# Sample Size calculation using T-test for four groups

I have a small study I am running to test feasibility. I have three intervention arms and one control. I am curious if one of the interventions is >>> placebo. The three interventions are: 1) Intervention A, 2) Intervention B, 3) Intervention A+B

I am wondering if to justify my sample size, I can run sample size calculations via t-test between two groups (intervention A+B vs placebo) instead of running sample size calculation via ANOVA. ANOVA sample size calculations with four groups are much too large. The other intervention arms I thought could be used as power analysis for a big study, and feasibility could be showed by comparing one intervention arm to placebo.

Would this be normal as I write up the sample size justification?

The sample size justification you use will be driven by:

1. The hypothesis or hypotheses you wish to test;
2. The assumed effect size you find meaningful;
3. The variation(s) in treatment differences;
4. The tolerance for Type I error;
5. The desired power to detect the treatment difference if it exists;
6. The allocation ratio(s) between treatment groups;
A popular justification in a multi-arm placebo controlled study is to consider the sample size needed to detect a particular treatment difference in a single active arm against placebo. In other words, suppose you have historical evidence to suggest that A and B behave synergistically, so that your largest effect size will be in the combined treatment arm against placebo. Then you can calculate a sample size based on this single contrast; e.g., a two-sample $t$-test of $n$ subjects per arm will have $90\%$ power to detect a treatment difference of $\delta$ with an assumed SD of $\sigma$ at a two-sided $\alpha$ level of $0.05$. Then you would use $N = 4n$ as your total sample size assuming an equal allocation ratio between treatment arms.
Or you can power your study for the smallest hypothesized treatment difference; e.g., assume the weakest difference you wish to detect is $\delta^*$. In such a case I would recommend an allocation ratio of $3:1$, as if you might pool the active treatment arms and compare that against placebo. You might not do that when you actually analyze the study; but for purposes of estimating a required sample size, the goal is to ensure that you enroll enough subjects to have at least the nominal power you set.
These are not the only possibilities. There exist more sophisticated ways to power such a study design, but almost invariably they make certain assumptions that may not hold, and the two-sample $t$-test is in many cases the conservative approach.