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 0.80\times (\log 0.5)^2}= 137 \end{equation}

  • $\begingroup$ I hope you don't mind, but I converted your question into latex. One thing, should $Z_{\alpha}$ not be $Z_{\alpha/2}$ $\endgroup$ Feb 22, 2011 at 17:04
  • $\begingroup$ If it is clear for folks I don't mind at all. You are right should be a 2-sided alpha. $\endgroup$
    – user712
    Feb 22, 2011 at 17:35
  • $\begingroup$ What are $\theta R$ and $\psi R$ ? Should they be $\theta_R$ and $\psi_R$? $\endgroup$
    – onestop
    Feb 22, 2011 at 19:02

1 Answer 1


Yes, your power will change based on the ratio of exposed to unexposed. For example, in a recent study I did the power calculations for, at an equal sample size, an Exposed:Unexposed ratio of 1:2 achieved power = 0.80 at a HR of ~1.3. It took until HR ~1.6 or so for a ratio of 1:10.

In your case, since the sample size will vary but your HR won't, the smaller the ratio, the larger your sample size will need to be.


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