Applied Statistics and Probability for Engineers

those specific levels. Now we briefly consider the situation where one or more of the factors in a

factorial experiment are random, using the two-factor factorial design as an illustration.

The Random-Effects Model

Suppose that we have two factors Aand Barranged in a factorial experiment in which the lev-

els of both factors are selected at random from larger populations of factor levels, and we wish

to extend our conclusion to the entire population of factor levels. The observations are repre-

sented by the model

(S14-1)

where the parameters i, j, ()ij, and ijkare normally and independently distributed random

variables with means zero and variances , , , and ^2 , respectively. As a result of these

assumptions, the variance of any observation Yijkis

and ^2 , ^2 , ^2 , and ^2 are called variance components.The hypotheses that we are interested in

testing are H 0 : ^2 0, H 0 : ^2 0, and H 0 : ^2 0. Notice the similarity to the single-factor

experiment random-effects model discussed in Chapter 13.

The basic analysis of variance remains unchanged; that is, SSA, SSB, SSAB, SST, and SSEare

all calculated as in the fixed-effects case. To construct the test statistics, we must examine the

expected mean squares. They are

and

(S14-2)

Note from the expected mean squares that the appropriate statistic for testing the

no-interaction hypothesis H 0 : ^2 0 is

E 1 MSE 2 ^2

E 1 MSAB 2 ^2 n^2 

E 1 MSB 2 ^2 n^2 an^2 

E 1 MSA 2 ^2 n^2 bn^2 

V 1 Yijk 2 ^2 ^2 ^2 ^2

^2  ^2 ^2 

Yijk ij 1  (^2) ijij k •
i1, 2,... , a
j1, 2,... , b
k1, 2,... , n
14-4
since if H 0 is true, both numerator and denominator of F 0 have expectation ^2 , and only if H 0
is false is E(MSAB) greater than E(MSE). The ratio F 0 is distributed as F(a 1)(b 1),ab(n 1).
Similarly, for testing that there is no main effect of factor A, or H 0 : ^2 0, we would use
F 0  (S14-3)
MSAB
MSE
F 0  (S14-4)
MSA
MSAB
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