Chapter 6.4, Problem 52E

The paper “Good for Women, Good for Men, Bad for People: Simpson’s Paradox and the Importance of Sex-Specific Analysis in Observational Studies” (Journal of Women’s Health and Gender-Based Medicine [2001]: 867-872) described the results of a medical study in which one treatment was shown to be better for men and better for women than a competing treatment. However, if the data for men and women are combined, it appears as though the competing treatment is better.

To see how this can happen, consider the accompanying data tables constructed from information in the paper. Subjects in the study were given either Treatment A or Treatment B, and survival was noted. Let S be the event that a patient selected at random survives, A be the event that a patient selected at random received Treatment A, and B be the event that a patient selected at random received Treatment B.

1. a. The following table summarizes data for men and women combined:

1. i. Find P(S).
2. ii. Find P(S|A).
3. iii. Find P(S|B).
4. iv. Which treatment appears to be better?
5. b. Now consider the summary data for the men who participated in the study:

1. v. Find P(S).
2. vi. Find P(S|A).
3. vii. Find P(S|B).
4. viii. Which treatment appears to be better?
5. c. Now consider the summary data for the women who participated in the study:

1. ix. Find P(S). looks like Treatment B is better. This is an
2. x. Find P(S|A).
3. xi. Find P(S|B).
4. xii. Which treatment appears to be better?
5. d. You should have noticed from Parts (b) and (c) that for both men and women, Treatment A appears to be better. But in Part (a), when the data for men and women are combined, it looks like Treatment B is better. This is an example of what is called Simpson’s paradox. Write a brief explanation of why this apparent inconsistency occurs for this data set. (Hint: Do men and women respond similarly to the two treatments?)

To determine

i. Compute $P\left(S\right)$.

ii. Obtain $P\left(S|A\right)$.

iii. Calculate $P\left(S|B\right)$.

iv. Find the better treatment.

Answer to Problem 52E

i. The value of $P\left(S\right)$ is 0.76.

ii. The value of $P\left(S|A\right)$ is 0.717.

iii. The value of $P\left(S|B\right)$ is 0.803.

iv. Treatment B is better than Treatment A.

Explanation of Solution

Calculation:

The given information is the summary table of the survey. Event S denotes the event that a patient selected at random and survives, event A denotes that a patient selected at random received Treatment A, and B denotes the event that a patient selected at random and received Treatment B.

i.

The probability of any event A is given below:

$P\left(A\right)=\frac{\text{Number of outcomes in }A}{\text{Total number of outcomes in the samplespace}}$

The total number of randomly selected patient is 600.

The total number of patient selected at random survives is 456.

The probability of a randomly selected patients and who survive is calculated as follows:

$\begin{array}{c}P\left(S\right)=\frac{456}{600}\\ =0.76\end{array}$

Thus, the probability of a randomly selected patients  who survive is 0.76.

ii.

Conditional rule:

The formula for probability of E given F is, $P\left(E|F\right)=\frac{n\left(E\cap F\right)}{n\left(F\right)}$.

The total number of patient selected at random and received Treatment A is 300.

The number of patient selected at random and received Treatment A and survive is 215.

The probability that the selected patients at random received Treatment A, given that the patient selected at random survives. It is calculated as follows:

$\begin{array}{c}P\left(S|A\right)=\frac{215}{300}\\ =0.717\end{array}$

Thus, the value of $P\left(S|A\right)$ is equal to 0.717.

iii.

The total number of patient selected at random and received Treatment B is 300.

The number of patient selected at random that received Treatment B and survive is 241.

The probability that the selected patient at random received Treatment B, given that the patient selected at random survives. It is calculated as follows:

$\begin{array}{c}P\left(S|B\right)=\frac{241}{300}\\ =0.803\end{array}$

Thus, the value of $P\left(S|B\right)$ is equal to 0.803.

iv.

The probability of patient who received Treatment B survived more than that of Treatment A.

Thus, Treatment B is better than Treatment A.

To determine

i. Compute $P\left(S\right)$.

ii. Obtain $P\left(S|A\right)$.

iii. Calculate $P\left(S|B\right)$.

iv. Find the better treatment.

Answer to Problem 52E

i. The value of $P\left(S\right)$ is 0.583.

ii. The value of $P\left(S|A\right)$ is 0.6.

iii. The value of $P\left(S|B\right)$ is 0.5.

iv. Treatment A is better than Treatment B.

Explanation of Solution

Calculation:

The given information is the summary table of the survey.

i.

The total number of randomly selected patient is 240.

The total number of patient selected at random and survives is 140.

The probability of a randomly selected patients who survive is calculated as follows:

$\begin{array}{c}P\left(S\right)=\frac{140}{240}\\ =0.583\end{array}$

Thus, the probability of a randomly selected patients who survive is 0.583.

ii.

Conditional rule:

The formula for probability of E given F is, $P\left(E|F\right)=\frac{n\left(E\cap F\right)}{n\left(F\right)}$.

The total number of patients selected at random that received Treatment A is 200.

The number of patient selected at random that received Treatment A and survives is 120.

The probability that the selected patient at random received Treatment A, given that the patient selected at random survives. It is calculated as follows:

$\begin{array}{c}P\left(S|A\right)=\frac{120}{200}\\ =0.6\end{array}$

Thus, the value of $P\left(S|A\right)$ is equal to 0.6.

iii.

The total number of patients selected at random that received Treatment B is 40.

The number of patient selected at random that received Treatment B and survive is 20.

The probability that the selected patient at random received Treatment B, given that the patient selected at random survives. It is calculated as follows:

$\begin{array}{c}P\left(S|B\right)=\frac{20}{40}\\ =0.5\end{array}$

Thus, the value of $P\left(S|B\right)$ is equal to 0.5.

iv.

The probability of patient who received Treatment A survived more than that of Treatment B.

Thus, Treatment A is better than Treatment B.

To determine

i. Compute $P\left(S\right)$.

ii. Obtain $P\left(S|A\right)$.

iii. Calculate $P\left(S|B\right)$.

iv. Find the better treatment.

Answer to Problem 52E

i. The value of $P\left(S\right)$ is 0.878.

ii. The value of $P\left(S|A\right)$ is 0.95.

iii. The value of $P\left(S|B\right)$ is 0.85.

iv. Treatment A is better than Treatment B.

Explanation of Solution

Calculation:

The given information is the summary table of the survey.

i.

The total number of randomly selected patient is 360.

The total number of patient selected at random that survive is 316.

The probability of a randomly selected patients who survive is calculated as follows:

$\begin{array}{c}P\left(S\right)=\frac{316}{360}\\ =0.878\end{array}$

Thus, the probability of a randomly selected patients who survive is 0.878.

ii.

Conditional rule:

The formula for probability of E given F is, $P\left(E|F\right)=\frac{n\left(E\cap F\right)}{n\left(F\right)}$.

The total number of patient selected at random that received Treatment A is 100.

The number of patient selected at random that received Treatment A and survive is 95.

The probability that the selected patients at random received Treatment A, given that the patient selected at random survives. It is calculated as follows:

$\begin{array}{c}P\left(S|A\right)=\frac{95}{100}\\ =0.95\end{array}$

Thus, the value of $P\left(S|A\right)$ is equal to 0.95.

iii.

The total number of patient selected at random that received Treatment B is 260.

The number of patient selected at random that received Treatment B and survive is 221.

The probability that the selected patients at random received Treatment B, given that the patient selected at random survives. It is calculated as follows:

$\begin{array}{c}P\left(S|B\right)=\frac{221}{260}\\ =0.85\end{array}$

Thus, the value of $P\left(S|B\right)$ is equal to 0.85.

iv.

The probability of patients who received Treatment A survived more than that of Treatment B.

Thus, Treatment A is better than Treatment B.

To determine

Explain the reason for the existence of apparent inconsistency in the data.

Explanation of Solution

From part (a), (b) and (c), it can be observed that Treatment A performs better than that of Treatment B, except part (a). In part (a), the data for men and women are combined. Thus, Treatment B performs better than that of Treatment A.