# Cohort Studies with Risk Data or Prevalence Data

Consider a cohort study of a dichotomous exposure, classified into exposed and unexposed. If the study followed all subjects for a fixed period of time and there were no important competing risks and no confounding, we could display the essential data as follows:

 Exposed Unexposed Cases a b People at risk N N

# Exposed Unexposed

Cases a b

People at risk N1 N0

From this table, it is easy to estimate the risk difference, RD, and the risk ratio, RR: To apply Equation 8-1 and 8-2 to get confidence intervals for the risk difference and the risk ratio, we need formulas for the standard error of the RD and the ln(RR): # Example: Confidence Limits for Risk Difference and Risk Ratio

As an example of risk data, consider Table 9-1, which describes recurrence risks among women with breast cancer treated either with tamoxifen or with a combination of tamoxifen and radiotherapy. From the data in Table 9-1, we can calculate a risk of recurrence of 321/686 = 0.47 among women treated with tamoxifen and radiotherapy and a risk of 411/689 = 0.60 among women treated with tamoxifen alone. The risk difference is 0.47 — 0.60 = -0.13, with the minus sign indicating that the treatment group receiving both tamoxifen and radiotherapy had the lower risk. To obtain a 90% confidence interval for this estimate of risk difference, we use Equation 8-1 and 9-2 as follows: This calculation gives 90% confidence limits around —0.13 of —0.17 and -0.08. The 90% confidence interval for the risk difference ranges from a risk that is 17% lower in absolute terms to a risk that is 8% lower in absolute terms for women receiving the combined tamoxifen and radiotherapy treatment.

We can also compute the risk ratio and its confidence interval from the same data. The risk ratio is (321/686)/(411/689) = 0.78, indicating that the group receiving combined treatment faces a risk of recurrence that is 22% lower (1 — 0.78) relative to the risk of recurrence among women receiving tamoxifen alone. The 90% lower confidence bound for the risk ratio is calculated as follows: Substituting a plus sign for the minus sign before the Z multiplier of 1.645 gives 0.85 for the upper limit. The 90% confidence interval for the risk ratio estimate of

Table 9-1 Risk of Recurrence of Breast Cancer in a Randomized Trial of Women Treated with Tamoxifen and Radiotherapy or Tamoxifen Alone

 Tamoxifen and Radiotherapy Tamoxifen Only Women with recurrence 321 411 Total women treated 686 689

Data from Feychting et al.1

0.78 is 0.72 to 0.85, which is equivalent to saying that the benefit of combined treatment ranges from a 28% lower risk to a 15% lower risk, measured in relative terms. (It is common when describing a reduced risk to convert the risk ratio to a relative decrease in risk by subtracting the risk ratio from unity; a lower limit for the risk ratio equal to 0.72 indicates a 28% lower risk because 1 — 0.72 = 0.28, or 28%.) Keep in mind that these percentages indicate a risk measured in relation to the risk among those receiving tamoxifen alone: the 28% lower limit refers to a risk that is 28% lower than the risk among those receiving tamoxifen alone.