Applied Statistics and Probability for Engineers

In this model, iis a fixed effect defined such that , is a random effect, the

interaction term ()ijis a random effect, and ijkis a normally and independently distributed

random variable with mean zero and variance ^2. The interaction elements ()ijare normal

random variables with mean zero and variance , and Because

the sum of the interaction effects over the levels of the fixed factor equals zero, this version of

the mixed model is called the restricted model.The interaction elements are not all

independent. For more details, see Montgomery (2001).

The expected mean squares in this case are

31 a
12 a 4 ^2 gai 11  (^2) ij0.
gai 1 i0, j
Therefore, the appropriate test statistic for testing H 0 : i0 is
14-7
(S14-8)
E 1 MSE 2 ^2
E 1 MSAB 2 ^2 n^2 
E 1 MSB 2 ^2 an^2 
E 1 MSA 2 ^2 n^2 
bna
a
i 1
^2 i
a
1
F 0  (S14-9)
MSA
MSAB
F 0  (S14-10)
MSB
MSE
for which the reference distribution is Fa 1,(a 1)(b 1). For testing H 0 : ^2 0, the test statistic is
for which the reference distribution is Fb 1,ab(n 1). Finally, for testing H 0 : ^2 0 we would use
F 0  (S14-11)
MSAB
MSE
for which the reference distribution is F(a 1)(b 1),ab(n 1).
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