Question

the following two - way table summarizes the number of cancer patients who died or survived at two cancer clinics. the three - way table breaks down the data further by whether the patients cancer was active or in remission.

	clinic a	clinic b	total
survived	390 (65%)	320 (80%)	710 (71%)
total	600 (100%)	400 (100%)	1000 (100%)

cancer	clinic	outcome	total
		survived	40 (89%)
		total	45 (100%)
	clinic b	died	65 (18%)
		survived	300 (82%)
		total	365 (100%)
	total	died	70 (17%)
		survived	340 (83%)
		total	410 (100%)
active	clinic a	died	205 (37%)
		survived	350 (63%)
		total	555 (100%)
	clinic b	died	15 (43%)
		survived	20 (57%)
		total	35 (100%)
	total	died	220 (37%)
		survived	370 (63%)
		total	590 (100%)

explain how clinic b has a lower overall percentage of patients that died, even though clinic a did better for both groups of patients (remission and active).

\\(\circ\\) patients got better care at clinic b, perhaps because it had better facilities or doctors.
\\(\circ\\) patients are simply more likely to die at clinic a.
\\(\circ\\) clinic a treated more patients than clinic b.
\\(\circ\\) clinic a treats many more patients with active cancer than clinic b, and more patients died with active cancer.

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Explanation:

To determine why Clinic B has a lower overall death percentage but Clinic A did better for both remission and active cancer groups, we analyze the patient distribution:

Step 1: Analyze Remission Group

• Clinic A: Died = 5, Survived = 40; Total = 45. Death rate = \( \frac{5}{45} \approx 11\% \)
• Clinic B: Died = 65, Survived = 300; Total = 365. Death rate = \( \frac{65}{365} \approx 18\% \)

Clinic A has a lower death rate in remission.

Step 2: Analyze Active Cancer Group

• Clinic A: Died = 205, Survived = 350; Total = 555. Death rate = \( \frac{205}{555} \approx 37\% \)
• Clinic B: Died = 15, Survived = 20; Total = 35. Death rate = \( \frac{15}{35} \approx 43\% \)

Clinic A has a lower death rate in active cancer.

Step 3: Analyze Overall Patient Mix

• Clinic A: Total patients = 600 (from two - way table). Remission patients: 45, Active patients: 555. So, proportion of active patients = \( \frac{555}{600} = 92.5\% \)
• Clinic B: Total patients = 400. Remission patients: 365, Active patients: 35. Proportion of active patients = \( \frac{35}{400}=8.75\% \)

Clinic A treats many more patients with active cancer (a more severe condition with higher natural death rates) than Clinic B. When calculating the overall death rate, Clinic B's lower proportion of active cancer patients (who have higher death rates) makes its overall death rate lower, even though Clinic A has better survival rates within each cancer status (remission and active) group.

Answer:

Clinic A treated many more patients with active cancer than Clinic B.