published on 45th anniversary of the article (August, 1970), with month -precision....
I've heard Andrew Noymer bring up Schoen's Δ (del) a couple times now in the context of 'good lifetable summary indicators that have been passed over, but without any obvious reason'. Usually it takes three to tip me over, but this time it was two mentions. Maybe because we're talking about demography tools. Well, it should have been one mention!
Δ is just the geometric mean of a mortality rate schedule, and it has lots of neat properties that you can read about in the short linked article. One of them is that ratios of Δ for two populations (or sexes, or years, etc) can be interpreted in a straightforward way: if the ratio of male Δ to female Δ is 2, then male mortality rates are twice as high. This is not the case for life expectancy or the age standardized death rates. If you halve mortality rates, you don't double life expectancy, and so forth. Age standardized death rates are arbitrary due to the use of a standard (even if some standards are common practice...).
So here are all Δ in the HMD (or their inverse, actually...), for both sexes (males blue, females red):
Talk about divergence!!!
If we want to study trends in divergence / convergence, and if we want to make group comparisons in mortality, there is a good argument that this is the measure we should be using. You can decompose differences in much the same way as we decompose differences in life expectancy, and so forth, partitioning the difference out to ages and causes. Just say'n.
That viz (can you call it a dataviz? there's no data... concept viz?) got me thinking. The basic notion is that the meaning of a year get's relativized to the amount of time you've lived. As we grow older, the proportion of our life that a given year takes up is less and less. This is all because our reference period (lived life) is growing. Anyway, it's a theoretical optimum, rather than actual perception, but it coincides with the anecdotes people have about time flying faster all the time. I googled a bit and found that plenty of people actually study the perception of time as a function of age. That's awesome.
I then thought, if you knew how long you'd live, then you'd know how long you have left, and this could be the reference rather than years lived. Let's call this, forward-looking relativized, as opposed to backward-looking relativized. Forward-looking relativization is symmetrical to backward-looking relativization. If you knew when you were going to die, you'd have both durations to relativize to. Then what would you think? How would this change the way you make decisions? Does running out of time make you savor? Does it make you slow down and notice details? At mid life, would we switch perspectives, always referring to the shorter segment of life (the one behind or the one in front). Maybe we'd take an average? But which kind of mean? When you don't know, use the arthmetic mean! And here's how the directionally-averaged perception of a unit of time looks by years lived, years left, and lifespan (in an ATL diagram).
You'd of course need to weight the lifelines in there, possibly using d(x) from the lifetable, or some other population weights. Plenty of imagining to do in this direction.
* In this case, the notions of fast and slow can easily flip, depending on what kind of mean you take ... In fact, even in the given arithmetic representation, you could swap slow and fast, and it'd still be legit. mental yoga.
"... the question arises: would not a treatment of demographic problems that based itself on hypotheses in order to extract necessary conclusions be of doubtful practical value? We would be powerfully misled in viewing matters that way. The conditions that present themselves in an actual population are always excessively complicated. Whoever has failed to grasp clearly the necessary relations among the characteristics of a theoretical population subject to simple hypotheses, will certainly be unable to manage in the much more complicated relations that exist in a real population. If one has wavered in the attack on a simple problem, he will assuredly stumble in the face of very serious complications. It is for this reason that authors who profess little respect for the application of mathematical analysis to demographic problems are those who in their writings present us with horrible examples of the confusion that results from striving to resolve by an avalanche of words problems whose complexity imposes on us the use of the condensed language of mathematics." - A. Lotka (1934, 1939, both in French) This translation was from D. P. Smith and H. Rossert (1998)
True then, true today. HT to VCR for sending me to this manuscript.
*It turns out to be useful to have a large number of such retorts on hand for the kind of work I do.
This post will be woefully short. Basically, you know how with APC you buy two and get the third for free? That is, you really only have two pieces of info with APC: well, if you had three pieces of info you get SIX indices! The six indices are chronological age (A), period (P), birth cohort (C), thanatological age (T), death cohort (D), and lifespan (L). In short, you only need 3 pieces of information to build out a 3-d temporal space. For example, with 1) birth cohort, 2) lifespan, and 3) a position in time (period), then we get chrono age, death cohort and thano age for free! Who doesn't like free things?! This is an ongoing project of mine to build out a 3-d Lexis-like space. The projection you see in this WebGL object follows the right-angles that are all around in demography, whereas an isotropic (time proportions are same in all directions) version of the same space ends up being a tetrahedral-octahedral honeycomb (say what?!). This beast of a diagram was done using the rgl package in R, which lets you save to WebGL, which lets me save the thing so you can see it in a browser. But if you come hang out then we can make one out of Zometool!
I'm excited to present this (and a proper buildup to it) at a lab talk next week for the Population and Health Lab at the MPIDR. I'm also excited to present it at the upcoming EAPS "Changing patterns of mortality and morbidity : age-, time-, cause- and cohort-perspectives" workshop in Prague. By then, Jonas Schöley will have been working on an interactive Shiny App to view data that permit the use of such coordinates, and Pancho Villavicencio will be helping my dot the i's and cross the t's when it comes to describing the geometry of all this. Can you say "let's calculate some new kinds of rates!"? Demography rules.