1. Joyful mood is a meritorious deed that cheers up people around you
like the showering of cool spring breeze.

2. Categorical Data Analysis
Chapter 8: Loglinear Models
for Contingency Tables
(SAS: Chapter 12)

3. Loglinear vs. Logit Models
Loglinear models treat categorical variables equally, focusing on associations and interactions in their joint distribution.
Logit models, by contrast, describe how a single categorical variable (response) depends on other (explanatory) variables.

4. Distributions/Models for Categorical Data
Binomial data (logit models)
When the sample size n is fixed and the response per subject is binary
Use logit models; sas link is logit
Multinomial data (general logit models)
When the sample size n is fixed and the response per subject is multinary
Use base-category logit models (sas link: glogit) for nominal response; use culmulative logit models/proportional odds models (sas link: clogit)
Poisson data (loglinear models)
When the sample size n is not fixed and the response per subject is binary/multinary
A Poisson model conditioned on a given n is a Binomial/ Multinomial model (Sec )
PROC LOGISTIC
PROC LOGISTIC
PROC GENMOD

5. Example: Z = Sex Response Sex Treatment Died Lived Men Standard 950 50
Partial table
Response
Sex
Treatment
Died
Lived
Men
Standard
950 50
New
9000
1000
Women
5000 5 95
Total
5950
5050
9005
1095
Marginal table

6. What Models to Use
If the row totals are pre-fixed (prospective study):
Can only study the column (response) distribution for a given row (factors)
Bi- or multi-nomial data (logit models)
If the grand total is pre-fixed (prospective study):
Can study the joint distribution of response and factors (all variables)
Can be treated as bi- or multi-nomial data (logit models)
If nothing is pre-fixed/ totally observational (respective study):
Poisson data/ loglinear models

7. Loglinear Models for Counts
Poisson counts: count ~ Poisson(u)
Qualitative factors: X, Y, …
Saturated Model:
As usual, the baseline (last level) effects are set as 0 for each term

8. Independence Model
No interaction effect between X and Y on counts; that is, X and Y are independent
As usual, the baseline effects are set as 0 for each term

9. Interpretation of Parameters
The effect of factor on log(odds) is:
For Ix2 table, level i vs. baseline level I
Without XY term:
With XY term:

10. Associations in 3-way Tables
Let Y be the response, X be the major factor and Z be nuisance factor
The observed marginal association of X on Y might be simply due to the other factor Z
In general we cannot collapse a 3-way table and interpret the 2-way marginal table

11. Example: Z = Sex Response Sex Treatment Died Lived Men Standard 950 50
Partial table
Response
Sex
Treatment
Died
Lived
Men
Standard
950 50
New
9000
1000
Women
5000 5 95
Total
5950
5050
9005
1095
Marginal table

12. Type of Independence of X, Y
Conditionally independent given Z
Mutually independent with Z
Jointly independent of Z
Marginally independent
weak
strong

13. Associations in 3-way Tables
Eg. 2x2xK tables
Conditional odds ratio
Marginal odds ratio
Marginal independence of X, Y:
marginal X-Y odds ratios are all 1
Conditional independence of X, Y given Z:
conditional X-Y odds ratios given Z are all 1;
Homogeneous association of X, Y given Z:
conditional X-Y odds ratios given Z are the same (no need to be 1)

14. Partial Association (Sec 2.3)
The associations in partial tables are called “partial” associations between X and Y given Z
They are measured by conditional odds ratios

15. Associations in 3-way Tables
We need to condition on all important variables; but it is not practical.
In randomized experiments this (confounding) problem is less likely to happen.
To study whether an association exists between a primary factor and the response variable AFTER controlling for other possibly confounding variables, such as
Different medical centers
Severity of Condition
Age

16. Loglinear Models for 3-way Tables
Saturated (also full) model:
Deviance is the Likelihood-Ratio test statistic of
Ho: current model vs. H1: saturated model; can be a measure of goodness of fit

17. Interpreting Model parameters
X: effect of X on (expected) counts
XY: the partial association between X and Y given Z
XYZ: significant  XY depends on Z
insignificant  XY does not depend on Z

18. Interpreting Models Loglinear Model Symbol Interpretation (X, Y, Z)
(Y, XZ)
(XY, XZ)
(YZ, XZ)
(XY, YZ, XZ)
(XYZ)

19. Inference for Loglinear Models
Goodness-of-fit tests
Residuals
Tests for partial associations
Confidence intervals for odds ratios

20. The Loglinear-Logit Connection
Loglinear symbol
Logit
symbol
(Y,XZ)
(--)
(XY,XZ)
(X)
(XZ,YZ)
(Z)
(XY,YZ,XZ)
(X+Z)
(XYZ)
(X*Z)
Using logit models to interpret loglinear models
Correspondence between loglinear and logit models

21. Connection with Logit Models
The loglinear model which corresponds to a logit model is the one with the most general interaction among explanatory variables from the logit model. It has the same association and interaction structure relating the explanatory variables to the response.