Applied Statistics and Probability for Engineers

and has a single degree of freedom. If the design is unbalanced, the comparison of treatment

means requires that and Equation S13-1 becomes

(S13-2)

A contrast is tested by comparing its sum of squares to the mean square error. The resulting

statistic is distributed as F, with 1 and N adegrees of freedom.

A very important special case of the above procedure is that of orthogonal contrasts.

Two contrasts with coefficients and are orthogonal if

or for an unbalanced design if

For atreatments a set of a 1 orthogonal contrasts will partition the sum of squares due to

treatments into a 1 independent single-degree-of-freedom sums of squares. Thus, tests

performed on orthogonal contrasts are independent.

There are many ways to choose the orthogonal contrast coefficients for a set of treat-

ments. Usually, something in the context of the experiment should suggest which comparisons

will be of interest. For example, if there are a3 treatments, with treatment 1 a control and

treatments 2 and 3 actual levels of the factor of interest to the experimenter, appropriate

orthogonal contrasts might be as follows:

Note that contrast 1 with ci   2, 1, 1 compares the average effect of the factor with the con-

trol, while contrast 2 with di0, 1, 1 compares the two levels of the factor of interest.

Contrast coefficients must be chosen prior to running the experiment, because if these

comparisons are selected after examining the data, most experimenters would construct tests that

compare large observed differences in means. These large differences could be due to the presence

of real effects, or they could be due to random error. If experimenters always pick the largest dif-

ferences to compare, they will inflate the type I error of the test, since it is likely that in an unusu-

ally high percentage of the comparisons selected, the observed differences will be due to error.

EXAMPLE S13-1 Consider the hardwood concentration experiment. There are four levels of hardwood concen-

tration, and possible sets of comparisons between these means and the associated orthogonal

comparisons are

H 0 :  1  3  2  4 e y 1 .y 2. y 3 .y 4.

H 0 :  1  2  3  4 d y 1. y 2 .y 3 .y 4.

H 0 :  1  4  2  3 c y 1. y 2. y 3 .y 4.

H 0 :  2  3  0

H 0 :   2  1  2  3  0

a

a

i 1

nicidi^0

a

a

i 1

cidi^0

5 ci 6 5 di 6

SSc

aa

a

i 1

ciyi.b

2

a

a

i 1

nic^2 i

gai 1 nici0,

13-3

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