Download - Chapter 2 Describing Contingency Tables
![Page 1: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/1.jpg)
Chapter 2 Describing Contingency Tables
Reported by Liu Qi
![Page 2: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/2.jpg)
Review of Chapter 1
• Categorical variable• Response-Explanatory variable• Nominal-Ordinal-Interval variable• Continuous-Discrete variable• Quantitative-Qualitative variable
![Page 3: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/3.jpg)
Review(cont.)
• Use binomial, multinomial and Poisson distribution
• Not normality distribution• Tow most used models: logistic regression(logit) log linear
![Page 4: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/4.jpg)
Binomial distribution
![Page 5: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/5.jpg)
Multinomial distribution
![Page 6: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/6.jpg)
Poisson distribution
![Page 7: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/7.jpg)
Poisson<->Multinomial
![Page 8: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/8.jpg)
Something unfamiliar
• Maximum likelihood estimation• Confidence intervals• Statistical inference for
binomial parametersmultinomial parameters
……
![Page 9: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/9.jpg)
Terminology and notation
CellContingency table
![Page 10: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/10.jpg)
Terminology and notation
• Subjective• Sensitivity and Specificity• Conditional distribution• Joint distribution• Marginal distribution• Independence =>
![Page 11: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/11.jpg)
Sampling Scheme
• Poisson the joint probability mass function:
• Multinomial independent/product multinomial
• Hyper geometric
![Page 12: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/12.jpg)
Example for sampling
![Page 13: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/13.jpg)
Types of studies
• Retrospective: case-control• Prospective:– Clinical trial observational study– Cohort study
• Cross-sectional: experimental study
![Page 14: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/14.jpg)
Comparing two proportions
• Difference • Relative risk• Odds ratio– Odds defined as – For a 2*2 table, odds ratio– Another name: cross-product ratio
![Page 15: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/15.jpg)
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.
![Page 16: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/16.jpg)
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
![Page 17: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/17.jpg)
Case-Control Studies and the Odds Ratio
![Page 18: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/18.jpg)
However(cont.)
![Page 19: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/19.jpg)
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
![Page 20: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/20.jpg)
Death penalty example
![Page 21: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/21.jpg)
Death penalty example(cont.)
![Page 22: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/22.jpg)
Death penalty example(cont.)
Simpson’s paradox
![Page 23: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/23.jpg)
Conditional and marginal odds ratios
• Conditional
• Marginal
![Page 24: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/24.jpg)
Conditional independence
• Conditional independence:
• Joint probability:
![Page 25: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/25.jpg)
Marginal independence
![Page 26: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/26.jpg)
Marginal versus Conditional
![Page 27: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/27.jpg)
Marginal versus Conditional(cont.)
• Marginal • conditional
![Page 28: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/28.jpg)
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.
![Page 29: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/29.jpg)
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
![Page 30: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/30.jpg)
Extensions for i*j Tables
For a 2*2 table• Odds ratio
An i*j table• Odds ratios
![Page 31: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/31.jpg)
Representation methods
• Method 1
![Page 32: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/32.jpg)
Method 2
![Page 33: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/33.jpg)
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.
![Page 34: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/34.jpg)
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:
![Page 35: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/35.jpg)
Measures of variation
• Entropy:• Goodman and Kruskal(1954) (tau)
• Lambda:
![Page 36: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/36.jpg)
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.
![Page 37: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/37.jpg)
Ordinal Trends
• An example:
![Page 38: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/38.jpg)
Three kinds of relationship
• Concordant• Discordant• Tied
![Page 39: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/39.jpg)
Example(cont.)
• D = 849• Define (C-D)/(C+D) as Gamma measure.• Here,
• A weak tendency for job satisfaction to increase as income increases.
![Page 40: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/40.jpg)
Generalized
![Page 41: Chapter 2 Describing Contingency Tables](https://reader035.vdocuments.us/reader035/viewer/2022081507/56815f78550346895dce7ff2/html5/thumbnails/41.jpg)
Properties of Gamma Measure
• Symmetric• Range [-1,1]• Absolute value of 1 means perfect linear• Monotonicity is required for• Independence => ,not vice-versa.