Comparing Risks With the Relative Risk

Why Compare Risks as a Ratio?

Suppose two groups experience the same outcome, such as an infection within 30 days of treatment. One group might have a risk of 3%, and another a risk of 6%. The difference is 3 percentage points, while the second risk is twice the first. These describe the same data on different scales: a difference gives an absolute change, and a ratio gives a relative comparison.

The relative risk (also called the risk ratio) is the ratio of the outcome risk in one population group to the outcome risk in another. Its interpretation depends on which group is placed in the numerator and which is used as the reference. It is therefore essential to name the groups in the same order when defining, calculating, and interpreting it.

A risk here means the probability or proportion of individuals who experience the specified outcome over a defined period or under a defined observation scheme. For example, “risk of infection within 30 days” names both an outcome and a time window. Comparisons are meaningful only when the groups’ outcomes and time windows are defined consistently.

Estimating Relative Risk From Samples

Population risks are usually unknown. Suppose a sample from group 1 contains \(x_1\) individuals with the outcome out of \(n_1\), and a sample from group 2 contains \(x_2\) individuals with the outcome out of \(n_2\). The sample risks are \(\hat{p}_1=x_1/n_1\) and \(\hat{p}_2=x_2/n_2\). The sample relative risk is obtained by replacing each population risk in the definition with its sample estimate.

For the usual two-group setup, each individual has a binary outcome—either the specified event occurs or it does not. Within each group, outcomes can be represented by Bernoulli observations with that group’s event probability. To interpret the sample proportions as estimates for defined populations, the samples or group assignments must represent the groups of interest; the two groups should also be formed and observed in a way that supports a valid comparison. The formula itself is descriptive, but these design features matter when drawing conclusions beyond the observed samples.

How to Read the Value

The value one is the benchmark for equal risks: dividing a positive risk by itself gives one. Values on either side of one indicate the direction of the comparison, as long as the group order is kept fixed.

For a value above one, the amount above one can be expressed as a proportional increase relative to the reference risk. For example, a relative risk of 1.5 means the first risk is 1.5 times the second risk, or 50% higher. For a value below one, the reduction relative to the reference risk is \(1-RR\) as a proportion. A relative risk of 0.80 means the first risk is 80% of the second, or 20% lower. These percentages compare risks proportionally; they are not percentage-point changes.

Why the Group Order Matters

A relative risk is directional in the sense that the numerator and denominator determine the question being answered. If \(p_1\) is divided by \(p_2\), the result compares group 1 with group 2. Reversing the comparison gives \(p_2/p_1\). When both risks are positive, this is the reciprocal of the original relative risk:

For example, if group 1’s risk is twice group 2’s, then group 2’s risk is half group 1’s. The values 2 and 0.5 do not conflict; they answer comparisons with opposite group orders. Always state the numerator group and reference group in words alongside the numerical result.

A Short Algebraic Connection to Risk Differences

Relative risk and risk difference use the same two population risks but summarize their relationship differently. The following identity makes the connection explicit. Starting from \(RR=p_1/p_2\), with \(p_2>0\), subtract one:

Thus, the relative amount by which \(p_1\) exceeds or falls below \(p_2\) is the risk difference divided by the reference risk. This also explains why the same absolute difference can correspond to different relative risks when the reference risk changes. For instance, a difference of 0.02 is half of a reference risk of 0.04, but only one-fifth of a reference risk of 0.10. The relative comparison depends on its denominator, so a relative risk should not be reported without identifying the reference group.

Common Mistakes and Careful Interpretation

• Reversing the groups without changing the wording. If the calculation is intervention risk divided by reference risk, describe the intervention group relative to the reference group. Reversing the order changes the result to its reciprocal when both risks are positive.
• Calling a relative change a percentage-point change. A relative risk of 2 means twice the risk, not a difference of two percentage points. Calculate the two risks separately if an absolute difference is needed.
• Treating a sample estimate as a known population value. The estimated relative risk summarizes observed sample proportions. Sampling variability means another sample may produce a different estimate; a point estimate alone does not quantify that uncertainty.
• Interpreting an association as a cause. A relative risk describes a comparison of outcome risks. It does not, by itself, show that membership in one group caused the outcome risk to change. Causal conclusions require a design and assumptions that support them.

The denominator must also be positive for the relative risk to be defined. If the reference-group sample risk is zero, the sample ratio cannot be calculated by ordinary division. That is a limitation of this estimate, not evidence that the relative risk has some finite numerical value.