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Test Time-To-Event Data

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Test Relative Incidence in Self Controlled Case Series Studies

You can use this calculator to perform power and sample size calculations for a time-to-event analysis, sometimes called survival analysis. A two-group time-to-event analysis involves comparing the time it takes for a certain event to occur between two groups.

For example, we may be interested in whether there is a difference in recovery time following two different medical treatments. Or, in a marketing analysis we may be interested in whether there is a difference between two marketing campaigns with regards to the time between impression and action, where the action may be, for example, buying a product.

Since 'time-to-event' methods were originally developed as 'survival' methods, the primary parameter of interest is called the hazard ratio.
The *hazard* is the probability of the event occurring in the next instant given that it hasn't yet occurred.
The *hazard ratio* is then the ratio of the hazards between two groups
Letting $\theta$ represent the hazard ratio, the hypotheses of interest are

$H_1:\theta\gt \theta_0$

or

$H_1:\theta\lt \theta_0$

where $\theta_0$ is the hazard ratio hypothesized under the null hypothesis; $\theta_0$ can also be viewed as the non-inferiority/superiority margin, just like in the other non-inferiority/superiority calculators here. The calculator above and the formulas below use the notation that

$\theta$ | is the hazard ratio |

$\ln(\theta)$ | is the natural logarithm of the hazard ratio, or the log-hazard ratio |

$p_E$ | is the overall probability of the event occurring within the study period |

$p_A$ and $p_B$ | are the proportions of the sample size allotted to the two groups, named 'A' and 'B' |

$n$ | is the total sample size |

This calculator uses the following formulas to compute sample size and power, respectively:
$$n=\frac{1}{p_A\;p_B\;p_E}\left(\frac{z_{1-\alpha}+z_{1-\beta}}{\ln(\theta)-\ln(\theta_0)}\right)^2$$

$$1-\beta= \Phi\left( z-z_{1-\alpha}\right) \quad
,\quad z=\left(\ln(\theta)-\ln(\theta_0)\right)\sqrt{n\;p_A\;p_B\;p_E}$$
where

- $n$ is sample size
- $\Phi$ is the standard Normal distribution function
- $\Phi^{-1}$ is the standard Normal quantile function
- $\alpha$ is Type I error
- $\beta$ is Type II error, meaning $1-\beta$ is power

R code to implement these functions:

hr=2 hr0=1 pE=0.8 pA=0.5 alpha=0.05/2 beta=0.20 (n=((qnorm(1-alpha)+qnorm(1-beta))/(log(hr)-log(hr0)))^2/(pA*(1-pA)*pE)) ceiling(n) # 82 (Power=pnorm((log(hr)-log(hr0))*sqrt(n*pA*(1-pA)*pE)-qnorm(1-alpha))) ## Note: divide alpha by 2 to get 2-sided test of the cited example

Chow S, Shao J, Wang H. 2008.Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series.page 177.

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