Reposting from the stats stackexchange.

John Cook, in his blog https://www.johndcook.com/blog/2015/03/09/why-isnt-everything-normally-distributed/, writes that many aren't:

Adult heights follow a Gaussian, a.k.a. normal, distribution [1]. The usual explanation is that many factors go into determining one’s height, and the net effect of many separate causes is approximately normal because of the central limit theorem.

If that’s the case, why aren’t more phenomena normally distributed? Someone asked me this morning specifically about phenotypes with many genetic inputs.

The central limit theorem says that the sum of many independent, additive effects is approximately normally distributed [2]. Genes are more digital than analog, and do not produce independent, additive effects. For example, the effects of dominant and recessive genes act more like max and min than addition. Genes do not appear independently—if you have some genes, you’re more likely to have certain other genes—nor do they act independently—some genes determine how other genes are expressed.

Height is influenced by environmental effects as well as genetic effects, such as nutrition, and these environmental effects may be more additive or independent than genetic effects.

I have two questions regarding this. First, I can't come up with any obvious example of phenotypes that don't follow a normal distribution. Could anyone help me? And second, his claims seem to contradict this other paper: https://www.uvm.edu/~dstratto/bcor102/readings/4_Evol_of_Phenotypes.pdf

To understand the genetic basis of quantitative traits, it is important to think about the effect of a particular allele, not simply its presence or absence. A single locus can produce three discrete phenotypes, but as more and more loci contribute to a trait the phenotypic distribution comes closer and closer to a normal (bell shaped) distribution

Who is right here? Does the fact that genes are digital rather than analog really make a difference? And what about the second argument that they aren't independent. Is that really necessary (second paper seems to indicate otherwise if I am understanding properly).

  • $\begingroup$ What about the most obvious Mendelian traits? Those would be phenotypes that follow a bimodal distribution, because they are caused by a single gene with just two alleles. Many other phenotypes also follow a bimodal distribution for the same reason. As you add in more modulatory or regulatory genetic variation you will smear the distribution and it will become more normally distributed. That is exactly what your references are telling you. $\endgroup$
    – Bryan Krause
    Jun 9 '17 at 21:53

Many phenotypes might be normally distributed assuming a large enough population size and many QTL (Quantitative Trait Loci) as the normal distribution is a simple result of the CTL (Central Limit Theorem). If it is unclear to you, you should definitely have a look at the central limit theorem.

Typically, if the number of loci is too low, the phenotype may for example follow some kind of continuous approximation to a Poisson distribution. Of course, if the allelic effects are correlated among loci or if there is some linkage disequilibrium or if the phenotype is bounded or discrete, or nominal, you may not end up with a normal distribution of phenotypes but all of that is a discussion for another time.

I can't come up with any obvious example of phenotypes that don't follow a normal distribution

As, the second quote says, think of any phenotype that is discrete. Here are a few examples

  • sex
  • right / left handed
  • eye color

Now if you want a quantitative phenotype that is not normally distributed, then height (which is the bad example of your first quote) is very slightly bimodal due to an average height difference between male and females (look at Are males taller than females in humans?). You can also think of any phenotypes that are bounded. For example running speed. Athletes will cause a very long right tail, while people in wheel chairs will make a big stack of probability mass in 0 km/h.

There is a lot of things to consider when trying to guess whether a is normally distributed or not in a population. It will take some good knowledge of statistics and good knowledge of the genetics of the population.

  • $\begingroup$ Thanks for this clear answer! I've never been convinced by the "height can't be normally distributed because it can't be negative" argument, since it can still be approximated reasonably well by a normal. Running speed seems like something you can't approximate with a normal then, but I didn't know it's considered a phenotype! Any explicit example for something that follows a poisson distribution? $\endgroup$
    – samlaf
    Jun 11 '17 at 14:47
  • $\begingroup$ Pretty much anything can be a considered a phenotype. A phenotype is whatever effect the genotype has on the world around it which mainly include all the characteristic (behaviour, anatomy, physiology, ...) of the body. I should have said "some kind of 'continuous approximation' to a Poisson distribution'" (now edited). It is just some theoretical reality of a model where there are few loci all biallelic with allelic effect 0 or 1, then you would see Poisson distributed trait. I can't give a specific example. $\endgroup$
    – Remi.b
    Jun 11 '17 at 14:53

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