Statistical Methods for Psychology

Looking at our original data in Table 17.1, we note that in the low fault condition

153 people were found guilty and 24 were found not guilty. Thus, the conditional odds

of being judged guilty giventhat the victim was seen as low on Fault is 153/24 5

6.3750. (This can be read to mean that in the low fault group the odds in favor of being

found guilty are 6.3750:1.) For every person in that group who is found not guilty, 6.375

are found guilty. These are equivalent ways of saying the same thing. The conditional odds

of being found guilty given that the victim is seen as having a highdegree of fault are only

105/76 5 1.3816.

If there had been no interaction between Fault and Verdict, the odds of being found

guilty would have been the same in the two Fault conditions. Therefore, the ratio

of the two odds would have been approximately 1.00. Instead, the ratio of the two con-

ditional odds, the odds ratio ( ),is 6.3750/1.3816 5 4.6142. The odds that a defen-

dant will be found guilty in the low fault condition are about 4.6 times greater than

in the high fault condition. The “blame the victim” strategy, whether fair or not, seems

to work.

An important feature of the odds ratio is that it is independent of the size of the sam-

ple, whereas is not. A second advantage is that within the context of a 2 3 2 table, a

test on the odds ratio would be equivalent to a likelihood ratio test of independence.

A third advantage of is that its magnitude will not be artificially affected by the pres-

ence of unequal marginal distributions. In other words, if we doubled the number of

cases in the high fault condition (but still held other things constant), (either Pear-

son’s or likelihood ratio) and phi would change. The odds ratio ( ), however, would

not be affected.

17.5 Treatment Effects (Lambda)

As we have already seen, log-linear models have a nice parallel with the analysis of variance,

and that parallelism extends to the treatment effects. In the analysis of variance, treatment

effects are denoted by terms like m, , , and , whereas in log-linear models we denote

these effects by , , , and. As you know, log-linear models work with the natural logs

of frequencies rather than with the frequencies themselves.

In Section 17.3 we saw that the independence model (the model without the inter-

action term) did not fit the data from Pugh’s study ( 5 37.35). To model the data ad-

equately, we are going to have to use a model that contains the interaction term:

. Remember that for the fully saturated model the observed
and expected frequencies are the same. Thus we will start with the logs of these frequen-
cies as the raw data, as shown in Table 17.7. Notice that the table also contains the row and
column marginal means and the grand mean.

ln(Fij)=l1lVi 1lFj 1lVFij

x^2

llVi lFj lVFij

ai bj abij

Æ

x^2

Æ

x^2

x^2

Æ

642 Chapter 17 Log-Linear Analysis

Table 17.7 Natural logs of cell frequencies

Verdict

Guilty Not Guilty Marginals

High 5.03043 3.17805 4.10424

Fault Low 4.65396 4.33073 4.49235

Marginals 4.84220 3.75439 4.29830

conditional odds

odds ratio ( )Æ