# Confidence Interval for a Risk Difference

As with the risk ratio and the rate ratio, a 95% confidence interval for the risk difference can be calculated to provide us with a range of plausible risk differences based on our sample. Recall that one of the assumptions in building a confidence interval is that , the measure you are building a confidence interval around, is normally distributed. Unlike the risk ratio and the rate ratio, the risk difference is normally distributed, so we can calculate the confidence interval for the risk difference directly and do not need to transform it to the log-scale. The formula for the 95% Confidence Interval for the risk difference is as follows:

Risk Difference + [1.96 x SE(Risk Difference)]

To illustrate we will use data from a prospective cohort study in which one of the associations that was examined was the association between smoking and lung cancer. To simplify we regarded anyone who smoked regularly as a "smoker."

Table - Incidence of Lung Cancer in Smokers and Non-Smokers

 Lung Cancer No Lung Cancer Total Smokers 40 20 60 Non-Smokers 10 130 140 Total 50 150 200

• Step 1: Calculate the standard error of the risk difference. Note: You may notice that this formula is different from the one in your textbook. The formula shown here is preferred; the one in the textbook is a formula that generates an approximation.

• Step 2: Calculate the lower and upper confidence bounds

- The lower bound of the 95% confidence interval of the risk difference is:
Risk Difference - [1.96 x SE[Risk Difference]] = 0.596 - [1.96 x 0.067] = 0.596 - 0.131 = 0.465

- The upper bound of the 95% confidence interval of the risk difference is:
Risk Difference + [1.96 x SE[Risk Difference]] = 0.596 + [1.96 x 0.067] = 0.596 + 0.131 = 0.727

• Step 3: Report and interpret the estimate and the confidence interval

There were 596 excess lung cancer cases per 1000 subjects in the group that smoked, compared to the group who did not smoke during the study period. Based on this sample, we are 95% confident that the true risk difference lies between 465/1000 and 727/1000 excess cases of lung cancer.

As a rule of thumb, you should use the following wording to interpret the 95% confidence interval for the risk difference: "Based on this sample, we are 95% confident that the true risk difference lies between [lower bound] and [upper bound] excess/fewer cases of [disease]."