多元变量统计方法入门（第4版）：经典英文教程

Statistical

Multivariate

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Authors

Bryan F.J. Manly was a professor of statistics at the University of Otago,

Dunedin, New Zealand until 2000 after which he moved to the United

States to work as a consultant for Western EcoSystems Technology Inc.

In 2015 he returned to New Zealand and is now again a professor at the

University of Otago in the School of Medicine.

Jorge A. Navarro Alberto is a mathematician and professor at the

Autonomous University of Yucatán, Mexico, where he has taught statis-

tics for biologists, marine biologists and natural resource management

students for more than 25 years. His research interests are centered in

ecological and environmental statistics, primarily in statistical modeling

of biological data, multivariate statistical methods applied to the study of

ecological communities, ecological sampling, and null models in ecology.

chapter one

The material of

multivariate analysis

1.1 Examples of multivariate data

The statistical methods that are described in elementary texts are mostly

univariate methods, because they are only concerned with analyzing

variation in a single random variable. On the other hand, the whole point

of a multivariate analysis is to consider several related variables simulta-

neously, with each one being considered to be equally important, at least

initially. The potential value of this more general approach can be seen by

considering a few examples.

Example 1.1: Storm survival of sparrows

After a severe storm on February 1, 1898, a number of moribund spar-

rows were taken to Hermon Bumpus’ biological laboratory at Brown

University, Rhode Island. Subsequently, about half of the birds died,

and Bumpus saw this as an opportunity to see whether he could nd

any support for Charles Darwin’s theory of natural selection. To this

end, he made eight morphological measurements on each bird and

also weighed the birds. The results for ve of the measurements are

shown in Table1.1, for females only.

From the data that he obtained, Bumpus (1898) concluded

“[that] the birds which perished, perished not through accident,

but because they were physically disqualied, and that the birds

which survived, survived because they possessed certain physical

characters.” Specically, he found that the survivors “are shorter

and weigh less … have longer wing bones, longer legs, longer

sternums and greater brain capacity” than the nonsurvivors. He

also concluded that “the process of selective elimination is most

severe with extremely variable individuals, no matter in which

direction the variation may occur. It is quite as dangerous to be

above a certain standard of organic excellence as it is to be con-

spicuously below the standard.” This was to say that stabilizing

selection occurred, so that individuals with measurements close to

the average survived better than individuals with measurements

far from the average.

In fact, the development of multivariate statistical methods had

hardly begun in 1898, when Bumpus was writing. The correlation

3Chapter one: The material of multivariate analysis

coefcient as a measure of the relationship between two variables

was devised by Francis Galton in 1877. However, it was another

56 years before Harold Hotelling described a practical method

for carrying out a principal components analysis, which is one of

the simplest multivariate analyses that can usefully be applied

to Bumpus’ data. Bumpus did not even calculate standard devia-

tions. Nevertheless, his methods of analysis were sensible. Many

authors have reanalyzed his data and, in general, have conrmed

his conclusions.

Taking the data as an example for illustrating multivariate meth-

ods, several interesting questions arise, in particular:

1. How are the various measurements related? For example, does

a large value for one of the variables tend to occur with large

values for the other variables?

2. Are there statistically signicant differences between survivors

and nonsurvivors for the mean values of the variables?

3. Do the survivors and nonsurvivors show similar amounts of

variation for the variables?

4. If the survivors and nonsurvivors do differ in terms of the dis-

tributions of the variables, then is it possible to construct some

function of these variables that separates the two groups? It

would then be convenient if large values of the function

tended to occur with the survivors, as the function would

then apparently be an index of the Darwinian tness of the

sparrows.

Table1.1 (Continued) Body measurements of female sparrows

Bird X

40 163 249 33.4 19.5 22.8

41 163 242 31.0 18.1 20.7

42 156 237 31.7 18.2 20.3

43 159 238 31.5 18.4 20.3

44 161 245 32.1 19.1 20.8

45 155 235 30.7 17.7 19.6

46 162 247 31.9 19.1 20.4

47 153 237 30.6 18.6 20.4

48 162 245 32.5 18.5 21.1

49 164 248 32.3 18.8 20.9

Source: Data from Bumpus, H.C., Biological Lectures, Marine Biology Laboratory,

Woods Hole, MA, 1898.

Note: X

= total length, X

= alar extent, X

= length of beak and head, X

= length

of humerus, and X

= length of keel of sternum (all in millimeters). Original

measurements were in inches and millimeters. Birds 1–21 survived and

birds 22–49 died.

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