Joyful mood is a meritorious deed that cheers up people around you
like the showering of cool spring breeze.
Categorical Categorical Data AnalysisData Analysis
Chapter 8: Loglinear Models Chapter 8: Loglinear Models
for Contingency Tablesfor Contingency Tables
(SAS: Chapter 12)(SAS: Chapter 12)
Loglinear Models for CountsLoglinear Models for Counts• Poisson counts: count ~ Poisson(u)
• Qualitative factors: X, Y, …
• Saturated Model:
As usual, the baseline effects are set as 0 for each term
ijjiij XYYX log
Independence ModelIndependence Model
jiij YX 0log
• 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
Interpretation of ParametersInterpretation of Parameters• The effect of factor on log(odds) is:
– Without XY term:
– With XY term:
Associations in 3-way TablesAssociations 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
Example: Z = Clinic Example: Z = Clinic Response
Clinic Treatment Success Failure
1 A 18 12
B 12 8
2 A 2 8
B 8 32
Total A 20 20
B 20 40
Example: Z = SexExample: Z = SexResponse
Sex Treatment Died Lived
Men Standard 950 50
New 9000 1000
Women Standard 5000 5000
New 5 95
Total Standard 5950 5050
New 9005 1095
Partial table
Marginal table
Type of Independence of X, YType of Independence of X, Y
Mutually independent with Z
Jointly independent with Z
Marginally independent
Conditionally independent with Z
kk
kkkZXY
2112
2211)(
2112
2211
XY
Eg. 2x2xK tables
Associations in 3-way TablesAssociations in 3-way 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
Partial Association (Sec 2.3)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
Associations in 3-way TablesAssociations 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
Loglinear Models for 3-way TablesLoglinear Models for 3-way Tables
• Saturated (also full) model:
ijkjkikij
kjiijk
XYZYZXZXY
ZYX
log
Interpreting Model parametersInterpreting 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
Interpreting ModelsInterpreting Models
Loglinear Model
Symbol Interpretation
(X, Y, Z)
(Y, XZ)
(XY, XZ)
(YZ, XZ)
(XY, YZ, XZ)
(XYZ)
Inference for Loglinear ModelsInference for Loglinear Models
• Goodness-of-fit tests
• Residuals
• Tests for partial associations
• Confidence intervals for odds ratios
The Loglinear-Logit ConnectionThe Loglinear-Logit Connection
• Using logit models to interpret loglinear models
• Correspondence between loglinear and logit models
Loglinear symbol
Logit symbol
(Y,XZ) (--)
(XY,XZ) (X)
(XZ,YZ) (Z)
(XY,YZ,XZ) (X+Z)
(XYZ) (X*Z)
Connection with Logit ModelsConnection 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.