Statistical Methods for Psychology

Now recall that when we calculated treatment effects in the analysis of variance we

took deviations of means around the grand mean. We will do something similar here.

Thus,

ANOVA Effects Log-linear Model Effects

Note the parallelism. Further,. Thus, we can

calculate all the rest of the effects directly.^5

17.6 Three-Way Tables

We now have all of the concepts that are necessary to move to more complex designs.

Log-linear models come into their own once we move to contingency tables of more than

two dimensions. These are the situations in which standard chi-square analyses are not able

to reveal a full understanding of the data. In this section we will concentrate on three-way

tables because they illustrate all of the essential points. Extrapolation to tables of higher

dimensionality is direct. Good examples of the analysis and interpretation of four- and

five-way tables can be found in Pugh (1983) and Tabachnick and Fidell (2007), respec-

tively. (If you try to duplicate the results in Pugh’s paper, instruct your program to add 0.5

to the cell frequencies in the four-way table before running any analyses. This is normally

done anyway on a temporary basis when the program deals with the highest interaction to

avoid problems of cell frequencies of zero; ln(0) is undefined. Pugh instructed BMDP4F to

leave the 0.5 in while computing all tables.)

When we move beyond a simple R 3 Ctable, the calculations of expected frequen-

cies, especially for interactions involving subsets of variables, become appreciably

more complex. Such calculations are usually carried out by an iterative process in which

initial estimates are continually refined until the result meets some specified criterion.

Most analyses at this level are solved by computers, and that is the approach adopted

here. This chapter will focus on analyses computed by SPSS GENLOG, though SPSS

has two other procedures for log-linear analysis—SPSS LOGLINEAR and SPSS

HILOGLINEAR. SYSTAT, or PROC CATMOD and GENMOD in SAS are also possi-

bilities. Results of different programs sometimes vary because they use different algo-

rithms for their solutions.

lFV 22 = 0.38229

lFV 21 =-0.38229

lFV 12 =-0.38229

lV 2 =-0.54390

lF 2 = 0.19406

al

F

i = al

V

j =ail

FV

ij = a

j

lFVij =0.00

=0.38229

aNNbij=Xij 2 Xi. 2 X.j 1 X.. lFV 11 =5.03043 2 4.10424 2 4.84220 1 4.29830

bNj=X.j 2 X.. lV 1 =4.84220 2 4.29830=0.54390

aN =Xi. 2 X.. lF 1 =4.10424 2 4.29830= 2 0.19406

mN =X.. l=4.29830

Section 17.6 Three-Way Tables 643

(^5) Because of the way SPSS codes categorical variables you will not obtain these values when you ask it to
estimate parameters, because those estimates will be taken with respect to the cell in the lower right. However
other software (e.g., R) uses a design matrix that will produce these values. The end result is the same, however.