Statistical Methods for Psychology

Mutual Dependence Model

We are now testing a model that assumes that two factors operate jointly, but independ-

ently, to produce expected cell frequencies. If the two variables are independent, then

where RTstands for the row total, CTfor the column total, GTfor the grand total, and the

“dot notation” is used to show that we have collapsed across that dimension. This is the

same formula for expected frequencies that we saw in Chapter 6.

We began this chapter by testing this hypothesis of independence. The expected frequen-

cies and the likelihood ratio are given on page 632. From those calculations we found that

which is significant on 1 df. Thus, we can further conclude, and importantly so in this case,

that a model that posits an independence between Fault and Verdict also does not fit the

data. The only conclusion remaining is that the likelihood of a jury convicting a defendant

of rape depends on an interaction between Fault and Verdict. Perceived guilt is, in part, a

function of the blame that is attributed to the victim.

If we use the model that includes both Verdict and Fault, as well as their interaction,

we will have what is called a saturated model. This is a model that has as many parame-

ters (an intercept, a row effect, a column effect, and an interaction effect) as it has cells,

and it is guaranteed to fit perfectly with.

The models that we have examined are listed below in Table 17.4—for the moment you

can ignore Column two. You can see that all but the last one fail to fit the data (i.e., have a

x^2 = 0

x^2 =37.3503

x^2

Fij=

RT 3 CT

GT

=

fi. 3 f.j

f..

Section 17.1 Two-Way Contingency Tables 635

Table 17.3 Observed and expected frequencies for the second conditional

equiprobability model

Verdict

Guilty Not Guilty Total

Low 153 24 177

(88.5) (88.5)

Fault High 105 76 181

(90.5) (90.5)

Total 258 100 358

=109.544

= 2 a153 ln

153

88.5

1 24 ln

24

88.5

1 105 ln

105

90.5

1 76 ln

76

90.5

b

x^2 = (^2) afija
fij
Fij
b
saturated model
Table 17.4 Five possible models for data in Table 17.1
Model Representation df p

1. Equiprobability ln(Fij) 5 109.5889 3 ,.05


2. Conditional Equiprobability ln(Fij) 5 37.3960 2 ,.05


3. Conditional Equiprobability ln(Fij) 5 109.5442 2 ,.05


4. Independence ln(Fij) 5 37.3503 1 ,.05


5. Saturated ln(Fij) 5 l 1 liV 1 lFj 1 lijVF 0.00 0 —

l 1 liV 1 lFj

l 1 ljF

l 1 liV

l

x^2