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Tag archive: Statistics

Follows from: Evolution is Sampling Error

It seems a lot of people either missed my footnotes in the last post about effective population size, or noticed them, read them and were confused1. I think the second response is reasonable; for non-experts, the concept of effective population size is legitimately fairly confusing. So I thought I’d follow up with a quick addendum about what effective population size is and why we use it. Since I’m not a population geneticist by training this should also be a useful reminder for me.

The field of biology that deals with the evolutionary dynamics of populations — how mutations arise and spread, how allele frequencies shift over time through drift and selection, how alleles flow between partially-isolated populations — is population genetics. “PopGen” differs from most of biology in that its foundations are primarily hypothetico-deductive rather than empirical: one begins by proposing some simple model of how evolution works in a population, then derives the mathematical consequences of those initial assumptions. A very large and often very beautiful edifice of theory has been constructed from these initial axioms, often yielding surprising and interesting results.

Population geneticists can therefore say a great deal about the evolution of a population, provided it meets some simplifying assumptions. Unfortunately, real populations often violate these assumptions, often dramatically so. When this happens, the population becomes dramatically harder to model productively, and the maths becomes dramatically more complicated and messy. It would therefore be very useful if we could find a way to usefully model more complex real populations using the models developed for the simple cases.

Fortunately, just such a hack exists. Many important ways in which real populations deviate from ideal assumptions cause the population to behave roughly like an idealised population of a different (typically smaller) size. This being the case, we can try to estimate the size of the idealised population that would best approximate the behaviour of the real population, then model the behaviour of that (smaller, idealised) population instead. The size of the idealised population that causes it to best approximate the behaviour of the real population is that real population’s “effective” size.

There are various ways in which deviations from the ideal assumptions of population genetics can cause a population to act as though it were smaller – i.e. to have an effective size (often denoted \(N_e\)) that is smaller than its actual census size – but two of the most important are non-constant population size and, for sexual species, nonrandom mating. According to Gillespie (1998), who I’m using as my introductory source here, fluctuations in population size are often by far the most important factor.

In terms of some of the key equations of population genetics, a population whose size fluctuates between generations will behave like a population whose constant size is the harmonic mean of the fluctuating sizes. Since the harmonic mean is much more sensitive to small values than the arithmetic mean, this means a population that starts large, shrinks to a small size and then grows again will have a much smaller effective size than one that remains large2. Transient population bottlenecks can therefore have dramatic effects on the evolutionary behaviour of a population.

Many natural populations fluctuate wildly in size over time, both cyclically and as a result of one-off events, leading to effective population sizes much smaller than would be expected if population size were reasonably constant. In particular, the human population as a whole has been through multiple bottlenecks in its history, as well as many more local bottlenecks and founder effects occurring in particular human populations, and has recently undergone an extremely rapid surge in population size. It should therefore not be too surprising that the human \(N_e\) is dramatically smaller than the census size; estimates vary pretty widely, but as I said in the footnotes to the last post, tend to be roughly on the order of \(10^4\).

In sexual populations, skewed sex ratios and other forms of nonrandom mating will also tend to reduce the effective size of a population, though less dramatically3; I don’t want to go into too much detail here since I haven’t talked so much about sexual populations yet.

As a result of these and other factors, then, the effective sizes of natural populations is often much smaller than their actual census sizes. Since genetic drift is stronger in populations with smaller effective sizes, that means we should expect populations to be much more “drifty” than you would expect if you just looked at their census sizes. As a result, evolution is typically more dominated by drift, and less by selection, than would be the case for an idealised population of equivalent (census) size.

  1. Lesson 1: Always read the footnotes. Lesson 2: Never assume people will read the footnotes. 

  2. As a simple example, imagine a population that begins at size 1000 for 4 generations, then is culled to size 10 for 1 generation, then returns to size 1000 for another 5 generations. The resulting effective population size will be:

    \(N_e = \frac{10}{\frac{9}{1000} + \frac{1}{10}} \approx 91.4\)

    A one-generation bottleneck therefore cuts the effective size of the population by an order of magnitude. 

  3. According to Gillespie again, In the most basic case of a population with two sexes, the effective population size is given by \(N_e = \left(\frac{4\alpha}{(1+\alpha)^2}\right)\times N\), where alpha is the ratio of females to males in the population. A population with twice as many males as females (or vice-versa) will have an \(N_e\) about 90% the size of its census population size; a tenfold difference between the sexes will result in an \(N_e\) about a third the size of the census size. Humans have a fairly even sex ratio so this particular effect won’t be very important in our case, though other kinds of nonrandom mating might well be. 

A common mistake people make about evolution is to think it’s all about natural selection and adaptation. In fact, random non-adaptive changes often dominate the evolutionary process.

Today I’m going to lay out a useful framework that I hope makes this fact more intuitive, which might in turn help non-experts build better intuitive models of evolutionary processes. This will come in handy when I try to explain non-adaptive theories of ageing later on.

Sampling error and genetic drift

We can think of evolution as sampling error: deviation in the genetic composition of the offspring in a population relative to their parents. To illustrate this, let’s imagine a simple, asexual population, evenly divided between two gene variants (alleles) which produce no difference in fitness1:

Ten dots representing individuals, stacked vertically and coloured to represent their genotype: five red and five blue

These individuals will reproduce, giving rise to the next generation. Since all the individuals are genetically identical and have the same chance of reproducing, we can think of these offspring being randomly sampled, with replacement, from the previous generation:

Two columns of ten dots, stacked vertically, with lines between the columns representing parentage. Some individuals produce no offspring, some one, and some more than one.

Since there is a great deal of randomness involved in who reproduces successfully and whose offspring survive, not all individuals will produce the same number of offspring in the next generation, even though they all had the same probability of reproducing to begin with. As a result, even in the absence of selection effects, the allele distribution of the new generation is likely to differ from that of the previous generation; this random, unbiased change in allele distribution is known as genetic drift.

As a result of genetic drift, the allele distribution will fluctuate up and down stochastically; sooner or later, one or the other will be eliminated from the population, resulting in fixation:

Twenty columns of ten dots, stacked vertically, with lines between the columns representing parentage. The red allele reaches fixation at generation 14.

The time to fixation depends on the population size2 and some other population parameters; here’s an example plot for a population with a carrying capacity of 200 instead of 10:

A plot of the allele frequency distribution of a larger population, reaching fixation at roughly generation 350.

Fairly dramatic genetic changes, then, can accumulate in a population based purely on genetic drift; there’s not necessarily any need to invoke selection to explain why genetic differences between populations accumulate over time. That said, what happens when we add selection into the mix?

Natural selection is sampling bias

Suppose that one of the starting alleles in the population is less fit than the other: individuals with that allele are less likely to produce reproductively-successful offspring. What happens now?

If one allele is much less fit than the other, the individuals bearing it will probably die without issue, producing a very boring plot:

Another twenty-column plot, this time in green and purple. The single initial purple individual produces no offspring, so the purple lineage dies out immediately.

So far, so trivial. The interesting cases occur when the fitness of one genotype is close to (either a bit higher or a bit lower than) the old one. In this case, thanks to genetic drift, the less-fit allele (here in purple) can persist in the population for a surprisingly long time…

Another twenty-stage green-and-purple plot, this time representing two alleles with only a small difference in fitness. The less-fit purple allele persists until stage 18, then is lost.

…or even fix!

An independent run of the scenario from the previous figure. This time, the slightly-less-fit purple allele reaches fixation at stage 6.

Overall, under these conditions (carrying capacity = 10, relative fitness ~ 0.9) the less-fit allele will reach fixation about a quarter of the time; more than enough for 100% the population to be bearing many deleterious alleles.

These results are a pretty trivial application of statistics, but they have very important implications for how we should view evolution. Thanks to genetic drift, beneficial mutations will often die out and deleterious ones reach fixation. How often this occurs depends on various factors, the most obvious of which is the magnitude of the mutation’s effect on fitness — the more dramatic the effect, the greater selection’s ability to overcome drift and eliminate the less-fit allele.

However, another crucial variable, underappreciated outside evolutionary biology, is population size.

Evolution and the law of large numbers

According to the law of large numbers, the average of a sample converges in probability towards its expected value as sample size increases: the larger the sample, the smaller the expected relative mean absolute difference between the sample mean and the expected value3. If you flip a coin ten times, the chance of deviating from the expected value (five heads) by at least 20% is more than 75%, whereas it’s only 5% if you flip 100 times and virtually zero if you flip 1000 times. The larger the sample, the more likely you are to see roughly what you expect.

In our framework of evolution as sampling error, natural selection determines the expected value: the number of offspring of each genotype we expect to see in the next generation, given the distribution in the current generation. But the smaller the population, the more likely it is to deviate substantially from this expectation – that is, for random genetic drift to overwhelm the bias imposed by natural selection.

If you combine this sample-size-dependent variability with the absorbing nature of fixation and elimination (that is, once an allele has been eliminated, it isn’t coming back), you obtain the result that the larger the population, the more likely it is that the fitter allele is actually the one that gets fixed, all else equal. We can see this in our toy model from earlier, where the green allele is 10% fitter than the purple allele and both start with 50% prevalence in the population:

A plot of population size vs probability of fixation for two competing alleles, one of which is 10% fitter than the other. The fitter allele fixes at just over 50% when population size is very small, rising to 100% at population sizes of 100 or larger.

When population size is very small, the chance that the fitter (green) allele is the one that eventually fixes is close to 50%; as population size increases, this probability increases, until for sufficiently-large populations it is virtually certain. Smaller differences in fitness would require larger population sizes to consistently fix the fitter allele4.

Population size, then, is a crucial factor affecting the optimisation power of evolution: the larger the population size, the greater the capacity of natural selection to select for beneficial mutations and eliminate deleterious ones. This is why bottlenecks and founder effects are so important in evolution: by reducing the size of the population, they both increase the relative prevalence of rare mutations and decrease the relative strength of natural selection, resulting in very powerful drift. The results of this can be quite striking: on the tiny Micronesian island of Pingelap, for example, almost 10% of the population are completely colourblind, a condition that is extremely rare elsewhere5. This is believed to be the result of a typhoon in 1775 that left only 20 survivors, one of whom was a carrier of the condition6.


What can we infer from all this? Firstly, when thinking about evolutionary processes it’s vital not to neglect genetic drift. Just because something spread throughout a population and reached fixation does not mean it is adaptive. Secondly, this is especially true when populations are small, and we should always pay careful attention to population size when thinking about how a population might evolve. In general, we should expect larger populations7 to be fitter than smaller ones, since (among other things) natural selection will be more effective at weeding out deleterious alleles and propagating beneficial ones.

Finally, it has not escaped my notice that this framework has obvious implications for thinking about analogous evolutionary processes that might occur outside of biology. More on this anon.

  1. I’m also assuming non-overlapping generations and a constant carrying capacity; relaxing these assumptions makes the maths more complicated but shouldn’t alter the basic conclusion. Similarly, while new genetic variants are capable of spreading much more quickly through sexual populations, the same basic phenomena still apply. 

  2. Actually, the time to fixation (and many other aspects of the population’s behaviour) depend on its effective population size, which depends not only on its actual population size but also on various demographic and genetic factors. This is an absolutely crucial distinction that I am eliding here for the sake of brevity (in my defence, population geneticists seem to also do the same thing when speaking casually). Effective population sizes are often much smaller than actual (“census”) sizes; for example, the usual estimate that gets bandied about for global human effective population size is roughly 10,000. 

  3. In fact, the RMD looks like it might vary as a power law of sample size: A log-log plot of relative mean absolute difference vs sample size, showing a very linear-looking relationship I noticed this from simulations and haven’t bothered to tease out the underlying mathematics here, but still, kinda cool. 

  4. See e.g. this plot for a 1% difference in fitness: A plot of population size vs probability of fixation for two competing alleles, one of which is 1% fitter than the other. The fitter allele fixes at roughly 50% when population size is very small, rising to about 95% at a population size of 300. The rise in fixation rate is much slower than when the fitness difference was 10%. 

  5. According to Wikipedia, the proportion of Americans with the same condition is 0.003%. 

  6. It probably didn’t hurt that the suspected carrier was also the chief of the island. 

  7. Again, I’m actually talking about effective population size here, not census size.