Power and sample size calculation

Some notes on the behaviour of sample size calculations and power for unequal group sizes

This information was omitted from the video and so is included here as text.

In many scenarios the ratio of participants in the control to treatment group, or in the exposed to not exposed group, is unequal. In such settings we need a bigger TOTAL sample size to get the SAME statistical power for a given effect size (δ) and level of significance (α).

There are many reasons why we might have unequal size groups. For example, in an observational study, this might just reflect the proportion of people exposed and not exposed to the factor of interest; in a case-control study, the outcome may be rare and so we choose to sample more controls to increase study power as they are easier and less costly to recruit; in a randomised controlled trial, the treatment may be expensive and so is only given to a smaller proportion of patients.

To illustrate the effect of unequal groups on power, let’s suppose we want to detect a difference of 250g in birth weight between two groups of babies born to overweight mums. One group of mothers received an intensive education and consultation programme to help them eat and exercise healthily during pregnancy. We assume the population SD in each group is 400g and the total sample size is 100.

Table 1 below shows that if the groups are of equal size (a 1:1 ratio), then the power is 0.87. The study has an 87% chance of detecting a true difference in birth weight of 250g. The power reduces as the group sizes become more and more unequal. So we get maximum power for a given Total sample size when the groups are of equal size.

Table 1.

Sample size		Ratio (intervention:control)	Power	N of controls needed to get the same   power as a 1:1 ratio
Intervention	Control
50	50	1:1	0.88
33	77	1:2	0.84	106
25	75	1:3	0.77	>5000
20	80	1:4	0.71

 

Another learning point about power comes from table 1. The last row column shows the number of controls needed to achieve the same power as that for the 1:1 ratio in the study of 100 mums (0.88). You can see that if we had 33 mums in the intervention group, then we need 106 mums in the control group to get 88% power. However if we had 25 mums in the intervention group then even if we were able to recruit over 25000 controls, we would still not achieve the same power as the 1:1 study of 100 patients. So there is a limit to how much power we can get by recruiting more controls. The behaviour of this is complex as it depends on the size of the smaller group and the SD.

As mentioned above, in a case-control study it is often easier to recruit more controls since we often use the case-control design to study rare outcomes. Directly following on from the previous paragraph, we can also work out the extra increase in power from increasing the ratio of controls to cases in a case control study. While this is also dependent the parameters that go into a power calculation, in particular the effect size and sample size of the case group, there is a useful rule of thumb that is worth remembering. The extra increase in power from increasing the case:control ratio falls dramatically once the ratio goes beyond 1:4, so there is little extra gain in power from recruiting more than 4 controls to one case.