### Example

## Risk and the Physicians’ Health Study

Researchers in the Physicians’ Health Study (1989) designed a randomized 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. This was a double-blind experiment. Here are the final results.

| **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 given that aspirin was taken every other day.
- The probability of a heart attack given that a placebo was taken every other day.

From the table we have

*P*(heart attack | aspirin) = 139 / 11,037 = 0.013
*P*(heart attack | placebo) = 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) and divide by the risk for the placebo group.

Here is the calculation:

**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.)

In the preceding example, we compared the difference in risk (how much the risk changed) to the risk for the placebo (nontreatment) 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 nontreatment (e.g., use of a placebo), or it could be an existing treatment that we hope to improve on. So we have:

The following table is used for the next Try It activity.

Let’s summarize our work with probability. We defined three kinds of probabilities related to a two-way table.

When we calculate the probability of a negative outcome like a heart attack, we often refer to the probability as a *risk*. We compare risk by calculating the percentage change: