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

Binomial distribution

Multinomial distribution

Poisson distribution

Poisson Multinomial

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

Terminology and notation Cell Contingency table

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

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

Example for sampling

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

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

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.

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

Case-Control Studies and the Odds Ratio

However(cont.)

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

Death penalty example

Death penalty example(cont.)

Simpsons paradox

Conditional and marginal odds ratios Conditional Marginal

Conditional independence Conditional independence: Joint probability:

Marginal independence

Marginal versus Conditional

Marginal versus Conditional(cont.) Marginal conditional

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.

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

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

Representation methods Method 1

Method 2

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.

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:

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

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.

Ordinal Trends An example:

Three kinds of relationship Concordant Discordant Tied

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

Generalized

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