Evolution, 4th Edition

A–6 APPENDIX

variables (FIGURE A.6). We saw in Chapter 6 that regression is used

to estimate selection gradients (see Fig ure 6.11) and heritability (see

Figure 6.14). A regression line is fit so that the sum of the squared dif-

ferences between the line and the values of the variable on the y-axis

is minimized. The slope of this line, which is called the regression

coefficient and is often symbolized as β, tells us how rapidly the value

of the second variable typically increases or decreases with the value

of the first variable. The regression coefficient of measurement y on

measurement x is defined as

β =

(^) σxy
(^) σ^2 x^ (A.4)
The range of possible values that a regression coefficient can take is
unbounded: in principle it can range from –∞ to +∞. The units of a
regression coefficient are units of the y variable divided by units of
the x variable. A regression predicts the average value of y for a given
value of x, while a correlation conveys how tight the association is
between x and y without making a prediction about one from the
other.
Principal components are another tool used to describe and ana-
lyze how different measurements vary together. The first principal
component, or PC1, is a line fit to the data so that its orientation is
along the direction that has the greatest amount of variation. (This
line is not equal to the regression line but often is close to it.) The
second principal component, or PC2, is fit using two rules: it must be
perpendicular to the first principal component, and it must run in the
direction that has the greatest amount of remaining variation. FIGURE
A.7 shows an example in which the measurements are arm length and
leg length, which are strongly correlated (r = 0.8).
Principal components have several uses. One is to simplify data
analysis by reducing the number of variables that need to be analyzed.
When two variables are strongly correlated, as in Figure A.7, the loca-
tion of any point can be quite accurately described by one number
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β = 0.9
11.0
10.0
9.0
8.0
8.0 9.0
Mean of bill depth of parents (mm)
Mean bill depth of offspring (mm)
10.0 11.0
G. fortis
1978
1976
FIGURE A.6 Regressions of the bill depth of offspring
on the bill depth of their parents in the Galápagos finch
Geospiza fortis in 1976 and 1978. Each point represents
a family, with the average value of the parents plot-
ted on the x-axis and the average of their offspring
on the y-axis. Although offspring were larger in 1978,
the regression coefficients (the slopes of the lines)
were nearly the same in both years: β = 0.9. (This value
estimates the heritability (h^2 ) of bill depth—see Chapter
6.) The bill depth of an offspring can be predicted from
that of its parents: find the mean of the two parents
on the x-axis, move upward from there to the regres-
sion line, then move left to the y-axis. That value is the
expected bill depth of the offspring. (After Grant 1986,
based on Boag 1983.)
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PC1
PC2
Leg length
Arm length
FIGURE A.7 Principal components describe how two or
more measurements vary together. This example shows the
relation between arm length and leg length in a hypotheti-
cal population of humans. The first principal component,
PC1, runs along the direction that has the greatest amount
of variation. In this case, it corresponds to overall size: indi-
viduals with small values of PC1 are small overall (lower left),
while those with large values of PC1 are large overall (upper
right). The second principal component, PC2, is perpen-
dicular to PC1. Here this axis of variation corresponds to
the relative sizes of the arm and leg: individuals with small
values of PC2 have long arms and short legs (lower right),
while those with large values have long legs and short arms
(upper left). The lengths of the lines for PC1 and PC2 are
proportional to the amount of variation (specifically, the
standard deviation) in that direction.
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