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 parametersmultinomial parameters

……

Terminology and notation

CellContingency 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=0.0171• 104/11037=0.0094• Relative risk:• 0.0171/0.0094=1.82

• Odds ratio:• (189*10933)/

(10845*104)=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.)

Death penalty example(cont.)

Simpson’s 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(<45,45-65,>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.