Section 8.3: Multivariate Analysis
145
more often in the Northern Hemisphere. As the variance estimates used
in the two tests are nearly independent, this, together with the very high
rejection rates, suggests that the SST anomaly really has an influence on the
northern hemisphere SLP. For some other atmospheric variables, however,
Hannoschöck and Frankignoul (1985) found that the rejection rate was lower
and no global assessment of significance could be made, lacking an estimate
of the effective number of independent tests.
8.3 Multivariate Analysis
8.3.1 Test on Means
of Multidimensional Normal Variables
If a sampIe of n independent observations i\ ... in of am-dimensional random vector X with the multinormal.N([1,~) distribution is available, with
n -1 ~ m, the null hypothesis Ho that [1 = [10 can be tested against the alternative hypothesis HA, say [1 =f:. [10, by considering the single sampIe Hotelling
T 2 statistic
(8.9)
where
=.
1 ~ ..
x = - L...JXi
n i=l
(8.10)
and
(8.11)
are unbiased estimates of the mean [1 and the covariance matrix ~. Indeed,
when the null hypothesis is true, the quantity F = m(;;~1)T2 has the F
distribution with m and m - n degrees of freedom (e.g., Morrison, 1976).
Thus, Ho is rejected at the a-Ievel if
T2 > m(n - 1) Fa;m,n-m
n-m
(8.12)
where Fa;m,n-m denotes the upper a quantile of the F distribution with
m and m - n degrees of freedom, and accepted otherwise. T2 is the direct
multivariate analogue of the univariate t-ratio (8.1), and it reduces to t 2
when m = 1. Except for skewed distributions, the test is rather robust
against departures from normality. However, the T 2 test has little power
unless the sampIe size n is much larger than m, which is seI dom the case in
GCM experiments. Note that, as n tends to 00, T 2 become asymptotically
distributed as X~ when Ho is true. When the true covariance matrix ~ is
145
more often in the Northern Hemisphere. As the variance estimates used
in the two tests are nearly independent, this, together with the very high
rejection rates, suggests that the SST anomaly really has an influence on the
northern hemisphere SLP. For some other atmospheric variables, however,
Hannoschöck and Frankignoul (1985) found that the rejection rate was lower
and no global assessment of significance could be made, lacking an estimate
of the effective number of independent tests.
8.3 Multivariate Analysis
8.3.1 Test on Means
of Multidimensional Normal Variables
If a sampIe of n independent observations i\ ... in of am-dimensional random vector X with the multinormal.N([1,~) distribution is available, with
n -1 ~ m, the null hypothesis Ho that [1 = [10 can be tested against the alternative hypothesis HA, say [1 =f:. [10, by considering the single sampIe Hotelling
T 2 statistic
(8.9)
where
=.
1 ~ ..
x = - L...JXi
n i=l
(8.10)
and
(8.11)
are unbiased estimates of the mean [1 and the covariance matrix ~. Indeed,
when the null hypothesis is true, the quantity F = m(;;~1)T2 has the F
distribution with m and m - n degrees of freedom (e.g., Morrison, 1976).
Thus, Ho is rejected at the a-Ievel if
T2 > m(n - 1) Fa;m,n-m
n-m
(8.12)
where Fa;m,n-m denotes the upper a quantile of the F distribution with
m and m - n degrees of freedom, and accepted otherwise. T2 is the direct
multivariate analogue of the univariate t-ratio (8.1), and it reduces to t 2
when m = 1. Except for skewed distributions, the test is rather robust
against departures from normality. However, the T 2 test has little power
unless the sampIe size n is much larger than m, which is seI dom the case in
GCM experiments. Note that, as n tends to 00, T 2 become asymptotically
distributed as X~ when Ho is true. When the true covariance matrix ~ is