Statistical Methods for Psychology

significant ) That means that unless you allow for an interaction of the variables, you

will not be able to fit the data adequately. Thus Verdict and Fault interact.

17.2 Model Specification

The models we have been discussing can be represented algebraically as well as

descriptively. The algebraic notation can seem awkward, but it allows us to learn a

great deal more about the data. It is somewhat confusing because we start out with one

set of parameters, represented as t(tau), usually with a superscript, and then shortly

convert to the natural logarithm of t, represented as l(lambda), with superscripts.

Both of these statistics strongly resemble the grand mean (m) and treatment effects

(a,b, and ab) that we saw in the analysis of variance. (You might think that we would

be satisfied with one or the other, but in fact both have their uses.) I would urge you

to read the next two sections fairly quickly just to see where we are heading, and then

come back to it after you see how such parameter estimates are used in more complex

models.

The following gets a bit confusing at first, but it’s not really that hard. Remember in

the analysis of variance that we had models like

All that I’m going to do is derive some terms that are parallel to these. First you’ll see t.

Think of it as m. Then you’ll see and. Think of these as and. I’ll make mention

of —you can guess what that is like. And finally, I’ll take logs of all this stuff. That’s

just so that I can add them up the same way we added m, , , and , to get an ex-

pected frequency.

In the simplest equiprobability model, all cell frequencies are explained by a single pa-

rameter t, where tis estimated by the geometric meanof the expected cell frequencies

given by the model. In other words,

(This model corresponds to the equiprobability model discussed in the previous section.)

A geometric mean is the nth root of the product of nterms, so in this case the geometric

mean of the four expected frequencies is

which is not a very exciting result.

For the first conditional equiprobable models we have to go further. We again define

as the geometric mean of the expected cell frequencies in that model, but here those

expected frequencies are different from the equiprobability model because they take differ-

ences due to Verdict into account.

We also define (where the superscript “V” stands for “Verdict”) as the ratio of the

geometric mean of the expected frequencies for the first (Guilty) column to the geometric

mean of the expected frequencies of all the cells (the grand mean) ( ). Then

NtV 1 =

1 (129)(129)

Nt =

129

80.3119

=1.6062

tN

tN 1 V

Nt= (^14) (129)(129)(50)(50)=80.3119
tN

14 (89.5)(89.5)(89.5)(89.5)= =89.5

Fij=tN

ai bj abij

tFVij

tFi tVj ai bj

Xijk=m1ai1bj1abij 1 eijk

x^2.

636 Chapter 17 Log-Linear Analysis

geometric mean