1 Chapter 5 Two-Way Tables Associations Between Categorical Variables.
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• Quantitative variables correlation [Ch 3] & regression [Ch 4]
• categorical variables two-way tables of frequency counts [Ch 5]
Associations between variables
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Two-Way Table of CountsR-by-C tables
Variables
AGE variable = column variable (3 levels)
EDUCATION variable = row variable (4 levels)
This is a 4-by-3 table
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Marginal Distributions
Variables
Row variablemarginal totals
37,786 81,435 56,008
27,85858,07744,46544,828
Column variablemarginal distribution
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Marginal Percents
• Relative frequencies (%s) for each variable separately
• Descriptive purposes only; does not not address association
• Illustrative Example (Distribution of education level)
– Statement: Describe the distribution of education levels in the population
– Plan: Calculate marginal percents for row variable “EDUCATION”
marginal countmarginal percent 100%
table total
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Marginal Percents Example
% not completing HS = 27,859 / 175,230 × 100% = 15.9%
% completing HS = 58,077 / 175,230 × 100% = 33.1%
% with 1-3 yrs college = 44,465 / 175,230 × 100% = 25.4%
% with 4+ yrs college = 44,828 / 175,230 × 100% = 25.6%
Step 3: “Solve”
Rowtotals
Tabletotal
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Marginal Percents (Example)
• 16% did not complete high school
• 33% completed high school
• 25% completed 1 to 3 years of college
• 26% completed 4+ years of college
Step 4: “Conclude”
Merely descriptive statements
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Association
• If the row variable is the explanatory variable → compare conditional row proportions
• If the column variable is the explanatory variable → compare conditional column proportions
cell countconditional row proportion
row total
cell countconditional column proportion
column total
Use conditional proportions to determine associations
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Example: Association between AGE & EDUCATION
State: Is AGE associated with EDUCATION level?
Plan: Since AGE is the explanatory variable calculate conditional column proportions. We do not need to calculate every conditional proportion. (Be selective.) Let us calculate the proportion completing 4+ years of college by AGE
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Example: “Solve” & “Conclude”
p̂ 25-34
11,071 = .2930 = 29%
37,786
55ˆ =p 10,597
.1892 = 19%56,008
35 54p̂ 23,160
= .2843 = 28%81,435
ˆLet p represent proportion completing 4+ years of college
Conclude: As age goes up, % completing college goes down Negative association between age and college completion
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• No association: conditional percents nearly equal at all levels of the explanatory variable
• Positive association: as explanatory variable rises conditional percentages increase
• Negative associations: as explanatory variable rises conditional percentages go down
Direction of association
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State: Is ACCEPTANCE into UC Berkeley graduate school (response variable) associated with GENDER (explanatory variable)?
Example: Gender bias?
Accepted Not accept. TotalMale 198 162 360Female 88 112 200Total 286 274 560
Plan: Since GENDER is the explanatory variable calculate row percents (acceptance “rates” by gender); compare % accepted by GENDER
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Example: “Gender bias?”
Accepted Not accept TotalMale 198 162 360Female 88 112 200Total 286 274 560
Conclude: positive association between “maleness” and acceptance
Step 3: Solve
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Simpson’s Paradox
• Lurking variable MAJOR applied to– Business school major (240 applicants) – Art school major (320 applicants)
• State: Does lurking variable explain association between maleness and acceptance?
• Plan: Subdivide (“stratify”) data into subgroups according to lurking variable MAJOR then calculate acceptance rates by gender within subgroups
Simpson’s Paradox ≡ lurking variable reverses direction of the association
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“Gender Bias” Data by MAJOR
Business School ApplicantsSuccess Failure Total
Male 18 102 120Female 24 96 120
Total 42 198 240
All ApplicantsSuccess Failure Total
Male 198 162 360Female 88 112 200Total 286 274 560
Art School ApplicantsSuccess Failure Total
Male 180 60 240Female 64 16 80
Total 244 76 320
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Business School ApplicantsSuccess Failure Total
Male 18 102 120
Female 24 96 120
Total 42 198 240
Conclude: Negative association with maleness
0.15 18
proportion accepted, males 120
240.20 proportion accepted, females
120
Conclude: Negative association with maleness
0.15 18
proportion accepted, males 120
240.20 proportion accepted, females
120
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Art School Applicants
Success Failure Total
Male 180 60 240
Female 64 16 80
Total 244 76 320
Conclude: Negative association with maleness
0.75240
180
proportion accepted, males
640.80 proportion accepted, females
80
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• Overall: higher acceptance rate for men
• Within Business school: higher acceptance rate for women
• Within Art school: higher acceptance rate for women
• Therefore, the lurking variable (MAJOR) reversed the direction of the association (Simpson’s Paradox)
• Acceptance to grad school at UC Berkeley favored women after “controlling for” MAJOR
Gender Bias Example Conclusion
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HIV vaccine boostHIV vaccine boost (Exercise 5.6)
State: Do data support that vaccine delivered by EP results in a higher proportion responding?Plan = ?Solution = ?Conclusion = ?
Kidney StonesKidney Stones (Exercise 5.7)
Small Stones
Open Surgery
Percutaneous
Success 81 234
Failure 6 36
Large Stones
Open Surgery
Percutaneous
Success 192 55
Failure 71 25
(a) Find % of kidney stones, combining the data for small and large stones, that were successfully removed for each of the two procedures. Which procedure had the higher overall success rate?(b) What % of all small kidney stones were successfully removed? What % of all large kidney stones…? Which type of kidney stone is easier to treat?
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