Loglinear Models for Independence and Interaction in Three-way Tables

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 μY = α + βx μY = 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 • μij = μαi βj • log μij = λ + λiX+ λjY • where λiXis row effect, and λjY is column effect • odds for column response independent of row • Saturated (Dependence) Model • terms logμij = λ + λiX+ λjY + λijXY • where λijXYare 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 {πijk} • Also apply to Poisson sampling with means {µijk}

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 • πijk = (πi++) (π+j+) (π++k) for all i, j, k • Loglinear Model for Expected Frequencies • log μijk = λ + λiX+ λjY + λkZ • 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: • πijk = (π+jk) (πi++) for all i, j, k • Loglinear Model for Expected Frequencies • log μijk = λ + λiX+ λjY + λkZ + λjkYZ • 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: • πijk = (πi+k) (π+jk) / π++kfor all i, j, k • Loglinear Model for Expected Frequencies • log μijk = λ + λiX+ λjY + λkZ + λikXZ + λjkYZ • 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: • πij+ = (πj++) (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 μijk = λ + λiX+ λjY + λkZ + λijXY + λikXZ + λjkYZ • Interpretation: • Homogenous association: • identical conditional odds ratios between any two variables over the levels of the third variable • θij(1) = θij(2) = … = θij(K) for all i and j

14. Saturated Model • Loglinear Model for Expected Frequencies • log μijk = λ + λiX+ λjY + λkZ + λijXY + λikXZ + λjkYZ+ λijkXYZ • 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 • χ2 or G2 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 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 Loglinear Model a A, alcohol use; C, cigarette use; M, marijuana use.

19. Estimated Odds Ratios for Loglinear Models

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.