Statistical Methods for Psychology

estimates agree, we have no reason to reject. If they disagree sufficiently, we conclude

that underlying treatment differences must have contributed to our second estimate, inflat-

ing it and causing it to differ from the first. Therefore, we reject.

Variance Estimation

It might be helpful at this point to state without proof the two values that we are really esti-

mating. We will first define the treatment effect,denoted , as ( ), the difference

between the mean of ( ) and the grand mean (m), and we will define as the

variation of the true populations’ means ( ).^3

In addition, recall that we defined the expected valueof a statistic [written E()] as its

long-range average—the average value that statistic would assume over repeated sam-

pling, and thus our best guess as to its value on any particular trial. With these two con-

cepts we can state

where is the variance within each population and is the variation^4 of the population

means ( ).

Now, if is true and , then the population means don’t

vary and 5 0,

and

and thus

Keep in mind that these are expected values; rarely in practice will the two sample-

based mean squares be numerically equal.

If is false, however, the will not be zero, but some positive number. In this case,

because MStreatwill contain a nonzero term representing the true differences among the .mj

E(MSerror) 6 E(MStreat)

H 0 u^2 t

E(MSerror)=E(MStreat)

E(MStreat)=s^2 e 1 n(0)=s^2 e

E(MSerror)=s^2 e

u^2 t

H 0 m 1 =m 2 = Á =m 5 =m

mj

s^2 e u^2 t

=s^2 e 1 nu^2 t

E(MStreat)=s^2 e 1

nat^2 j

k 21

E(MSerror)=s^2 e

u^2 t=

a(mj2m)

2

k 21

=

at

(^2) j
k 21
m 1 , m 2 ,... , m 5
treatmentj mj u^2 t
tj mj2m

H 0

H 0

Section 11.3 The Logic of the Analysis of Variance 323

(^3) Technically, is not actually a variance, because, having the actual parameter ( ), we should be dividing by k
instead of k 2 1. Nonetheless, we lose very little by thinking of it as a variance, as long as we keep in mind pre-
cisely what we have done. Many texts, including previous editions of this one, represent as to indicate that it
is very much like a variance. But in this edition I have decided to be honest and use.
(^4) I use the wishy-washy word “variation” here because I don’t really want to call it a “variance,” which it isn’t, but
want to keep the concept of variance.
ut^2
ut^2 st^2
ut^2 m
treatment effect
expected value