Evolution, 4th Edition

A STATISTICS PRIMER A–11

where x! stands for x × (x – 1) ... 3 × 2 × 1. (This equation comes

from the binomial distribution, which often appears in statis-

tics.) With that formula, we find that if the actual frequency of

A 1 in the population is p 1 = 0.5, then the likelihood of our data

is L = 0.0046.

FIGURE A.12 shows how the likelihood (that is, probability of

the observed data) varies as p 1 ranges from 0 to 1. This is called

the likelihood function. The likelihood function reaches its greatest

value, L = 0.22, with p 1 = 0.2. That is, the data are most likely if

that is the true frequency of A 1 in the population we sampled.

This is called the maximum likelihood estimate of the allele

frequency. In this example, the maximum likelihood estimate

corresponds to common sense: it equals the frequency of A 1 in

our actual sample of genes (4/20 = 0.2). In other situations, the

maximum likelihood estimate cannot be found from an average

or other simple summary statistic.

The complete likelihood function gives us more information

than just its maximum. It also conveys the range of values of p 1 that

are plausible. The maximum likelihood estimate suggests that the

frequency of A 1 is somewhere near 0.2, but it is almost certainly not

exactly equal to 0.2. It is often useful to consider the confidence

interval, which is the range of values in which the real value of p 1 is

very likely to lie. A rule commonly used is to determine the range of

values of p 1 for which the likelihood L is no more than seven times

smaller than the maximum likelihood. We can be 95 percent cer-

tain that the true value of p 1 lies within that range. In our example,

the maximum likelihood is L = 0.22, so we seek the value of p 1 that

gives a likelihood that is at least equal to 0.22 / 7 = 0.031. Figure A.12 shows that

range of values is from p 1 = 0.07 to p 1 = 0.41. We are 95 percent confident that the true

value of the allele frequency lies somewhere in that range.

This example illustrates two of the major applications of the likelihood

approach: using maximum likelihood to estimate something about the population

(such as its mean), and finding the confidence interval for that quantity. Likelihood

is used for a broad range of problems in evolutionary biology, such as estimating

phylogenies and effective population sizes. The key requirement is that we be able

to calculate the probability of the data given assumptions about how they were

produced.

Bayesian Inference

An alternative to likelihood that is increasingly used in many areas of evolutionary

biology is Bayesian inference. The goal here is to find the probability that the allele

frequency (or other variable) in the population is equal to any given value. There

are two main motivations for using the Bayesian approach. The first is to make use

of information that we already have. Likelihood has no way of combining prior

information with new data, but Bayesian inference does. With little or no new

data, Bayesian estimates rely heavily on the prior information. But as more and

more new data are gathered, they are given more and more weight. With enough

new data, the prior information has a negligible effect on our estimate.

Say, for example, that after sampling ten alleles from platypuses living in the

first river, we move to a second river nearby. We think that platypuses migrate

back and forth between the rivers, so we expect allele frequencies in the two

populations to be similar. We can therefore use our first sample to form a prior

Futuyma Kirkpatrick Evolution, 4e

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0.031

0.05

0.10

0.15

0.20

Maximum likelihood:

L = 0.22

Maximum likelihood

estimate of p 1 = 0.2

Condence interval:

L 0.07 < p 1 < 0.41

, the likelihood of

p^1

0.2 0.4 0.6

Value of p 1

FIGURE A.12 The likelihood function for p 1 , the frequency

of allele A 1 in the population, given that we have a sample

with 4 copies of allele A 1 and 16 copies of allele A 2. The

likelihood reaches a maximum value of L = 0.22 when

p 1 = 0.2. The confidence interval is the range of values in

which we are 95 percent sure that the true value of the

allele frequency lies (the shaded box). It corresponds

approximately to values of p 1 that give likelihoods no less

than seven times smaller than the maximum likelihood, that

is, L greater than 0.22 / 7 = 0.031. The confidence interval

ranges from p 1 = 0.07 to p 1 = 0.41.

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