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 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

4. 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

5. Interpretation of Parameters The effect of factor on log(odds) is: –Without XY term: –With XY term:

6. 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

7. Example: Z = Clinic Response ClinicTreatmentSuccessFailure 1A1812 B 8 2A28 B832 TotalA20 B 40

8. Example: Z = Sex Response SexTreatmentDiedLived MenStandard95050 New90001000 WomenStandard5000 New595 TotalStandard59505050 New90051095 Partial table Marginal table

9. Type of Independence of X, Y Mutually independent with Z Jointly independent with Z Marginally independent Conditionally independent with Z

10. Eg. 2x2xK tables Associations 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

11. 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

12. 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

13. Loglinear Models for 3-way Tables Saturated (also full) model:

14. 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

15. Interpreting Models Loglinear Model SymbolInterpretation (X, Y, Z) (Y, XZ) (XY, XZ) (YZ, XZ) (XY, YZ, XZ) (XYZ)

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

17. The 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)

18. 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.