ynegbinompowersim {NBDesign}

Two-sample sample size calculation for negative binomial distribution with variable follow-up

Description

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.

Usage

ynegbinompowersim(nsize=200,r0=1.0,r1=0.5,shape0=1,shape1=shape0,pi1=0.5,
   alpha=0.05,twosided=1,fixedfu=1,type=1,u=c(0.5,0.5,1),ut=c(0.5,1.0,1.5),
   tfix=ut[length(ut)]+0.5,maxfu=10.0,tchange=c(0,0.5,1),
   ratec1=c(0.15,0.15,0.15),ratec0=ratec1,rn=10000)

Arguments

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 fixedfu; type=2: same as 1 but subject to censoring; type=3: depending on entry time, minimum fu is fixedfu and maximum fu is maxfu; type=4: same as 3 but subject to censoring

u

recruitment rate

ut

recruitment interval, must have the same length as u

tfix

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

maxfu

maximum follow-up time, should not be greater than tfix

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 tchange must have the same length.

ratec1

piecewise constant drop-out rate for the treatment

ratec0

piecewise constant drop-out rate for the control

rn

Number of repetitions

Details

Let \tau_{min} and \tau_{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=\tau_{min}. For type 2 follow-up T_f=min(C,\tau_{min}). For type 3 follow-up, T_f=min(R+\tau_{min}-E,\tau_{max}). For type 4 follow-up, T_f=min(R+\tau_{min}-E,\tau_{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,\ldots,n. Suppose that Y_i follows a negative binomial distribution such that

P(Y_i=y\mid Z_i=j)=\frac{\Gamma(y+1/k_j)}{\Gamma(y+1)\Gamma(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)+\beta_0+\beta_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".

Value

power

simulation power (in percentage)

Author(s)

Xiaodong Luo

Examples

##calculating the sample sizes
abc=ynegbinompowersim(nsize=200,r0=1.0,r1=0.5,shape0=1,
        pi1=0.5,alpha=0.05,twosided=1,fixedfu=1,
        type=4,u=c(0.5,0.5,1),ut=c(0.5,1.0,1.5),
        tchange=c(0,0.5,1),
        ratec1=c(0.15,0.15,0.15),rn=10)
###Power
abc$power