# ynegbinomsize: Two-sample sample size calculation for negative binomial... In NBDesign: Design and Monitoring of Clinical Trials with Negative Binomial Endpoint

## 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

 1 2 3 4 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 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

\log(u_i)=\log(t_i)+β_0+β_1 Z_i.

Let \hat{β}_0 and \hat{β}_1 be the MLE of β_0 and β_1. The varaince of \hat{β}_1 is

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

where

\tilde{a}_j(r)=∑_{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)=∑_{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)≥ \tilde{a}_j(r), which means Zhu and Lakkis's method will underestimate variance of \hat{β}_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}=π_1 r_1+π_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.

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

 1 2 3 4 5 6 7 8 9 ##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 

NBDesign documentation built on May 2, 2019, 2:09 a.m.