Although some sample size programs are able to estimate unequal sizes, most calculate sample size assuming that the two groups are of equal size. This page provides algorithms to convert sample size for equal size groups to sample sizes for unequal size groups that have the same power. Two calculations are offered.

**Adjustment of sample sizes by ratio**. This converts sample size per group of two equal size groups to two samples sizes
that has a nominated ratio, retaining the same statistical power. The data required are the original sample size
(assuming equal size groups), and the nominated ratio.
**An example** We wish to compare the prevalence of depression between male and female residents in retirement communities.
We anticipate that about 10% of men (proportion = 0.1) and 5% (proportion = 0.05) of women in those communities may be depressed.
Using power of 0.8 and α of 0.05, we found from https://www.statstodo.com/SSiz2Props.php that we
need 435 cases of each sex for the study.

Unfortunately, in retirement communities, women outnumbers men by 2 to 1, and we wanted the same sex ratio in our study. Using the
program in the next panel to adjust the original sample size of 435 each group to a ratio of 2 to 1, we
estimate that the sample sizes required to achieve the same statistical power are 653 women and 327 men.

**Adjustment by the sample size in one group** This converts sample size per group of two equal size groups to two samples
sizes with a nominated number in one group, retaining the original statistical power. The data required are the original
sample size per group, and the nominated sample size for one of the groups. **Please note : ** that the nominated sample size
cannot be <= half of the original samle size, or an error in calculation will result.
**An example** We wish to study the cost of delivering babies with breech presentation, comparing
vaginal delivery and Caesarean section. From experience, we estimate the standard deviation of costs
for delivery to be $1000, and we would conclude that costs are different if they differ more than $500
(effect size = diff / sd = 0.5). Using power of 0.8 and α of 0.05, we found from https://www.statstodo.com/SSiz2Means.php that we need 64 cases for each of the two methods of delivery.

Unfortunately, most of the babies with breech were delivered by Caesarean section, and we can only find 40 records of vaginal
delivery. We therefore need to know how many records of Caesarean Section we will need to be able to compare with the
same statistical power as the original model hat assumed equal size groups. Using the program in the next panel and adjust the original sample size of 64 each group to 40 for vaginal delivery, we
estimated that we will require 160 records of Caesarean Section to enable a comparison with the same power as 64 per group.

**Adjustment from unequal sample sizes to equal sample size**. After data collection is completed, the sample sizes in the two groups may not be equal. This can be planned, or related to missing data, cases dropped out, and so on. When power calculation is required, and if the power calculation is based on sample size of equal groups, there is a need to translate the unequal sample sizes to an equal sample size that has the same power. The data required are the two unequal sample sizes, and the calculation produces the single sample size for both groups with equivalent power.
**An Example** In the example comparing costs of vaginal delivery and Caesarean Section, we managed to collected 56 vaginal deliveries and 180 Caesarean sections. For a power analysis, we wish to know what equal sample sizes have the same power. Using the program in the next panel, we determined that 86 cases in each group has the same statistical power as 56 in one group and 180 in the other

### References

Altman, DG. Practical Statistics for Medical Research London, UK; Chapman & Hall; 1991.

Altman, DG. How large a sample? In: Gore SM, Altman DG. , editor.
Statistics in Practice. London, UK: British Medical Association; 1982.

Whitley E and Ball J (2002) Statistics review 4: Sample size calculations
Crit Care. 6(4): 335-341.

Gerald van Belle (2002). Statistical Rules of Thumb. John Wiley and Sons,
New York. ISBN 0-471-40227-3. p. 45-46