Statistical Methods for Psychology

often adopt a strategy suggesting that the victim is in some way responsible for the crime. By

attacking the victim’s past behavior, the victim is put on trial instead of the defendant. Pugh’s

study varied the gender of the juror, the level of stigma attached to the victim, and the degree

to which the juror could assign fault to the victim, and then looked at the degree to which the

defendants were judged guilty or not guilty. For our first example we will collapse over two of

those variables and look at the relationship between the degree to which the victim was be-

lieved to be “at fault” and the verdict. These data are shown in Table 17.1. (Expected frequen-

cies for the standard test of the independence of these two variables are shown in parentheses.)

If we ran the standard Pearson chi-square test on these data, we would find, with a mi-

nor change in notation,

which is significant at a5.05. The change in notation, equating with the observed fre-

quency in and with the expected frequency in that cell, was instituted to bring the

notation in line with the standard notation used with log-linear analysis.

If we calculate the likelihood ratio (see Section 6.8) instead of the Pearson’s

chi-square, we would have

which is also approximated by the distribution on 1 degree of freedom.^1 Again we

would reject the null hypothesis of independence of rows and columns. We would conclude

that in making a judgment of guilt or innocence the jurors base that judgment, in part, on

the perceived fault of the victim.

The use of the chi-square test, whether using Pearson’s statistic or the likelihood ratio sta-

tistic, focuses directly on hypothesis testing. We are asking if there is a relationship between

assignment of fault and the juror’s decision. (From this point on, all statistics will be like-

lihood ratio s unless otherwise noted.^2 ) But we can look at these data from a different

perspective—the perspective of model building. We saw the modeling approach clearly in the

analysis of variance where we associated a two-way factorial design with the model

Xijk=m1ai1bj1abij 1 eijk

x^2

x^2

x^2

=37.3503

= 2 a153 ln

153

127.559

1 24 ln

24

49.441

1 105 ln

105

130.441

1 76 ln

76

50.559

b

x^2 = (^2) afij lna
fij
Fij
b
x^2
cellij Fij
fij
x^21 = a

(O 2 E)^2

E

= a

(fij 2 Fij)^2

Fij

=35.93

632 Chapter 17 Log-Linear Analysis

Table 17.1 Data from Pugh (1983) collapsed across two variables

Verdict

Guilty Not Guilty Total

Low 153 24 177

(127.559) (49.441)

Fault High 105 76 181

(130.441) (50.559)

Total 258 100 358

(^1) In this chapter we will frequently refer to natural logarithms. These are normally abbreviated as ln or as loge.
We will use ln throughout.
(^2) If you use SAS the likelihood ratio chi-square will be labeled as the “deviance.”