Question

Suppose we are interested in investigating the relationship between high salt intake and death from cardiovascular disease (CVD). One possible study design is to identify a group of high- and low-salt users then follow them over time to compare the relative frequency of CVD death in the two groups. In contrast, a less expensive study design is to look at death records, identify CVD deaths from non-CVD deaths, collect information about the dietary habits of the deceased, then compare salt intake between individuals who died of CVD versus those who died of other causes. This design is called a retrospective design.

Suppose a retrospective study is done in a specific county of Massachusetts; data are collected on men ages 50-54 who died over a 1-month period. Of 35 men who died from CVD, 5 had a diet with high salt intake before they died, while of the 25 men who died from other causes, 3 had a diet with high salt intake. These data are summarized in the following table.
CVD Death Non-CVD Death Total
High Salt Diet 5 2 7
Low Salt Diet 30 23 53
Total 35 25 60

Under the null hypothesis of no association, what are the expected cell counts?

Answer 1

Under the null hypothesis, the expected counts for the relationship between salt intake and CVD death are determined using the formula (Row total × Column total) / Grand total. The expected counts would be approximately 4.08 for high salt intake and CVD death, 2.92 for high salt intake and non-CVD death, 30.92 for low salt intake and CVD death, and 22.08 for low salt intake and non-CVD death.

Step-by-step explanation:

Under the null hypothesis of no association between salt intake and death from cardiovascular disease (CVD), we would expect the cell counts to show no difference in the proportion of high salt diets between those who died of CVD and those who died of other causes. To calculate the expected counts under the null hypothesis, we use the formula:

Expected count = (Row total × Column total) / Grand total

• For high salt intake and CVD death: Expected count = (7 × 35) / 60 = 4.08
• For high salt intake and non-CVD death: Expected count = (7 × 25) / 60 = 2.92
• For low salt intake and CVD death: Expected count = (53 × 35) / 60 = 30.92
• For low salt intake and non-CVD death: Expected count = (53 × 25) / 60 = 22.08

These values are the expected frequencies for each cell if there was no association between salt intake and CVD death.