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Chapter 2 Describing Contingency Tables

Chapter 2 Describing Contingency Tables. Reported by Liu Qi. Review of Chapter 1. Categorical variable Response-Explanatory variable Nominal-Ordinal-Interval variable Continuous-Discrete variable Quantitative-Qualitative variable. Review(cont.) .

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Chapter 2 Describing Contingency Tables

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  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. • logθis 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*104)=1.83

  17. Case-Control Studies and the Odds Ratio

  18. However(cont.)

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

  20. Death penalty example

  21. Death penalty example(cont.)

  22. Death penalty example(cont.) Simpson’s 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(<45,45-65,>65) • And Age is an Effect Modifier

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

  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.

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