Applied Statistics and Probability for Engineers

where is an overall mean, iis the effect of the ith treatment, jis the effect of the jth

block, and ijis the random error term, which is assumed to be normally and independently

distributed with mean zero and variance ^2. Treatments and blocks will initially be consid-

ered as fixed factors. Furthermore, the treatment and block effects are defined as deviations

from the overall mean, so and. We also assume that treatments and

blocks do not interact. That is, the effect of treatment iis the same regardless of which block

(or blocks) it is tested in. We are interested in testing the equality of the treatment effects.

That is

Testing the hypothesis that all the treatment effects iare equal to zero is equivalent to

testing the hypothesis that the treatment means are equal. To see this, note that the mean of the

ith treatment is i, defined as

and since , we have the mean of the ith treatment defined as

Therefore, testing the hypothesis that the atreatment means are equal is equivalent to testing

that all the treatment effects iare equal to zero.

The analysis of variance can be extended to the randomized complete block design. The

procedure uses a sum of squares identity that partitions the total sum of squares into three

components.

ii, i1, 2,p, a

g

b

j 1 j^0



1

ba

b

j 1

E^1 ij^2 i

1

ba

b

j 1

j

iE

°a

b

j 1

Yij

b

¢



1

ba

b

j 1

E^1 Yij^2 

1

ba

b

j 1

E^1 ijij^2

H 1 : i 0 at least one i

H 0 :  1  2 pa 0

gai 1 i 0 gbj 1 j 0

13-4 RANDOMIZED COMPLETE BLOCK DESIGN 493

The sum of squares identity for the randomized complete block designis

(13-27)

or symbolically

SSTSSTreatmentsSSBlocksSSE

a

a

i 1

a

b

j 1

1 yijy.jyi.y..^2

2

a

a

i 1

a

b

j 1

1 yijy..^2

(^2) b
a
a
i 1
1 yi.y..^2
(^2) a
a
b
j 1
1 y.jy..^2
2
Furthermore, the degrees of freedom corresponding to these sums of squares are
ab 1  1 a 12  1 b 12  1 a 121 b 12
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