Conditional Probability and Independence: Basics 3

When comparing the conditional probabilities of two events it is important to consider influential variables. A lurking variable is a variable which has an important effect on outcomes, but which has not been accounted for in the data.

This table compares the death rates at two Hospitals:

Hospital Deaths	Hospital U	Hospital R
Lived	2350	485
Died	150	15
Total	2500	500

The death rates at: Hospital U = 6%, at Hospital R = 3%.

Does this mean Hospital R is better?

Hospital U is a large urban hospital with a trauma unit treating injured and very sick patients, hospital R is a regional hospital which primarily performs non-emergency elective proceedures. Patient condition may be an important factor to consider.

This table compares the condition of the patients at both hospitals:

Hospital Deaths	Hospital U		Hospital R
Patient Condition	Good	Poor	Good	Poor
Lived	495	1855	441	44
Died	5	145	9	6
Total	500	2000	450	50

For Patients in Good Condition: U's deaths= 1%, R's deaths = 2%

For Patients in Poor Condition: U's deaths= 7.25%, R's deaths = 12%

Hospital U has better results in both categories, yet worse results overall due to the disproportionate number of its patients in poor condition. The condition of the patient is a lurking variable.

The situation described above illustrates a principle known as Simpson's Paradox:
When data are aggregated over a lurking variable, the results may reverse.

Page 3 - Lurking Variables and Simpson's Paradox