J ’1

tests about σB and the b j s without making further + I σB + σC + σE .

2 2 2

¯ i j—¦

J

assumptions about the ¬xed parts of the block

effect.

Thus, if H0 : σB = 0 has not been rejected, the null

2

Hypothesis (9.29) is tested by comparing

hypothesis that there is not a con¬guration effect

(H0 : b1 = b2 +···+b J ) can be tested by comparing

SSB/(J ’ 1)

F= J ’1

SSI/((I ’ 1)(J ’ 1))

J SSBF /(J ’ 1)

against F(J ’ 1, (I ’ 1)(J ’ 1)) critical values. F = SSI/((I ’ 1)(J ’ 1))

Again, the test is identical to that for block

effects reported in [9.3.3]. Figure 9.4 showed weak with critical values from F(1, (I ’ 1)(J ’ 1)).

evidence for a block effect, which appears to be When there is evidence that σB > 0, the no

2

associated with a change in computing hardware con¬guration effect hypothesis should be tested by

comparing

part way through the experiment.

Further dissection of the block effect is possible J SSBF /(J ’ 1)

if we assume that only the computer type and F =

SSBR /(J ’ 2)

source of initial data affect the ¬xed part of the

block effect (i.e., if we assume b2 = · · · = with critical values from F(1, J ’ 2). As noted

b J ).Then, using linear contrasts, SSB (9.24) previously, there were signi¬cant con¬guration

can be partitioned into statistically independent effects in the CCCma AMIP ensemble.

components as:

9.4.7 Testing the Interaction Effect: Potential

SSB = SSBF + SSBR

Predictability from Internal Sources. The in-

teraction effects in this experiment are particularly

where

interesting because they represent slow, and hence

J

J ’1 ¯ 1 2

¯ potentially predictable, processes in the simulated

SSBF = nI Y1—¦—¦ ’ Y j—¦—¦

J ’1

J climate of CCC GCMII that are internal to the

j=2

9: Analysis of Variance

188

against F((I ’ 1)(J ’ 1), IJ ) critical values.

climate system. An earlier investigation with the

Note that (9.27) and (9.29) can still be tested with

predecessor model to CCC GCMII (see Zwiers

the full data set.

[440]) found evidence for such variations in a sim-

Application of the ˜rough and ready™ method

ulated climate when the sea-surface temperatures

to 850 hPa temperature from the six simulation

and sea-ice boundaries follow a ¬xed annual cycle.

CCCma AMIP experiment demonstrates weak evi-

It will be necessary to account for the effects

dence for interaction effects (the null hypothesis is

of dependence within seasons to test the null

rejected over 14% of the globe). What makes the

hypothesis

result interesting is that most of these rejections

H0 : σC = 0

2

(9.30)

occur over land. They are apparently related to

that there are no interaction effects. It is easily land surface processes that evolve differently from

shown that the expected value of the sum of simulation to simulation. We return to the interac-

squared errors in our application is given by tion effects in this experiment in [9.4.11].

E(SSE) = IJ (3 ’ T0 )σE ,

2

9.4.9 A More Re¬ned Test for Interaction

where T0 is given by (9.26). This is smaller than Effects. The ˜rough and ready test™ is not

the expected value of SSE when errors are fully entirely satisfactory for a couple of reasons. An

independent and not a convenient quantity to use aesthetic objection is that the problem of within

in a test of (9.30). Item 5 in [9.4.2] indicates that a season dependence has been avoided rather than

suitable test statistic should be of the form solved. More troubling is the loss of one-third of

SSI/((I ’ 1)(J ’ 1)) the data available for estimating error variability.

F= 2 We therefore embark on a path that results in full

nσ E¯ i j—¦

use of the data.

Our goal is to ¬nd factors C and n — such that

= T0 σE .

2 2 2

where nσ E is an estimator of nσE

¯ ¯

i j—¦ i j—¦

The distribution of F under (9.30) is most easily

A. C—SSE/(n — IJ ) is an approximately unbiased

found if nσ E is also independent of SSI and

2

estimator of T0 σE ,

¯ 2

i j—¦

distributed as a χ 2 random variable because F will

B. C—SSE/(T0 σE ) is approximately distributed

2

then be F distributed under H0 .

χ 2 (n — IJ ), and

C. C — SSE/(n — IJ ) is independent of variance

9.4.8 A Rough and Ready Interaction Test.

Two solutions are available to the problem of components SSA, SSB, and SSI.

testing for interaction effects in the presence of

As with T0 , factors C and n — are implicitly

within season dependence.

functions of the within season dependence.

A rough and ready solution is based on the

Once these results are obtained, it is possible to

argument that the correlation within seasons is

test (9.30) by comparing

small, and that it is negligible if monthly means

are separated by at least a month. We could SSI/((I ’ 1)(J ’ 1))

F= (9.31)

therefore drop the middle month in each season

CSSE/(n — IJ )

when computing SSE and adjust the degrees of

with F((I ’ 1)(J ’ 1), n — IJ ) critical values.

freedom for error accordingly. That is, we compute

Our ¬rst step in developing a test like (9.30) is

I J 2

Yi, j,1 + Yi, j,3 to note that, in our application, SSE contains IJ

—

SSE = Yi, j,1 ’

2 statistically independent terms of the form

i=1 j=1

2

Yi, j,1 + Yi, j,3 3

¯

+ Yi, j,3 ’ . Si j = (Ei jl ’ Ei j—¦ )2 .

2

l=1

Each of the IJ terms in this sum consists of the

We ¬nd an approximating distribution for each

sum of two squared deviations that are constrained

individual Si j . We then use this result together with

to add to zero. Thus each term contributes only 1

independence arguments to obtain items A“C in

df for a total of IJ df. The effect of within season

order to test (9.30) using (9.31).

dependence can then be ignored and a test of (9.30)

Zwiers [444], using a method similar to that

can be conducted by comparing

outlined in [8.3.20], shows that Si j can be written

SSI/((I ’ 1)(J ’ 1))

F= Si j = Z2 + Z2 ,