temperature for December, January, and February

SSE = (Yi jl ’ Yi j—¦ )2 . (9.24)

from which the annual cycle common to all

i=1 j=1 l=1

six simulations has been removed (see Zwiers

[444] for details of the procedure used). For each

Assuming ¬xed effects and iid N (0, σE ) errors,

2

DJF season we regard the three monthly means

the following can be shown.

obtained for the season as three replicates of the

I

• E(SSA) = n J + (I ’ 1)σE .

2 2

i=1 ai treatment (i.e., sea-surface temperature) and block

(i.e., simulation) combination that corresponds to

• If H0 : a1 = · · · = a I = 0 is true, then that season. Although these replicates are not quite

SSA/σE ∼ χ 2 (I ’ 1).

2

independent of one another, we operate, for now,

as if they were.

J

• E(SSB) = n I + (J ’ 1)σE .

2 2

j=1 b j

• If H0 : b1 = · · · = b J = 0 is true, then 9.4.1 The Model. The model we use to

SSB/σE ∼ χ 2 (J ’ 1). represent this data is a two way model with

2

interaction in which some effects are ¬xed and

I J

• E(SSI) = n 2 + (I ’ 1)(J ’ others are random. The model is given by

j=1 ci j

i=1

2

1)σE . Yi jl = µ + ai + B j + Ci j + Ei jl , (9.25)

• If H0 : c1,1 = · · · = c IJ = 0 is true, then where i = 1979, . . . , 1987 indicates the year of

SSI/σE ∼ χ 2 ((I ’ 1)(J ’ 1)).

2

the December month in each DJF season, j =

1, . . . , 6 indicates the member of the ensemble of

• E(SSE) = IJ (n ’ 1)σE . 2

simulations, l = 1, 2, 3 indicates the ˜replicate™

(i.e., December, January or February).

• SSE/σE ∼ χ 2 (IJ (n ’ 1)).

2

We treat the year effects ai as ¬xed effects

• SSA, SSB, SSI, and SSE are independent. because every simulation was forced with the

same sea-surface temperature and sea-ice record as

Tests for treatment, block, and interaction effects dictated by the AMIP protocol (see Gates [137]).

as well as tests of linear contrasts among A ¬xed mean response to a given sea-surface

treatments, blocks, and interactions follow in the temperature and sea-ice regime is anticipated

usual way. in each simulation. This is not to say that

As in [9.3.2], the power of the test for treatment each simulation is identical, since low-frequency

effects can be enhanced if the block and/or variations from internal sources ensures that the

interaction sums of squares can be pooled with simulations are different. However, the ¬xed

the sum of squared errors. For example, if the sea-surface temperature and sea-ice signal are

null hypothesis that there is no block/treatment assumed to induce the same amount of interannual

interaction is accepted, then an improved estimator variability in each simulation.

9.4: Two Way ANOVA with Mixed Effects 185

There are certainly problems with this last

The block effects B j are treated as random ef-

assumption that should make us cautious about the

fects and assumed to be independently distributed

N (b j , σB ) where the ¬xed parts of the block

2 subsequent inferences we make. For example, our

assumptions imply that the amount of variability

effects, b j , are constrained to sum to zero. That

is, we represent the block effect as B j = b j + B— at high frequencies is not affected by either

j

— s are iid N (0, σ 2 ). The idea is that the the imposed sea-surface temperature and sea-

where the B j B

ice regime or by the state of slowly varying

¬xed part of the block effect represents variation in

internal processes. Also note that we were careful

the simulation con¬guration (such as the source of

not to make the assumption that the errors are

initialization data) and the random part represents

independent, because they are actually weakly

excess intersimulation variability caused by the

correlated within seasons. We therefore assume

particular choice of initial conditions. Variations

only that errors Ei jl and Ei j l are independent for

in initial conditions might cause CCC GCMII to

(i, j) = (i , j ). Errors Ei jl and Ei jl for l = l are

produce simulations that occupy distinctly differ-

not assumed to be independent.

ent parts of the model™s phase space if the model

has more than one stable regime. Rejection of H0 :

σB = 0 might be evidence of this. However, except

2

9.4.2 Partition of the Total Sum of Squares.

for the possibilities of this sort of chaotic be- With all these assumptions, we are able to partition

haviour and computing glitches, we do not expect the total sum of squares into treatment, block,

block effects to contribute signi¬cantly to total interaction, and error components as in [9.3.4] (see

variability. We will see below that it is possible to(9.19)“(9.24)). Because model (9.25) has mixed

separate the ¬xed and random components of the effects and some dependence amongst errors,

block effects in model (9.25) provided additional the interpretation of the variance components is

assumptions are made about the structure of the somewhat different from that in [9.3.4]. By taking

¬xed components. expectations and making arguments such as those

The interaction effects Ci j are treated as pure in [9.2.4] and [9.2.6] we obtain the following.

random effects and are assumed to be iid N (0, σC )

2 I

1 E(SSA) = n J i=1 ai2 + n(I ’ 1)(σAB + 2

random variables that are independent of the σE ).

2

¯ i j—¦

block effects. The interaction effects represent

interannual variations that are not common to all 2 If H0 : a1 = · · · = a I = 0 is true, then

runs. That is, this term in (9.25) represents the

SSA

effects of slow processes in the climate system that

∼ χ 2 (I ’ 1).

do not evolve the same way in every simulation. 2 + σ2 )

n(σC ¯

For example, CCC GCMII contains a simple E i j—¦

land surface processes model (see McFarlane et

3 E(SSB) = n I J b2 + n I (J ’ 1)σB + 2

al. [270]). The evolution of the soil moisture j=1 j

¬eld in this land surface model will certainly n(J ’ 1)(σC + σE ).

2 2

¯ i j—¦

be affected by the prescribed evolution of sea-

surface temperature and sea ice, but it will 4 If H0 : b1 = · · · = b J = σB = 0 is true, then

2

not be completely determined by these forcings.

Therefore about 30% of the lower boundary of SSB

∼ χ 2 (J ’ 1).

the simulated climate evolves differently from 2 + σ2 )

n(σC ¯

one simulation to the next. The effects of these E i j—¦

variations in the lower boundary over land, and

5 E(SSI) = n(I ’ 1)(J ’ 1)(σAB + σE ).2 2

other slow variations generated internally by the ¯ i j—¦

GCM, are not common to all simulations and will

therefore be re¬‚ected in the interaction term. 6 If H0 : σC = 0 is true, then

2

The noise terms Ei jl represent the effects

SSI

of intra-seasonal variations caused by processes ∼ χ 2 ((I ’ 1)(J ’ 1)).

2 + σ2 )

n(σC

(such as daily weather) that operate on shorter ¯

E i j—¦

than interannual time scales (see Zwiers [444]

and the discussion of potential predictability in 7 SSA, SSB, SSI, and SSE are independent.

Section 17.2). We assume that the errors are