A research team is investigating the relationship between preoperative smoking status and … — USMLE Step 3

A research team is investigating the relationship between preoperative smoking status and the incidence of post-operative respiratory complications in patients undergoing elective abdominal surgery. The researchers categorize $500$ patients into three groups: current smokers ( $n=100$ ), former smokers ( $n=150$ ), and never smokers ( $n=250$ ). During the $30$ -day post-operative period, the following number of patients developed respiratory complications:

- Current smokers: $15$
- Former smokers: $10$
- Never smokers: $5$

The researchers plan to perform a Chi-square test of independence to analyze this categorical data. Assuming the null hypothesis of no association is true, what is the expected count of current smokers who develop post-operative respiratory complications?

Answer: 6 patients

Answer

In a Chi-square test of independence, the expected frequency for any given cell in a contingency table is calculated by multiplying its corresponding row marginal total by its column marginal total, and then dividing by the grand total. For current smokers who develop complications, the row total (all current smokers) is

100

, the column total (all patients with complications) is

30

, and the grand total is

500

. Therefore, the expected count is

\frac{100 \times 30}{500} = 6

Step-by-Step Solution

Calculate the total number of patients who developed respiratory complications (the column marginal total).

15 + 10 + 5 = 30

patients

The expected frequency formula requires the marginal total for the column of interest to represent the overall complication rate.

Identify the total number of current smokers (the row marginal total) and the grand total of patients.

Row total =

100

patients; Grand total =

500

patients

These values are required to complete the expected frequency formula.

Calculate the expected count using the formula:

\frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}

\frac{100 \times 30}{500} = \frac{3000}{500} = 6

This calculation determines the expected number of current smokers with complications assuming the null hypothesis (that smoking status and complications are completely independent) is true.

Key Concept

Calculating expected cell frequencies in a Chi-square test of independence

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