Steve Kass

I’ve often been puzzled by the contradictory statistics about lifespan and smoking.

According to many reports, smoking shortens lifespan by 13.2 years for men and 14.5 years for women. (Google smoking 13.2 14.5)

Studies of smoking and life expectancy, however, tend to find that non-smokers can expect to live only about five years longer than smokers.

What’s going on?

According to standard life expectancy tables, a living 70-year-old has a remaining life expectancy of about 14 years. An 80-year-old can expect to live about 8 or 9 more years. At any age, there’s an actuarial estimate for remaining life expectancy, and it’s always a positive number.

I haven’t done the calculation, but my guess is that Americans die, on average, with an average remaining life expectancy of about 10 years.

Does this mean that "death shortens lifespan by 10 years"? No.

Consider skydivers. Death by skydiving is likely to occur at a relatively young age. Say the average age of skydivers in fatal skydiving accidents is 40. Since 40-year-olds have an average life expectancy of 39 years, is it reasonable to say that "skydiving shortens lifespan by 39 years"?

Consider (any) surgery on 75-year-olds. Some of them die from the surgery, and the life expectancy of a 75-year-old is about 11 years. Does geriatric surgery shorten lifespan by 11 years? No. Not for those who don’t die, and not for those who do, either, since on average they were probably less healthy than average to begin with.

Consider being born prematurely. The average actuarial life expectancy of a 0-year-old is about 77 years. Some premature babies die (77 years earlier than the actuarial estimate). Does being born prematurely shorten lifespan by 77 years? Not for those who live.

The often-quoted 13.2 and 14.5 year figures follow this methodology. Those numbers are the average (for men and women respectively) actuarial life expectancy estimates for people who die from a disease directly attributable to their smoking.

Since death alone, by this logic, shortens lifespan by about 10 years, death by smoking probably only knocks of a few years, not 13 or 14. And if you don’t die from smoking, who knows – maybe your death only shortens your life expectancy by 8 or 10 years.

Go figure.

A lot of people are racking their brains trying to explain some wrong numbers that two Harvard School of Public Health researchers graphed. The real explanation of the surprising result is that the authors got their crosstabs backwards. They were looking for these numbers (the ones circled in green), which show the percentage of self-identified strong Democrats who say they are in poor health (9.1%), and the percentage of self-identified strong Republicans say they are in poor health (5.0%).

Unfortunately, they graphed these numbers (the ones circled in red) which represent something else:

Among many other places, the wrong results (which contained several other wrong numbers caused by the same mistake) were reported here, here, and here, except that it was reported as surprising research, not wrong research. These reports have generated lively discussions about why Democrats are in such poor health. Except they aren’t. With luck, many of the people reading about this will eventually find the answer here.

Under the headline “Rise in TB Is Linked to Loans From I.M.F.”, Nicholas Bakalar writes for the New York Times today that “The rapid rise in tuberculosis cases in Eastern Europe and the former Soviet Union is strongly associated with the receipt of loans from the International Monetary Fund, a new study has found.”

The study, led by Cambridge University researcher David Stuckler, was published in PLoS Medicine and is online at (URL may wrap):

http://medicine.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pmed.0050143&ct=1.

Cambridge, Schmambridge. First clue: the Times quotes Stuckler: “When you have one correlation, you raise an eyebrow,” Mr. Stuckler said. “But when you have more than 20 correlations pointing in the same direction, you start building a strong case for causality.”

In twenty post-communist countries, the variable “participated in an IMF loan program in year Y” was significantly negatively associated with “TB rate per 100000 people,” whether using rate of cases, of deaths, or of new cases.

After reading the paper and looking at much of the source data, I agree with William Murray, an IMF spokesman also quoted in the article: “This is just phony science.”

Why do I agree with Murray?

Take the supporting table below, for example. It shows all the TB mortality data from “did not participate in an IMF loan program” years: year-to-year percentage changes in TB mortality rates (based on Logs) [sic]. Among the 45 values are 31 0.00s and nothing else close to zero. Almost half the nonzero values are from Poland and Hungary, but—oddly—the change is nonzero in odd-numbered years and zero in even-numbered years. There are four -22.31s, two -18.23s, a 15.42 and a -15.42, a 13.35 and a -13.35, and four stray values, one of which is -69.31. Now I know -0.6931 from calculus (the natural log of ½), and I googled 0.2231: it’s the natural log of 0.8. (There were about four times as many “did participate” country-years, for a total of 200+ data points.)

If you haven’t guessed, the data here, which mostly express stable or declining TB mortality, and which found the entire study, and which the authors attribute significantly to non-participation in IMF loan programs, are 4-significant-digit percentage changes between logs of adjacent very small positive integers. The small integers are from the Global Tuberculosis Database, queryable here: http://www.who.int/globalatlas/dataQuery/default.asp. This WHO data is rounded to whole numbers and for the countries and years studied, ranged between 1 and 20.

While this data is crude, I don’t doubt the study’s main finding: among post-communist countries, “participated in an IMF loan program in year Y” was significantly negatively associated with “TB rate per 100000 people.” What I doubt is that the relationship has anything to do with the IMF loan program.

The timeframe studied was 1989 to 2003, and a quick look at the data reveals a pattern to which are the “in an IMF loan program” years for the countries studied. Most countries began participating in 1991, 1992, or 1993, and most countries continued their participation through 2003, the end of the study timeframe. During this time, TB was on the rise, and there’s no question the mid nineties were not a typical period.

While the authors mention many correction strategies and tests to avoid one or another kind of bias, they didn’t mention the way in which “in program” years were distributed as one potential confounder. I can’t see how they ruled it out. There data isn’t there. From 1994 to 1997, there are only 10 “not participating” data points, mostly from Czech Republic, Slovenia, and Poland, which countries were anomolous in having shown no increase in TB during their IMF years. Some countries, Bosnia for example, seem to have been omitted from this part of the analysis, despite having participated in an IMF loan program and data being available from WHO.

The countries studied included Russia, with 140,000,000 people, as well as Estonia, Latvia, Macedonia, Slovenia, Albania, Armenia, Bosnia, Lithuania, counted together having less than 20% of Russia’s population. The authors acknowledge the possibility of ecological fallacy with little investigation. Summary statistics, such as the mean and standard deviation of TB rate among the countries, are unweighted by population, and fail to reflect the real situation. Over one time period quoted, the number resulting from taking the average of each countries TB rate, unweighted for population, went up 30%, but the TB rate among the population under study in fact doubled. Whether this changes any interpretation, I can’t say, but it does make a difference.

Not only am I not a statistician, I’m not an economist, and I have no idea whether the IMF did great things or not in mid-nineties eastern europe and former Soviet Union. But Stuckler and colleagues haven’t convinced me of anything.

… that teaching mathematical concepts with confusing real-world examples is not a good idea.

Last month, the journal Science published an article about learning mathematics. Many newspapers and magazines picked up the story and pitched it at readers, summarizing the research result more or less like ars technica did [here]: “[S]tudents who learn through real-world examples have a difficult time applying that knowledge to other situations.”

This doesn’t agree with my experience in the classroom, and I sought the source. The full text of the Science article isn’t free, and the following remarks are based on what seems to be an earlier report of the same research by the same authors: “Do Children Need Concrete Instantiations to Learn an Abstract Concept?,” in the Proceedings of the XXVIII Annual Conference of the Cognitive Science Society [accessed at http://cogdev.cog.ohio-state.edu/fpo644-Kaminski.pdf on May 6, 2008.] Here the Ohio State researchers described their experiment.

In the first phase, subjects learned a mathematical concept.

[The concept] was a commutative group of order three. In other words, the rules were isomorphic to addition modulo three. The idea of modular arithmetic is that only a finite number of elements (or equivalent [sic] classes) are used. Addition modulo 3 considers only the numbers 0,1, and 2. Zero is the identity element of the group and is added as in regular addition: 0 + 0 = 0, 0 + 1 = 1, and 0 + 2 = 2. Furthermore, 1 + 1 = 2. However, a sum greater than or equal to 3 is never obtained. Instead, one would cycle back to 0. So, 1 + 2 = 0, 2 + 2 = 1, etc.

Subjects learned this concept through one of two scenarios: a scenario using geometric symbols with “no relevant concreteness” or a scenario “with relevant concreteness” using measuring cups containing various levels of liquid.

Ok, I’ll bite. I was a New Math child, and Mrs. Szeremeta taught us modular arithmetic with (analog) clocks. A “liquid left over” scenario with measuring cups ought to work, too. Most students “know” the idea of measuring cups.

Mrs. Szeremeta’s clocks worked, because the clock scenario contained “relevant concreteness,” and because its concreteness was familiar. For a concrete example to be a good teaching tool for an abstract concept, the concreteness has to be both relevant and familiar. Relevant means the concrete instantiation has to behave in real life more or less according to the rules of the abstract concept. Familiar means students know or can quickly learn how it works in real life. As the authors correctly observe, “the perceptual information communicated by the symbols themselves can act as reminders of the structural rules.”

Oops.

In addition to scenarios “with no concreteness” and “with relevant concreteness,” other kinds of scenario exists: ones “with confounding concreteness,” or ones with “irrelevant concreteness,” or ones with “distracting concreteness.” A scenario with confounding concreteness is one that draws on the familiar, but where the familiar behavior works contrarily to the rules of the abstract concept.

Here is the authors’ concrete scenario:

To construct a condition that communicates relevant concreteness, a scenario was given for which students could draw upon their everyday knowledge to determine answers to test problems. The symbols were three images of measuring cups containing varying levels of liquid. Participants were told they need to determine a remaining amount when different measuring cups of liquid are combined.

So far so good, but unfortunately, the three images used to represent the equivalence classes were those of a 1/3-full, a 2/3 -full, and a full measuring cup, representing the equivalence classes [1], [2], and [0] respectively.

Yes, the authors used a full measuring cup, not an empty one, to represent the additive identity, zero. The upshot of the experiment, then, in my opinion, was to compare an abstract implementation (geometric symbols with no concreteness), with a second implementation having both relevant and confounding concreteness. Relevant, because combining liquid and considering remainders works like addition in this group, but confounding, because one notion of the abstract concept is the idea of an additive identity (zero, nothing), which students learned to equate with a full measuring cup, which contains something, not nothing.

Students were expected to report the amount of liquid remaining after combining two amounts (but the result could not be “none”). In both the “cups” domain and the “no relevant concreteness” domain where squiggle = 0, disk = 1, and rhombus = 2, students learned to report correct answers (called “remainders” or “results”, depending on the domain) from the action of combining items.

The now combining-savvy students were challenged to learn similar rules in a new domain. The new symbols were images of a round ladybug, a tallish vase, and (I think) a cabochon ring viewed so its projection on the page is an eccentric ellipse with larger axis horizontal. They were told that the rules of this new system were similar to what they had previously learned, and that they should figure them out using their new knowledge.

Coincidentally, or perhaps not, the three images somewhat resembled baroque renditions of the earlier disk, rhombus, and squiggle, but bore little similarity to variously-filled measuring cups. They were taught a “game” where two students each pointed to an item, and a “winner” then pointed to one. Subjects were expected to learn which object the winner would point to. Students might have discovered that the visual similarities disk~ladybug, rhombus~vase, and squiggle~cabochon explained the new behavior.

Certainly the new domain was an abstract one. A pointing game like this, or real-world “ways” vases, ladybugs, and cabochons combine, are not part of the real world. I’d consider this new domain a purely symbolic one. When we teach mathematics, we want students to be able to transfer their knowledge to new domains both applied (with relevant concreteness) and abstract (symbolic, or with no relevant concreteness).

It doesn’t surprise me that students masters a new abstract domain more easily if they’d previously mastered one; and I’d expect it easier to master a new domain with relevant concreteness if you’d previously mastered one of those.

The short of it? Interesting research, but the experimental design is flawed as far as answering the question posed. Certainly this is no reason to give up finding creative, relevant, and familiar examples for abstract mathematical ideas.