2

E i j—¦

independent of the block and interaction effects. mean error.

9: Analysis of Variance

186

9.4.3 Variance of the Seasonal Mean Error. Squaring and summing, we obtain

In our speci¬c application, in which n = 3, the I I

variance of the seasonal mean error is SSA = n J ai + n J (Ci—¦ ’ C—¦—¦ )2

2

Ei, j,1 + Ei, j,2 + Ei, j,3 i=1 i=1

σ¯2 = Var

E i j—¦ I

3

+ nJ (Ei—¦—¦ ’ E—¦—¦—¦ )2

= T0 σE /3,

2

i=1

+ cross-terms.

where σE /3 is the variance of the mean of three

2

When taking expectations, we see that the

iid errors and T0 is a factor that re¬‚ects how expected values of the cross-terms in this

the dependence between the errors in¬‚ates the expression are zero (some cross-terms are products

variance. T0 is called the decorrelation time (see of independent, zero mean random variables;

Sections 17.1 and 17.2 for a detailed discussion of others are products between constants and zero

the decorrelation time and its estimation). In this mean random variables). Therefore, the expected

value of SSA reduces to

case it is easily shown that

2 I

T0 = 1 + (2ρ1 + ρ2 ), (9.26) E(SSA) = n J ai2

3

i=1

where ρ1 is the correlation between errors in

I

+ nJE i=1 (Ci—¦ ’ C—¦—¦ )2

adjacent months, that is,

ρ1 = Cor Ei, j,1 , Ei, j,2 = Cor Ei, j,2 , Ei, j,3 , ¯ ¯ —¦—¦—¦

I

+ nJE i=1 (Ei—¦—¦ ’ E2 ) .

and ρ2 is the correlation between errors separated

by a month, that is,

Therefore, using (4.5) and (4.6), we see that

ρ2 = Cor Ei, j,1 , Ei, j,3 .

I

E(SSA) = n J ai2 + n J (I ’ 1)σC2

We will analyse the effect of the correlated errors

shortly. i=1

+ n J (I ’ 1)σE .

2

¯ i—¦—¦

9.4.4 Distribution of the Variance Components.

However, we ¬rst illustrate how items 1“7 in

[9.4.2] are obtained by considering items 1 and 2 Finally, we note that σE i—¦—¦ = σEi j—¦ /J . Assertion 1

2 2

¯

follows.

in detail.

Recall from (9.21) that

9.4.5 Testing the Year Effect: Potential Pre-

I

SSA = n J (Yi—¦—¦ ’ Y—¦—¦—¦ ) .

2 dictability from External Sources. Items 1“7

in [9.4.2] provide us with suf¬cient information to

i=1

The χ 2 assertion (item 2) is veri¬ed by using construct tests about year and block effects. As in

arguments similar to those in [8.3.20] to show that [9.2.5] and [9.3.3], a test of

(9.21) can be rewritten as a sum of I ’ 1 squared H0 : a 1 = · · · = a I = 0 (9.27)

independent normal random variables with mean

determines whether there is a detectable signal

zero. Hence SSA, when scaled by the variance

attributable to the external boundary forcing. If

of these normal random variables, is distributed

so, the climate may be predictable on seasonal

χ 2 (I ’ 1).

time scales because we believe the lower boundary

The scaling variance (item 1) is obtained as

conditions (i.e., sea-surface temperature and sea-

follows. Using model (9.25) we see that

ice extent) to be predictable on these time scales

Yi—¦—¦ = µ + ai + B—¦ + Ci—¦ + Ei—¦—¦ due to the much large thermal inertia of the upper

Y—¦—¦—¦ = µ + B—¦ + C—¦—¦ + E—¦—¦—¦ ocean and cryosphere.

From items 1, 2, and 5“7 in [9.4.2] we see that

where the over-bar and —¦ notation have the usual

hypothesis (9.27) is tested against the alternative

meaning, given in [9.2.3]. Taking differences, we

that some of the year effects are nonzero by

see that

comparing

(Yi—¦—¦ ’ Y—¦—¦—¦ ) = ai + (Ci—¦ ’ C—¦—¦ ) SSA/(I ’ 1)

F= (9.28)

+ (Ei—¦—¦ ’ E—¦—¦—¦ ). SSI/((I ’ 1)(J ’ 1))

9.4: Two Way ANOVA with Mixed Effects 187

with critical values of F(I ’ 1, (I ’ 1)(J ’ 1)). J J

1 2

¯ ¯

SSBR = n I Y j—¦—¦ ’ Y j—¦—¦ .

Note that this F-ratio was also used to test J ’1

j=2 j=2

this hypothesis in the two way model without

interaction that was applied to the seasonal means

SSBF is proportional to the squared difference

in [9.3.2] and [9.3.3]. The numerical values of

between the mean state simulated in the Cray

the ratios are also identical because only seasonal

and the mean state simulated in the ¬ve NEC

means are used in the calculation of (9.28).

simulations, and SSBR can be recognized as a

As reported in [9.3.3], there is a signi¬cant

scaled estimate of the intersimulation variance

sea-surface temperature effect. These effects are

that is computed from those simulations that are

potentially predictable (see Section 17.2 and also

assumed to have the same con¬guration effects.

[9.4.7“11]). Hindcast experiments (see Zwiers

Taking expectations, we can show that

[444]) demonstrate that in this case potential

predictability is actual predictability.

E(SSBR ) = n(J ’ 2) I σB + σC + σE

2 2 2

¯ i j—¦

9.4.6 Testing the Block Effect. Using items

and, using now familiar arguments, we can

3“7 in [9.4.2] we can construct a test of the null

demonstrate that H0 : σB = 0 can be tested by

2

hypothesis that there is not a block effect

comparing

H0 : b1 = · · · = b J = σB = 0

2

(9.29) SSBR /(J ’ 2)

F=

SSI/((I ’ 1)(J ’ 1))

against the alternative hypothesis that there is

a block effect. This particular form of the null with F(J ’ 2, (I ’ 1)(J ’ 1)) critical values. No

hypothesis comes about because we assumed, in evidence was found to suggest that σ 2 > 0 in the

B

[9.4.1], that the block (i.e., simulation) effect has CCCma ensemble of AMIP simulations.

both a ¬xed and a random component. That is, we Again, taking expectations, it can be shown that

assumed that B j ∼ N (b j , σB ) with the constraint

2

J

= 0. The ¬xed and random J

that j=1 b j 1 2

E(SSBF ) = n I b1 ’ bj

J ’ 1 j=2

components are confounded in our experimental