91Ó°ÊÓ

The paper "Good for Women, Good for Men, Bad for People: Simpson's Paradox and the Importance of Sex-Spedfic 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 \(\mathrm{A}\) or Treatment \(\mathrm{B}\), 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 \(\mathrm{A}\), and \(B\) be the event that a patient selected at random received Treatment \(\mathrm{B}\). a. The following table summarizes data for men and women combined: $$ \begin{array}{l|ccc} & \text { Survived } & \text { Died } & \text { Total } \\ \hline \text { Treatment A } & 215 & 85 & \mathbf{3 0 0} \\ \text { Treatment B } & 241 & 59 & \mathbf{3 0 0} \\ \text { Total } & \mathbf{4 5 6} & \mathbf{1 4 4} & \\ \hline \end{array} $$ i. Find \(P(S)\). ii. Find \(P(S \mid A)\). iii. Find \(P(S \mid B)\). iv. Which treatment appears to be better? b. Now consider the summary data for the men who participated in the study: $$ \begin{array}{l|rrr} & \text { Survived } & \text { Died } & \text { Total } \\ \hline \text { Treatment A } & 120 & 80 & \mathbf{2 0 0} \\ \text { Treatment B } & 20 & 20 & 40 \\ \text { Total } & \mathbf{1 4 0} & \mathbf{1 0 0} & \\ \hline \end{array} $$ i. Find \(P(S)\). ii. Find \(P(S \mid A)\). iii. Find \(P(S \mid B)\). iv. Which treatment appears to be better? c. Now consider the summary data for the women who participated in the study: $$ \begin{array}{l|rrc} & \text { Survived } & \text { Died } & \text { Total } \\ \hline \text { Treatment A } & 95 & 5 & \mathbf{1 0 0} \\ \text { Treatment B } & 221 & 39 & \mathbf{2 6 0} \\ \text { Total } & \mathbf{3 1 6} & \mathbf{1 4 4} & \\ \hline \end{array} $$ i. Find \(P(S)\). ii. Find \(P(S \mid A)\). iii. Find \(P(S \mid B)\). iv. Which treatment appears to be better? 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 \(\mathrm{B}\) 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?)

Step by step solution

01

Calculate P(S) - combined data

Calculate P(S) as the sum of survivors divided by total subjects ( 456 / 600 = 0.76 ).

02

Calculate P(S|A) and P(S|B) - combined data

Calculate P(S|A) as the number of survivors from treatment A divided by total receiving treatment A ( 215 / 300 = 0.7167 ). Similarly, calculate P(S|B) as number of survivors from treatment B divided by total receiving treatment B ( 241 / 300 = 0.8033 ).

03

Evaluate treatments - combined data

By comparing the values of P(S|A) and P(S|B), it appears that treatment B is better since P(S|B) > P(S|A).

04

Calculate P(S) - men only

Calculate P(S) as the sum of male survivors divided by total male subjects ( 140 / 240 = 0.5833 ) .

05

Calculate P(S|A) and P(S|B) - men only

Calculate P(S|A) as the number of male survivors from treatment A divided by total males receiving treatment A ( 120 / 200 = 0.6 ). Similarly, calculate P(S|B) as number of male survivors from treatment B divided by total males receiving treatment B ( 20 / 40 = 0.5 ).

06

Evaluate treatments - men only

By comparing the values of P(S|A) and P(S|B), it appears that treatment A is better since P(S|A) > P(S|B). Again, this is using the data from men only.

07

Calculate P(S) - women only

Calculate P(S) as the sum of female survivors divided by total female subjects ( 316 / 360 = 0.8778 ) .

08

Calculate P(S|A) and P(S|B) - women only

Calculate P(S|A) as the number of female survivors from treatment A divided by total females receiving treatment A ( 95 / 100 = 0.95 ). Similarly, calculate P(S|B) as number of female survivors from treatment B divided by total females receiving treatment B ( 221 / 260 = 0.85 ).

09

Evaluate treatments - women only

By comparing the values of P(S|A) and P(S|B), it appears that treatment A is better since P(S|A) > P(S|B). Again, this is using the data from women only.

10

Simpson's Paradox Explanation

The paradox here is that while Treatment A appears to be better for both men and women separately, Treatment B appears to be better when we considered the combined data. This apparent inconsistency occurs because men and women respond differently to the treatments and the proportions of men and women who received the treatments are also uneven. Specifically, a larger number of women, who generally had higher chances of surviving, received Treatment B which made Treatment B appear more effective when data was combined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

conditional probability

Conditional probability allows us to determine the likelihood of one event occurring, given that another event has already occurred. This is often represented as \( P(A \mid B) \) which reads as "the probability of event A occurring given that B is true". In the context of our study, it helps us evaluate the effectiveness of each treatment given a specific condition, such as the treatment received or the group (e.g., men or women) being studied.

Let's take a step-by-step look at how this was calculated in the study:

• To find \( P(S \mid A) \), which is the probability of survival given Treatment A, we divide the number of survivors in Treatment A by the total number of participants in Treatment A. For the combined data, this was calculated as \( 215 / 300 = 0.7167 \).


• Similarly, \( P(S \mid B) \) represents the probability of surviving given Treatment B. This was calculated as \( 241 / 300 = 0.8033 \).

By examining these probabilities, we can analyze which treatment appears more effective under different conditions, which is crucial for understanding the prevalence of Simpson's Paradox in the observational study.

observational study

An observational study involves observing subjects and measuring variables of interest without assigning treatments to the subjects. In the context of medical studies, participants are not controlled by the researchers but are observed in a real-world setting where naturally occurring variations are examined.

Our example stems from such an observational study where patients received either Treatment A or Treatment B based on conditions not manipulated by the study designers. These conditions may include patient preference, physician recommendation, or pre-existing health factors. This makes observational studies particularly interesting as they reveal insights into real-life situations, but also come with challenges like potential confounding factors.

Observational studies, like the one discussed, are valuable for showing how different treatments perform across various groups. Yet, they can lead to apparent contradictions such as Simpson's Paradox because underlying variables, like gender, might affect outcomes without being evenly distributed across treatment groups.

sex-specific analysis

Sex-specific analysis involves breaking down data by gender to understand how male and female subjects respond differently to treatments or conditions. This type of analysis is crucial for studies, especially in medicine, as physiological differences can lead to varying responses between the sexes. In the given study, sex-specific analysis helped unravel the seeming contradiction posed by Simpson's Paradox.

When examining the results separately for men and women:

• It was found that Treatment A is superior for both groups individually, as both \( P(S \mid A) \) values were higher compared to \( P(S \mid B) \).


• However, when the figures are combined, the results seemed to favor Treatment B due to unequal gender distribution in treatment groups.

This highlights the importance of sex-specific analysis to ensure accurate conclusions are drawn, preventing misleading interpretations that could arise from aggregated data.