1. Chapter 2 Describing Contingency Tables Reported by Liu Qi

2. Review of Chapter 1 Categorical variable Response-Explanatory variable Nominal-Ordinal-Interval variable Continuous-Discrete variable Quantitative-Qualitative variable

3. Review(cont.) Use binomial, multinomial and Poisson distribution Not normality distribution Tow most used models: logistic regression(logit) log linear

4. Binomial distribution

5. Multinomial distribution

6. Poisson distribution

7. Poisson Multinomial

8. Something unfamiliar Maximum likelihood estimation Confidence intervals Statistical inference for binomial parameters multinomial parameters ……

9. Terminology and notation Cell Contingency table

10. Terminology and notation Subjective Sensitivity and Specificity Conditional distribution Joint distribution Marginal distribution Independence =>

11. Sampling Scheme Poisson the joint probability mass function: Multinomial independent/product multinomial Hyper geometric

12. Example for sampling

13. Types of studies Retrospective: case-control Prospective: – Clinical trial observational study – Cohort study Cross-sectional: experimental study

14. Comparing two proportions Difference Relative risk Odds ratio – Odds defined as – For a 2*2 table, odds ratio – Another name: cross-product ratio

15. Properties of the Odds Ratio 0=<θ <, θ=1 means independence of X and Y the farther from 1.0, the stronger the association between X and Y. log θ is convenient and symmetric Suitable for all direction No change when any row/column multiplied by a constant.

16. Aspirin and Heart Attacks Revisited 189/11034=0.0171 104/11037=0.0094 Relative risk: 0.0171/0.0094=1.82 Odds ratio: (189*10933)/(10845*1 04)=1.83

17. Case-Control Studies and the Odds Ratio

18. However(cont.)

19. Partial association in stratified 2*2 tables Experimental studies We hold other covariates constant to study the effect of X on Y. Observational studies Control for a possibly confounding variable Z Partial tables=>conditional association Marginal table

20. Death penalty example

21. Death penalty example(cont.)

22. Simpsons paradox

23. Conditional and marginal odds ratios Conditional Marginal

24. Conditional independence Conditional independence: Joint probability:

25. Marginal independence

26. Marginal versus Conditional

27. Marginal versus Conditional(cont.) Marginal conditional

28. Homogeneous Association For a 2*2*K table, homogeneous XY association defined as: A symmetric property: – Applies to any pair of variables viewed across the categories of the third. – No interaction between two variables in their effects on the other variable.

29. Homogeneous Association(cont.) Suppose: – X=smoking(yes, no) – Y=lung cancer(yes, no) – Z=age( 65) – And Age is an Effect Modifier

30. Extensions for i*j Tables For a 2*2 table Odds ratio An i*j table Odds ratios

31. Representation methods Method 1

32. Method 2

33. For I*J tables (I-1)*(J-1) odds ratios describe any association All 1.0s means INDEPENDENCE! Three-way I*J*K tables, Homogeneous XY association means: any conditional odds ratio formed using two categories of X and Y each is the same at each category of Z.

34. Measures of Association Two kinds of variables: – Nominal variables – Ordinal variables Nominal variables: Set a measure for X and Y: – V(Y),V(Y|X) Proportional reduction:

35. Measures of variation Entropy: Goodman and Kruskal(1954) (tau) Lambda:

36. About Entropy Uncertainty coefficient: U=0=>INDEPENDENCE U=1=>π(j|i)=1 for each i, some j. Drawbacks: No intuition for such a proportional reduction.

37. Ordinal Trends An example:

38. Three kinds of relationship Concordant Discordant Tied

39. Example(cont.) D=849 Define (C-D)/(C+D) as Gamma measure. Here, A weak tendency for job satisfaction to increase as income increases.

40. Generalized

41. Properties of Gamma Measure Symmetric Range [-1,1] Absolute value of 1 means perfect linear Monotonicity is required for Independence =>,not vice-versa.