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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.). - PowerPoint PPT Presentation
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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.