Statistical Methods for Psychology

As we will see later, under certain conditions the assumption of homogeneity of

variance can be relaxed without substantially damaging the test, though it might alter the

meaning of the result. However, there are cases where heterogeneity of variance,or

“heteroscedasticity” (populations having differentvariances), is a problem.

Normality

A second assumption of the analysis of variance is that the recall scores for each condition

are normally distributed around their mean. In other words, each of the distributions in

Figure 11.1 is normal. Since represents the variability of each person’s score around the

mean of that condition, our assumption really boils down to saying that error is normally

distributed within conditions. Thus you will often see the assumption stated in terms of

“the normal distribution of error.” Moderate departures from normality are not usually

fatal. We said much the same thing when looking at the t test for two independent samples,

which is really just a special case of the analysis of variance.

Independence of Observations

Our third important assumption is that the observations are independent of one another.

(Technically, this assumption really states that the error components [ ] are independent, but

that amounts to the same thing here.) Thus for any two observations within an experimental

treatment, we assume that knowing how one of these observations stands relative to the treat-

ment (or population) mean tells us nothing about the other observation. This is one of the im-

portant reasons why subjects are randomly assigned to groups. Violation of the independence

assumption can have serious consequences for an analysis (see Kenny & Judd, 1986).

The Null Hypothesis

As we know, Eysenck was interested in testing the researchhypothesis that the level of

recall varies with the level of processing. Support for such a hypothesis would come from

rejection of the standard nullhypothesis

The null hypothesis could be false in a number of ways (e.g., all means could be differ-

ent from each other, the first two could be equal to each other but different from the last

three, and so on), but for now we are going to be concerned only with whether the null hy-

pothesis is completely true or is false. In Chapter 12 we will deal with the problem of

whether subsets of means are equal or unequal.

11.3 The Logic of the Analysis of Variance

The logic underlying the analysis of variance is really very simple, and once you under-

stand it the rest of the discussion will make considerably more sense. Consider for a mo-

ment the effect of our three major assumptions—normality, homogeneity of variance, and

the independence of observations. By making the first two of these assumptions we have

said that the five distributions represented in Figure 11.1 have the same shape and disper-

sion. As a result, the only way left for them to differ is in terms of their means. (Recall that

the normal distribution depends only on two parameters, mand s.)

We will begin by making no assumption concerning —it may be true or false. For

any one treatment, the variance of the 10 scores in that group would be an estimate of the

H 0

H 0 : m 1 =m 2 =m 3 =m 4 =m 5

eij

eij

Section 11.3 The Logic of the Analysis of Variance 321

heterogeneity of
variance

heteroscedas-
ticity