Evolution, 4th Edition

A–10 APPENDIX

A final point is that it is critical to distinguish between statistical significance and

biological significance. Two populations of deer might have very different mean sizes,

but with a small sample size we would not be able to prove statistically that they

are different. Conversely, with enormous sample sizes it is possible to prove that

two populations have different mean sizes, even if the difference is so small that

it is irrelevant to the biological question of interest. Deciding how large an effect

must be in order to qualify as “biologically significant” is the job of the investigator,

and no statistical analysis can determine that. The most useful inferences are made

when an effect is statistically significant and also large enough to be biologically

interesting.

Likelihood

Likelihood is an important branch of statistics used to estimate properties of a

population and to test hypotheses. In statistics, “likelihood” is defined as the

probability of observing the data that we have, given assumptions for how the

data were generated.

Imagine that we sample ten platypuses from a river and find that they have 4

copies of allele A 1 and 16 copies of allele A 2. We can use likelihood to find the prob-

ability of that sample if the actual frequency of allele A 1 in the population is a given

value, for example p 1 = 0.5. Probability theory tells us that, if the allele frequency in

the population is p 1 , then the likelihood that in a random sample we would get n 1

copies of A 1 and n 2 copies of A 2 is:

(A.5)

Futuyma Kirkpatrick Evolution, 4e

Sinauer Associates

Troutt Visual Services

Evolution4e_A.11.ai Date 02-02-2017 03-01-2017

50

TX CO

60

70

80

90

100

(A) (B)

Weight (kg)

–10

XCO – XTX (in kg)

0 10

Frequency

Note: Please conrm color usage on bars and arrows. I didn’t have gure caption

to conrm how (A) and (B) are related.

FIGURE A.11 Randomization is a powerful way to test statistical

hypotheses. In this example, we ask whether deer in Colorado

are heavier than deer in Texas. (A) The weights of 14 deer from

Texas and 19 deer from Colorado are shown. The mean weight

of the Texas deer is x

_

TX = 67 kg, and the mean weight of the

Colorado deer is x

_

CO = 77 kg (means indicated by the two arrows).

(B) The null hypothesis is that that the distribution of weights is

the same in Texas and Colorado. Randomizing the data 10^6 times

produces the distribution of the difference between the means

(x

_

CO – x

_

TX) under that null hypothesis. The actual difference ob-

served, shown by the arrow, is extremely unlikely. The probability

of a difference greater than what is actually seen in the data is

given by the area under the histogram to the right of the arrow.

That is much less than 5 percent, the standard threshold for statis-

tical significance. We reject the null hypothesis that deer in Texas

and Colorado have the same weight on average, and conclude

that the population of deer in Colorado is heavier on average

than the population in Texas.

L(n 1 , n 2 | p 1 ) =

(^) (n 1 + n 2 )!
(^) n 1 !n 2! p 1
n (^1) (1 – p n 2
1 )
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