## B. Confidence Intervals for the Risk Ratio (Relative Risk)

The risk difference quantifies the absolute difference in risk or prevalence, whereas the relative risk is, as the name indicates, a relative measure. Both measures are useful, but they give different perspectives on the information. A cumulative incidence is a proportion that provides a measure of risk, and a relative risk (or risk ratio) is computed by taking the ratio of two proportions, p1/p2. By convention we typically regard the unexposed (or least exposed) group as the comparison group, and the proportion of successes or the risk for the unexposed comparison group is the denominator for the ratio. The parameter of interest is the relative risk or risk ratio in the population, RR=p1/p2, and the point estimate is the RR obtained from our samples.

The relative risk is a ratio and does not follow a normal distribution, regardless of the sample sizes in the comparison groups. However, the natural log (Ln) of the sample RR, is approximately normally distributed and is used to produce the confidence interval for the relative risk. Therefore, computing the confidence interval for a risk ratio is a two step procedure. First, a confidence interval is generated for Ln(RR), and then the antilog of the upper and lower limits of the confidence interval for Ln(RR) are computed to give the upper and lower limits of the confidence interval for the RR.

The data can be arranged as follows:

 With Outcome Without Outcome Total Exposed Group (1) x1 n1-x1 n1 Non-exposed Group (2) x2 n2-x2 n2

### Computation of a Confidence Interval for a Risk Ratio

RR = p1/p2  1. Compute the confidence interval for Ln(RR) using the equation above.
2. Compute the confidence interval for RR by finding the antilog of the result in step 1, i.e., exp(Lower Limit), exp (Upper Limit).

Note that the null value of the confidence interval for the relative risk is one. If a 95% CI for the relative risk includes the null value of 1, then there is insufficient evidence to conclude that the groups are statistically significantly different.

Example:

[Based on Belardinelli R, et al.: "Randomized, Controlled Trial of Long-Term Moderate Exercise Training in Chronic Heart Failure - Effects on Functional Capacity, Quality of Life, and Clinical Outcome". Circulation. 1999;99:1173-1182].

These investigators randomly assigned 99 patients with stable congestive heart failure (CHF) to an exercise program (n=50) or no exercise (n=49) and followed patients twice a week for one year. The outcome of interest was all-cause mortality. Those assigned to the treatment group exercised 3 times a week for 8 weeks, then twice a week for 1 year. Exercise training was associated with lower mortality (9 versus 20) for those with training versus those without.

 Died Alive Total Exercised 9 41 50 No Exercise 20 29 49 29 70 99

The cumulative incidence of death in the exercise group was 9/50=0.18; in the incidence in the non-exercising group was 20/49=0.4082. Therefore, the point estimate for the risk ratio is RR=p1/p2=0.18/0.4082=0.44. Therefore, exercisers had 0.44 times the risk of dying during the course of the study compared to non-exercisers. We can also interpret this as a 56% reduction in death, since 1-0.44=0.56.

The 95% confidence interval estimate for the relative risk is computed using the two step procedure outlined above.  Substituting, we get:  This simplifies to  So, the 95% confidence interval is (-1.50193, -0.14003).

A 95% confidence interval for Ln(RR) is (-1.50193, -0.14003). In order to generate the confidence interval for the risk, we take the antilog (exp) of the lower and upper limits:

exp(-1.50193) = 0.2227 and exp(-0.14003) = 0.869331

Interpretation: We are 95% confident that the relative risk of death in CHF exercisers compared to CHF non-exercisers is between 0.22 and 0.87. The null value is 1. Since the 95% confidence interval does not include the null value (RR=1), the finding is statistically significant. Consider again the randomized trial that evaluated the effectiveness of a newly developed pain reliever for patients following joint replacement surgery. Using the data in the table below, compute the point estimate for the relative risk for achieving pain relief, comparing those receiving the new drug to those receiving the standard pain reliever. Then compute the 95% confidence interval for the relative risk, and interpret your findings in words.

 Treatment Group n # with Reduction of 3+ Points Proportion with Reduction of 3+ Points New Pain Reliever 50 23 0.46 Standard Pain Reliever 50 11 0.22