Statistical Methods for Psychology

serious alternative to the saturated model. But the approach will be very useful when we

come to more complex designs.

From Table 17.4 (on p. 635) we see that the first four models all have significant val-

ues. This means that for each of these models there is a significant difference between ob-

served and expected values; none of them fits the obtained data. From such results we must

conclude that only a model that incorporates the interaction term can account for the re-

sults. Thus, as we have previously concluded, Fault and Verdict interact and, within the

context of Pugh’s experiment, we cannot model the data without taking this interaction into

account. Because, for Pugh, Verdict is a dependent variable, we conclude that decisions

about guilt or innocence are dependent on perceptions about Fault. (This is one of the few

places in inferential statistics where we actually seek nonsignificant results.)

From the point of view of fitting models, these results suggest that we should conclude

that

But when viewed from the perspective of the analysis of variance, something is missing in

such a conclusion. In the analysis of variance we start out with (and generally retain) a

model such as this, but we also test the individual elements of the model. In other words

we asked, “Within the complete model are there significant effects due to V, to F, and to

their interaction?” That is a question we haven’t really asked here. When we tested, for ex-

ample, the model , we were asking whether such a model fit the data, but

we were not asking the equally important question, When we adjust for other effectsis

there a difference attributable to Fault?

There are two ways of asking these questions using log-linear models—the easy way

and the harder way, paralleling what we did in the equal-ncase of the analysis of variance

in Chapter 16. The advantage of the more complicated way is that it generalizes to the

process we will use on interactions in more complex designs.

Let’s start with the easy way because it supplies a frame of reference. If you want to

know whether there is a difference in the data attributable to Verdict (i.e., are there sig-

nificantly more decisions of Guilty than Not Guilty), why not just ask that question

directly by looking at the marginal totals? In other words, just run a one-dimensional

likelihood ratio , as shown in Table 17.5. The 5 72.1929 is a significant result on 1df,

and we would conclude that there is a difference in the number of cases judged guilty

and not guilty.

Now let’s ask the same question about low and high levels of Fault (see Table 17.6).

This effect (0.0447) is clearly not significant—nor would Pugh have expected it to be given

x^2 x^2

ln(Fij)=l1lFj

ln(Fij)=l1lVi 1lFj 1lVFij

x^2

Section 17.3 Testing Models 639

Table 17.5 Test on differences due to Verdict

Verdict

Guilty Not Guilty

fij 258 100
Fij 179 179

=72.1929

= 2 a258 ln

258

179

1 100 ln

100

179

b

x^2 = (^2) afij ln a
fij
Fij
b