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Follow the three R’s: Respect for self,

Respect for others and Responsibility for all your actions

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Categorical Categorical Data AnalysisData Analysis

Chapter 1: IntroductionChapter 1: Introduction

Chapter 2: Two Way TablesChapter 2: Two Way Tables

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A variable

Categorical/Qualitative

Numerical/Quantitative

Nominal variable Ordinal variable Discrete variable Continuous variable

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The Role of a VariableThe Role of a Variable• Explanatory/predictor/independent

variable

• Response/outcome/dependent variable

Q: Will women smoking during pregnancy have higher probability of miscarriage?

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Statistical Tools vs. Variable TypesStatistical Tools vs. Variable Types

Response (output)

Explanatory var (input)

Numerical Categorical/Mixed

Numerical•Simple and Multiple Regression

•Analysis of Variance (ANOVA)•Analysis of Covariance (ANCOVA)

Categorical Categorical data analysis

Distributions for Categorical DataDistributions for Categorical Data

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• Binomial data• When the sample size n is fixed and the response per subject is binary

• Multinomial data• When the sample size n is fixed and the response per subject is multinary

• Poisson data• When the sample size n is not fixed and the response per subject is binary/multinary

• Poisson model conditional on a given n is Binomial/ Multinomial model (Sec. 1.2.5)

Distributions for Categorical DataDistributions for Categorical Data

• These models have overdispersion problem because they assume all subjects having the same probability of responding (from the same population) and ignore the covariates

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Review: Maximum Likelihood Review: Maximum Likelihood Inference MethodInference Method

• Likelihood functions• MLE (maximum likelihood estimate)• Asymptotical distribution of mle• Wald inference (test and confidence

interval)• Example: binomial data

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Two-Way TablesTwo-Way Tables• A table summarizes two categorical

variables• Cells of a two-way table are

frequency counts of combined outcomes of the two variables

• Called IxJ tables for I rows and J columns

Eg. Do you believe “afterlife”?

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Probability & IndependenceProbability & Independence• Joint probability• Marginal probability• Conditional probability

– Sensitivity = P(+ | disease)– Specificity = P( - | no disease)

• Independence of X and Y

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Three Sampling MethodsThree Sampling Methods• Poisson sampling: no fixed total

• Single multinomial sampling: fixed grand total

• Independent multinomial sampling: fixed row or column totals

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Comparing Proportions in Comparing Proportions in 2x2 Tables2x2 Tables

• Difference of proportions: pi1-pi2

• Relative risk: pi1/pi2

• Odds Ratio: odds1/odds2odds1=pi1/(1-pi1)odds2=pi2/(1-pi2)

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More on Odds RatioMore on Odds Ratio• Properties

• Inference

• Relationship to relative risk

Comparing Proportions in Comparing Proportions in stratified 2x2 Tablesstratified 2x2 Tables

• Find the “conditional odds rations” describing the partial association in stratified 2x2 tables

• Marginal vs. conditional independence

• Homogeneous association• Same idea applying to IxJ tables

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Table 2.7Table 2.7Response

Clinic Treatment Success Failure

1 A 18 12

B 12 8

2 A 2 8

B 8 32

Total A 20 20

B 20 40

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