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