9.4: Two Way ANOVA with Mixed Effects 189

where Z1 and Z2 are independent zero mean Equating means and variances and solving for c

normal random variables with variances σE »1 and and n — yields

2

σE »2 , respectively. Parameters »1 and »2 , which

2

2 »1 + »2

2 2

c = σE

characterize the within season dependence, are the

»1 + »2

nonzero eigenvalues of the matrix AT RA where

(»1 + »2 )2

Ri j = ρ|i’ j| , and —

n= ,

« »2 + »2

2 ’1 ’1 1 2

1

2 ’1 .

’1

A= which, after substitution for »1 and »2 , yields

3 ’1 ’1 2

9 ’ 12ρ1 + 8ρ1 ’ 6ρ2 ’ 4ρ1 ρ2 + 5ρ2

2 2

c=

Here ρ0 = 1, ρ1 is the correlation between Ei, j,l 9 ’ 6ρ1 ’ 3ρ2

and Ei, j,l+1 for l = 1, 2, and ρ2 is the correlation (9.32)

between Ei, j,1 and Ei, j,3 . The eigenvalues are

2(3 ’ 2ρ1 ’ ρ2 )2

n— = .

given by

9 ’ 12ρ1 + 8ρ1 ’ 6ρ2 ’ 4ρ1 ρ2 + 5ρ2

2 2

»1 = 1 ’ ρ2 (9.33)

4 1

» 2 = 1 ’ ρ1 + ρ2 . We can check our work by testing these

3 3

expressions when within season errors are iid; that

Because Z1 and Z2 do not generally have equal is, when ρ = ρ = 0. We see we get the right

1 2

variances, the exact distribution of Si j is dif¬cult answers, c = 1 and n — = 2, by substituting

to ¬nd. In fact, the exact distribution can neither ρ = ρ = 0 into (9.32) and (9.33). When 1 >

1 2

be expressed analytically nor tabulated ef¬ciently. ρ > ρ ≥ 0, we see that c ¤ 1 (as expected,

1 2

We therefore need to ¬nd an approximating because » ¤ 1 and » ¤ 1) and n — ¤ 2.

1 2

distribution. Because the components Si j of SSE are

It is reasonable to select the χ 2 distribution independent, (9.32) and (9.33) provide us with the

as the approximating distribution because Si j ∼ result that

χ 2 (2) when Z1 and Z2 have equal variances (i.e.,

when »1 = »2 = 1) and Si j ∼ χ 2 (1) when SSE/c ∼ χ 2 (n — IJ ).

one of the eigenvalues is zero.6 A χ 2 distribution

Therefore the constant C required by items A“C

with a fractional number of degrees of freedom

above is given by

somewhere between these two extremes should

T0 σE2

therefore work well. Thus we need to ¬nd a

constant c and equivalent degrees of freedom n — C = c

such that cχ 2 (n — ) approximates the distribution of

χ (3 + 4ρ1 + 2ρ2 )(3 ’ 2ρ1 ’ ρ2 )

= .

Si j . We do this by matching the mean and variance

9 ’ 12ρ1 + 8ρ1 ’ 6ρ2 ’ 4ρ1 ρ2 + 5ρ2

2 2

2 (n — ) random variable.

χ

of Si j with that of a cχ

2 (n — ) random variable, then the (9.34)

χ

If is a cχ

mean and variance of are given by In summary, we account for within sea-

E( ) = cn — son dependence in our test of H0 (9.30) by

computing F as in (9.31), and comparing

Var( ) = 2c2 n — . with F((I ’ 1)(J ’ 1), n — IJ ) critical values. The

˜shrinkage factor™ C is given by (9.34). The ˜equiv-

The mean and variance of Si j are given by

alent degrees of freedom™ for the denominator are

n — IJ , where n — is given by (9.33).

E Si j = σE (»1 + »2 )

2

2 1

= 2σE (1 ’ ρ1 ’ ρ2 )

2

9.4.10 Estimating Within Season Dependence.

3 3

We need to know the within season correlations ρ1

= 2σE (»1 + »2 )

42 2

Var Si j

and ρ2 to perform the test derived above. Since we

4 82 2

= 4σE (1 ’ ρ1 + ρ1 ’ ρ2

4 do not know them, they must be estimated, and we

3 9 3 must be careful to do this in such a way that items

4 52

’ ρ1 ρ2 + ρ2 ). A“C are not seriously compromised.

Unfortunately, ρ1 and ρ2 can not be estimated

9 9

directly from the monthly data because, in this

6 Recall that if Z , . . . , Z are iid N (0, σ 2 ), then

n

1

n Z2 )/σ 2 ∼ χ 2 (n).

( context, the usual estimator has extremely large

i=1 i

9: Analysis of Variance

190

bias and variability. Instead, ρ1 and ρ2 are obtained

by ¬tting a parametric time series model (see

1.9

Chapter 10) to the daily data after they have been

Equivalent df

adjusted for the annual cycle, and then inferring ρ1

and ρ2 from the ¬tted model.

1.7

Because the parameters of the ¬tted time series

model are estimated from a very large number

of days of data (4860 in case of the CCCma

1.5

AMIP experiment), they have very little sampling

variability. Consequently, the derived estimates of

0.0 0.2 0.4 0.6 0.8 1.0

ρ1 and ρ2 also have very little sampling variability,

Lag-1 day correlation

and therefore items A“C will not be seriously

compromised provided that the ¬tted time series

model ¬ts the daily data well (see Zwiers [444] for