Design the appropriate 2x2 table, calculate and interpret the appropriate measure of association.

You suspect that the association between alcohol use and CHD might be confounded by smoking. You collect the following data:
Smokers Non-Smokers
CHD No CHD CHD No CHD
Alcohol Use 80 40 10 20
No Alcohol Use 20 10 40 80
How would you calculate the measure of association?

Have you heard of the Chi-squared (X^2) test?

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To calculate the measure of association between alcohol use and CHD, we can use the chi-square test for independence. This test is often used to determine if there is a significant association between categorical variables.

First, let's create a 2x2 table using the data provided:

Smokers Non-Smokers
CHD 80 40
No CHD 10 20

Next, we need to calculate the expected frequencies for each cell in the table. The expected frequency is the count we would expect to see in each cell if there was no association between the variables.

To calculate the expected frequency for each cell, we can use the following formula:

Expected Frequency = (Row Total * Column Total) / Grand Total

For example, the expected frequency for the cell representing smokers with CHD would be:

Expected Frequency = (120 * 90) / 300 = 36

Calculate the expected frequencies for all cells in the table using this formula. The resulting table would look like this:

Smokers Non-Smokers
CHD 36 84
No CHD 54 66

Now, we can calculate the chi-square statistic. The formula for chi-square is:

Chi-square = Σ((Observed - Expected)^2 / Expected)

where Σ represents the sum of all cells in the table.

Using this formula, calculate the chi-square statistic. In this case, it would be:

Chi-square = ((80-36)^2 / 36) + ((40-84)^2 / 84) + ((10-54)^2 / 54) + ((20-66)^2 / 66)

Once you calculate the chi-square statistic, you would compare it to the critical value from the chi-square distribution with (number of rows - 1) * (number of columns - 1) degrees of freedom. If the calculated chi-square value is larger than the critical value, it suggests a significant association between alcohol use and CHD.

Finally, interpret the measure of association based on the chi-square test. A significant association would indicate that alcohol use and CHD are not independent of each other and that there is a relationship between them. However, the chi-square test does not provide information about the strength and direction of the association. To further understand the relationship, additional measures such as Cramer's V or phi coefficient can be calculated.