# Sample size / power calculations for Kaplan-Meier survival curves

The problem is simple: we have two groups of animals, treated and controls. Around 20% of the untreated animals will die during the course of the experiment, and we would like to be able to detect effect such that instead of 20%, 80% of animals will die in the treated group, with power 0.8 and alpha=0.05. Group sizes are equal and no other parameters are given.

What is the necessary group size?

I used the ssizeCT.default function from the powerSurvEpi R package. Based on the explanation in the package manual, this calculates (in my simple case) the required sample size in a group as follows:

$n = \frac{m}{p_E + p_C}$

where $p_E$ and $p_C$ are, respectively, probabilities of failure in the E(xpermiental) and C(ontrol) groups. I assume that in my case I should use 0.8 and 0.2, respectively, so $n=m$. The formulas here are simplified in comparison with the manual page of ssizeCT.default, simply because the group sizes are identical.

$m$ is calculated as

$m=\big(\frac{RR+1}{RR-1}\big)^2(z_{1-\alpha/2}+z_{1-\beta})^2$

RR is the minimal effect size that we would like to be able to observe with power 0.8 and at alpha 0.05. That means, if the real effect size is RR or greater, we have 80% chance of getting a p-value smaller than 0.05 if the group sizes are equal to $m$. To calculate RR, I first calculate $\theta$, the hazard ratio, and for this I use the same approximate, expected mortality rates (20% and 80%):

$\theta = \log(\frac{\log(0.8)}{\log(0.2)}) = -1.98$

Since $RR=exp(\theta)=0.139$; thus $m=18.3$. This seems reasonable (based on previous experience).