# Sample Size Calculation in the Analysis of Survival Data for Clinical Trials

### Description

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

### Usage

1 | ```
ssizeCT.default(power, k, pE, pC, RR, alpha = 0.05)
``` |

### Arguments

`power` |
power to detect the magnitude of the hazard ratio as small as that specified by |

`k` |
ratio of participants in group E (experimental group) compared to group C (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. |

### Details

This is an implementation of the sample size 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 number of participants needed in each group to
achieve a power of *1-β* is

*n_E=\frac{m k}{k p_E + p_C}, n_C=\frac{m}{k p_E + p_C}*

where

*m=\frac{1}{k}≤ft(\frac{k RR + 1}{RR - 1}\right)^2≤ft(
z_{1-α/2}+z_{1-β}
\right)^2,*

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

### Value

A two-element vector. The first element is *n_E* and the second
element is *n_C*.

### Note

(1) The sample size formula assumes that the central-limit theorem is valid and hence is appropriate for large samples.
(2) *n_E* and *n_C* will be rounded up to integers.

### References

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.

### See Also

`ssizeCT`

### Examples

1 2 3 4 | ```
# Example 14.42 in Rosner B. Fundamentals of Biostatistics.
# (6-th edition). (2006) page 809
ssizeCT.default(power = 0.8, k = 1, pE = 0.3707, pC = 0.4890,
RR = 0.7, alpha = 0.05)
``` |