Description Usage Arguments Details Value Author(s) Examples

This will calculate the power for the negative binomial distribution for the 2-sample case under different follow-up scenarios: 1: fixed follow-up, 2: fixed follow-up with drop-out, 3: variable follow-up with a minimum fu and a maximum fu, 4: variable follow-up with a minimum fu and a maximum fu and drop-out.

1 2 3 4 |

`nsize` |
total number of subjects in two groups |

`r0` |
event rate for the control |

`r1` |
event rate for the treatment |

`shape0` |
dispersion parameter for the control |

`shape1` |
dispersion parameter for the treatment |

`pi1` |
allocation prob for the treatment |

`alpha` |
type-1 error |

`twosided` |
1: two-side, others: one-sided |

`fixedfu` |
fixed follow-up time for each patient |

`type` |
follow-up time type, type=1: fixed fu with fu time |

`u` |
recruitment rate |

`ut` |
recruitment interval, must have the same length as |

`tfix` |
fixed study duration, often equals to recruitment time plus minimum follow-up |

`maxfu` |
maximum follow-up time, should not be greater than |

`tchange` |
a strictly increasing sequence of time points starting from zero at which the drop-out rate changes. The first element of tchange must be zero. The above rates and |

`ratec1` |
piecewise constant drop-out rate for the treatment |

`ratec0` |
piecewise constant drop-out rate for the control |

`rn` |
Number of repetitions |

Let *τ_{min}* and *τ_{max}* correspond to the minimum follow-up time `fixedfu`

and the maximum follow-up time `maxfu`

. Let *T_f*, *C*, *E* and *R* be the follow-up time, the drop-out time, the study entry time and the total recruitment period(*R* is the last element of `ut`

). For type 1 follow-up, *T_f=τ_{min}*. For type 2 follow-up *T_f=min(C,τ_{min})*. For type 3 follow-up, *T_f=min(R+τ_{min}-E,τ_{max})*. For type 4 follow-up, *T_f=min(R+τ_{min}-E,τ_{max},C)*. Let *f* be the density of *T_f*.
Suppose that *Y_i* is the number of event obsevred in follow-up time *t_i* for patient *i* with treatment assignment *Z_i*, *i=1,…,n*. Suppose that *Y_i* follows a negative binomial distribution such that

*P(Y_i=y\mid Z_i=j)=\frac{Γ(y+1/k_j)}{Γ(y+1)Γ(1/k_j)}\Bigg(\frac{k_ju_i}{1+k_ju_i}\Bigg)^y\Bigg(\frac{1}{1+k_ju_i}\Bigg)^{1/k_j},*

where *k_j, j=0,1* are the dispersion parameters for control *j=0* and treatment *j=1* and

*\log(u_i)=\log(t_i)+β_0+β_1 Z_i.*

The data will be gnerated according to the above model. Note that the piecewise exponential distribution and the piecewise uniform distribution are genrated using R package PWEALL functions "rpwe" and "rpwu", respectively.

The parameters in the model are estimated by MLE using the R package MASS function "glm.nb".

`power` |
simulation power (in percentage) |

Xiaodong Luo

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