The table below summarizes the results of an experiment evaluating three treatments for Type 2 Diabetes in patients aged 10-17 who were being treated with metformin. The three treatments considered were continued treatment with metformin (met), treatment with metformin combined with rosiglitazone (rosi), or a lifestyle intervention program. Each patient had a primary outcome, which was either lacked glycemic control (failure) or did not lack that control (success).

Failure Success
met 120 112
rosi 90 143
lifestyle 109 125

What is the expected cell count for the patients treated with a lifestyle intervention program who lacked glycemic control?

Question

The table below summarizes the results of an experiment evaluating three treatments for Type 2 Diabetes in patients aged 10-17 who were being treated with metformin. The three treatments considered were continued treatment with metformin (met), treatment with metformin combined with rosiglitazone (rosi), or a lifestyle intervention program. Each patient had a primary outcome, which was either lacked glycemic control (failure) or did not lack that control (success).

Failure	Success
met	120	112
rosi	90	143
lifestyle	109	125

What is the expected cell count for the patients treated with a lifestyle intervention program who lacked glycemic control?

BlackTom AI · Accepted Answer

To determine the expected cell count for lifestyle intervention with lack of glycemic control, we use the standard formula for expected counts in a contingency table: (row total × column total) ÷ grand total.

First, identify the row totals:
- Met: 120 + 112 = 232
- Rosi: 90 + 143 = 233
- Lifestyle: 109 + 125 = 234
Next, identify the column totals:
- Failure: 120 + 90 + 109 = 319
- Success: 112 + 143 + 125 = 380
Then, compute the grand total: 319 + 380 = 699
Now compute the expected count for Lifestyle and Failure: (Lifestyle row total × Failure column total) ÷ Grand total = (234 × 319) ÷ 699.

Calculate the numerator: 234 × 319 = 234 × (300 + 19) = 70,200 + 4,446 = 74,646.
Divide: 74,646 ÷ 699 ≈ 106.85.
Rounding conventions typically give 106.8 (to one decimal place).

Evaluating the answer options:
- 127.2: This is notably higher than the calculated expectation; it would require a much larger interaction between Lifestyle and Failure than the data support.
- 114.5: This is closer but still overestimates the expected count by several units; the precise calculation yields about 106.8.
- 107.4: This is near the computed value but not as close as 106.8 when rounding to one decimal place; the exact ratio gives around 106.85, which rounds to 106.8.
- 106.8: This matches the computed expected count when rounded to one decimal place, and aligns with the standard (row total × column total) ÷ grand total calculation.

In summary, the expected count for Lifestyle and Failure is derived from the proportional distribution across row and column totals, yielding approximately 106.85, which corresponds to 106.8 when rounded. The other options deviate from this precise calculation due to either overestimation or rounding differences.

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