Categorical Data Analysis

Rates

	HD	No HD	Row Total
Smoker	38	14962	15000
Non-Smoker	44	84956	85000
Column Total	82	99918	100000

• Marginal Rate
  • % of population that satisfies the condition
  • Rate(Smoker) $=\frac{15000}{100000}$
• Conditional Rate
  • A rate based on part of the population that satisfies a certain condition
  • % of people out of population that satisfies “given” condition
  • Rate(HD | Smoker) → Rate of HD given Smoker
  • Rate(HD | Smoker) $=\frac{38}{15000}$
• Joint Rate
  • % of population that satisfies both conditions
  • Rate(HD & Smoker) $=\frac{38}{100000}$

Symmetry Rule

If rate(A | B) [op] rate(A | not B), then rate(B | A) [op] rate(B | not A). Where [op] is $\ne ,>,<,=$

Basic Rule of Rates

1. Given subgroups A, B and C, the overall rate(A) is always between rate(A | B) and rate(A | C)
2. From 1, when C = “not B”, rate(A) is always between rate(A | B) and rate(A | not B)
3. The closer rate(B) gets to 100%, the closer rate(A) gets to rate(A | B)
4. Rate(A) is exactly between rate(A | B) and rate(A | not B) if rate(B) = $50\%$

Association

	B	Not B	Row Total
A
Not A
Col. Total

• Categorical variables A and B are associated to each other if rate(A | B) $\ne$ rate(A | not B)
• A and B are positively associated if rate(A | B) $>$ rate(A | not B)
• Negatively associated if rate(A | B) $<$ rate(A | not B)
• Comparing rate(A | B) vs rate(A | not B) is exactly the same as comparing rate(B | A) vs rate(B | not A)
• Association does not establish causation
• Whether the association can be generalised from sample to population is based on Generalisability

Confounder

A confounder is a third variable, associated with both the dependent and independent variables It causes any association determined between the dependent and independent variable to be unreliable

Find Confounders

	Male, HD	Male, No HD	Female, HD	Female, No HD	Row Total
Smoker	25	9582	13	5380	15000
Non Smoker	30	34954	14	50002	85000
Column Total	55	44536	27	55382	100000

To determine if sex is a confounding variable, check for association between sex vs smoking and sex vs HD e.g. since rate(Smoker | Male) $>$ rate(Smoker | Female) and rate(HD | Male) $>$ rate(HD | Female), sex is a confounder

Control for Confounders

• Split the data
  • If there is an association observed in both groups, then there is an association

Simpson’s Paradox

• When a trend appears in more than half of the groups of data, but disappears or reverses when the groups are combined
  • Relationship between rates in subgroups disappear/reverse when the groups are combined
• Simpson’s paradox $\Rightarrow$ there is a confounding variable (confounder $\nRightarrow$ Simpson’s paradox)

Major	No. of Applications (Male)	No. of Successful Applications (Male)	Rate (Male)	No. of Applications (Female)	No. of Successful Applications (Female)	Rate (F)
A	2000	800	40%	10	2	20%
B	10	8	80%	1000	600	60%
C	10	8	80%	1000	600	60%
Overall	2020	816	40.3%	2010	1202	59%

Paradox: In a majority of majors, rate of successful male applications is higher than female, but overall, rate of successful male applications is lower