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Being abnormal

Gene ExpressionBy Razib KhanJune 29, 2006 2:52 PM


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In Unequal by nature: A geneticist's perspective on human differences, James F. Crow states:

Two populations may have a large overlap and differ only slightly in their means. Still, the most outstanding individuals will tend to come from the population with the higher mean.

This is a trivial observation. It is biologically relevant because heritable quantitative traits are to a great extent the raw material for evolution, and, they generally follow an approximate normal distribution. The reasoning is simple, many loci of small independent additive effects are a good approximation of the genetic architecture of many phenotypes, and this structure simulates, roughly, the independent random variables which result in a normal distribution because of the central limit theorem. Obviously two of the most important parameters in the normal distribution are the mean (which is also the mode & median in a perfectly ideal distribution) and variance around that mean. Unfortunately, human minds are not unbiased statistical inference devices. Otherwise, cognitive psychologists would be shorted many interesting questions. It seems that the implications of the normal distribution and its most famous parameters (the mean and the variance) should be obvious to all college educated individuals. But my experience is that this isn't true, unfortunately. Experience indicates that principles are often more profitably imparted visually, so I took 10 minutes and cranked out a pretty picture via Excel that you can view below the fold. For the record, what I'm trying to show here is a comparison between two populations. In one, the mean height is 70 inches, 5'10 (about the American mean for males, 1.78 meters for the rest of the world). I assume the standard deviation (square root of the variance) is 4 inches, so in a perfect normal distribution 68% of the population will be within 1 standard deviation, 95% within 2, and 99% within 3. Obviously you chop the remainder in half for the tails of the distribution, so that for 2 standard deviations about 2.5% is at the top and 2.5% at the bottom. Surely uninteresting to the moment warriors out there, but I wanted to state it plainly and clearly. Now imagine a second population, the same sample size, also normally distributed. Let's keep the standard deviation the same, 4 inches, but let's move one parameter, the mean. When comparing the populations let us fix one at 70 as the other moves, so one population is of "average" height, while the second is "tall"(er). I simply incremented the second population by 0.5 inches until its mean was 6 inches above the first (i.e., 76 inches - 70 inches is an difference between means of 6 inches). So, at the end of the process, the mean of the "tall" population was 76 inches, so the population average in this group would be in the 6.5th percentile of the "average" population (mean 70 inch population). The graph below is self-explanatory, on the x axis you have the difference between the heights of the two populations. On the y axis you have the ratio of the numbers of individuals in the "tall" population to the "average" that are above the respective

standard deviations of the average population

. In other words, I'm displaying (assuming the populations are the same size) the difference in the number above an absolute threshold between the two populations. It is clear that as the mean of the "tall" population increases, its numerical advantage deviated above the norm of the "average" population increases at an exponential rate. Not only that, but as you increase the deviations about the norm the extent of the exponent of growth also increases as the "tall" population gains further advantange. At 8 inches above the "average" population norm you are only 6'6. This would be a medium sized individual in the National Basketball League in the United States, someone who could play either of the two guard slots. To be a center you probably have to be at least 6'10 inches, which is a little above 3 standard deviations above the mean of the "average" population. The graph below shows that when the "tall" population has a mean 6 inches above the "average" population, 3 standard deviations above the mean of the "average" population there will be more 40 times as many individuals from the "tall" population.


All trivial of course. Now, some of you may ask about biological relevance. As I said, to a great extent this sort of variation is the raw material for microevolutionary processes. Selection upon a population occurs at a rate proportional to the regression coefficient of the value of the trait in the parents on the offspring, the narrow sense heritability (see the breeder's equation). As Jim Crow noted, "Nature seems to follow least-squares principles." In more plain language, if you plotted the values of height of parents on the x axis, and offspring on the y axis, the slope of the fitted trendline would equal the extent of additive genetic variance. A slope of 0 would imply that there was no heritable relationship, a slope of 1 implies 100% (perfect) heritability. Here is Francis Galton's regression of heights which illustrates the principle:


Of course, biological variation doesn't follow the normal distribution necessarily to a close approximation. Until recently, the famed population biologist Luigi Luca Cavalli-Sforza could state that beyond 4 loci a polygenic trait exhibits a normal distribution experimentally (see The Genetics of Human Populations). In other words, 4 random variables, often slightly interactive and of non-equal phenotypic contributions might be able to pass themselves off as gaussian for biological work! Of course, the reality is that there might be non-genetic variables generating the distribution, but you get the picture. Gene-gene interactions and a whole host of other factors probably lead to the fact that the "tails" of biological distributions tend to be "fat." Selection results in the increase of rare gene frequencies whose combinations may result in non-independent (epistatic) effects, just as one example. The overall point is that when I say heritable, I mean something mathematically simple and trivial (the regression noted above on a normally distributed trait), but biologically rather subtle and the object of great disputation. I end with David Hume:

In every system of morality, which I have hitherto met with, I have always remark'd, that the author proceeds for some time in the ordinary ways of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surpriz'd to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, 'tis necessary that it shou'd be observ'd and explain'd; and at the same time that a reason should be given; for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.

(A Treatise of Human Nature)

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