Applied Statistics and Probability for Engineers

indicate that treatment 4 (20% hardwood) produces paper with higher mean tensile strength

than do the other treatments, and treatment 1 (5% hardwood) results in lower mean tensile

strength than do the other treatments. The means of treatments 2 and 3 (10 and 15% hard-

wood, respectively) do not differ.

This simple procedure is a rough but very effective multiple comparison technique. We

now briefly describe two other procedures: orthogonal contrasts and Tukey’s method.

Orthogonal Contrasts

Many multiple comparison procedures use the idea of a contrast. Consider the hardwood con-

centration experiment presented in Example 13-1. Since the hypothesis H 0 :  1  2  3 

 4 0 was rejected, we know that some hardwood concentrations produce different tensile

strengths than others, but which ones actually cause this difference? At the outset of the

experiment, we might suspect that hardwood concentrations 3 and 4 produce the same tensile

strength. This implies that we would like to test the hypothesis

This hypothesis could be tested by using a linear combination of treatment totals, say,

If we had suspected that the averageof hardwood concentrations 1 and 3 did not differ from

the average of hardwood concentrations 2 and 4, the hypothesis would have been

which implies using the linear combination of treatment totals

In general, the comparison of treatment means of interest will imply a linear combination

of treatment totals such as

with the restriction that These linear combinations are called contrasts.The sum

of squares for any contrast is

gai 1 ci0.

c a

a

i 1

ciyi.

y 1 .y 3. y 2. y 4.

H 1 :  1  3 

 2  4

H 0 :  1  3  2  4

y 3. y 4.

H 1 :  3 

 4

H 0 :  3  4

13-2

(S13-1)

SSc

aa

a

i 1

ciyi.b

2

na

a

i 1

c

2

i

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