# 10.8 - Additional Sample Size Topics

## Ratio of Cases to Controls Section

Another consideration for sample size is if the same number of cases and controls should be used.
Power increases but at a decreasing rate as the ratio of controls/cases increases. Little additional power is gained at ratios higher than four controls/cases. There is little benefit to enrolling a greater ratio of controls to cases.

Under what circumstances would it be recommended to enroll a large number of controls compared to cases?

Perhaps the small gain in power is worthwhile if the cost of a Type II error is large and the expense of obtaining controls is minimal, such as selecting controls with covariate information from a computerized database. If you must physically locate and recruit the controls, set up clinic appointments, run diagnostic tests, and enter data, the effort of pursuing a large number of controls quickly offsets any gain. You would use a one-to-one or two-to-one range. The bottom line is there is little additional power beyond a four-to-one ratio.

## Cohort v Case-control sample sizes Section

Sample sizes for cohort studies depend upon the rate of the outcome, not the prevalence of exposure. Sample size for case-control studies is dependent upon prevalence of exposure, not the rate of outcome. Because the rate of outcome is usually smaller than the prevalence of the exposure, cohort studies typically require larger sample sizes to have the same power as a case-control study.

The example below is from a study of smoking and coronary heart disease where the background incidence rate was 0.09 events per person-year among the non-exposed group and the prevalence of the risk factor was 0.3.

The sample size requirements to detect a given relative risk with the 90% power using two-sided 5% significance tests for cohort and case-control studies are listed below:

Relative Risk Cohort study Case-Control study
1.1 44,398 21,632
1.2 11,568 5,820
1.3 5,346 2,774
1.4 3,122 1,668
1.5 2,070 1,138
2 602 376
3 188 146
from Woodward, M. Epidemiology Study Design and Analysis. Boca Raton: Chapman and Hall:, 2013, p.321

In such a situation, with a relative risk of 1.1, more than twice the number of subjects are required for a cohort study as for a case-control study. In every study in the table, the case-control design requires a smaller sample than does the cohort study to detect the same level of increased risk. This is generally true. There is also a dependence upon the rate of the outcome, but in general, case-control studies involve less sampling.

Furthermore, in designing a cohort study, loss-to-follow-up is important to consider. Based on your own experience or that of the literature, any sample size calculation should be inflated to account for the expected drop-outs. For example, if the drop-out rate is expected to be 5%, multiply n by 1/(1-0.05) and recruit the increased number of subjects.