Applied Statistics and Probability for Engineers

488 CHAPTER 13 DESIGN AND ANALYSIS OF SINGLE-FACTOR EXPERIMENTS: THE ANALYSIS OF VARIANCE

interpretation. If the variance of the treatment effects iis by independence the variance of

the response is

The variances and ^2 are called variance components,and the model, Equation 13-19, is

called the components of variance modelor the random-effects model.To test hypotheses

in this model, we assume that the errors ijare normally and independently distributed with

mean 0 and variance ^2 and that the treatment effects iare normally and independently dis-

tributed with mean zero and variance .*

For the random-effects model, testing the hypothesis that the individual treatment effects

are zero is meaningless. It is more appropriate to test hypotheses about. Specifically,

If 0, all treatments are identical; but if 0, there is variability between treatments.

The ANOVA decomposition of total variability is still valid; that is,

(13-20)

However, the expected values of the mean squares for treatments and error are somewhat

different than in the fixed-effect case.

SSTSSTreatmentsSSE

^2  ^2 

H 1 : ^2  0

H 0 : ^2 ^0

^2 

^2 

^2 

V 1 Yij 2 ^2 ^2

^2 ,

In the random-effects model for a single-factor, completely randomized experiment,

the expected mean square for treatments is

(13-21)

and the expected mean square for error is

^2 (13-22)

E 1 MSE 2 E c

SSE

a 1 n 12

d

^2 n^2 

E 1 MSTreatments 2 E a

SSTreatments

a 1

b

*The assumption that the {i} are independent random variables implies that the usual assumption of

from the fixed-effects model does not apply to the random-effects model.

gai 1 i 0

From examining the expected mean squares, it is clear that both MSEand MSTreatments

estimate ^2 when H 0 : 0 is true. Furthermore, MSEand MSTreatmentsare independent.

Consequently, the ratio

F 0  (13-23)

MSTreatments

MSE

^2 

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