Power calculation for the Comparison of Survival Curves Between Two Groups under the Cox Proportional-Hazards Model for clinical trials.

1 | ```
powerCT.default(nE, nC, pE, pC, RR, alpha = 0.05)
``` |

`nE` |
number of participants in the experimental group. |

`nC` |
number of participants in the control group. |

`pE` |
probability of failure in group E (experimental group) over the maximum time period of the study (t years). |

`pC` |
probability of failure in group C (control group) over the maximum time period of the study (t years). |

`RR` |
postulated hazard ratio. |

`alpha` |
type I error rate. |

This is an implementation of the power calculation method described in Section 14.12 (page 807) of Rosner (2006). The method was proposed by Freedman (1982).

Suppose we want to compare the survival curves between an experimental group (*E*) and
a control group (*C*) in a clinical trial with a maximum follow-up of *t* years.
The Cox proportional hazards regression model is assumed to have the form:

*h(t|X_1)=h_0(t)\exp(β_1 X_1).*

Let *n_E* be the number of participants in the *E* group
and *n_C* be the number of participants in the *C* group.
We wish to test the hypothesis *H0: RR=1* versus *H1: RR* not equal to 1,
where *RR=\exp(β_1)=*underlying hazard ratio
for the *E* group versus the *C* group. Let *RR* be the postulated hazard ratio,
*α* be the significance level. Assume that the test is a two-sided test.
If the ratio of participants in group
E compared to group C *= n_E/n_C=k*, then the power of the test is

*power=Φ(√{k*m}*|RR-1|/(k*RR+1)-z_{1-α/2}),*

where

*m=n_E p_E+n_C p_C,*

and *z_{1-α/2}*
is the *100 (1-α/2)* percentile of
the standard normal distribution *N(0, 1)*, *Φ* is the cumulative distribution function (CDF)
of *N(0, 1)*.

The power of the test.

The power formula assumes that the central-limit theorem is valid and hence is appropriate for large samples.

Freedman, L.S. (1982).
Tables of the number of patients required in clinical trials using the log-rank test.
*Statistics in Medicine*. 1: 121-129

Rosner B. (2006).
*Fundamentals of Biostatistics*. (6-th edition). Thomson Brooks/Cole.

1 2 3 4 | ```
# Example 14.42 in Rosner B. Fundamentals of Biostatistics.
# (6-th edition). (2006) page 809
powerCT.default(nE = 200, nC = 200, pE = 0.3707, pC = 0.4890,
RR = 0.7, alpha = 0.05)
``` |

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