I presume that there is a control group as one of the groups. If this is the case, we can use this as the basis for the calculation.
The second question is the nature of the design. If you have two different interventions, then you can power the trial as for a 2-group trial. You are essentially running two two-group trials.
But if the two intervention groups are different intensities of the same intervention (say, different doses) then you will need a power calculation for a clinically significant difference between them.
So – please describe the study hypothesis(es) for us!
Prior responses are correct--the most important thing is to state all hypotheses clearly. Once that's done, determine the number. One simple way of adjusting for multiple comparisons is by incorporating Bonferroni correction. Once you've stated the hypotheses you can tailor the overall Type I error according to the hypotheses by distributing portions of the Type I error to each hypothesis (note: they don't have to be equal, but when in doubt, divide the overall Type 1 error by the number of tests of hypotheses, and using that in your sample size software.
The cohort study would involve the following groups:
G1: a control group of "healthy" patients receiving a dental treatment that has a 95% success rate according to recent systematic reviews;
G2 : a group of patients at medium risk receiving the same dental treatment;
G3: a group of patients at high risk receiving the same dental treatment.
The increased risk of the last two groups is (hypothetically) dictated by general health conditions characterized by different severity. So, given that the expected incidence of a failure is 0.05 in G1, how many subjects per group do I need assuming a relative risk equal to 3 in G2 and 5 in G3? I would like to set the confidence level and power to 0.95 and 0.8 respectively.
If you can work with groups of 150 individuals each, it is rather indicated to highlight a relative risk of 3. With groups of 50 individuals each, you can only show a relative risk of 5.
Attached is a file that warrants such sizes with OpenEpi.