ynegbinomsize {NBDesign}

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

Description

This will calculate the sample size 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

ynegbinomsize(r0=1.0,r1=0.5,shape0=1,shape1=shape0,pi1=0.5,
      alpha=0.05,twosided=1,beta=0.2,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,eps=1.0e-03)

Arguments

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

beta

tyep-2 error

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 fixedfu

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.

ratec1

piecewise constant drop-out rate for the treatment. The rate and tchange must have the same length.

ratec0

piecewise constant drop-out rate for the control. The rate and tchange must have the same length.

eps

error tolerance for the numerical intergration

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

\log(u_i)=\log(t_i)+\beta_0+\beta_1 Z_i.

Let \hat{\beta}_0 and \hat{\beta}_1 be the MLE of \beta_0 and \beta_1. The varaince of \hat{\beta}_1 is

\mbox{var}(\hat{\beta}_1)=1/\tilde{a}_0(r_0)+1/\tilde{a}_1(r_1)

where

\tilde{a}_j(r)=\sum_{i=1}^n I(Z_i=j)k_jrt_i/(1+k_jrt_i), \hspace{0.5cm}j=0,1,

and k_j, j=0,1 are the dispersion parameters for control j=0 and treatment j=1. Note that Zhu and Lakkis (2014) use

a_j(r)=\sum_{i=1}^n I(Z_i=j)k_jrE(t_i)/\{1+k_jrE(t_i)\},

to replace \tilde{a}_j(r), j=0,1. Using Jensen's inequality, we can show a_j(r)\ge \tilde{a}_j(r), which means Zhu and Lakkis's method will underestimate variance of \hat{\beta}_1, which leads to either smaller than required sample size or inflated power. For comparison, I provide sample sizes under both \tilde{a}_j(r) and a_j(r).

Zhu and Lakkis (2014) discuss three types of the variance under the null. The first way is to set \tilde{r}_0=\tilde{r}_1=r_0, using event rate from the control group. The second way is to set \tilde{r}_0=r_0, \tilde{r}_1=r_1, using true event rates. The third way is to set \tilde{r}_0=\tilde{r}_1=\tilde{r}, where \tilde{r}=\pi_1 r_1+\pi_0 r_0, using maximum likelihood estimation.

Therefore, for each type of follow-up, there are 3 sample sizes calculated (because there are 3 varainces under the null) for with and without approximation of Zhu and Lakkis (2014).

Note that PASS14.0 provides 3 ways of null varaince with the default being the MLE. PASS does not allow different dispersion parameters between treatmetn and control. EAST only provides the second way of null varaince but allows for different dispersion parameters. Both of these softwares base on the approximatin method of Zhu and Lakkis (2014), which underestimate the required sample sizes.

Value

tildeXN

sample sizes based on current approach, i.e. not based on the Zhu and Lakkis's approximation

XN

sample sizes based on the Zhu and Lakkis's approximation

Exposure

mean exposure under different follow-up types with element 1 for control, element 2 for treatment and element 3 for overall.

SDExp

Sd of the exposure under different follow-up types with element 1 for control, element 2 for treatment and column 3 for overall.

Author(s)

Xiaodong Luo

References

Zhu~H and Lakkis~H. Sample size calculation for comparing two negative binomial rates. Statistics in Medicine 2014, 33: 376-387.

Examples

##calculating the sample sizes
abc=ynegbinomsize(r0=1.0,r1=0.5,shape0=1,pi1=0.5,alpha=0.05,twosided=1,
    beta=0.2,fixedfu=1,type=4,u=c(0.5,0.5,1),ut=c(0.5,1.0,1.5),
    tfix=1.5,maxfu=1,tchange=c(0,0.5,1),ratec1=c(0.15,0.15,0.15),
    eps=1.0e-03)
###Zhu and Lakkis's sample sizes (i.e. with approximation) 
abc$XN
###Our sample sizes (i.e. without approximation)
abc$tildeXN