From: rusin@moriarty.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Mortality Rate
Date: 30 Nov 1996 08:37:09 GMT
In article <329A150F.1C65@execonn.com>,
Marshall Dudley wrote:
>I was reading the other day that the mortality rate for white women in
>the US is about 400 per 100,000 per year. A little quick calculation
>shows that white women must live an average of about 250 years. This is
>obviously wrong, but I cannot figure out why it is wrong. Is not the
>mortality rate the inverse of the average life span?
I'm no statistician, but I can see several things wrong: you need to
define your terms carefully, and make clear the assumptions of your
model. Let's see how this happens, by trying to model population growth.
Assume a population (e.g. that of US white women) experiences a birth rate
of b (i.e., b new individuals per existing individual per year) and
a death rate of d (the poster suggests d=.004). Then the population
follows a function P = P0 exp( (b-d)t ), if we interpret these "rates" as
instantaneous rates of change. Note that we are already assuming these
numbers b and d are constant, and that they account for all changes
in P (e.g. changes in nationality, race, or gender (!) so as to move
into or out of the population.)
Note that one _cannot_ determine life expectancy from these data
alone. I will illustrate this by considering some extreme situations.
For example, suppose b=d=.01; one expects then a life expectancy of
1/d=100 yr, I suppose. But suppose we start with P0=100, so each year
one person dies and one is born. We can experiment with extreme cases
like these: starting in year 2, the death is of the individual born the
previous year; or, starting in year 3, it's of the individual born two
years earlier. After a million years there have of course been
1 000 000 *P0 years lived, by 1 000 100 individuals. But how do we count the
life expectancy? If we look only at those so who have died so far, we
get life expectancies of 1 yr, or 2 yrs, respectively. (well, + epsilon
I guess for the deaths in the first year or two). Or we could average all the
years lived so far per person, which is, essentially by definition, about
100 yr; but this would have to be a definite under-representation of the
life expectancy since those still alive at the end will grow older.
Lest you think this is the fault of extreme assumptions, let me offer the
reminder that: _in an exponentially increasing population, the number of
people still alive is a (roughly fixed) fraction of all those who ever
lived_. Thus the "edge effect" arising from the question of how to count
those still alive at the end of the observations will require a
nontrivial argument.
Of course, it seems unreasonable to assume all deaths come from the
very young; the point is really to illustrate that some assumptions
about when the deaths do occur is necessary. If we assume for example
that death is just as likely to occur at any age, then the
number of individuals remaining in a group of size M after t years
will be M exp( - d t). Thus in a given year, the members of this
cohort contributed M exp( -d t) [1-exp(-d)] deaths, all of people
aged t years. Adding up age-of-death for each individual and dividing
by their number M shows their average lifespan to be
1/[exp(d)-1], which is indeed roughly d. (It's exactly d if we average
the true age-at-death rather than the age rounded down to an integer.)
Perhaps, though, it makes even more sense to assume that death strikes
only the very old. Suppose all deaths occur only to those of age N.
Then the number of deaths at time t has to equal the number of
births at time t-N, giving an equation exp( (b-d)N ) = b/d. We can
solve for N = ln(1 + (b-d)/d) / (b-d) ~ 1/d, again, but this
assumes b-d is small (compared to d); if b is twice as large as
d, for example, then the life expectancy is only ln2 / d.
Note that in this last model, we could just as well compute "life expectancy"
via a median rather than a mean: at what age have half the members of
an age cohort died? Using this definition in the steady-death model
also gives a "life expectancy" of ln2 / d. (69 years, if d=.01)
In any case, we find that different models have given different
statements about life expectancy.
Actually some extreme cases can explain anomalies without much of a
model. For example, suppose there were suddenly many deaths this
year, or many births. Then in future years, the population would
overall be younger and thus we expect fewer deaths per 1000 per year
-- at least for a while. This is relevant since in the population
under consideration there are some striking anomalies in age
distribution. I have shown below some data taken from the "Information
Please" Almanac; the source is the US Gov't.
These are some 1990 data for US white women. The first column shows
age of the group for which data are given; when columns refer to an
age range, it's [age-5, age+4]. The second column shows the expected
number of years of life remaining for persons of the given age. Third
column is the mortality probability, the probable number of deaths per
1000 individuals in the age range, per year. (Note: this doesn't even
begin to tell the tale of high US infant mortality!). Fourth column
shows percent of US population which is of that age range; I couldn't
find the data for white women only -- as a group they are noticeably
older, in general. I have in the last column the size of the 1-year
cohorts for selected years, for comparison; note that this shows
number of total US births and does not reflect deaths nor immigration,
nor does it break down by gender or race.
Age Expectancy Mortality Population Cohort (millions)
0 79.4 7.0(<1yr) 4.2(1990)
1 78.9 0.4(1-4yr) 7.60% 4.0(1989)
10 70.1 0.2(5-14yr) 14.22 3.6(1980)
20 60.3 0.5(etc) 14.42 3.7(1970);3.1(1973-6)
30 50.6 0.6 17.35 4.3(1957-60)
40 41.0 1.2 15.57 3.6(1950)
50 31.6 3.1 10.18 2.6(1940);2.9(1945)
60 23.0 8.2 8.32 2.6(1930)
70 15.4 19.2 7.25 3.0(1920)
80 9.0 48.4 4.08 2.8(1910)
90 (6.4for 85yr) 144.0(for 85+) 1.27(for 85+)
Thus you can see that (no surprise here) deaths mostly strike the
elderly. But it's also true that it's the Depression-era and WWII-era
babies who are entering the period of most rapid decrease; these are
much smaller age cohorts than most of the ones which followed, so a
comparatively small portion of the population is dying now. If I have
computed the population-weighted average of the mortality rates
correctly, it comes to about 6.7 per 1000 US white women per year,
substantially more than the 4 per 1000 reported by the first poster.
Presumably curving the population percentages more correctly toward
the older age groups would bring the overall mortality rate closer to
the 8.6 per 1000 reported for the nation. As 1000/8.6=116, this would
still suggest a long lifespan, but only if using one of the more naive
models.
US birth rates, as low as b=.014 in the mid-1970s (but twice that around
the turn of the century) have been well above mortality rates (d=.0086
as above, also only half the turn-of-the century rate). As noted in
one of the models of population growth, this easily reduces the
life-expectancy prediction from that model by 20% or so. Models which
produce an answer of "ln2/d" appear to be right on target here.
(ln2*1000/8.6 = 81 yrs; a 1-year-old US white female can expect to
live to be 79.9, from the table above).
Of course if you really want to do this right, you need to consider
immigration, more up-to-date statistics, and so on; but perhaps you get
the idea now how to nuance the "1/d" calculation into something reasonable.
dave