Applied Statistics and Probability for Engineers

The afactor levels in the experiment could have been chosen in two different ways.

First, the experimenter could have specifically chosen the atreatments. In this situation, we

wish to test hypotheses about the treatment means, and conclusions cannot be extended to

similar treatments that were not considered. In addition, we may wish to estimate the treat-

ment effects. This is called the fixed-effects model.Alternatively, the atreatments could be

a random sample from a larger population of treatments. In this situation, we would like to

be able to extend the conclusions (which are based on the sample of treatments) to all treat-

ments in the population, whether or not they were explicitly considered in the experiment.

Here the treatment effects iare random variables, and knowledge about the particular ones

investigated is relatively unimportant. Instead, we test hypotheses about the variability of

the iand try to estimate this variability. This is called the random effects,or components

of variance,model.

In this section we develop the analysis of variancefor the fixed-effects model. The

analysis of variance is not new to us; it was used previously in the presentation of regression

analysis. However, in this section we show how it can be used to test for equality of treatment

effects. In the fixed-effects model, the treatment effects iare usually defined as deviations

from the overall mean , so that

(13-2)

Let yi.represent the total of the observations under the ith treatment and represent the average

of the observations under the ith treatment. Similarly, let represent the grand total of all obser-

vations and represent the grand mean of all observations. Expressed mathematically,

(13-3)

where Nanis the total number of observations. Thus, the “dot” subscript notation implies

summation over the subscript that it replaces.

We are interested in testing the equality of the atreatment means  1 ,  2 ,..., a.Using

Equation 13-2, we find that this is equivalent to testing the hypotheses

(13-4)

Thus, if the null hypothesis is true, each observation consists of the overall mean plus a

realization of the random error component ij. This is equivalent to saying that all N

observations are taken from a normal distribution with mean and variance ^2. Therefore,

if the null hypothesis is true, changing the levels of the factor has no effect on the mean

response.

The ANOVA partitions the total variability in the sample data into two component parts.

Then, the test of the hypothesis in Equation 13-4 is based on a comparison of two independ-

ent estimates of the population variance. The total variability in the data is described by the to-

tal sum of squares

SST a

a

i 1 a

n

j 1

1 yijy.. 22

H 1 : i 0 for at least one i

H 0 :  1  2 pa 0

y..a

a

i 1

(^) a
n
j 1

yij y..y.. N

yi. a

n

j 1

yij yi.yi. n i1, 2,... , a

y..

y..

yi.

a

a

i 1

i^0

13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 473

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