Statistical Methods for Psychology

Now let’s move from our physical model of height to one that more directly underlies

our example. We will look at this model in terms of Eysenck’s experiment on the recall of

verbal material. Here represents the score of in (e.g., represents

the third person in the Rhyming condition). We let mrepresent the mean of all subjects who

could theoretically be run in Eysenck’s experiment, regardless of condition. The symbol

represents the population mean of (e.g., is the mean of the Rhyming condi-

tion), and is the degree to which the mean of deviates from the grand mean

( ). Finally, is the amount by which in deviates from the

mean of his or her group ( ). Imagine that you were a subject in the memory

study by Eysenck that was just described. We can specify your score on that retention test

as a function of these components.

This is the structural modelthat underlies the analysis of variance. In future chapters

we will extend the model to more complex situations, but the basic idea will remain the

same. Of course we do not know the values of the various parameters in this structural

model, but that doesn’t stop us from positing such a model.

Assumptions

As we know, Eysenck was interested in studying the level of recall under the five condi-

tions. We can represent these conditions in Figure 11.1, where and represent the

mean and variance of whole populations of scores that would be obtained under each of

these conditions. The analysis of variance is based on certain assumptions about these pop-

ulations and their parameters. In this figure the fact that one distribution is to the right of

another does not say anything about whether or not its mean is different from others.

Homogeneity of Variance

A basic assumption underlying the analysis of variance is that each of our populations has

the same variance. In other words,

where the notation is used to indicate the common value held by the five population

variances. This assumption is called the assumption of homogeneity of variance, or, if you

like long words, homoscedasticity.

The subscript “e” stands for error, and this variance is the error variance—the vari-

ance unrelated to any treatment differences, which is variability of scores within the same

condition. Homogeneity of variance would be expected to occur if the effect of a treatment

is to add a constant to everyone’s score—if, for example, everyone who thought of adjec-

tives in Eysenck’s study recalled five more words than they would otherwise have recalled.

s^2 e

s^21 =s^22 =s^23 =s^24 =s^25 =s^2 e

mj s^2 j

=m1tj1 ́ij

Xij=m1(mj2m)1 ́ij

́ij=Xij2mj

tj=mj2m ́ij Personi Conditionj

tj Conditionj

Conditionj m 2

mj

Xij Personi Conditionj X 32

320 Chapter 11 Simple Analysis of Variance

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Figure 11.1 Graphical representation of populations of recall scores

structural model

homogeneity of
variance
homoscedasticity
error variance