2

from one experiment to the next. However, it is

there is again a one-to-one relationship between

easy to conceive of experiments in which the 2

response to the treatments is random and therefore R and F. In this case

1 + (n ’ 1)R 2

F’1

can not be replicated from one experiment to 2

R= and F = .

the next. Treatments that have this property add F + (n ’ 1) a’R2

9: Analysis of Variance

178

While the form of R 2 is similar to that of Consequently, the F test conducted by compar-

2 , ing

the adjusted coef¬cient of determination Ra

the interpretation is quite different because

SSA/(J ’ 1)

speci¬cation is impossible in the random effects F =

SSE/(N ’ J )

setup. R 2 simply estimates the proportion of

variance that is induced by the ˜treatment™ against critical values from F(J ’ 1, N ’ J )

variations. is approximate rather than exact; the exact

signi¬cance level of the test will be somewhat

9.2.10 Unequal Sample Sizes. Although exper- different from the speci¬ed signi¬cance level.

iments may be planned so that all treatments are Another consequence of unequal sample sizes

replicated the same number of times, an experi- is that the power of the test (recall [6.2.1]) is

ment often yields samples of unequal size. Also, determined primarily by the size of the smallest

we must often adapt analysis of variance tech- sample. Thus, even when the same total number

niques to data that were not originally gathered for of experimental units are used, experiments

ANOVA purposes. We therefore brie¬‚y consider with unequal sample sizes are generally less

ef¬cient than experiments with equal sample sizes.

one way models with unequal sample sizes:

However, if variations in sample size are not

Yi j = µ + a j + Ei j enormous and all other assumptions implicit in

for i = 1, . . . , n j , and j = 1, . . . , J. the analysis are satis¬ed, the loss of power and

precision usually do not pose a serious problem.

As usual, we assume that errors Ei j are iid

N (0, σE ). The treatment effects can be either ¬xed

2

or random. The number of replicates subjected to 9.2.11 Relationships Between Treatments. We

now return to the ¬xed effects model of (9.1).

treatment j is denoted n j .

The total sum of squares can still be partitioned The only inferential consideration so far has been

into treatment and error components as in [9.2.3]. whether the treatment effects a j are jointly zero.

However, once this hypothesis has been rejected

We have

one would like to extract additional information

J nj

from the data. Tools that can be used for this

SST = (Yi j ’ Y—¦—¦ )2

purpose are called linear contrasts.

j=1 i=1

J

SSA = n j (Y—¦ j ’ Y—¦—¦ )2 9.2.12 Linear Contrasts. Linear contrasts are

used to test hypotheses about speci¬c relationships

j=1

between treatment means that may have arisen

nj

J

SSE = (Yi j ’ Y—¦ j )2 . from physical considerations. For example, the

AMIP period included the strongest El Ni˜ o event

n

j=1 i=1

on record (1982/83) and a relatively weak El Ni˜ on

As in the equal sample size case, SSA and SSE event (1986/87). Thus we might ask, within the

con¬nes of our one way setup, whether the mean

are statistically independent, and

anomalous response to 1982/83 lower boundary

SSE/σE ∼ χ 2 (N ’ J ),

2

conditions is similar to the response to the 1986/87

lower boundary conditions.

where

These kinds of questions can be asked using

J

linear contrasts. Tests of simple contrasts, which

N= n j.

compare only two treatments or samples, are

j=1

similar to the tests employed in composite analysis

A dif¬culty, however, is that SSA/σE is not

2 (see Section 17.3). However, the tests of contrasts

may be more powerful than tests of composite

distributed χ 2 (J ’ 1) under the null hypothesis

differences because the test of the contrast uses

that there is no treatment effect, either ¬xed or

more information about within sample variability.

random. This violation of the usual distributional

A linear contrast is any linear combination of

theory occurs because SSA can not be rewritten as

the treatment (or sample) means

a sum of (J ’ 1) squared normal random variables

that all have the same variance. In this case the J

block mean errors E—¦ j are independent, zero mean wc = cjµj

normal random variables with variance σE /n j .

2

j=1

9.2: One Way Analysis of Variance 179

for which J c j = 0. Questions such as that

j=1

discussed above are expressed as null hypotheses

about linear contrasts:

J

c j µ j = 0.

H0 : (9.11)

j=1

In the AMIP example we might set c j = 0 for

all j except 1982/83, for which we might choose

c82/83 = 1, and 1986/87, for which we might

choose c86/87 = ’1. This contrast would satisfy

the requirement that the coef¬cients sum to zero, Figure 9.3: The natural log of the F-ratios

and the null hypothesis would read ˜the mean for the contrast comparing 1982/83 DJF 850

response in 1982/83 is equal to that in 1986/87.™ hPa temperature with 1986/87 DJF 850 hPa

temperature in the CCCma six run ensemble of

9.2.13 Testing Linear Contrasts. The test of AMIP simulations. The shading indicates ratios

the linear contrast is constructed in the now that are signi¬cantly greater than 1 at the 10%

familiar fashion. First, we construct an estimator signi¬cance level.

of the contrast

J

then the resulting tests are statistically indepen-

wc = c j Y—¦ j . (9.12)

dent.

j=1

Finally note that J ’ 1 orthonormal contrasts

We substitute the model (9.1) into (9.12), and could be used to partition the treatment sum of

compute the expectation of wc . We learn that squares into (J ’1) independent components, each

2

with one degree of freedom, and each independent

J c2

J 2

of the sum of squared errors SSE.

j

E wc = c j a j + σE .

2

2

n

j=1 j

j=1

9.2.14 The Response of the CCC GCMII

This suggests that a suitable test of (9.11) is based

˜

to the 1982/83 El Nino Using the Method

on

of Linear Contrasts. The F-ratios comparing