Calculating and Interpreting Relative Risk

1. Defining and Computing Relative Risk

Just like with the difference in proportions, we start by comparing the risk of an outcome in two different groups: one that is "exposed" to a certain factor and one that is not.

Let's use our familiar definitions for the two conditional probabilities (risks):

• Risk in the exposed group (${\pi }_1$): $P(\text{Outcome}|\text{Exposed})=\frac{a}{a+b}$
• Risk in the unexposed group (${\pi }_2$): $P(\text{Outcome}|\text{Unexposed})=\frac{c}{c+d}$

Where $a,b,c,d$ are the cell counts in a standard 2x2 table.

Instead of subtracting these risks, the Relative Risk (RR), also known as the Risk Ratio, is calculated by dividing them:

This simple ratio tells us how many times more (or less) likely the outcome is in the exposed group compared to the unexposed group.

To see this calculation in action, let's watch a short video that walks through an example from a cohort study.

As you saw in the video, the calculation is straightforward:

1. Calculate the risk for the exposed group: In the example, the risk of lung cancer for smokers was $30/100=0.30$.
2. Calculate the risk for the unexposed group: The risk of lung cancer for non-smokers was $10/100=0.10$.
3. Compute the ratio: $RR=\frac{0.30}{0.10}=3.0$.

Now, what does this number "3.0" actually mean?

2. Interpreting Relative Risk

The interpretation of the relative risk hinges on its value relative to 1. Since it's a ratio, a value of 1 means the numerator and denominator are the same—implying no difference in risk.

The following video provides an excellent summary of how to interpret the different possible values of a relative risk.

Let's formalize those interpretations:

• RR = 1: The risk of the outcome is identical in both the exposed and unexposed groups. There is no association between the exposure and the outcome. This is the null value.
• RR > 1: The risk of the outcome is higher in the exposed group. The exposure is considered a risk factor for the outcome. For example, an RR of 3.0 means the risk is 3 times higher in the exposed group.
• RR < 1: The risk of the outcome is lower in the exposed group. The exposure is considered a protective factor against the outcome. For example, an RR of 0.5 means the risk in the exposed group is half that of the unexposed group.

To ensure your interpretation is always precise, it's helpful to use a standard template. The following reading provides a robust sentence structure for this.

Using the template from the reading, our interpretation for the smoking example (RR = 3.0) would be:

"The risk of developing lung cancer was 3 times as high in smokers compared to non-smokers over the 20-year study period."

Notice the phrase "times as high." This wording works correctly whether the RR is greater or less than 1. For a protective factor with RR = 0.4, we would say the risk is "0.4 times as high," which correctly implies a lower risk.

3. When to Use Relative Risk (and When Not To)

Your goal is a functional understanding, which means knowing when a tool is right for the job. The calculation of relative risk relies on having valid estimates of incidence—the rate of new cases in a population.

When It Works Best: Cohort Studies

Relative risk is the natural measure of association for study designs where you start with a group of people who are initially free of the outcome and follow them over time to see who develops it. This is the definition of a cohort study.

To understand why this study design is key, please read the following brief section.

Because cohort studies (and randomized controlled trials) allow us to directly measure the risk in both the exposed ($a/(a+b)$) and unexposed ($c/(c+d)$) groups, calculating their ratio (the RR) is meaningful.

When It Fails: Case-Control Studies

What if we use a different study design? In a case-control study, we start by finding people who already have the disease ("cases") and a comparable group who do not ("controls"). Then, we look backward to determine their past exposure.

In this design, the number of cases and controls is fixed by the researcher. The proportion of exposed people in the "unexposed" row of our 2x2 table, $c/(c+d)$, does not represent the true risk in the general non-exposed population. Since we can't calculate the true risks ${\pi }_1$ and ${\pi }_2$, we cannot directly calculate the relative risk from a case-control study. This is a critical limitation and a primary reason why we need another measure of association, which we will discuss in the next lesson.

Other Limitations

1. Relative vs. Absolute Impact: Just as we discussed in the last lesson, a large RR can be misleading if the baseline risk is very low. An RR of 2.0 sounds significant, but if it represents a risk increasing from 1-in-a-million to 2-in-a-million, the absolute change (a risk difference of 0.000001) is tiny. Always consider both relative and absolute measures to get the full picture.
2. Sample Size: An RR of 2.5 might seem important, but if it comes from a very small study, the result could be due to chance. As discussed in the article by Sauerbrei and Blettner (Interpreting Results in 2 × 2 Tables), a larger sample size will produce a narrower confidence interval around the RR estimate, giving us more certainty. An RR of 2.5 from a study of 225 people might not be statistically significant, while the same RR from a study of 900 people would be highly significant.

Conclusion

In this lesson, you learned about our second major measure of association for 2x2 tables, the relative risk. It complements the difference in proportions by providing a multiplicative, or relative, comparison of risk.

Key Takeaways:

• Definition: Relative Risk (RR) is the ratio of the risk in an exposed group to the risk in an unexposed group ($RR={\pi }_1/{\pi }_2$).
• Computation: It is calculated by dividing the proportion of outcomes in the exposed group by the proportion of outcomes in the unexposed group.
• Interpretation:
  • RR > 1 indicates a risk factor.
  • RR < 1 indicates a protective factor.
  • RR = 1 indicates no association (the null value).
• Appropriate Use: Relative risk is the preferred measure for cohort studies and randomized controlled trials, where incidence can be directly measured. It cannot be calculated directly from case-control studies.

Preview of the Next Lesson:

We've now covered an absolute measure (difference in proportions) and a relative measure (relative risk). But what do we do in situations like case-control studies where calculating risk directly is impossible? In the next lesson, we will introduce our third and final key measure of association for 2x2 tables: the odds ratio, which solves this very problem.