How to Calculate a Confidence Interval for Relative Risk

---

We often calculate relative risk when analyzing a 2×2 table, which takes on the following format:

The relative risk tells us the probability of an event occurring in a treatment group compared to the probability of an event occurring in a control group.

It is calculated as:

• Relative risk = [A/(A+B)] / [C/(C+D)]

We can then use the following formula to calculate a confidence interval for the relative risk (RR):

• Lower 95% CI = e^{ln(RR) – 1.96√1/a + 1/c – 1/(a+b) – 1/(c+d)}
• Upper 95% CI = e^{ln(RR) + 1.96√1/a + 1/c – 1/(a+b) – 1/(c+d)}

The following example shows how to calculate a relative risk and a corresponding confidence interval in practice.

Example: Calculating a Confidence Interval for Relative Risk

Suppose a basketball coach uses a new training program to see if it increases the number of players who are able to pass a certain skills test, compared to an old training program.

The coach recruits 50 players to use each program. The following table shows the number of players who passed and failed the skills test, based on the program they used:

We can calculate the relative risk as:

• Relative Risk = [A/(A+B)] / [C/(C+D)]
• Relative Risk = [34/(34+16)] / [39/(39+11)]
• Relative Risk = 0.8718

We would interpret this to mean that the probability that a player passes the test by using the new program are just 0.8718 times the probability that a player passes the test by using the old program.

In other words, the probability that a player passes the test are actually lowered by using the new program.

We can then use the following formulas to calculate the 95% confidence interval for the relative risk:

• Lower 95% CI = e^{ln(.8718) – 1.96√(1/34 + 1/39 – 1/(34+16) – 1/(39+11)} = 0.686
• Upper 95% CI = e^{ln(.8718) + 1.96√(1/34 + 1/39 + 1/(34+16) – 1/(39+11)} = 1.109

Thus, the 95% confidence interval for the relative risk is [0.686, 1.109].

We are 95% confident that the true relative risk between the new and old training program is contained in this interval.

Since this confidence interval contains the value 1, it is not statistically significant.

This should make sense if we consider the following:

• A relative risk greater than 1 would mean that the probability that a player passes the test by using the new program is higher than the probability that a player passes the test by using the old program.
• A relative risk less than 1 would mean that the probability that a player passes the test by using the new program is lower than the probability that a player passes the test by using the old program.

So, since our 95% confidence interval for the relative risk contains the value 1, it means the probability of a player passing the skills test using the new program may or may not be higher than the probability of the same player passing the test using the old program.

Additional Resources

The following tutorials provide additional information on odds ratios and relative risk:

How to Interpret Odds Ratios
How to Interpret Relative Risk
How to Calculate Odds Ratio and Relative Risk in Excel