Statistical Methods for Psychology

when you go to other sources to look up nested (or random) designs you will often find the

two discussed together. A final point to keep in mind is that in all of the between-subjects

designs in this book subjects are nested within other factors and are considered to be a ran-

dom factor. All of our Fstatistics are computed taking that into account.

13.9 Measures of Association and Effect Size

We can look at the magnitude of an effect in two different ways, just as we did with the

one-way analysis. We can either calculate an r-family measure, such as , or we can cal-

culate a d-family measure such as d. Normally when we are examining an omnibus F, we

use an r-family measure. However, when we are looking at a contrast between means it is

usually more meaningful to calculate an effect size estimate (d). We have seen both types

of measures in previous chapters.

r-Family Measures

As with the one-way design, it is possible to calculate the magnitude of effect associated

with each independent variable. The easiest, but also the most biased, way to do this is to

calculate. Here we would simply take the relevant sum of squares and divide by. Thus,

the magnitude of effect for variable Ais 5 and for variable Bis

5 , whereas the magnitude of effect for the interaction is 5.

There are two difficulties with the measure that we have just computed. In the first

place is a biased estimate of the true magnitude of effect in the population. To put this

somewhat differently, is a very good descriptive statistic, but a poor inferential statistic.

Second, , as we calculated it here, may not measure what we want to measure. We will

speak about that shortly when we discuss partial.

Although is also biased, the bias is much less than for. In addition, the statistical

theory underlying allows us to differentiate between fixed, random, and mixed models

and to act accordingly.

To develop for two-way and higher-order designs, we begin with the set of expected

mean squares given in Table 13.8, derive estimates of , and then form

ratios of each of these components relative to the total variance. Rather than derive the for-

mulae for calculating for the three different models, as I have done in previous editions

of this book, I will present the results in a simple table. I strongly suspect that no student

remembered the derivation five minutes after he or she read it, and that many students were

so numb by the end of the derivation that they missed the final result.

For a factorial analysis of variance the basic formula to estimate remains the same

whether we are looking at fixed or random variables. The only difference is in how we cal-

culate the components of that formula. We will start by letting refer to the estimate

of the variance of the independent variable we care about at the moment, such as A, B, or

AB, and by letting refer to the sum of all sources of variance. (If an effect is fixed,

replace by ) Then if we know the value of these terms we can estimate as

For the main effect of A, for example, this becomes

All we have to know is how to calculate the variance components (s^2 effect).

v^2 a=

sN^2 a

sN^2 total

=

sN^2 a

sN^2 a1sN^2 b1sN^2 ab1sN^2 e

vN^2 effect=

sN^2 effect

sN^2 total

s^2 u^2. v^2 effect

sN^2 total

sN^2 effect

v^2

v^2

s^2 a, s^2 b, s^2 ab, and s^2 e

v^2

v^2

v^2 h^2

h^2

h^2

h^2

h^2

hb^2 SSB>SStotal hab^2 SSAB>SStotal

ha^2 SSA>SStotal

h^2 SStotal

h^2

438 Chapter 13 Factorial Analysis of Variance