Here's a really simple "Hospital Mortality" example using data from a 2x2 table. It illustrates an unfair (biased) overall treatment comparison while the corresponding comparisons made within sub-groups of similar patients (columns) are both unbiased!

A "Simpson's Paradox" Example

Where would you want to be treated? Your two choices are a World Class hospital or a Nearby hospital with the following statistics on in-house mortality by disease severity:

Cardiac Mortality

**Note that the Nearby hospital could truthfully
advertise that it's overall cardiac mortality rate is lower (by 6 tenths of one
percent) than that of the World Class hospital. Unfortunately, this claim
is actually meaningless in light of the above detail on how mortality varies
with disease severity!**

**The inescapable reality here is that the World
Classis hospital has a 2% lower cardiac mortality rate for low severity patients
and a 3% lower cardiac mortality rate for high severity patients. The LC
"adjusted" main-effect (if all patients were to choose World Class hospitals
over Nearby hospitals) would be a mean reduction in cardiac mortality of 2.55%.
**

**This particular World Class hospital has a relatively high
overall mortality rate simply because more than two-thirds of the patients it
treats have high cardiac disease severity! Society should be happy that the
above
Nearby hospital is relatively low-volume, treating only 22% of the patients
tabulated here. **

**TECHNICAL NOTES:**

Years after the above sort of phenomenon was
first discussed in statistical literature, Blyth(1972) declared it to represent
a **paradox! ** In reality, this is a relatively common form of
**CONFOUNDING.** In this example,
**IMBALANCE** between treatment cohorts results
because cardiac disease severity (low versus high) is associated with both
hospital preferences ("World Class" versus "Nearby") and the ultimate outcome
(mortality rate.)

The data are presented twice above. The first table displays cell percentages, where the third row represents hospital differences (World Class minus Nearby.) The second table displays mortality fractions (fatalities divided by sample size), where the third row represents total fatalities divided by total sample sizes within columns.

**Historical Statistical References:**

Yule GU. Notes on the Theory of
Association of Attributes in Statistics. ** Biometrika** 1903; 2:
121–134.

Simpson EH. The Interpretation of
Interaction in Contingency Tables. ** J Roy Stat Soc B** 1951; 13:
238–241.

Blyth CR. On Simpson's Paradox and the
Sure-Thing Principle. ** J Amer Stat Assoc** 1972; 67: 364–366.

Lopiano KK, Obenchain RL and Young SS. Fair Treatment
Comparisons in Observational Research. ** Statistical Analysis and Data Mining** 2014; 7:
376–384.