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Risk

When we calculate the probability of a negative outcome like a heart attack, we often refer to the probability as a risk. For example, we talk about the probability of winning the lottery but the risk of getting struck by lightning. Whenever you see the word risk, keep in mind it's just another word for probability.

EXAMPLE Risk and the Physician's Health Study

Researchers in the Physicians’ Health Study (1989) conducted a clinical trial to determine whether aspirin reduces the risk of heart attack. Researchers randomly assigned a large sample of healthy male physicians (22,071) to one of two groups. One group took a low dose of aspirin (325 mg every other day). The other group took a placebo (a sugar pill that looked like aspirin.) Here are the final results.

Physicians’ Health Study on aspirin and heart attacks

| Heart Attack | No Heart Attack | Row Totals
Aspirin | 139 | 10,898 | 11,037
Placebo | 239 | 10,795 | 11,034
Column Totals | 378 | 21,693 | 22,071

Note that the categorical variables in this case are

Explanatory variable: Treatment (aspirin or placebo)

Response variable: Medical outcome (heart attack or no heart attack)

Question: Does aspirin lower the risk of having a heart attack?

To answer this question, we compare two conditional probabilities:

The probability of a heart attack for those taking aspirin every other day.

The probability of a heart attack for those taking placebo every other day.

From the table we have

The probability of a heart attack in the aspirin group = 139 / 11,037 = 0.013

The probability of a heart attack in the placebo group = 239 / 11,034 = 0.022

The result shows that taking aspirin reduced the risk from 0.022 to 0.013.

We often compare two risks by calculating the percentage change. We calculate the difference (how much the risk changed with the new treatment) and divide by the risk when there is no treatment (or a currently accepted standard treatment.)

EXAMPLE Comparing two risks

Here is the calculation of the percent change in risk for the aspirin study from the previous example:

Therefore, we conclude that taking aspirin results in a 41% reduction in risk.

As reported in the New England Journal of Medicine, “This trial of aspirin for the primary prevention of cardiovascular disease demonstrates a conclusive reduction in the risk of myocardial infarction (heart attack).” (Source: "Final Report on the Aspirin Component of the Ongoing Physicians’ Health Study," New England Journal of Medicine 321(3):129–35, 1989.)

Comment

In the preceding example, we compared the difference in risk (how much the risk changed) to the risk for the placebo (non-treatment) group:

In general, we are interested in determining how much a new treatment reduces the risk compared to a reference risk. The reference may be non-treatment (e.g., use of a placebo), or it could be an existing treatment that we hope to improve on. So we have:

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