Yule-Simpson Effect
Theorem
The is a veridical paradox in the theory of contingency tables.
An association that is significant in each of two contingency tables may disappear or be reversed if the two tables are combined.
Examples
Arbitrary Example
Suppose that a new treatment $\text N$ is being compared with a standard treatment $\text S$, in an experiment with $3000$ patients.
Let a record be kept of the number of cures $\text C$ and failures $\text F$.
The numbers in each category are shown here:
- $\begin{array}{r|cc|c} & \text S & \text N & \text {Totals} \\ \hline \text C & 450 & 530 & 980 \\ \text F & 850 & 1170 & 2020 \\ \hline \text {Totals} & 1300 & 1700 & 3000 \end{array}$
A $\chi$-squared test indicates a differential effect of treatments with the standard giving a greater proportion of cures.
However, when the results are broken down into patients living in urban and rural areas, the data are as given in the following tables:
- $\begin{array}{r|cc|c} \text {Urban:} & \text S & \text N & \text {Totals} \\ \hline \text C & 100 & 350 & 450 \\ \text F & 500 & 1050 & 1550 \\ \hline \text {Totals} & 600 & 1400 & 2000 \end{array} \qquad \begin{array}{r|cc|c} \text {Rural:} & \text S & \text N & \text {Totals} \\ \hline \text C & 350 & 180 & 530 \\ \text F & 350 & 120 & 470 \\ \hline \text {Totals} & 700 & 300 & 1000 \end{array}$
Now it is seen that in both urban and rural areas, the new drug gives a better cure rate.
Source of Name
This entry was named for George Udny Yule and Edward Hugh Simpson.
Historical Note
The was established by George Udny Yule and Edward Hugh Simpson, and published by Simpson in $1951$.
Sources
- 1951: E.H. Simpson: The Interpretation of Interaction in Contingency Tables (J.R. Stat. Soc. Ser. B Vol. 13: pp. 238 – 241)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Simpson's paradox
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Simpson's paradox