Simpson’s Paradox

Simpson’s paradox, or the Yule–Simpson effect, is a paradox in probability and statistics, in which a trend appears in different groups of data but disappears or reverses when these groups are combined.

UC-Berkeley Example

One of the best known real-life example of Simpson’s paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate school.

Gender	Applicants	Admitted
Male	8442	44%
Female	4321	35%

# load 'my.cmh'
source('~/Dropbox/Teaching/STAT766/mycmh.R', encoding = 'UTF-8')
my.cmh(3714,4728,1512,2809)

##        Test Cochran Mantel.Haenszel
## 1 Statistic 95.7927         95.7852
## 2   P-value  0.0000          0.0000

This 1973 data shows that men were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to random chance.

However, when the data are stratified by department:

Department	Male Applicants	Male Admitted	Female Applicants	Female Admitted
A	825	62%	108	82%
B	560	63%	25	68%
C	325	37%	593	34%
D	417	33%	375	35%
E	191	28%	393	24%
F	272	6%	341	7%

Data2 <- as.table(array(c(511,314,89,19,353,207,17,8,120,205,202,391,138,279,131,244,53,138,94,299,16,256,24,290),
                   dim = c(2, 2, 6),
                   dimnames =
                   list(Response = c("+", "-"), Gender=c("M","F"),
                        Ulcer = c("A","B","C","D","E","F"))))
mantelhaen.test(Data2,correct=F)

## 
##  Mantel-Haenszel chi-squared test without continuity correction
## 
## data:  Data2
## Mantel-Haenszel X-squared = 1.6722, df = 1, p-value = 0.196
## alternative hypothesis: true common odds ratio is not equal to 1
## 95 percent confidence interval:
##  0.7671426 1.0556245
## sample estimates:
## common odds ratio 
##          0.899897