Yule-Simpson Effect/Examples

Examples of Yule-Simpson Effect

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.

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