1. Log-Linear Models & Dependent Samples Feng Ye, Xiao Guo, Jing Wang

2. Outline  Symmetry, Quasi-independence & Quasi- symmetry.  Marginal Homogeneity & Quasi-symmetry.  Ordinal Quasi-symmetry Model.  Conclusion.

3. Quasi-independence Model  Model structure  Assumption: Independent model holds for off diagonal cells.  Model fit df = (I-1)^2-I

4. Symmetry  Model structure  Assumption: The off diagonal cells have equal expected counts.  Model fit: df = I^2-[I+I(I-1)/2]

5. Symmetry 123Total 1205817 25201020 38102023 Total172023

6. Quasi-symmetry  Model structure  Model fit df = (I-2)(I-1)/2 Symmetry: Independence:

7. Application  Each week Variety magazine summarizes reviews of new movies by critics in several cities. Each review is categorized as pro, con, or mixed, according to whether the overall evaluation is positive, negative, or a mixture of the two. April 1995 through September 1996 for Chicago film critics Gene Siskel and Roger Ebert.

8. Application  Reviews of new movies by critics. Ebert SiskelConMixedPro Con24813 Mixed81311 Pro10964

9. Output & Interpretation Model df G2 p-value Quasi-Independence10.00610.938 Symmetry30.59280.9 Quasi-symmetry10.00610.938

10. Quasi-independence  Con-0.9603  Mixed-0.6239  pro-1.5069 e.g exp(0.9603+0.6239)=4.41

11.  Symmetry = Quasi-symmetry + Marginal homogeneity  Quasi-symmetry + Marginal homogeneity = Symmetry  Fit statistics for marginal homogeneity Marginal Homogeneity & Quasi-symmetry

12.  Symmetry Model  Criteria For Assessing Goodness Of Fit  Criterion DF Value Value/DF  Deviance 3 0.5928 0.1976  Scaled Deviance 3 0.5928 0.1976  Pearson Chi-Square 3 0.5913 0.1971  Scaled Pearson X2 3 0.5913 0.1971  Log Likelihood 351.2829   Quasi-Symmetry Model  Criteria For Assessing Goodness Of Fit  Criterion DF Value Value/DF  Deviance 1 0.0061 0.0061  Scaled Deviance 1 0.0061 0.0061  Pearson Chi-Square 1 0.0061 0.0061  Scaled Pearson X2 1 0.0061 0.0061  Log Likelihood 351.5763  G 2 (S/QS) = 0.5928 - 0.0061 = 0.5867 with df = 2, showing marginal homogeneity is plausible.

13. Marginal Homogeneity Testing

14. Marginal homogeneity is the special case β j =0. Specifying design matrix to produce expected frequency {µ ab }. Using G 2 and X 2 tests marginal homogeneity, with df=I-1

15.  d a = p +a – p a+ ; d’ =( d 1,….d I-1 )  Covariance matrix V with elements:  V ab = -(p ab + p ba ) – (p +a – p a+ )(p +b – p b+ )  V aa = p +a + p a+ -2p aa – (p +a – p a+ ) 2  Under marginal homogeneity, E(d) = 0.  W is asymptotically chi-squared with df = I-1.

16. Marginal Models  Criteria For Assessing Goodness Of Fit  Criterion DF Value Value/DF  Deviance 2 0.5868 0.2934  Scaled Deviance 2 0.5868 0.2934  Pearson Chi-Square 2 0.5855 0.2927  Scaled Pearson X2 2 0.5855 0.2927  Log Likelihood 351.2859   Analysis of Variance  Source DF Chi-Square Pr > ChiSq  --------------------------------------------  Intercept 2 191.15 <.0001  review 2 0.59 0.7455  Residual 0..

17. Ordinal Quasi-symmetry Model  Quasi-independence, quasi-symmetry, symmetry models for square tables treat classifications as nominal.  Changing the constraints for log linear model to obtain reduced model for ordered response.  Model structure has a linear trend. Where is the ordered scores

18. Ordinal Quasi-symmetry Model  Parameter estimation & interpretation 1. Fitted marginal counts ==(?) observed marginal counts 2. Dividing the first two equations by n indicates the same means.  Goodness of fit: Checking the distance between reduced model and saturated model.

19. Logit Representation  Logit model  Interpretation 1. difference between marginal distribution 2. marginal homogeneity 3. Identify as binomial with trials, and fit a logit model with no intercept and predictor

20. Marginal Homogeneity  Marginal model (cumulative logits) marginal homogeneity :  Ordinal quasi-sym model 1. At the condition of ordinal quasi-symmetry marginal homogeneity is equivalent to symmetry 2. Fit statistic

21. Application Data 1  Reviews of new movies by critics. Ebert SiskelConMixedPro Con24813 Mixed81311 Pro10964

22. Output & Interpretation  Output  Marginal homogeneity? 1. No meaning to check if when ordinal quasi-symmetry fits poorly. 2. Using marginal model is a good way. 3. Check the symmetry under the condition of ordinal quasi-symmetry.

23. Conclusion  Summary statistics provide an overall picture of square tables. Kappa & Percentage  Log-linear provides a valuable addition even an alternative to summary statistic. 1. Quasi-symmetry is the most general model for square table. 2. Adding or deleting variables from log-linear models provides different useful models. 3. Quasi-symmetry models proposes a good instrument for marginal homogeneity.