The exercise below is designed to guide you through the calculations for'determining whether or not a variable is a confounding variable. These calculations can get detailed, and it is strongly suggested that you show all your work, including writing the formulas you are using. (Please note that the data, although plausible, are hypothetical.) To determine whether baldness causes coronary heart disease (CHD) in men, a hypothetical cohort study was carried out. The epidemiologist in charge of the study recruited 10,000 bald men and 10,000 men with hair into the study and followed all of them for 10 years to determine whether they developed CHD. Results are shown below.

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To determine whether baldness is a confounding variable in the relationship between baldness and coronary heart disease (CHD), we need to analyze the data collected from the cohort study. Here are the steps we will follow:

1. **Define the Variables**:
   - Let $$ B $$ represent baldness (1 for bald, 0 for not bald).
   - Let $$ C $$ represent coronary heart disease (1 for CHD, 0 for no CHD).

2. **Collect the Data**:
   - We need the number of bald men who developed CHD and the number of men with hair who developed CHD. Let's denote:
     - $$ n_{B1} $$: Number of bald men with CHD
     - $$ n_{B0} $$: Number of bald men without CHD
     - $$ n_{H1} $$: Number of men with hair who developed CHD
     - $$ n_{H0} $$: Number of men with hair who did not develop CHD

3. **Calculate the Proportions**:
   - The proportion of bald men who developed CHD:
     $$
     P(B=1 | C=1) = \frac{n_{B1}}{n_{B1} + n_{B0}}
     $$
   - The proportion of men with hair who developed CHD:
     $$
     P(B=0 | C=1) = \frac{n_{H1}}{n_{H1} + n_{H0}}
     $$

4. **Determine the Odds Ratios**:
   - The odds of CHD in bald men:
     $$
     O(B=1) = \frac{n_{B1}}{n_{B0}}
     $$
   - The odds of CHD in men with hair:
     $$
     O(B=0) = \frac{n_{H1}}{n_{H0}}
     $$
   - The odds ratio (OR) comparing bald men to men with hair:
     $$
     OR = \frac{O(B=1)}{O(B=0)} = \frac{n_{B1}/n_{B0}}{n_{H1}/n_{H0}}
     $$

5. **Assess Confounding**:
   - A confounding variable is one that is associated with both the exposure (baldness) and the outcome (CHD) and can distort the perceived relationship between them. We will need to check if the odds ratio significantly changes when controlling for potential confounders (e.g., age, smoking status, etc.).

6. **Statistical Testing**:
   - We may perform statistical tests (e.g., Chi-square test) to determine if the differences in proportions are statistically significant.

Please provide the specific data from the study (e.g., the number of bald men with CHD, the number of men with hair with CHD, etc.) so that we can proceed with the calculations.
To determine if baldness is a confounding variable for CHD, we need to: 1. **Define Variables**: - $$ B $$: Baldness (1 = bald, 0 = not bald) - $$ C $$: Coronary Heart Disease (1 = CHD, 0 = no CHD) 2. **Collect Data**: - Number of bald men with CHD ($$ n_{B1} $$) - Number of bald men without CHD ($$ n_{B0} $$) - Number of men with hair with CHD ($$ n_{H1} $$) - Number of men with hair without CHD ($$ n_{H0} $$) 3. **Calculate Proportions**: - Proportion of bald men with CHD: $$ \frac{n_{B1}}{n_{B1} + n_{B0}} $$ - Proportion of men with hair with CHD: $$ \frac{n_{H1}}{n_{H1} + n_{H0}} $$ 4. **Determine Odds Ratios**: - Odds of CHD in bald men: $$ \frac{n_{B1}}{n_{B0}} $$ - Odds of CHD in men with hair: $$ \frac{n_{H1}}{n_{H0}} $$ - Odds Ratio (OR): $$ \frac{n_{B1}/n_{B0}}{n_{H1}/n_{H0}} $$ 5. **Assess Confounding**: - Check if controlling for other factors (like age or smoking) changes the OR significantly. 6. **Statistical Testing**: - Use tests like the Chi-square test to see if the differences are statistically significant. Provide the specific data to perform these calculations.

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