The following table shows hypothetical data for two experimental treatments (Treatment A and Treatment B) against two stages of cancer (Early and Advanced). "Remission" indicates a successful treatment. "Rate" is the proportion of remissions.

Question

\begin{tabular}{|l|r|r|r|r|r|r|}
\hline & \multicolumn{3}{|c|}{ Early } & \multicolumn{3}{|c|}{ Advanced } \\
\cline { 2 - 7 } Treatment & Remissions & Death & Rate & Remissions & Death & Rate \\
\hline A & 5 & 1 & 0.833 & 4 & 6 & 0.400 \\
\hline B & 10 & 4 & 0.714 & 1 & 4 & 0.200 \\
\hline
\end{tabular}

This table shows the same data aggregated into a two-way table.

\begin{tabular}{|l|r|r|r|}
\hline Treatment & Remissions & Deaths & Rate \\
\hline A & 9 & 7 & 0.563 \\
\hline B & 11 & 8 & 0.579 \\
\hline
\end{tabular}

Which is true for this scenario?

Answer 1

To analyze the data provided in the tables, we need to consider the remission rates for each treatment, both when comparing the two stages of cancer separately and when looking at the aggregate data.

1. Individual Treatment Rates for Early and Advanced Stages:

   • For Treatment A:
     • Early: 5 remissions out of (5+1) = 6 total patients, giving a rate of 0.833.
     • Advanced: 4 remissions out of (4+6) = 10 total patients, giving a rate of 0.400.
   • For Treatment B:
     • Early: 10 remissions out of (10+4) = 14 total patients, giving a rate of 0.714.
     • Advanced: 1 remission out of (1+4) = 5 total patients, giving a rate of 0.200.

2. Aggregate Data:

   • For Treatment A: 9 remissions total out of (9+7) = 16 total patients, resulting in a rate of \( \frac{9}{16} = 0.563 \).
   • For Treatment B: 11 remissions total out of (11+8) = 19 total patients, resulting in a rate of \( \frac{11}{19} \approx 0.579 \).

3. Comparison of Rates:

   • In the Early stages, Treatment A (0.833) has a better remission rate than Treatment B (0.714).
   • In the Advanced stages, Treatment A (0.400) has a better remission rate than Treatment B (0.200).
   • In the aggregate data, Treatment B (0.579) has a slightly better remission rate than Treatment A (0.563).

Conclusion

While Treatment A consistently outperforms Treatment B in terms of remission rates for both early and advanced stages of cancer, it appears that the aggregate results show Treatment B having a marginally better overall remission rate due to a larger number of remissions in the Early stage that balance out against fewer patients in the Advanced stage.

Key Points:

• Treatment A is more effective in both stages separately.
• In terms of overall performance across all patients, Treatment B shows a slightly higher aggregate remission rate.
• This scenario illustrates how overall performance can be affected by the distribution of patients between groups.

Overall, it is true that although Treatment A is more effective in both individual stages, the aggregate remission rate is higher for Treatment B.