# Solved – Power analysis for survival analysis

If I hypothesize that a gene signature will identify subjects at a lower risk of recurrence, that is decrease by 0.5 (hazard ratio of 0.5) the event rate in 20% of the population and I intend to use samples from a retrospective cohort study does the sample size need to be adjusted for unequal numbers in the two hypothesised groups?

For example using Collett, D: Modelling Survival Data in Medical Research, Second Edition – 2nd Edition 2003. The required total number of events, d, can be found using,

begin{equation}
d = frac{(Z_{alpha/2} + Z_{beta/2})^2}{p_1 p_2 (theta R)^2}
end{equation}

where \$Z_{alpha/2}\$ and \$Z_{beta/2}\$ are the upper \$alpha/2\$ and upper \$beta/2\$ points, respectively, of the standard normal distribution.

For the particular values,

• \$p_1 = 0.20\$
• \$p_2 = 1 – p_1\$
• \$theta R = -0.693\$
• \$alpha = 0.05\$ and so \$Z_{0.025}= 1.96\$
• \$beta = 0.10\$ and so \$Z_{0.05} = 1.28\$,

and taking \$theta R = log psi R = log 0.50 = -0.693\$, the number of events required (rounded up) to have a 90% chance of detecting a hazard ratio of 0.50 to be significant at the two-sided 5% level is then given by

begin{equation}
d = frac{(1.96 + 1.28)^2}{0.20 times 0.80times (log 0.5)^2}= 137
end{equation}

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