Encyclopedia of Environmental Science and Engineering, Volume I and II

1132 STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE

variance of X i , and will be designed as sij The generalization

of the concept of variance to a multidimensional variable

then becomes the matrix of variances and covariances. This

matrix will be called the covariance matrix. The covariance

matrix for the population is given as

∑

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

ss s

ss s

ss s

11 12 1

21 22 2

122

Κ

Κ

ΚΚΚΚΚ

...

n

n

nn nn

.

(30)

A second useful matrix is the matrix of correlations

rr

rr

rr

n

n

nnn

11 1

21 2

1

...

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(31)

If the assumption is made that each of the individual vari-

ables is described by a normal distribution, then the distri-

bution of X may be described by the multivariate normal

distribution. This assumption will be made in subsequent

discussion, except where noted to the contrary.

Tests on Means

Suppose that measures have been obtained on several vari-

ables for a sample, and it is desired to determine whether that

sample came from some known population. Or there may be

two samples; for example, suppose data have been gathered

on physiological effects of two concentrations of SO 2 for

several measures of physiological functioning and the inves-

tigator wishes to know if they should be regarded as samples

from the same population. In such situations, instead of using

t -tests to determine the significance of each individual differ-

ence separately, it would be desirable to be able to perform

one test, analogous to the t -test, on the vectors of the means.

A test, known as Hotelling’s T^2 test, has been developed

for this purpose. The test does not require that the popula-

tion covariance matrix be known. It does, however, require

that samples to be compared come from populations with

the same covariance matrix, an assumption analogous to the

constant variance requirement of the t -test.

To understand the nature of T^2 in the single sample case,

consider a single random variable made up of any linear

combination of the n variables in the vector X (all of the

variables must enter into the combination, that is, none of the

coefficients may be zero). This variable will have a normal

distribution, since it is a sum of normal variables, and it can

be compared with a linear combination of elements from

the vector for the population with the same coefficients, by

means of a t -test. We then adopt the decision rule that the null

hypothesis will be accepted only if it is true for all possible

linear combinations of the variables. This is equivalent to

saying that it is true for the largest value of t as a function of

the linear combinations. By maximizing t^2 as a function of

the linear combinations, it is possible to derive T^2. Similar

arguments can be used to derive T^2 for two samples.

A related function of the mean is known as the linear

discriminant function. The linear discriminant function is

defined as the linear compound which generates the largest

T^2 value. The coefficients used in this compound provide

the best weighting of the variables of a multivariate obser-

vation for the purpose of deciding which population gave

rise to an observation. A limitation on the use of the linear

discriminant function, often ignored in practice, is that it

requires that the parameters of the population be known, or

at least be estimated from large samples. This statistic has

been used in analysis of data from monitoring stations to

determine whether pollution concentrations exceed certain

criterion values.

Other statistical procedures employing mean vectors are

useful in certain circumstances. See Morrison for a further

discussion of this question.

Multivariate Analysis of Variance (MANOVA)

Just as the concepts underlying the t -test could be general-

ized to the comparison of more than two means, the concepts

underlying the comparison of two mean vectors can be gen-

eralized to the comparison of several vectors of means.

The nature of this generalization can be understood in

terms of the linear model, considered previously in connec-

tion with analysis of variance. In the multivariate situation,

however, instead of having a single observation which is

hypothesized to be made up of several components com-

bined additively, the observations are replaced by vectors of

observations, and the components by vectors of components.

The motivation behind this generalization is similar to that

for Hotelling’s T^2 test: it permits a test of the null hypothesis

for all of the variables considered simultaneously.

Unlike the case of Hotelling’s T^2 , however, various

methods of test construction do not converge on one test sta-

tistic, comparable to the F test for analysis of variance. At

least three test statistics have been developed for MANOVA,

and the powers of the various tests in relation to each other

are very incompletely known.

Other problems associated with MANOVA are similar in

principle to those associated with ANOVA, though computa-

tionally they are more complex. For example, the problem of

multiple comparison of means has its analogous problem in

MANOVA, that of determining which combinations of mean

vectors are responsible for significant test statistics. The

number and type of possible linear models can also ramify

considerably, just as in the case of ANOVA. For further dis-

cussion of MANOVA, see Morrison (1967) or Seal.

Extensions of Correlation Analysis

In a number of situations, where multivariate measurements

are taken, the concern of the investigator centers on the

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