1. Loglinear Models for Independence and Interaction in Three-way Tables Veronica Estrada Robert Lagier

2. Quick Review from Agresti, 4.3 Poisson Loglinear Models are based on Poisson distribution of Y counts and employ log link function: log μ Y = α + βx μ Y = exp(α + βx)

3. Value of Loglinear Models? Used to model cell counts in contingency tables where at least 2 variables are response variables Specify how expected cell counts depend on levels of categorical variables Allow for analysis of association and interaction patterns among variables

4. Models for Two-way Tables Independence Model –μ ij = μα i β j –log μ ij = λ + λ i X + λ j Y –where λ i X is row effect, and λ j Y is column effect –odds for column response independent of row Saturated (Dependence) Model –terms logμ ij = λ + λ i X + λ j Y + λ ij XY –where λ ij XY are association that represent interactions between X and Y –odds for column response depends on row

5. Loglinear Models for Three-way (I x J x K) Tables Describe independence and association patterns Assume a multinomial distribution of cell counts with cell probabilities {π ijk } Also apply to Poisson sampling with means {µ ijk }

6. Types of Independence for Cell Probabilities in I x J x K Tables Mutual Independence Joint Independence Conditional Independence Marginal Independence

7. Mutual Independence π ijk = (π i++ ) (π +j+ ) (π ++k ) for all i, j, k Loglinear Model for Expected Frequencies –log μ ijk = λ + λ i X + λ j Y + λ k Z Interpretation: –X independent of Y independent of Z independent of X –No association between variables

8. Joint Independence X jointly independent of Y and Z: –π ijk = (π +jk ) (π i++ ) for all i, j, k Loglinear Model for Expected Frequencies –log μ ijk = λ + λ i X + λ j Y + λ k Z + λ jk YZ Interpretation: –X independent of Y and Z –Partial association between variables Y and Z 3 Joint Independence Models

9. Conditional Independence X and Y conditionally independent of Z: – π ijk = (π i+k ) (π +jk ) / π ++k for all i, j, k Loglinear Model for Expected Frequencies –log μ ijk = λ + λ i X + λ j Y + λ k Z + λ ik XZ + λ jk YZ Interpretation: –X and Y independent given Z –Partial association between X,Z and Y,Z 3 Conditional Independence Models

10. Marginal Independence X and Y marginally independent of Z: – π ij+ = (π j++ ) (π +j+ ) for all i, j, k Interpretation: –X and Y independent in the two-way table that has been collapsed over the levels of Z –Variables may have different strength of marginal association than conditional (partial) association - Simpson’s Paradox

11. Partial v. Marginal Tables

12. Relationships Among Types of XY Independence

13. Homogenous Association Model Loglinear Model for Expected Frequencies –log μ ijk = λ + λ i X + λ j Y + λ k Z + λ ij XY + λ ik XZ + λ jk YZ Interpretation: –Homogenous association: identical conditional odds ratios between any two variables over the levels of the third variable θ ij(1) = θ ij(2) = … = θ ij(K) for all i and j

14. Saturated Model Loglinear Model for Expected Frequencies –log μ ijk = λ + λ i X + λ j Y + λ k Z + λ ij XY + λ ik XZ + λ jk YZ + λ ijk XYZ Interpretation: –Each pair of variables may be conditionally dependent –Odds ratios for any pair of variables may vary over levels of the third variable –perfect fit to observed data

15. Inference for Loglinear Models Interpretation of Loglinear model parameters is at the level of the highest- order terms χ 2 or G 2 Goodness of Fit Tests can be used to select best fitting model Parameter estimates are log odds ratios for associations

16. Example: Alcohol, Cigarette, and Marijuana Data Alcohol UseCigarette Use Marijuana Use: Yes Marijuana Use: NO Yes No 911 44 538 456 NoYes No 3232 43 279 Source: Data courtesy of Harry Khamis, Wright State University

17. SAS Code data drugs; input a c m count; cards; 1 1 1 911 1 1 2 538 1 2 1 44 1 2 2 456 2 1 1 3 2 1 2 43 2 2 1 2 2 2 2 279 ; proc genmod; class a c m; model count = a c m / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m c*m / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m a*m / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m a*c / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m a*c a*m / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m a*c c*m / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m a*c a*m c*m / dist=poi link=log obstats; run; proc genmod; class a c m; model count = a c m a*c a*m c*m a*c*m/ dist=poi link=log obstats; run;

18. Fitted Values for Loglinear Models Alcohol Use Cigarette Use Marijuan a Use (A, C, M) (AC, M)(AM, CM) (AC, AM, CM) (ACM) Yes No 540.0 740.2 611.2 837.8 909.24 438.84 910.4 538.6 911 538 NoYes No 282.1 386.7 210.9 289.1 45.76 555.16 44.6 455.4 44 456 NoYes No 90.6 124.2 19.4 26.6 4.76 142.16 3.6 42.4 3 43 NoYes No 47.3 64.9 118.5 162.5 0.24 179.84 1.4 279.6 2 279 Loglinear Model A, alcohol use; C, cigarette use; M, marijuana use. a

19. Estimated Odds Ratios for Loglinear Models Model Conditional Association Marginal Association AC AM CM AC AM CM (A,C,M) 1.0 1.0 1.0 1.0 1.0 1.0 (AC,M) 17.7 1.0 1.0 17.7 1.0 1.0 (AM,CM) 1.0 61.9 25.1 2.7 61.9 25.1 (AC,AM,CM) 7.8 19.8 17.3 17.7 61.9 25.1 (ACM) 13.8 24.3 17.5 17.7 61.9 25.1

20. Computation of the Odds Ratio

21. Model (AC, AM, CM) permits all pairwise associations but maintains homogeneous odds rations between two variables at each level of the third. The previous table shows that estimated odds ratios are very dependent on the model, and from this we can only say that the model fits well.

22. Conditional independence has implications regarding marginal (in) dependence; however, marginal (in) dependence does not have implications regarding conditional (in) dependence. Conditional independence->marginal independence Conditional independence->marginal dependence Marginal independence does not ->conditional independence Marginal dependence does not ->conditional dependence.