**Relative Risk**

**Relative risk (RR)**, also known as the risk ratio, is a ratio of proportions. Specifically, it is a ratio of the rate of exposure in the individuals who have the condition (or response) divided by the rate of exposure in the individuals who do not have the condition. Consider the example in table 17.2, in which an experimental warm-up program is examined to determine whether it could decrease the risk of knee injuries. Subjects were randomized to either the warm-up group or the control group. Here, the warm-up program is the exposure (the independent variable) and knee injury is the condition (the dependent variable). If the warm-up program is effective, we would expect the rate of knee injuries to be lower in those who do the warm-up (those exposed to the warm-up) than is those in the control group (those who were not exposed to the warm-up). (It may be that the warm-up is dangerous and therefore makes knee injuries more likely, in which case we would expect the rate of knee injury to be higher in the warm-up subjects than in the controls.) Conversely, if the warm-up is ineffective, the rate of knee injuries should be about the same in the exposed and nonexposed groups.

If the rate of knee injuries is about the same in the two groups, then the relative risk should be around 1.0. This means that the null hypothesis value for relative risk is 1.0, and we need to statistically assess how different from 1.0 our sample data have to be for us to be confident that the exposure influences the rate of the condition. In practice, the formal null hypothesis significance test is not typically performed on relative risk calculations; instead, confidence intervals are constructed about the calculated relative risk. If the 95% CI does not include 1.0, that is equivalent to rejecting H0 at an α level of .05.

The relative risk value is an effect size value. If the relative risk is 3.0, then we are estimating that the rate of condition is three times higher in the exposed group than in the unexposed group and, therefore, the exposure is associated with increased risk. If the relative risk is 0.33, then we are estimating that the rate of the condition is about one-third of that in the unexposed group and, therefore, the exposure is associated with decreased risk.

In the example in table 17.2, 100 subjects were randomized to a structured program in which they warmed up before practice (the exposure), and 100 did not warm up. Of those in the exposed group, 14 experienced a knee injury (cell A) and 86 did not experience a knee injury (cell B). In the unexposed group, 27 experienced a knee injury (cell C) and 73 did not experience a knee injury (cell D). To calculate the numerator for relative risk, we determine the proportion of knee injuries in the exposed group, which in this example is 14/100 = 0.14 (14% of the subjects in the warm-up group experienced a knee injury). This is simply the ratio of the number of subjects in the exposed group with a knee injury (cell A) divided by the total number of subjects in the exposed group (cell A + cell B), so that numerator for relative risk is A/(A + B). Similarly, to calculate the denominator of relative risk, we calculate the proportion of knee injuries in the unexposed group. Here, this is equal to 27/73 = 0.27 (27% of subjects in the unexposed group experienced a knee injury), which is the ratio of the number of subjects in the unexposed group with a knee injury (cell C) divided by the total number of subjects in the unexposed group (cell C + cell D). Therefore the denominator of relative risk is C/(C + D).

Collectively, the equation for relative risk is

RR = [A/(A + B)]/[C/(C + D)], (17.01)

which for the data in table 17.2 is [14/(14 + 86)]/[27/(27 + 73)] = 0.14/0.27 = 0.519 ~ 0.52.What does a relative risk of 0.52 tell us? One way to think of this value is that the warm-up program was associated with a reduction of risk of approximately 48%. Another is that the warm-up program was associated with about 52% of the risk of the program in which subjects did not warm up.

The calculation of the confidence interval is not shown here because it a bit more involved than confidence intervals based on the normal distribution, but statistical software calculates the confidence interval automatically. From the data in table 17.2, the 95% CI is 0.29 to 0.93. Because the interval excludes 1.0, this is tantamount to rejecting the null hypothesis at α = .05. Our best estimate of the relative risk is 0.52, and we are 95% confident that the true relative risk is somewhere between 0.29 and 0.93. Collectively, we would conclude that the warm-up program appears to reduce the risk of injury by somewhere between 7% and 71%. Note that the cause–effect inference stems from the design of the study (an experiment) and not from the statistical outcome per se.

From the data in table 17.2, we can calculate two other indices. First is absolute risk reduction (ARR; also known as attributable risk). With absolute risk reduction, instead of making a ratio we take the difference between the proportion of injured subjects in the exposed group and the proportion of injured subjects in the unexposed group. In equations form,

ARR = [A/(A + B)] − [C/(C + D)]. (17.02)

We simply take the difference between the numerator and denominator of the relative risk calculation. From the example data, ARR = 0.14 − 0.27 = −0.13. We can interpret this as meaning that about 13 injuries will be prevented for every 100 individuals who perform the warm-up program. The other index we can calculate is the more commonly reported value of the number needed to treat (NNT). The number needed to treat is simply the inverse of the absolute risk reduction. Here we ignore the sign and NNT = 1/0.13 = 7.69. A number needed to treat value of 7.69 indicates that we need to treat about 7.7 individuals with a warm-up program to prevent one injury.