estimates σE when H0 is true, and that it estimates

2

Treatments. As in regression analysis (Chap-

a number larger than σE when H0 is false. It may

2

ter 8) it is possible to compute a coef¬cient of

therefore be possible to construct a test of H0 if

multiple determination

another statistic can be found that estimates only

σE regardless of whether or not H0 is true. An

2

R 2 = SSA/SST (9.7)

argument similar to the one we just completed

shows that SSE/((n ’ 1)J ) has this property. that diagnoses the proportion of the response

Hence variable variance that is explained by the ¬tted

SSA/(J ’ 1) model. As with regression, this is a somewhat

F= (9.6) optimistic estimate of the ability of the model to

SSE/(J (n ’ 1))

specify the response given the treatment.

may be a suitable statistic for testing H0 . An adjustment that attempts to reduce the

In order to use F in a test we must ¬nd its tendency for R 2 to be optimistic is derived as

distribution under the null hypothesis. Methods follows. The expected value of the total sum of

like those of [8.3.20] can be used to demonstrate squares is

that

J

• SSA/σE ∼ χ 2 (J ’ 1), under H0 ,

2

E(SST ) = n a 2 + (n J ’ 1)σE .

2

j

j=1

• SSE/σE ∼ χ 2 ((n ’ 1)J ), and

2

Therefore the proportion of the expected total sum

• SSA is independent of SSE. of squares that is due to the treatments is

Therefore, using [2.7.10], we ¬nd that J 2

n j=1 a j

. (9.8)

F ∼ F(J ’ 1, (n ’ 1)J ) J

+ (n J ’ 1)σE

2 2

n j=1 a j

under H0 . Thus we ¬nally obtain the result that

However, (9.5) shows that the numerator of (9.7)

H0 can be tested at the (1 ’ p) signi¬cance level

˜

is a biased estimator of the numerator of (9.8). We

by comparing F computed from (9.6) against the

therefore adjust SSA in (9.7) so that it becomes an

p-quantile of F(J ’ 1, (n ’ 1)J ) obtained from

˜

Appendix G. 3 Often, a pattern analysis approach (see Chapters 13-16)

provides richer and more insightful results. A pattern analysis

technique is used to obtain patterns representing the dominant

9.2.5 Application of a One Way Fixed Effects modes of variation. The ¬elds are then projected onto these

Model to the CCCma AMIP Experiment. The patterns. The loadings, or pattern coef¬cients, are subsequently

results of the one way analysis of variance of DJF analysed in an ANOVA.

9.2: One Way Analysis of Variance 177

variability to the response variable rather than

changing its mean. Their effect is modelled using

the random effects version of (9.1), which is given

by

Yi j = µ + A j + Ei j ,

where the errors are iid N (0, σE ) and the ˜random

2

effects™ A j are iid N (0, σA ). Random variables A j

2

are assumed to be independent of the errors. With

.

these assumptions we see that

2

Yi j ∼ N (0, σA + σE ).

Figure 9.2: The adjusted proportion Ra of 2 2

the total (i.e., interannual plus intersimulation)

Rather than testing that the treatment changes

variance of DJF mean 850 hPa temperature that

the mean of the response variable, we are now

is explained by the imposed lower boundary

interested in testing the null hypothesis that

conditions in the six member CCCma ensemble of

the treatments do not induce between block (or

2

AMIP simulations. Shading indicates values of Ra

between sample) variability, that is,

greater than 0.2.

H0 : σA = 0.

2

(9.9)

unbiased estimator of the numerator in (9.8). The The statistic (9.6), used to test (9.3) in the ¬xed

resulting adjusted R 2 is effects case, is also used to test (9.9) in the

random effects case. The statistic also has the same

(J ’1)

SSA ’ J (n’1) SSE distribution under the null hypothesis.

Ra = .

2

SST The differences between the ¬xed and random

Note that sampling variability occasionally causes effects cases lie only in the interpretation of

the model and the treatment sum of squares.

2

Ra to be negative.

The model tells us only that the treatments may

increase interblock (or intersample) variability.

9.2.7 AMIP Example: Adjusted R 2 . The The treatment sum of squares is an estimator of

2

spatial distribution of Ra for our AMIP example this variability. In fact,

2

is illustrated in Figure 9.2. Notice that Ra is large

primarily over the tropical oceans. Note also that E(SSA/(J ’ 1)) = nσA + σE . 2 2

(9.10)

2

there is a one-to-one correspondence between Ra

9.2.9 R 2 for Random Effects Models. When

and F. In fact, we may write

random effects are assumed, we see from (9.10)

1 + J (n’1) Ra 2

F’1 J ’1 that the variance of the random treatment effect

Ra = and F = .

2

J (n’1) 1 ’ Ra 2

F + J ’1 can be estimated as [336]

SSA/(J ’ 1) ’ SSE/(J (n ’ 1))

Thus both statistics convey the same information, σ A = .

2

n

and critical values of F are easily expressed as

2

critical values of Ra . None the less, the messages The proportion of variance of the response variable

conveyed by Figures 9.1 and 9.2 are not the same. that is caused by the treatment effects is therefore

The latter gives a much clearer picture of the estimated as

physical relevance of the response to the forcing σA2

2

R=2

imposed by the bottom boundary conditions.

σ A + σE2

(J ’1)

SSA ’ J (n’1) SSE

9.2.8 A One Way Random Effects Model. The

= .

SST ’ SSE/J

one way model given by (9.1) and discussed above

regards the treatment effects a j , for j = 1, . . . , J ,

Note again that sampling variability may result in

as ¬xed (non-random) effects that can be replicated