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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: IntroductionChapter 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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