Page 171 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 6

against
H1 : P1 #Po
The null hypothesis states that the mean vector of the parent population of the
first sample is the same as the mean vector of the parent population from which
the second sample was drawn.
The test we must use is a multivariate equivalent of Equation (2.48) on p. 73.
In that two-sample t-test, we used a pooled estimate of the population variance
based on both samples. Accordingly, we must compute a pooled estimate, S,, of
the common variance-covariance matrix from our two multivariate samples. This
is done by calculating a matrix of sums of squares and products for each sample.
We can use the terminology of discriminant functions and denote the matrix of
sums of squares and cross products of sample A as SA; similarly, the matrix from
sample B is SB. The pooled estimate of the variance-covariance matrix is

S, = (nA + nB - 2)-l (sA + sB) (6.32)

We must next find the difference between the two mean vectors, D = EA - XB. Our
T2 test has the form
T2 = (6.33)
nAnB D‘S, lD
nA + nB
The significance of the T2 test statistic can be determined by the F-transformation:
nA + nB - m - 1
F= T2 (6.34)
(nA + nB - 2)m

which has m and (nA + nB - m - 1) degrees of freedom (Morrison, 1990).

Eq ua I ity of varia nce-covaria nce matrices
An underlying assumption in the two preceding tests is that the samples are drawn
from populations having the same variance-covariance matrix. This is the multi-
variate equivalent of the assumption of equal populationvariances necessary to per-
form t-tests of means. In practice, an assumption of equality may be unwarranted,
because samples which exhibit a high mean often will also have a large variance.
You will recall from Chapter 4 that such behavior is characteristic of many geologic
variables such as mine-assay values and trace-element concentrations. Equality of
variance-covariance matrices may be checked by the following “test of generalized
variances” which is a multivariate equivalent of the F-test (Morrison, 1990).
Suppose we have k samples of observations, and have measured m variables
on each observation. For each sample a variance-covariance matrix, Sk, may be
computed. We wish to test the null hypothesis

against the alternative
H1 Xi #Ej
The null hypothesis states that all k population variance-covariance matrices
are the same. The alternative is that at least two of the matrices are different.
Each variance-covariance matrix Si is an estimate of a population matrix Xi. If the
parent populations of the k samples are identical, the sample estimates may be

484