups
which share many but not all covariates tend to have very similar rates, and
regression lets us "partially pool" information from similar groups.

Once we have (estimates of) success rates across groups, we can calculate
differences (or ratios or whatever) across groups, and begin to do statistical
inference to get measures of uncertainty in those differences, test whether
they might be zero or exceed thresholds of concern, etc.

\clearpage

# Rates of disease

All of this applies straightforwardly to looking at the rates of different
medical conditions, especially diseases[^accidentsetc], across a population or groups within a
population.  The biggest wrinkle is now that "success", $Y_i=1$, represents
individual $i$ _having_ the disease or condition.  This is not what we would
ordinarily think of as "success", but it makes a bit more sense, perhaps,
if we adopt the viewpoint of the disease.

[^accidentsetc]: I am going to say "disease" to be definite, but this kind of reasoning gets applied to lots of things which are not, strictly speaking, diseases, like accidents of various kinds, homicide, etc.

At this point we need to distinguish between two similar but distinct kinds of
rates for disease.  One is called the **prevalence** of the disease, and is
simply the probability that a random member of the population has the disease
at a particular time (e.g., the probability that a random person in Allegheny
County had Covid-19 at 12:01 am on Thursday, 7 October), or, more
realistically, has the disease at any time during a particular interval (e.g.,
the week beginning 4 October).  The natural estimator is simply the number of
cases divided by the population size, in symbols $\frac{N_t}{n}$.  The **incidence** of the disease, on the
other hand, is the probability of _coming down with_ the disease during that
time interval, naturally estimated by the number of _new_ cases divided by the
population size[^denominatorofincidence], in symbols $\frac{\nu_t}{n}$, where I'm using $\nu_t$ (with $\nu$ "nu" for "new") to count the number of new cases between time $t-1$ and $t$.

[^denominatorofincidence]: You might wonder why the denominator is the total population size, rather than the population size _minus_ the number of old cases, $n-N_{t-1}$.  This is largely a matter of convention among the epidemiologists, which I'd guess reflects the fact that for a long time they've mostly worried about conditions which _don't_ affect large fractions of the population.

Prevalence is _generally_ higher than incidence; the exceptions are somewhat unusual[^butnotalways].

[^butnotalways]: Think about how the numerator for prevalence changes: $N_t = N_{t-1} + \nu_t - \rho_t$.  (That is, the number sick at time $t$ is the number sick at time $t-1$, _plus_ the number getting sick between $t-1$ and $t$, _minus_ the number removed between $t-1$ and $t$.  If the amount of time that passes between measurements is short compared to how long people typically have the disease, then $\rho_t < N_{t-1}$ and so $N_t > \nu_t$.  But if the progress of the disease is very rapid, or the time interval between measurements is very long, then more people might have acquired the disease _and passed through it_ than have it at either time $t-1$ or time $t$.  To use a contemporary example, Covid-19 lasts a few weeks (one way or another), so if we measure time in days or weeks, incidence does indeed have to be less than prevalence.  But if we measured time in intervals of 3 months (a "quarter"), the number of people who _got_ Covid-19 and either got better or died during that period might well be higher than the number who still had it.  (It's true that if we look at the _cumulative_ prevalence over the quarter, the number of people who had Covid-19 at any point during those months, that has to be at least as great as the number of people who came down with it during those months.  But then there's a big gap between cumulative and instantaneous prevalence.)  _Usually_, though, people try to collect data on a given disease fast enough that incidence will be less than prevalence.

To sum up these definitions, a disease's prevalence is (basically) the
probability of _having_ it, and its incidence is (basically) the probability of
_getting_ it.  Both are probabilities or rates, so we know how to estimate
them, the same way we'd estimate admissions rates or hiring rates.

At a large scale, both disease incidence and disease prevalence show a lot of
variation across social groups.  Plenty of diseases, for instance, are
age-specific: there are twenty-year-olds with bad knees, heart disease,
diabetes, arthritis and/or cancer, but all of these are conditions which become
more common as we age.  Some of this is due to unrepaired wear-and-tear on the
body[^geneticsofaging], some due to continued environmental exposures, etc.,
and some perhaps due to social inequalities.  Similarly, differences in disease
rates across racial and ethnic groups, across sexes, across educational levels,
can all be subject to the same kind of analysis with control variables, and
distinguishing between correlates and mechanisms, that we've previously
examined.

[^geneticsofaging]: This raises the question of _why_ the body doesn't always repair itself.  One theory is that genetic variants which increase fitness in the young, but have bad effects later on, will tend to be favored by evolution if the problems they cause don't show up until their carriers are past their reproductive years (or, in social species, their help-out-the-kids years).  Indeed, this could explain the origin of "reproductive years" in the first place, since even an organism that didn't age would always be subject to some risk of accidental death, so variations that increased fitness early but decreased it later could be evolutionary favorable in expectation, and so selected for.  This is an interesting but necessarily rather speculative theory.  TODO citations to Medawar 1952, Williams 1957.

## Spatial variation in disease rates

One issue which is especially important in understanding disease rates is that
of spatial variation.  This shows up for the kind of "successes" we looked at
in the last lecture as well, but it's even bigger for disease, for two reasons.
One is that lots of diseases have environmental causes, which are spatially
correlated.  (People have known since ancient times that some areas are more
healthy than others.)  The other is that lots of disease are infectious, which
_also_ produces spatial correlations.  But space is big, which means that even
large national surveys, when spread out over a whole country, often have only a
few individuals sampled from any given location.  Special techniques have been
developed for estimating disease rates over space which take these issues into
account.  There are, once again, issues about control and adjustment --- it is
often no accident who ends up living next to the toxic waste dump.

Dealing with this properly would involve going over a lot about spatial
statistics which we don't have time for.  So I will pacify my conscience by
just noting the issue, and offering some pointers to an exemplary paper
[@Kafadar-smoothing-geographical] and to a standard textbook
[@Lawson-spatial-epi-2nd], and plugging the special-topics class "Data Over
Space and Time" (36-467/667).

\clearpage

# Survival Curves, Life Tables, and Mortality Rates

We will now switch gears a little bit, from thinking about disease
rates to thinking how long people live.  Let's think about total lifespan
as a random variable, say $T$.  It's conventional to look at the upper
or complementary[^complementary] cumulative distribution function of $T$:
\[
\Prob{T \geq t} = \Prob{\text{survive to at least age}\ t} = \SurvivalCurve(t)
\]
Especially in this context, this is often called the **survival function**
or **survival curve**.

[^complementary]: If we think of $T$ as a continuous variable, we can say $\Prob{T \geq t} = \Prob{T > t}$, and this really is the complement of $\Prob{T \leq t}$.

## Survival curves

Let's establish some basic properties of survival curves:

- The curve starts at 1: $\SurvivalCurve(0) = 1$
- The curve tends to 0: $\SurvivalCurve(t) \rightarrow 0$ as $t\rightarrow \infty$
- The curve is non-increasing: $\SurvivalCurve(t+h) \leq \SurvivalCurve(t)$ for any $h > 0$
    + In fact the curve is _usually_ decreasing, $\SurvivalCurve(t+h) < \SurvivalCurve(t)$.  This is true for any biological survival curve I can think of, but I suppose there might be exceptions.
- The curve determines the probability distribution
    + If $t$ is discrete, then $\Prob{T=t} = \SurvivalCurve(t) - \SurvivalCurve(t+1)$
    + If $t$ is continuous, then the pdf $f(t) = - \frac{\SurvivalCurve}{dt}$ (why the minus sign?)
    + Since $T$ is non-negative, $\Expect{T} = \int_{0}^{\infty}{\SurvivalCurve(t) dt}$ (see [Lecture 2](http://www.stat.cmu.edu/~cshalizi/ineq/22/lectures/02/lecture-02.pdf), sec. 6.2, "Expected values and CDFs"; I will leave working out the corresponding discrete formula as an exercise.)
	
```{r}
# Slightly crude, but I just copied-and-pasted from [https://ftp.cdc.gov/pub/Health_Statistics/NCHS/Publications/NVSR/69-12/Table01.xlsx]
survival.data.all.2018 <- c(
    100000, 99435, 99399, 99372, 99354, 99339, 99325, 99312, 99301, 99291, 99282, 99273, 99263, 99250, 99232, 99208, 99175, 99135, 99087, 99030, 98964, 98890, 98807, 98717, 98620, 98519, 98413, 98303, 98188, 98070, 97946, 97818, 97686, 97548, 97404, 97254, 97097, 96934, 96764, 96589, 96408, 96222, 96028, 95823, 95603, 95368, 95115, 94843, 94549, 94233, 93890, 93519, 93118, 92679, 92198, 91673, 91102, 90488, 89829, 89124, 88371, 87566, 86708, 85798, 84838, 83827, 82763, 81637, 80444, 79180, 77833, 76402, 74887, 73246, 71502, 69629, 67631, 65482, 63181, 60738, 58141, 55397, 52483, 49408, 46191, 42800, 39306, 35776, 32166, 28529, 24923, 21411, 18059, 14927, 12070, 9530, 7334, 5490, 3992, 2814, 1921)
plot(0:100, survival.data.all.2018/1e5, type="l", xlab="age (t)",
     ylab="Pr(T >= t)", main="Survival curve for the US (total population, 2018)")
```
	
### Life expectancy

**Life expectancy** is simply the expected value of $T$, perhaps conditional
on certain other events.  Unmodified, it usually means life expectancy at birth,
which is
\[
\Expect{T} = \sum_{t=0}^{\infty}{t \Prob{T=t}}
\]

We can of course condition $T$ on other variables, such as membership in
a social group and/or covariates,
\[
\Expect{T|X=x, Z=z}  = \sum_{t=0}^{\infty}{t \Prob{T=t|X=x, Z=z}}
\]

We can _also_ condition $T$ on having lived to at least a particular age, say $t_0$:
\[
\Expect{T|T \geq t_0} = \sum_{t=t_0}^{\infty}{t \Prob{T=h|T \geq t_0}}
\]
This gives us life expectancy at age 18, or 25, or 65, or anything else we
want, depending on how we set $t_0$.

We can also look at what's called the "expectation of life" at a given age,
\[
\Expect{T-t_0|T \geq t_0} = \sum_{h=0}^{\infty}{h \Prob{T=t_0+h|T \geq t_0}}
\]
which is the number of _additional_ years, beyond $t_0$, someone can expect to
live.

## Life tables

Among actuaries, demographers, etc., it's conventional present the information
plotted in a survival curve as instead a table, called a **life table**.
The idea is that in this table we will track how many people remain alive
at each age, from an original (imaginary) population of a certain size,
conventionally 100,000 people.  So a table might look like this:

Age    | Surviving
-------|-------------------------
0      | 100,000
1      |  99,435
2      |  99,399
3      |  99,372
...    |  ...
44     |  95,603
...    |  ...
$t$    | 100,000 $\times \SurvivalCurve(t)$
...    |  ...

(I have taken these numbers from the life table for the whole US population
in 2018 [https://ftp.cdc.gov/pub/Health_Statistics/NCHS/Publications/NVSR/69-12/Table01.xlsx], as you worked with in the after-class exercise.)

Actuaries like to work with life tables because they simplify certain sorts of
calculation.  For instance, if you want to find the marginal probability of
dying at a certain age, you just take the difference between successive rows.
There is also a neat way to calculate life expectancy, and the additional
expectancy of life, from the table, which you can see illustrated in
the CDC life table for 2018.  However, there is clearly no information
in the life table that isn't also in the survival function (and vice versa).


## Death rates

There are two ways we can define the death rate at age $t$.  One is the
unconditional or marginal probability that we reach age $t$, but no higher age,
i.e., $\Prob{T=t} = \SurvivalFunction(t) - \SurvivalFunction(t+1)$.  (If we're
working in continuous time, this would be the probability _density_ at $T=t$.)
The other is the probability that we die by age $t+1$, _given that_ we have
reached age $t$: $\Prob{T=t+1|T \geq t}$.  The latter is often a more
appropriate notion of the immediate hazards to life.  We can estimate this
age-specific mortality rate simply by tracking everyone who is age $t$ for a
year, seeing how many of them die before the year is out, and dividing by the
number we started with.  (Some people would divide by the number left at the
middle of the year.)  Sometimes you see this aggregated across a small group
of years, say ages 41--45.


```{r}
marginal.deaths.all.2018 <- -diff(c(survival.data.all.2018, 0))
plot(0:100, marginal.deaths.all.2018/1e5, ylab="Marginal mortality rate",
     xlab="Age", type="l", log="y")
conditional.deaths.all.2018 <- marginal.deaths.all.2018/survival.data.all.2018
plot(0:100, conditional.deaths.all.2018, ylab="Conditional mortality rate",
     xlab="Age", type="l", log="y")
```

(The apparent uptick in the marginal probability of death from 99 to
100 is because everyone who dies at or above age 100 is lumped together in that
last bin.)

As discussed in class, age-specific mortality rates start out comparatively
high for infants, then (under modern conditions) plunge down to their minimum
in childhood, climb a bit for young people, and then begin a slow, steady rise
which accelerates as we pass from middle age to old age, steepening as we go.
That first year of life is risky, more so than any year we'll face again until
about fifty or so.

\clearpage

# Basic facts about inequality in survival curves and life expectancy

## Inequalities

- Women live longer than men, on average.  (This is true in almost every
country, and the apparent exceptions are places with _really_ bad data.)
- More-educated people live longer, on average, than less-educated people.
- In the contemporary US, black people are shorter-lived than white people, who
  are shorter-lived than Hispanics.
  
The interactions among these are complicated.  (Black women live longer than
white men, but not as long as Hispanic men.)  You will look at some of those
complications in Homework 6, but in the meanwhile, here are survival
curves subdivided by race and sex for the US in 2018 [@NVSR-2020, Fig. 4, p. 7]:

![](survival-curves-us-2018.jpg)

## Lengthening lives

In developed countries, life expectancies have been increasing for pretty
much all groups, pretty much all of the time, since around 1900 (and even
before, for some countries).  Here, for instance, is a plot of life
expectancy since 1900 in the US [@NVSR-2020, Fig. 1, p. 5]:

![](us-le-over-time.jpg)

### Shortening lives from catastrophes

There _are_ events which interrupt this steady increase.  The most important
are wars and epidemic diseases.  You can see the influenza pandemic that began
in 1918 in the plot above.  The US has been fortunate that its 20th and 21st
century wars didn't kill enough Americans to have a big impact on our life
expectancy.  If we could extend this back to the 1860s, we _would_ see the
impact of the US Civil War.  If we looked at other countries, the wars of the
20th century would definitely show up --- Russia, say, or France or Germany or
China[^famine].

[^famine]: Beyond wars and epidemics, the other kind of catastrophe which shortens life expectancy at the level of an entire population is famine.  This is not a problem in democracies, for reasons explained long ago by the great economist Amartya Sen [@Sen-dev-as-freedom].  (At least, not under modern conditions where there's global trade in food.)  If we were took look at non-democratic countries, however, we'd certainly see the "collectivization" famine in the Soviet Union in the 1930s, or the famines brought on by the Great Leap Forward in China in the 1950s, in their historical life expectancies.

\clearpage

# The demographic transition and changing survival curves

While it's necessarily hard to get mortality data and survival curves for
historical populations, we do know something about them, and what we know is
not pretty.  Here, for example, from the excellent website "Our World In Data"
is a compilation of historical survival curves for England and Wales, starting in 1851:

![](Survival-Curves-UK-Period-measure-scaled.jpeg)


You can see that survival to age 20 was only about 65%.  That is, roughly
a third of all those born did not reach adulthood.  This is a fact with
profound consequences for the history of our species.  Suppose that a
population is going to just reproduce itself, and not shrink.  Then every adult
woman needs to have, on average, at least two children who reach reproductive
age and go on to have children themselves.  Take reproductive age to be 20.
(It may have been a bit lower, but comparatively-poor diets delayed puberty and
the onset of fertility.)  A 65% survival rate means that there need to be, on
average, $2/0.65 = `r signif(2/0.65,2)`$ children born per woman.  If some
women don't have children (whether by choice or necessity), those who do need to
have even more to make up the difference.  If, say, 10% of women don't have
children at all, the mean number of children for the rest has to go up by a
factor of $1/0.9=`r signif(1/0.9,2)`$ to compensate, so we're at
$`r signif((2/0.65)/(0.9),2)`$ children per reproductive female.  Moreover,
we're  assuming that having offspring who survive to reach age 20 is enough, but
that might not give _them_ time to reproduce, which calls for even more
offspring initially.  (Child-birth itself was very hazardous for mothers.)  We
are also making no allowances for failed pregnancies, since the survival curve
starts at birth.  We are easily looking at something like _at least_ four
pregnancies per reproductive woman just to keep the population constant.

The upshot of all this arithmetic is that when survival curves were as bad as
they were in England and Wales in 1851, most women who could reproduce at all
_had_ to have lots of children, in order for enough of their children to reach
adulthood to reproduce the population.  (This in turn implied spending a lot of
time pregnant, trying to get pregnant, or, for lack of alternatives,
breastfeeding.)  And England and Wales in 1851 were actually, by the standards
of world history, a favorable time and place for survival.  For many long ages,
all over the world, the fact that so many children died so young dictated that
families had to _try_ to be large in order to reproduce themselves at all, and
so essentially all cultures have spent millennia encouraging fertility.

This had tremendous consequences when survival curves began to change,
especially, at first, at young ages.  You can see from the figure that the 1871
curve is essentially the same as the 1851 curve, but that by 1891 there is
already a difference of about 10% in the probability of reaching age 20, and it
had gone up by about 20% by 1931[^foodwatermedicine].  If almost 90% of
children will live to reach reproductive age, replacement fertility is more
like `r signif(2/0.9,2)` than `r signif(2/0.65,2)`.  But initially, people kept
having large families, only, unlike all previous ages, the children lived.  The
result was a population boom, accompanied by people learning that they didn't
_need_ large families.  (Indeed, in industrial, education-dependent societies,
they could give their children better lives by having fewer of them and putting
more into each child.)  Also, women discovered that they might like to do things
other than constantly raise children[^oversimplifying].

This pattern --- declining infant mortality plus a cultural norm of high
fertility leads to a population boom _and_ to declining fertility --- is
something that has been observed all over the world; social scientists call it
the **demographic transition**.  Different countries have gone through the
demographic transition at different times; England was one of the first, with
its population going from 8.6 million in 1800 to 30 million in 1900 to 41
million in 1950
[https://ourworldindata.org/grapher/population-of-england-millennium].  Many of
the countries which have fully gone through the transition (e.g., Japan, parts
of western Europe) now have lifetime fertility rates below replacement level,
which also has all kinds of consequences.



[^foodwatermedicine]: The most important causes for the initial upward movement of the survival curve, especially for the young, was improved nutrition from getting enough to eat, followed by such notable, and widely-opposed, public health measures as "sewers" and "clean drinking water", and then early vaccinations and pasteurizing foods.  More complicated medical treatments didn't really start to become effective until well in to the 20th century.

[^oversimplifying]: In case it's not obvious that I'm simplifying and exaggerating for rhetorical effect: I am simplifying and exaggerating for rhetorical effect.  But it's no coincidence that while people had been [proposing political and social equality between the sexes for thousands of years](http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Apage%3D451), those ideas didn't _go_ anywhere until the 19th century.

\clearpage

# Hey, wait a minute (or a decade)...

It may have occurred to you to wonder how we can know the life expectancy
of those now alive, or indeed the shape of the survival curve.  There would
seem to be two possibilities, both with serious drawbacks:

1. We can look at the **cohort** of all those born in a particular year (1851,
   or 1951, or 2001) and track how many of them are left in each succeeding
   year, getting a cohort-specific survival curve.  This is accurate for that
   cohort, but only valid retrospectively.  For those born in 2001, the
   survival curve today only goes up to age 20; for those born in 1971, only up
   to age 50.  When it comes to saying how long today's 20 year olds will live,
   we don't really have _data_.  So we have a cohort approach, but that doesn't
   let us extrapolate.
2. We can look at the **period** or **cross-section** of (say) 2021, counting how many people
   there were at each age at the start of the year, and then how many of them
   died over the course of that year, to get age-specific death rates for each
   age.  We can then combine these to create an over-all survival curve or
   life table.  This has probabilities for each age, but it doesn't reflect
   what actually happens to people as they age.

The truth is that the data we started with is a _period_ life table / survival
curve.  There is no actual cohort of people which has experienced or will
experience precisely this pattern of lifespans and mortality.  But it does show
the pattern of mortality confronting the population at a particular moment.  We
can be pretty sure that today's 20 year olds will, in 30 years, have a
_different_ death rate than today's 50 year olds, but it's hard to say exactly
how different and in what direction.  So extrapolating forward as though the
future would resemble the present is a sensible, or at least widely-used,
procedure.

Once you are willing to contemplate "synthetic" or imaginary populations, of
course, there are all sorts of possibilities.  You might, for instance, want to
extrapolate forward various improvements in death rates.  This can easily get
very conjectural, and small differences in assumptions can compound to big
differences in predictions several decades hence.



\clearpage

# The "Deaths of Despair" Controversy

I said above that in modern, developed countries, it is very rare for life
expectancy to decrease, and the exceptions are mostly attributable to
particular catastrophic events.  One notable exception has occurred in the
United States in the 21st century, where life expectancy for (non-Hispanic)
white people without college degrees has stagnated and even decreased, while it
has continued to increase for other groups.  While they were not the first to
notice this, the economists Anne Case and Angus Deaton did pioneering work on
this topic, helping to show that this was due specifically to an increase in
death rates among the middle aged.  They also drew attention to rising death
rates from suicide, alcohol and drugs.  They gave this
trio the memorable name of "deaths of despair", and speculated that they were
rising because of economic and political changes which adversely affected this
group, either absolutely or in relative terms.

This has led to a substantial scholarly controversy, with a number of
strands.

- Case and Deaton's initial study looked at death rates among 45--54 year olds.
One worry was that the distribution of ages _within_ that bracket had shifted,
so that there were relatively fewer 45 year olds and relatively more 54 year
olds, and older people have higher death rates.  More careful studies showed
that _some_ of the effect was due to this change in composition, but not all or
even most.
- Another was how much of the increase in death rates, and the decrease in
life expectancy, could be explained _specifically_ by the "deaths of despair".
- A third was whether the "deaths of despair" really belonged together, with critics suggesting that the changes in suicide and alcoholism were dwarfed by those due to drug abuse, specifically opioid abuse.  The idea there was that the issue wasn't general despair and social alienation, but a drug epidemic.  (One could of course ask _why_ that population was susceptible to a drug epidemic at that time.)

Notice that these are all controversies about explanations, adjustments and
mechanisms, just as we went over in Lecture 10.

The next homework will introduce you to one strand of this controversy, by
having you read a later paper by Case and Deaton.  Their book,
@Case-Deaton-despair, has a lot to say about how they have modified their
positions in response to criticism, and how they have responded to criticisms.


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# Further reading

Survival curves, life tables and mortality rates are all part of the science of
**demography**, which studies the dynamics of populations.  This is a vast
and intricate subject, intimately connected to statistics, but with its
own details and (alas) jargon.  @Alho-Spencer-statistical-demography is a
very good introduction to demography for statisticians.

On the history of population and the demographic transition,
@Cipolla-on-world-population is older but brief, clear, and not too superseded
by more recent research.  @FPearce-crash is a readable journalistic book on the
probable consequences of below-replacement-level fertility in an increasing
number of countries around the world.


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# References

