2

approach is also useful in analysis of variance

N (0, σE )) and the coef¬cients a j are constrained

2

because it provides a direct means of obtaining

to sum to zero.

distributional properties for con¬dence intervals

and test statistics. However, here we use a more

9.2.2 Where Do the Data Come From? Data of intuitive approach to the analysis of variance that

this sort might be a result of a planned experiment begins with the partitioning of variability.

that examined the effects of J treatments by Before beginning, let us introduce a little

applying each treatment to n experimental units. notation. Let

The experimenter would have made sure that the

1nJ

experimental units (e.g., people, rats, plots of

land, climate simulations, etc.) were representative Y—¦—¦ = n J Yi j

i=1 j=1

of the population from which they were drawn

and that the treatments were applied to the be the mean of all the observations and let

experimental units in random order.

1n

However, data of this sort might also have

Y—¦ j = Yi j

been obtained with somewhat less attention n i=1

to experimental design. Suppose, for example,

that we wish to use an ensemble of AMIP be the mean of all the observations that were

simulations to determine whether the speci¬ed the result of the jth treatment. The ˜—¦™ notation

sea-surface temperatures and sea-ice boundaries indicates averaging over the missing subscript. By

have an effect on the interannual variability of substituting the model (9.1) into these expressions

the simulated December, January, February (DJF) and taking expectations, it is easily shown that Y—¦—¦

climate. The 10-year AMIP period (January 1979 is an unbiased estimator of µ and that Y—¦ j is an

to December 1988) includes nine complete DJF unbiased estimator of µ + a j . Therefore Y—¦ j ’ Y—¦—¦

seasons. Each DJF season can be thought of as the is an unbiased estimator of a j .

9.2: One Way Analysis of Variance 175

The total sum of squares SST , given by Source Sum of Squares df

SSA J ’1

Treatment

n J

SSE J (n ’ 1)

SST = (Yi j ’ Y—¦—¦ )2 , Error

SST Jn ’ 1

i=1 j=1 Total

can be partitioned as follows. First, subtract and

add Y—¦ j inside the squared difference to obtain 9.2.4 Testing for a Treatment Effect. The

effect of the jth treatment is represented by

n J

coef¬cient a j in model (9.1). Thus the no treatment

2

SST = (Yi j ’ Y—¦ j ) + (Y—¦ j ’ Y—¦—¦ ) .

effect hypothesis can be expressed as

i=1 j=1

H0 : a1 = · · · = a J = 0, (9.3)

Then square and sum the individual terms to obtain

J or, equivalently, as

SST = n (Y—¦ j ’ Y—¦—¦ ) 2

J

j=1

a 2 = 0.

H0 : j

n J

j=1

+ (Yi j ’ Y—¦ j )2

i=1 j=1 We wish to test H0 against the alternative

n J hypothesis that at least some of the coef¬cients a j

’2 (Y—¦ j ’ Y—¦—¦ )(Yi j ’ Y—¦ j ). are different from zero. That is, we test H0 against

i=1 j=1

J

The sum of the cross-products is zero because a 2 > 0.

Ha : j

n

i=1 (Yi j ’ Y—¦ j ) = 0 for each j. Thus we have j=1

SST = SSA + SSE, We have already noted that Y—¦ j ’ Y—¦—¦ is

an unbiased estimator of a j , so it would seem

where

reasonable that a test of H0 should be based on

J

SSA, since it is proportional to the sum of squared

SSA = n (Y—¦ j ’ Y—¦—¦ )2 , (9.2)

coef¬cient estimates. Therefore let us examine the

j=1

treatment sum of squares SSA, given in (9.2),

and more closely.

Substituting the model (9.1) into (9.2) we obtain

n J

SSE = (Yi j ’ Y—¦ j )2 . J

i=1 j=1

SSA = n (µ + a j + E—¦ j ’ (µ + E—¦—¦ ))2

SSA is often referred to as the treatment sum j=1

of squares or the between blocks sum of squares. J J

=n +n (E—¦ j ’ E—¦—¦ )2 .

a2

SSE is referred to as the sum of squared errors (9.4)

j

j=1 j=1

or within blocks sum of squares. The latter names

are particularly descriptive of the calculations that Now note that the second term in (9.4) estimates

were performed. (J ’ 1)σE . We can show this by means of (4.6)

2

The treatment sum of squares is taken over J after noting the following.

deviations that sum to zero, thus it has J ’ 1

degrees of freedom (df). The sum of squared errors 1 E—¦ j is the average of n iid errors that have

variance σE . Therefore, using (4.4), we see

2

is taken over n J deviations such that deviations

that the variance of E—¦ j is σE /n.

within a particular block (or sample) must sum to 2

zero. That is, the sum of squared errors is taken

2 All errors Ei j are independent. Therefore

over deviations that are subject to J constraints.

Consequently, SSE has (n ’ 1)J df. The total sum the within block mean errors E—¦ j are also

of squares is summed over n J deviations which independent.

are subject to only one constraint (i.e., that they

It follows that the expected value of SSA is

sum to zero) and therefore the total sum of squares

have n J ’1 df. In summary, we have the following J

partition of the total sum of squares and degrees of E(SSA) = n a 2 + (J ’ 1)σE .

2

(9.5)

j

freedom. j=1

9: Analysis of Variance

176

mean 850 hPa temperature conducted with the six

member ensemble of CCCma AMIP simulations

are shown in Figure 9.1. In this case the variance

components and F-ratio were computed at every

point on the model™s grid.3 The F-ratio (9.6)

is plotted on a log scale in such a way that a

one contour increment indicates a factor of two

increase in f. The no treatment effect hypothesis is

rejected at the 10% signi¬cance level over 65.7%

of the globe. Experience with ¬elds that have

spatial covariance structure similar to that of 850

Figure 9.1: The natural log of the F-ratios for hPa temperature indicates that this rejection rate is

the year effect obtained from a one way analysis certainly ¬eld signi¬cant (see Section 6.8).

Note that very large F-ratios (i.e., f > 8 or

of variance of DJF mean 850 hPa temperature

simulated in the six member ensemble of CCCma ln(f) > 2.77) cover the entire tropical Paci¬c

AMIP simulations. The shading indicates ratios and Indian Oceans. Signi¬cantly large F-ratios are

that are signi¬cantly greater than 1 at the 10% also found over the North Paci¬c, the midlatitude

North and South Atlantic, and the southern Indian

signi¬cance level.

Oceans.

Equation (9.5) shows that SSA/(J ’ 1)