Introduction to Categorical Data Analysis

206 LOGLINEAR MODELS FOR CONTINGENCY TABLES
of response in column 1 equal exp(α) = exp(λY 1 − λY 2 ). In model (7.1), differences
between two parameters for a given variable relate to the log odds of making one
response, relative to another, on that variable.
For the independence model, one of {λX i } is redundant, and one of {λY j } is redundant.
This is analogous toANOVA and multiple regression models with factors, which
require one fewer indicator variable than the number of factor levels. Most software
sets the parameter for the last category equal to 0. Another approach lets the parameters
for each factor sum to 0. The choice of constraints is arbitrary. What is unique
is the difference between two main effect parameters of a particular type. As just
noted, that is what determines odds and odds ratios.
For example, in the 2000 General Social Survey, subjects were asked whether
they believed in life after death. The number who answered “yes” was 1339 of the
1639 whites, 260 of the 315 blacks and 88 of the 110 classified as “other” on race.
Table 7.1 shows results of fitting the independence loglinear model to the 3 × 2 table.
The model fits well. For the constraints used, λY 1 = 1.50 and λY 2 = 0. Therefore, the
estimated odds of belief in life after death was exp(1.50) = 4.5 for each race.
Table 7.1. Results of Fitting Independence Loglinear Model to
Cross-Classification of Race by Belief in Life after Death
Criteria For Assessing Goodness Of Fit
Criterion DF Value
Deviance 2 0.3565
Pearson Chi-Square 2 0.3601
Parameter DF Estimate
Standard
Error
Intercept 1 3.0003 0.1061
race white 1 2.7014 0.0985
race black 1 1.0521 0.1107
race other 0 0.0000 0.0000
belief yes 1 1.4985 0.0570
belief no 0 0.0000 0.0000
7.1.3 Saturated Model for Two-Way Tables
Variables that are statistically dependent rather than independent satisfy the more
complex loglinear model,
log μij = λ + λ X i + λY j
+ λXY
ij
(7.2)
The {λXY ij } parameters are association terms that reflect deviations from independence.
The parameters represent interactions between X and Y , whereby the effect of one
variable on the expected cell count depends on the level of the other variable. The
independence model (7.1) is the special case in which all λXY ij = 0.