B. Confidence Intervals for the 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

 

 

 

 

 

Confidence Interval for RR = p1/p2

CI-RR.png

  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.

 CI_RiskRatio_CHF.png

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

 

Thus 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.

 

Thinking.gif 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

Answer 

Confidence Intervals for the Odds Ratio

In case-control studies it is not possible to estimate a relative risk, because the denominators of the exposure groups are not known with a case-control sampling strategy. Nevertheless, one can compute an odds ratio, which is a similar relative measure of effect.6 (For a more detailed explanation of the case-control design, see the module on case-control studies in Introduction to Epidemiology). 

Consider the following hypothetical study of the association between pesticide exposure and breast cancer in a population of 6, 647 people. If data were available on all subjects in the population the the distribution of disease and exposure might look like this:

 

Diseased

Non-diseased

Total

Pesticide Exposure

7

1,000

1,007

Non-exposed

6

5,634

5,640

If we had such data on all subjects, we would know the total number of exposed and non-exposed subjects, and within each exposure group we would know the number of diseased and non-disease people, so we could calculate the risk ratio. In this case RR = (7/1,007) / (6/5,640) = 6.52, suggesting that those who had the risk factor (exposure) had 6.5 times the risk of getting the disease compared to those without the risk factor.

However, suppose the investigators planned to determine exposure status by having blood samples analyzed for DDT concentrations, but they only had enough funding for a small pilot study with about 80 subjects in total. The problem, of course, is that the outcome is rare, and if they took a random sample of 80 subjects, there might not be any diseased people in the sample. To get around this problem, case-control studies use an alternative sampling strategy: the investigators find an adequate sample of cases from the source population, and determine the distribution of exposure among these "cases". The investigators then take a sample of non-diseased people in order to estimate the exposure distribution in the total population. As a result, in the hypothetical scenario for DDT and breast cancer the investigators might try to enroll all of the available cases and 67 non-diseased subjects, i.e., 80 in total since that is all they can afford. After the blood samples were analyzed, the results might look like this:

 

Diseased

Non-diseased

 

Pesticide Exposure

7

10

 

Non-exposed

6

57

 

With this sampling approach we can no longer compute the probability of disease in each exposure group, because we just took a sample of the non-diseased subjects, so we no longer have the denominators in the last column. In other words, we don't know the exposure distribution for the entire source population. However, the small control sample of non-diseased subjects gives us a way to estimate the exposure distribution in the source population. So, we can't compute the probability of disease in each exposure group, but we can compute the odds of disease in the exposed subjects and the odds of disease in the unexposed subjects.

The Difference Between "Probability" and "Odds"?

  • The probability that an event will occur is the fraction of times you expect to see that event in many trials. Odds.Probability.png Probabilities always range between 0 and 1.
  • The odds are defined as the probability that the event will occur divided by the probability that the event will not occur.

If the probability of an event occurring is Y, then the probability of the event not occurring is 1-Y. (Example: If the probability of an event is 0.80 (80%), then the probability that the event will not occur is 1-0.80 = 0.20, or 20%.

The odds of an event represent the ratio of the (probability that the event will occur) / (probability that the event will not occur). This could be expressed as follows:

Odds of event = Y / (1-Y)

So, in this example, if the probability of the event occurring = 0.80, then the odds are 0.80 / (1-0.80) = 0.80/0.20 = 4 (i.e., 4 to 1).

  • If a race horse runs 100 races and wins 25 times and loses the other 75 times, the probability of winning is 25/100 = 0.25 or 25%, but the odds of the horse winning are 25/75 = 0.333 or 1 win to 3 loses.
  • If the horse runs 100 races and wins 5 and loses the other 95 times, the probability of winning is 0.05 or 5%, and the odds of the horse winning are 5/95 = 0.0526.
  • If the horse runs 100 races and wins 50, the probability of winning is 50/100 = 0.50 or 50%, and the odds of winning are 50/50 = 1 (even odds).
  • If the horse runs 100 races and wins 80, the probability of winning is 80/100 = 0.80 or 80%, and the odds of winning are 80/20 = 4 to 1.

NOTE that when the probability is low, the odds and the probability are very similar.

With the case-control design we cannot compute the probability of disease in each of the exposure groups; therefore, we cannot compute the relative risk. However, we can compute the odds of disease in each of the exposure groups, and we can compare these by computing the odds ratio. In the hypothetical pesticide study the odds ratio is

OR= (7/10) / (5/57) = 6.65

Notice that this odds ratio is very close to the RR that would have been obtained if the entire source population had been analyzed. The explanation for this is that if the outcome being studied is fairly uncommon, then the odds of disease in an exposure group will be similar to the probability of disease in the exposure group. Consequently, the odds ratio provides a relative measure of effect for case-control studies, and it provides an estimate of the risk ratio in the source population, provided that the outcome of interest is uncommon.

We emphasized that in case-control studies the only measure of association that can be calculated is the odds ratio. However, in cohort-type studies, which are defined by following exposure groups to compare the incidence of an outcome, one can calculate both a risk ratio and an odds ratio.

If we arbitrarily label the cells in a contingency table as follows:

 

Diseased

Non-diseased

 

Exposed

a

b

 

Non-exposed

c

d

 

then the odds ratio is computed by taking the ratio of odds, where the odds in each group is computed as follows:

OR = (a/b) / (c/d)

As with a risk ratio, the convention is to place the odds in the unexposed group in the denominator. In addition, like a risk ratio, odds ratios do not follow a normal distribution, so we use the log transformation to promote normality. As a result, the procedure for computing a confidence interval for an odds ratio is a two step procedure in which we first generate a confidence interval for Ln(OR) and then take the antilog of the upper and lower limits of the confidence interval for Ln(OR) to determine the upper and lower limits of the confidence interval for the OR. The two steps are detailed below.

 

Confidence Interval for OR

CI-Ln(OR).png

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

The null, or no difference, value of the confidence interval for the odds ratio is one. If a 95% CI for the odds ratio does not include one, then the odds are said to be statistically significantly different. We again reconsider the previous examples and produce estimates of odds ratios and compare these to our estimates of risk differences and relative risks.

 

Example

Consider again the hypothetical pilot study on pesticide exposure and breast cancer:

 

Diseased

Non-diseased

 

Pesticide Exposure

7

10

 

Non-exposed

6

57

 

We noted above that

OR= (7/10) / (5/57) = 6.65

 

We can compute a 95% confidence interval for this odds ratio as follows:

 CI_OR_DDTexample.png

Interpretation:

The odds of breast cancer in women with high DDT exposure are 6.65 times greater than the odds of breast cancer in women without high DDT exposure. We are 95% confident that the true odds ratio is between 1.85 and 23.94. The null value is 1, and because this confidence interval does not include 1, the result indicates a statistically significant difference in the odds of breast cancer women with versus low DDT exposure.

Note that, while this result is considered statistically significant, the confidence interval is very broad, because the sample size is small. As a result, the point estimate is imprecise. Notice also that the confidence interval is asymmetric, i.e., the point estimate of OR=6.65 does not lie in the exact center of the confidence interval. Remember that we used a log transformation to compute the confidence interval, because the odds ratio is not normally distributed. Therefore, the confidence interval is asymmetric, because we used the log transformation to compute Ln(OR) and then took the antilog to compute the lower and upper limits of the confidence interval for the odds ratio.

 

Thinking.gif Remember that in a true case-control study one can calculate an odds ratio, but not a risk ratio. However, one can calculate a risk difference (RD), a risk ratio (RR), or an odds ratio (OR) in cohort studies and randomized clinical trials. Consider again the data in the table below from the randomized trial assessing the effectiveness of a newly developed pain reliever as compared to the standard of care. Remember that a previous quiz question in this module asked you to calculate a point estimate for the difference in proportions of patients reporting a clinically meaningful reduction in pain between pain relievers as (0.46-0.22) = 0.24, or 24%, and the 95% confidence interval for the risk difference was (6%, 42%). Because the 95% confidence interval for the risk difference did not contain zero (the null value), we concluded that there was a statistically significant difference between pain relievers. Using the same data, we then generated a point estimate for the risk ratio and found RR= 0.46/0.22 = 2.09 and a 95% confidence interval of (1.14, 3.82). Because this confidence interval did not include 1, we concluded once again that this difference was statistically significant. We will now use these data to generate a point estimate and 95% confidence interval estimate for the odds ratio.

We now ask you to use these data to compute the odds of pain relief in each group, the odds ratio for patients receiving new pain reliever as compared to patients receiving standard pain reliever, and the 95% confidence interval for the odds ratio.

 

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

Answer 

 

LightBulb.jpg

When the study design allows for the calculation of a relative risk, it is the preferred measure as it is far more interpretable than an odds ratio. The odds ratio is extremely important, however, as it is the only measure of effect that can be computed in a case-control study design. When the outcome of interest is relatively rare (<10%), then the odds ratio and relative risk will be very close in magnitude. In such a case, investigators often interpret the odds ratio as if it were a relative risk (i.e., as a comparison of risks rather than a comparison of odds which is less intuitive).