What is Relative Risk?
Relative risk is the ratio of the probability of an adverse outcome in an exposure group divided by its likelihood in an unexposed group. This statistic indicates whether exposure corresponds to increases, decreases, or no change in the probability of the adverse outcome. Use relative risk to measure the strength of the association between exposure and the outcome. Analysts also refer to this statistic as the risk ratio.
Exposure can be a protective factor that reduces the chances of an adverse outcome (such as a vaccine or education program) or a risk factor that increases its likelihood (such as a toxin or a harmful environment). In this manner, the risk ratio determines whether exposure is a protective or risk factor.
Given its ability to identify protective and risk factors, analysts frequently use relative risk in medical, intervention, and ecological studies. Additionally, it can intuitively convey the treatment effect in randomized controlled trials with binary outcomes, such as infected or not infected.
In this post, learn the relative risk formula, how to calculate and interpret it, and work through a risk ratio example with real-world data.
Related posts: Probability Fundamentals and Understanding Ratios.
How to Calculate Relative Risk
In statistics, risk is a probability, usually relating to an adverse event (i.e., something bad). To calculate relative risk (RR), you must know all subjects’ exposure statuses and outcomes. Before learning how to calculate it, you first need to know about absolute risk.
Absolute risk (AR) is simply the number of events divided by the number of people in the group. In the context of RR, we’re working with two groups, those who were and were not exposed to something.
For example, if 1 in 10 people exposed to a substance gets sick, the exposed AR is 0.1. If 1 in 100 people who are not exposed get sick, the unexposed AR is 0.01.
In its simplest form, the relative risk formula is the ratio of AR for the two exposure groups, as shown below:
Using the example values above, let’s plug the exposed and unexposed ARs into the formula:
The relative risk result indicates that people exposed to the substance are ten times more likely to get sick! That’s the relative increased probability associated with exposure.
Relative Risk Formula
Now, let’s expand the relative risk calculation to show the formula in more detail. The table below shows a standard format for RR in a two-way contingency table. Learn more about Contingency Tables.
Events | Non-Events | Absolute Risk | |
Exposed | A | B | A / (A + B) |
Unexposed | C | D | C / (C + D) |
Each letter represents a count of events or non-events in the exposed and unexposed groups. A row represents a group defined by exposure status, exposed or unexposed. In an experiment, the exposed group receives a treatment, while the unexposed group is the control. Learn more about Control Groups.
Again, the RR is the ratio of those two event probabilities. The relative risk formula below uses the letter notation from above:
This relative risk calculation emphasizes how it is a ratio of two ARs.
How to Interpret Relative Risk
Because the relative risk formula is a ratio, that tells us how to interpret it. The value of 1 becomes an important benchmark because it indicates that the exposed and unexposed groups have equal absolute risks. Consequently, analysts compare their risk ratio results to one during interpretation. As the ratio moves away from one in either direction, the relationship between exposure and the outcome strengthens.
Relative Risk = 1: The risk ratio equals one when the numerator and denominator are equal. This equivalence occurs when the probability of the event occurring in the exposure group equals the likelihood of it happening in the unexposed group. There is no association between exposure and the outcome.
Relative Risk > 1: The numerator is greater than the denominator in the risk ratio. Therefore, the event’s probability is greater in the exposed group than in the unexposed group. This result identifies a risk factor because exposure corresponds with a greater probability of an adverse outcome.
Relative Risk < 1: The numerator is less than the denominator in the risk ratio. Consequently, the probability of the event is lower for the exposed group than for the unexposed group. This exposure is a protective factor because it corresponds with a lower probability of an adverse outcome.
Interpretation Cautions
Relative risk is a kind of correlation between exposure status and the outcome. As you have undoubtedly heard, correlation outside randomized experiments does not necessarily represent causal relationships! Use proper experimental designs to uncover causal relationships. Learn more about Experimental Design.
Additionally, relative risk doesn’t suggest anything about absolute risk. Hence, you should report both AR and RR to provide a complete picture. For example, the following table displays both types for two hypothetical studies. The relative risks are equivalent, but the ARs paint a very different picture of the problem’s prevalence!
Study | AR Exposed | AR Unexposed | Relative Risk |
1 | 0.75 | 0.25 | 3 |
2 | 0.09 | 0.03 | 3 |
Relative Risk Example
Finally, let’s work through a relative risk calculation example using real-world data. In this case, we’ll use data from a flu vaccination study (Beran et al., 2009). This study was a randomized controlled trial, the gold standard for identifying causal relationships.
In this experiment, exposure relates to receiving the flu vaccine. The vaccinated group is exposed, and the placebo control group is unexposed. The table contains the count of infections (events) and non-infections for both groups. We want to learn whether the vaccine protects against infections.
Treatment | Flu infections | Non-infections |
Vaccine | 49 | 5054 |
Placebo | 74 | 2475 |
Now, let’s plug these numbers into the relative risk formula:
The risk ratio is 0.3310, indicating the vaccine is a protective factor. The vaccinated are about 1/3 as likely to catch the flu as the unvaccinated.
Read about relative risk in the context of Poisson Regression Analysis.
Note that RR is different than vaccine effectiveness, although they’re related. Read my post about Flu Vaccination Effectiveness for more information.
RRs are similar to several other measures of relative probabilities. Learn about Odds Ratios and Hazard Ratios.
For a broader look at various types of risk, read my post Risk Calculations: Relative vs. Absolute & Risk Reduction.
Reference
Beran J, Vesikari T, Wertzova V, Karvonen A, Honegr K, Lindblad N, Van Belle P, Peeters M, Innis BL, Devaster JM. Efficacy of inactivated split-virus influenza vaccine against culture-confirmed influenza in healthy adults: a prospective, randomized, placebo-controlled trial. J Infect Dis 2009;200(12):1861-9.
Comments and Questions