[EMAIL PROTECTED] (Robert J. MacG. Dawson) wrote in news:[EMAIL PROTECTED]:
> No, it isn't! The mean of the daily proportions is not in > general the > same thing as the overall mean. This is closely related to Simpson's > paradox. > > Drive Table Proprortion > through DT > > Tue 30 10 .75 > Wed 30 10 .75 > Thu 30 10 .75 > Fri 30 90 .25 > Sat 30 90 .25 > Sun 30 90 .25 > ___________________________ > # 0.5 > ======= > Total 180 300 | 0.375 It's interesting how most "ecological fallacies" really come down to forgetting that you can't add fractions by taking the sum of the numerators and dividing it by the sum of the denominators. Another factor is the tendency to reify averages; if, say, a very short kid transfers into a class, the average of the heights of the students will go down, for sure, but we tend to use sloppy language and say "the height of the average student has gone down." In this case, nobody would seriously think that the transfer made any of the kids shorter, but when dealing with less familiar examples we often do the equivalent. As another example, let's say that the population of Slobovia is divided into two ethnic groups, Danierians and Pirenians. We observe that the average height, at the present time, of adult Pirenians is lower than that of adult Danierians by a statistically significant amount. We also know that adult height changes little if at all over the course of an individual's adulthood; if Jack is shorter than Joe at age 25, he will still be shorter than Joe at age 55. But we *cannot* magically slide from this to the conclusion that the difference in average heights for the two groups is therefore "immutable," because the "average Danierian" and the "average Pirenian" are *not* individuals whose height is under the control of individual biological processes that maintain stability of height over time; they are abstractions, namely aggregate measures taken over a population whose composition will change over time. Jack at 55 is the *same* individual as Jack at 25, but the "average Pirenian" of 2034 is *not* the same as the "average Pirenian" of 2004; the latter two are aggregate measures of *different* populations. In fact, we can say that there's a source of variation in the average height of either group that is *not* present in the measured height of any individual from either group, namely the effect of variation in group composition, something that simply doesn't exist at the individual level. Therefore, we will often make wrong inferences if we take group averages (or other composite measures) and plug them into models that are valid for individual measurements. As you've probably guessed by now, the above example is simply a translation of some aspects of the race/IQ debate. However, a few months ago I actually ran into a confusion between average heights and individual heights. It was in a debate over the causes of the "obesity epidemic" and I observed that the average weight of any developed country's population has been going up steadily for decades, well before there was much increase in obesity, and, at least in the US and the UK, there were no sudden "jumps" in average weight that could be accounted for by an Event. However, for much of that period the average *height* of the populations in question was also increasing, so the increased weight didn't translate into a higher average Body Mass Index (BMI), since the BMI is an increasing (linear) function of weight and a decreasing (quadratic) function of height. However, the increase in average adult height has levelled off in most developed countries, so now any increase in average weight translates into an increase in average BMI. And in fact the levelling-off of the increase in average height occurred earlier in the US than in the UK, and, surprise surprise, so did the increase in obesity. Anyway, someone else in the debate, who was strongly attached to the notion that there *was* an Event that caused a jump in weights (I think he was asserting that it was the introduction of high-fructose corn syrup), said I had to be incredibly stupid to believe that in the past adults kept growing taller past the age of 20 or so. He took an assertion about averages and treated it as an assertion about individuals, with absurd results. My point, BTW, was not that the increase in obesity wasn't real, but that everyone seemed to be asking "what did people do differently to cause it?" and not considering the possibility that the cause was something they *failed* to do differently. Of course nutritional issues are subject to passions that make the race/IQ debate look very tame by comparison, and there seems to be a strong preference for exotic explanations over mundane ones. One last statistical issue about the increase in obesity; obesity is defined as a BMI of 30 or over, which is to say that it's defined by dichotomizing a continuous variable, the cutpoint being out in the right tail of the distribution. That means that P(BMI>=30) is *not* a linear function of average BMI; if, say, BMI is distributed approximately normal, then as the curve slides to the right, the area to the right of the cutpoint will increase slowly at first (since what's sliding under the cutpoint is all tail) and then much more rapidly as the "body" of the curve starts bumping into the cutpoint. Thus small increases in average weight can lead to large increases in obesity, and again a steady trend starts to look like an Event. . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
