# ANCOVA controlling baseline in RCT - power analysis and sample size questions

I am performing a power analysis/sample size estimation for an RCT. We will be controlling for baseline symptoms and using post-treatment or change scores as our outcome variable. This design is referred to as "ANCOVA" and designs with these analyses are more robust than change-score analyses such as the paired t-test: https://trialsjournal.biomedcentral.com/articles/10.1186/s13063-019-3671-2

Is there a way to estimate sample size for estimation of such a model with a given power?

Yes.

ANCOVA is just linear regression. Given a smallest clinically meaningful effect for a variable of interest, the sample size required is

$$n = \dfrac{(z_{1-\alpha/2} + z_\gamma)^2 \sigma_{y \vert x}}{(\beta \sigma_x)^2(1-\rho^2)}$$

Here,

• $$z_{1-\alpha/2}$$ is the z quantile for your desired alpha rate. If you are using the standard 5% alpha, then this is 1.96.
• $$z_\gamma$$ is the z quantile for your desired power. If you are using 80% power, this is roughly 0.84
• $$\sigma_{y \vert x}$$ is the noise in the measurement. Its akin to the standard deviation of the residuals, where you to have the perfect model.
• $$\beta$$ is the smallest effect you wish to detect
• $$\sigma_x$$ is the standard deviation of the predictor
• $$(1-\rho^2)$$ is the variance inflation factor, which can be tough to know a priori.

I got these equations from Regression Methods in Biostatistics by Vittinghoff and colleagues. More information is included at the end of chapter 4 on linear regression.

ANCOVA is not more powerful than a paired t-test as you originally said, it is simply that if the change from baseline is not constant in the treatment / control groups respectively, ANCOVA is more robust to detect an effect than a paired t-test.

You will often find in power/sample size calculations that you are dealing with simplified modeling assumptions. If we begin with the assumptions of the paired t-test, the resulting sample size is the same as that of the ANCOVA minus one for the degree of freedom lending to estimating the "baseline" parameter. I am assuming the power/sample size calculation for the t-test is something you can easily access.

If the model effect is something different than that, you will need to specify exactly what the departure is, and probably use some form of simulation to find the power.