Nothing
R/fdr-sampsize-v1.0e.R
In FDRsampsize: Compute Sample Size that Meets Requirements for Average Power and FDR
Defines functions
fdr.sampsize
print.fdr.sampsize
fdr.power
alpha.fdr
n.fdr
average.power
afdr
alpha.power
pi.star
power.cox
power.onesampt
power.twosampt
power.oneway
power.tcorr
power.ranksum
power.signtest
power.hart
power.li
Documented in
afdr
alpha.fdr
alpha.power
average.power
fdr.power
fdr.sampsize
n.fdr
pi.star
power.cox
power.hart
power.li
power.onesampt
power.oneway
power.ranksum
power.signtest
power.tcorr
power.twosampt
print.fdr.sampsize
#> BEGIN DESCRIPTION
#> Package: FDRsampsize
#> Type: Package
#> Title: Compute Sample Size that Meets Requirements for Average Power and FDR
#> Version: 1.0
#> Author: Stan Pounds <stanley.pounds@stjude.org>
#> Maintainer: Stan Pounds <stanley.pounds@stjude.org>
#> Depends: R (>= 2.15.1)
#> Imports: stats
#> Date: 2016-01-06
#> Description: Defines a collection of functions to compute average power and sample size for studies that use the false discovery rate as the final measure of statistical significance.
#> License: GPL-2
#> END DESCRIPTION
#############################################################
#> BEGIN NAMESPACE
#> export(afdr,alpha.fdr,alpha.power,average.power,fdr.power,fdr.sampsize,pi.star,power.cox,power.hart,power.li,power.onesampt,power.oneway,power.ranksum,power.signtest,power.tcorr,power.twosampt)
#> importFrom("stats","pf","pnorm","qf","qnorm")
#> S3method(print,fdr.sampsize)
#> END NAMESPACE
#############################################################
#> BEGIN FDRsampsize-package
#> \alias{FDRsampsize-package}
#> \alias{FDRsampsize}
#> \docType{package}
#> \title{An R package to Perform Power and Sample Size Calculations for Microarray Studies}
#> \description{ A general approach to performing power and sample size calculations for microarray
#> studies has been developed in the literature. However, the software associated with those articles
#> implements the approach only for studies that will perform the t-test or one-way ANOVA to compare
#> gene expression across two or more groups. Here, we describe a set of R routines that implement
#> the general method for power and sample size calculations for a wider variety of statistical tests.
#> These routines accept the name of a function that computes the power for the statistical test of
#> interest and thus have the flexibility to perform calculations for virtually any statistical test
#> with a known power formula.}
#> \details{
#> \tabular{ll}{
#> Package: \tab FDRsampsize\cr
#> Type: \tab Package\cr
#> Version: \tab 1.0\cr
#> Date: \tab 2016-01-06\cr
#> License: \tab GPL(>=2) \cr }
#> }
#> \author{Stan Pounds <stanley.pounds@stjude.org> }
#> \references{A Onar-Thomas, S Pounds. FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance, Manuscript 2016.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END FDRsampsize-package
######################################
#> BEGIN fdr.sampsize
#> \alias{print.fdr.sampsize}
#> \title{Determine sample size required to achieve a desired average power while controlling the FDR at a specified level.}
#> \description{Determines the sample size needed to achieve a desired average power while controlling the FDR at a specified level.}
#> \arguments{
fdr.sampsize=function(fdr, #> \item{fdr}{Desired FDR (scalar numeric)}
ave.pow, #> \item{ave.pow}{Desired average power (scalar numeric)}
pow.func, #> \item{pow.func}{Character string name of function to compute power; must accept n, alpha, and eff.size as its first three arguments. Other optional arguments may also be provided.}
eff.size, #> \item{eff.size}{Numeric vector of effect sizes; interally, this will be provided as the third argument of pow.func}
null.effect, #> \item{null.effect}{Scalar value of the effect size under the null hypothesis. This may be 0 or 1 for tests that respectively use differences or ratios for comparisons.}
max.n=500, #> \item{max.n}{Maximum n to consider}
min.n=5, #> \item{min.n}{Minimum n to consider}
tol=0.00001, #> \item{tol}{Tolerance for bisection calculations}
eps=0.00001, #> \item{eps}{Epsilon for numerical differentiation}
lam=0.95, #> \item{lam}{Lambda for computing anticipated pi0 estimate, see Storey 2002.}
...) #> \item{...}{additional arguments for pow.func} }
{
# Compute alpha needed to give desired power at maximum sample size
maxpow=alpha.power(ave.pow,
max.n,
pow.func,
eff.size,
null.effect,
tol,...)
# Determine the anticipated FDR with that sample size and alpha
minfdr=afdr(max.n,
maxpow$alpha,
pow.func,
eff.size,
lam,eps,...)
pi0=mean(eff.size==null.effect) # Get the true value of pi0
if (minfdr>fdr) # Indicates that max.n cannot achieve specified FDR level
{
pi.hat=pi.star(max.n,pow.func,eff.size,lam=lam,eps=eps,...)
act.fdr=pi0*maxpow$alpha/(pi0*maxpow$alpha+(1-pi0)*maxpow$ave.pow)
res=list(OK=F,
n=max.n,
alpha=maxpow$alpha,
fdr.hat=minfdr,
act.fdr=act.fdr,
ave.pow=maxpow$ave.pow,
act.pi=pi0,
pi.hat=pi.hat,
eff.size=eff.size)
class(res)="fdr.sampsize"
return(res) # Return what is achievable with maximum sample size
}
# Determine FDR and power for minimum sample size
minpow=alpha.power(ave.pow,min.n,pow.func,eff.size,null.effect,tol,...)
maxfdr=afdr(min.n,minpow$alpha,pow.func,eff.size,lam,eps,...)
if (maxfdr<fdr)
{
pi.hat=pi.star(min.n,pow.func,eff.size,lam=lam,eps=eps,...)
act.fdr=pi0*minpow$alpha/(pi0*minpow$alpha+(1-pi0)*minpow$ave.pow)
res=list(OK=T,
n=min.n,
alpha=minpow$alpha,
fdr.hat=maxfdr,
act.fdr=act.fdr,
ave.pow=minpow$ave.pow,
act.pi=pi0,
pi.hat=pi.hat,
eff.size=eff.size)
class(res)="fdr.sampsize"
return(res)
}
res0=n.fdr(ave.pow,fdr,pow.func,eff.size,null.effect,
lam=lam,eps=eps,n0=min.n,n1=max.n,tol=tol,...)
res=list(OK=T,
n=res0$n,
alpha=res0$alpha,
fdr.hat=res0$fdr.hat,
act.fdr=res0$act.fdr,
ave.pow=res0$ave.pow,
act.pi=res0$act.pi,
pi.hat=res0$pi.hat,
eff.size=eff.size)
class(res)="fdr.sampsize"
return(res)
}
#> \details{This function checks the technical conditions regarding whether the desired FDR can be
#> achieved by min.n or max.n before calling n.fdr. Thus, for most applications,
#> fdr.sampsize should be used instead of n.fdr.}
#> \value{\code{fdr.sampsiz}e returns an object of class 'FDRsampsize', which is a list with the following components:
#> \item{OK}{indicates if the requirement is met}
#> \item{n}{the computed sample size}
#> \item{alpha}{the p-value threshold that gives the desired FDR}
#> \item{fdr.hat}{anticipated value of the FDR estimate given n and effect size }
#> \item{act.fdr}{actual expected FDR given n and effect size}
#> \item{ave.pow}{average power}
#> \item{act.pi}{actual value of pi0, the proportion of tests with a true null hypothesis.}
#> \item{pi.hat}{expected value of the pi0 estimate}
#> \item{eff.size}{input effect size vector}
#> }
#> \references{A Onar-Thomas, S Pounds. "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> \examples{
# # Load the package
#> power.twosampt # show the power.cox function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.twosampt,
#> eff.size=rep(c(1,0),c(10,990)),
#> null.effect=0)
#> res
#> }
#> END fdr.sampsize
# BEGIN print.FDRsampsize
# \title{Print the result object returned by fdr.sampsize.}
# \description{Print an object of class 'FDRsampsize', which is returned by fdr.sampsize.}
# \arguments{
print.fdr.sampsize=function(x,...) # \item{x}{an FDRsampsize object, which is the result of \link{fdr.sampsize}}
# \item{...}{additional arguments for print}
# }
{
object=x
res=c(n=round(object$n,2),
alpha=object$alpha,
ave.pow=object$ave.pow,
fdr.hat=object$fdr.hat,
act.fdr=object$act.fdr)
print(res,...)
}
# \value{prints a vector with the sample size estimate,
# the p-value threshold alpha, the average power,
# the anticipated FDR estimate, and the
# anticipated actual FDR}
# END print.FDRsampsize
######################################
#> BEGIN fdr.power
#> \title{Compute the average power at a specific FDR control level}
#> \description{Compute the average power at a specific level of FDR control for a given effect size and sample size}
#> \arguments{
fdr.power=function(fdr, #> \item{fdr}{Desired FDR, scalar}
n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{name of R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector; will be provided as the third argument of pow.func}
null.effect, #> \item{null.effect}{value of effect size that corresponds to null hypothesis}
lam=0.95, #> \item{lam}{name of R function to compute statistical power}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
tol=0.0001, #> \item{tol}{tolerance for bisection solution to alpha}
...) #> \item{...}{additional agruments for the functions} }
{
a0=alpha.fdr(fdr,n,pow.func,eff.size,null.effect,lam,eps,tol,...)
if (!a0$OK) return(0)
res=average.power(n,a0$alpha,pow.func,eff.size,null.effect=null.effect,...)
return(res)
}
#> \value{average power (scalar) of the tests with a false null hypothesis}
#> \examples{
#> fdr.power(fdr=0.10,n=50,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#>}
#> \references{A Onar-Thomas, S Pounds "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2016.
#>
#> Gadbury GL, et al. (2004) Power and sample size estimation in high dimensional biology. Statistical Methods in Medical Research 13(4):325-38.
#>
#> Pounds S and Cheng C (2005) Sample size determination for the false discovery rate. Bioinformatics 21(23): 4263-71.}
#> END fdr.power
######################################
#> BEGIN alpha.fdr
#> \title{Find the fixed p-value threshold that controls the FDR at a specified level}
#> \description{Find the p-value threshold that satisfies an FDR requirement (if such a threshold exists)}
#> \arguments{
alpha.fdr=function(fdr, #> \item{fdr}{Desired FDR, scalar}
n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{an R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{value of effect size that corresponds to the null hypothesis}
lam=0.95, #> \item{lam}{the lambda parameter in computing the pi0 (proportion of tests with a true null) estimate of Storey (2002)}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
tol=0.0001, #> \item{tol}{tolerance for bisection solution to alpha}
...) #> \item{...}{additional agruments for the functions} }
{
pi.hat=pi.star(n,pow.func,eff.size,lam,eps,...)
pow.temp=pow.func(n,eps,eff.size,...)
pr.rej=mean(pow.temp)
pdf0=exp(log(pr.rej)-log(eps))
minpossfdr=pi.hat/pdf0
if (minpossfdr>fdr) return(list(alpha=0,fdr=minpossfdr,OK=F))
a0=0
a1=1
fdr0=minpossfdr
fdr1=mean(eff.size==null.effect)
k=ceiling(-log(tol)/log(2))
for (i in 1:k)
{
a.mid=(a0+a1)/2
fdr.mid=afdr(n,a.mid,pow.func,eff.size,lam=lam,eps=eps,...)
if (fdr.mid>fdr)
{
a1=a.mid
fdr1=fdr.mid
}
else
{
a0=a.mid
fdr0=fdr.mid
}
}
return(list(alpha=a0,fdr=fdr0,OK=T))
}
#> \value{a list with the following components:
#> \item{fdr}{the FDR at that alpha}
#> \item{alpha}{the determined alpha}
#> \item{OK}{indicates if the requirement is met}
#> }
#> \examples{
#> alpha.fdr(fdr=0.1,n=50,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#> }
#> \references{A Onar-Thomas, S Pounds "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END alpha.fdr
######################################
#> BEGIN n.fdr
#> \title{Find the sample size that meets desired FDR and power criteria}
#> \description{Find smallest sample size that meets requirements for average power and FDR}
#> \arguments{
n.fdr=function(ave.pow, #> \item{ave.pow}{required average power (scalar)}
fdr, #> \item{fdr}{required FDR (scalar)}
pow.func, #> \item{pow.func}{name of R function that computes statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{Value of effect size that indicates null}
lam=0.95, #> \item{lam}{p-value at which to evaluate ensemble PDF}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
n0=5, #> \item{n0}{smallest sample size to be considered for bisection}
n1=500, #> \item{n1}{maximum sample size to be considered for bisection}
tol=0.000001, #> \item{tol}{tolerance for solving for alpha in iterations}
...) #> \item{...}{additional agruments for the functions} }
{
a0=alpha.power(ave.pow,n0,pow.func,eff.size,null.effect,tol,...)
a1=alpha.power(ave.pow,n1,pow.func,eff.size,null.effect,tol,...)
afdr0=afdr(n0,a0$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
afdr1=afdr(n1,a0$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
#print(afdr0)
k=ceiling((log(n1-n0)-log(tol))/log(2))
i=0
cl.eq=(ceiling(n1)==ceiling(n0))
while((i<k)&(!cl.eq))
{
i=i+1
n.mid=(n1+n0)/2
a.mid=alpha.power(ave.pow,n.mid,pow.func,eff.size,null.effect,tol,...)
afdr.mid=afdr(n.mid,a.mid$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
if (afdr.mid>fdr)
{
n0=n.mid
afdr0=afdr.mid
}
else
{
n1=n.mid
afdr1=afdr.mid
}
cl.eq=(ceiling(n1)==ceiling(n0))
}
n=ceiling(n1)
a=alpha.power(ave.pow,n,pow.func,eff.size,null.effect,tol,...)
pow.res=average.power(n,a$alpha,pow.func,eff.size,null.effect=null.effect,...)
fdr.hat=afdr(n,a$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
pi.hat=pi.star(n,pow.func,eff.size,lam=lam,eps=eps,...)
act.pi0=mean(eff.size==null.effect)
act.fdr=act.pi0*a$alpha/(act.pi0*a$alpha+(1-act.pi0)*pow.res)
res=list(n=n,
alpha=a$alpha,
fdr.hat=fdr.hat,
ave.pow=pow.res,
act.fdr=act.fdr,
pi.hat=pi.hat,
act.pi=act.pi0)
class(res)="FDRsampsize"
return(res)
}
#> \details{This performs the sample size calculation without checking whether the
#> minimum or maximum sample size satisfy the desired requirements. The fdr.sampsize
#> function checks these conditions and then calls n.fdr. Thus, many users will
#> may prefer to use the sampsize.fdr function instead of n.fdr.}
#> \value{a list with the following components:
#> \item{n}{a sample size estimate}
#> \item{alpha}{the p-value cut-off}
#> \item{fdr.hat}{an approximation to the expected value of the FDR estimate given n }
#> \item{ave.pow}{the average power}
#> \item{fdr.act}{the actual FDR given n}
#> \item{pi.hat}{expected value of the pi.hat estimator given n}
#> \item{act.pi}{actual pi0}
#> }
#> \references{A Onar-Thomas, S Pounds. "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END n.fdr
######################################
#> BEGIN average.power
#> \title{Compute average power for a given sample size}
#> \description{Compute average power for given sample size, effect size, and p-value threshold}
#> \arguments{
average.power=function(n, #> \item{n}{sample size}
alpha, #> \item{alpha}{p-value cut off (scalar)}
pow.func, #> \item{pow.func}{an R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{value of effect size that corresponds to null hypothesis}
...) #> \item{...}{additional agruments for the functions} }
{
pi=mean(eff.size==null.effect)
pr.rej=mean(pow.func(n,alpha,eff.size,...))
res=exp(log(pr.rej-pi*alpha)-log(1-pi))
return(res)
}
#> \value{average power (scalar)}
#> \examples{
#> average.power(n=50,alpha=0.01,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#> }
#> \references{
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#> Gadbury GL, et al. (2004) Power and sample size estimation in high dimensional biology. Statistical Methods in Medical Research 13(4):325-38.
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END average.power
######################################
#> BEGIN afdr
#> \title{Compute the anticipated FDR}
#> \description{Compute the anticipated FDR for given sample size, p-value threshold, and effect size.}
#> \arguments{
afdr=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value cut-off (scalar)}
pow.func, #> \item{pow.func}{an R function that computes statistical power}
eff.size, #> \item{eff.size}{effect size vector}
lam=0.95, #> \item{lam}{p-value at which to evaluate ensemble PDF (for pi.star)}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
...) #> \item{...}{additional agruments for the functions} }
{
pi.hat=pi.star(n,pow.func,eff.size,lam,eps,...)
pow.temp=pow.func(n,alpha,eff.size,...)
pr.rej=mean(pow.temp)
return(pi.hat*alpha/pr.rej)
}
#> \value{the aFDR}
#> \details{The aFDR is defined by Pounds and Cheng (2005) as the anticipated false discovery rate
#> incurred by performing all tests with p-value threshold alpha given the same size
#> effect size and power function.}
#> \examples{
#> afdr(n=50,alpha=0.01,pow.func=power.twosampt,eff.size=rep(c(1,0),c(100,900)))
#> }
#> \references{Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END afdr
######################################
#> BEGIN alpha.power
#> \title{Find the p-value threshold that gives a specified average power}
#> \description{Find p-value cut-off that yields desired average power given n and effect size}
#> \arguments{
alpha.power=function(ave.pow, #> \item{ave.pow}{desired average power (scalar)}
n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{name of R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{value of effect size that corresponds to null hypothesis}
tol=0.000001, #> \item{tol}{tolerance for bisection solution to alpha}
...) #> \item{...}{additional agruments for the functions} }
{
pow0=0
pow1=1
a0=0
a1=1
k=ceiling(-log(tol)/log(2))
for (i in 1:k) # Bisection loop
{
a.mid=(a0+a1)/2
pow.mid=average.power(n,a.mid,pow.func,eff.size,null.effect=null.effect,...)
#print(pow.mid)
if (pow.mid>ave.pow)
{
a1=a.mid
pow1=pow.mid
}
else
{
a0=a.mid
pow0=pow.mid
}
}
return(list(alpha=a1,ave.pow=pow1))
}
#> \value{a list with the following components:
#> \item{alpha}{desired value of alpha}
#> \item{ave.pow}{average power at that alpha}
#> }
#> \examples{
#> alpha.power(ave.pow=0.8,n=50,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#> }
#> \references{A Onar-Thomas, S Pounds. "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END alpha.power
######################################
#> BEGIN pi.star
#> \title{Compute the anticipated null proportion estimate}
#> \description{Compute an approximation of the expected value of the null proportion estimate given the sample size and effect size.}
#> \arguments{
pi.star=function(n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{an R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
lam=0.95, #> \item{lam}{p-value at which to numerically evaluate p-value pdf (scalar)}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
...) #> \item{...}{additional agruments for the functions} }
{
pow.lam=pow.func(n,lam,eff.size,...)
pow.lam.eps=pow.func(n,lam-eps,eff.size,...)
res=exp(log(mean(pow.lam)-
mean(pow.lam.eps))-
log(eps))
return(res)
}
#> \value{scalar value for approximated E(pi.hat)}
#> \references{#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.}
#> END pi.star
######################################
#> BEGIN power.cox
#> \title{Compute the power of a single-predictor Cox regression model}
#> \description{Use the formula of Hseih and Lavori (2000) to compute the power of a single-predictor Cox model.}
#> \arguments{
power.cox=function(n, #> \item{n}{number of events (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
logHR, #> \item{logHR}{log hazard ratio (vector)}
v) #> \item{v}{variance of predictor variable (vector)}
#> }
{
pnorm(qnorm(alpha/2),sqrt(n*v)*logHR)+
1-pnorm(qnorm(1-alpha/2),sqrt(n*v)*logHR)
}
#> \value{vector of power estimates for two-sided test}
#> \references{
#> Hsieh, FY and Lavori, Philip W (2000) Sample-size calculations for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials 21(6):552-560.
#> }
#> \examples{
# # Load the package
#> power.cox # show the power.cox function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.cox,
#> eff.size=rep(c(log(2),0),c(100,900)),
#> null.effect=0,
#> v=1)
#> res
#>}
#> END power.cox
######################################
#> BEGIN power.onesampt
#> \title{Compute power of the one-sample t-test}
#> \description{Estimate power of the one-sample t-test;Uses classical power formula for one-sample t-test}
#> \arguments{
power.onesampt=function(n, #> \item{n}{number of events (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
delta, #> \item{delta}{difference of actual mean from null mean (vector)}
sigma=1) #> \item{sigma}{standard deviation (vector or scalar, default=1)}
#> }
{
1-pf(qf(1-alpha,1,n-1),1,n-1,ncp=n*(delta/sigma)^2)
}
#> \value{vector of power estimates for two-sided test}
#> \examples{
# # Load the package
#> power.onesampt # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.onesampt,
#> eff.size=rep(c(2,0),c(100,900)),
#> null.effect=0,
#> sigma=1)
#> res
#>}
#> END power.onesampt
######################################
#> BEGIN power.twosampt
#> \title{Compute power of the two-samples t-test}
#> \description{Estimate power of the two-samples t-test;Uses classical power formula for two-sample t-test;Assumes equal variance and sample size }
#> \arguments{
power.twosampt=function(n, #> \item{n}{per-group sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
delta, #> \item{delta}{difference between population means (vector)}
sigma=1) #> \item{sigma}{standard deviation (vector or scalar)}
#> }
{
1-pf(qf(1-alpha,1,2*(n-1)),1,2*(n-1),ncp=n*(delta/sigma)^2/2)
}
#> \value{vector of power estimates for two-sided test}
#> \details{For many applications, the null.effect is zero difference of means.}
#> \examples{
# # Load the package
#> power.twosampt # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.twosampt,
#> eff.size=rep(c(2,0),c(100,900)),
#> null.effect=0,
#> sigma=1)
#> res
#>}
#> END power.twosampt
######################################
#> BEGIN power.oneway
#> \title{Compute power of one-way ANOVA}
#> \description{Compute power of one-way ANOVA;Uses classical power formula for ANOVA;Assumes equal variance and sample size }
#> \arguments{
power.oneway=function(n, #> \item{n}{per-group sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
theta, #> \item{theta}{sum of ((group mean - overall mean)/stdev)^2 across all groups for each hypothesis test (vector)}
k=2) #> \item{k}{the number of groups to be compared, default k=2}
#> }
{
1-pf(qf(1-alpha,k-1,k*(n-1)),k-1,k*(n-1),ncp=n*theta)
}
#> \value{vector of power estimates for test of equal means}
#> \details{For many applications, the null effect is zero for the parameter theta described above.}
#> \examples{
# # Load the package
#> power.oneway # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.oneway,
#> eff.size=rep(c(2,0),c(100,900)),
#> null.effect=0,
#> k=3)
#> res
#>}
#> END power.oneway
######################################
#> BEGIN power.tcorr
#> \title{Compute Power of the t-test for non-zero correlation}
#> \description{Estimate power of t-test for non-zero correlation;Uses classical power formula for t-test}
#> \arguments{
power.tcorr=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
rho) #> \item{rho}{population correlation coefficient (vector)}
#> }
{
1-pf(qf(1-alpha,1,n-2),1,n-2,ncp=(n-2)*rho^2/(1-rho^2))
}
#> \value{vector of power estimates for two-sided tests}
#> \details{For many applications, the null.effect is rho=0.}
#> \examples{
# # Load the package
#> power.tcorr # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.tcorr,
#> eff.size=rep(c(0.3,0),c(100,900)),
#> null.effect=0)
#> res
#>}
#> END power.tcorr
######################################
#> BEGIN power.ranksum
#> \title{Compute power of the rank-sum test}
#> \description{Compute power of rank-sum test;Uses formula of Noether (JASA 1987)}
#> \arguments{
power.ranksum=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
p) #> \item{p}{Pr (Y>X), as in Noether (JASA 1987)}
#> }
{ mu0=0.5*n*n
mu1=p*n*n
delta=(mu1-mu0)
sig2=n*n*(2*n+1)/12
sig=sqrt(sig2)
z.reject=-abs(qnorm(alpha/2,0,sig))
pow=pnorm(z.reject,delta,sig)+pnorm(-z.reject,delta,sig,lower.tail=F)
return(pow)
}
#> \value{vector of power estimates for two-sided tests}
#> \details{In most applications, the null effect size will be designated by p = 0.5, which indicates that
# the probability that a randomly selected subject from group A will have a
# greater value of the variable than a randomly selected subject from group B.
#> Thus, in the example below, the argument null.effect=0.5 is specified in the call to fdr.sampsize.}
#> \examples{
# # Load the package
#> power.ranksum # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.ranksum,
#> eff.size=rep(c(0.8,0.5),c(100,900)),
#> null.effect=0.5)
#> res
#>}
#> \references{Noether, Gottfried E (1987) Sample size determination for some common nonparametric tests.
#> Journal of the American Statistical Association, 82:645-647.}
#> END power.ranksum
######################################
#> BEGIN power.signtest
#> \title{Compute power of the sign test}
#> \description{Use the Noether (1987) formula to compute the power of the sign test}
#> \arguments{
power.signtest=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
p) #> \item{p}{Pr (Y>X), as in Noether (JASA 1987)}
#> }
{
mu0=0.5*n
mu1=p*n
sig0=sqrt(n*0.25)
sig1=sqrt(n*p*(1-p))
pow=pnorm(qnorm(alpha/2,mu0,sig0),mu1,sig1)+
pnorm(qnorm(1-alpha/2,mu0,sig0),mu1,sig1,lower.tail=F)
return(pow)
}
#> \value{vector of power estimates for two-sided tests}
#> \details{In most applications, the null effect size will be designated by p = 0.5 instead of p = 0.
#> Thus, in the call to fdr.sampsize, we specify null.effect=0.5 in the example below.}
#> \references{Noether, Gottfried E (1987) Sample size determination for some common nonparametric tests.
#> Journal of the American Statistical Association, 82:645-647.}
#> \examples{
# # Load the package
#> power.signtest # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.signtest,
#> eff.size=rep(c(0.8,0.5),c(100,900)),
#> null.effect=0.5)
#> res
#>}
#> END power.signtest
#> BEGIN power.hart
#> \title{Compute Power for RNA-seq Experiments Assuming Negative Binomial Distribution.}
#> \description{Use the formula of Hart et al (2013) to compute power for comparing RNA-seq expression across two groups assuming a negative binomial distribution.}
#> \arguments{
power.hart=function(n, #> \item{n}{per-group sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
log.fc, #> \item{log.fc}{log fold-change (vector), usual null hypothesis is log.fc=0}
mu, #> \item{mu}{read depth per gene (vector, same length as log.fc)}
sig) #> \item{sig}{coefficient of variation (CV) per gene (vector, same length as log.fc)}
#> }
{
z.alpha=qnorm(alpha/2)
res=pnorm(z.alpha,sqrt(n*log.fc^2/(2*(1/mu+sig^2))))+
pnorm(-z.alpha,sqrt(n*log.fc^2/(2*(1/mu+sig^2))),lower.tail=F)
return(res)
}
#> \value{vector of power estimates for the set of two-sided tests}
#> \references{SN Hart, TM Therneau, Y Zhang, GA Poland, and J-P Kocher (2013).
#> Calculating Sample Size Estimates for RNA Sequencing Data.
#> Journal of Computational Biology 20: 970-978.}
#> \details{This function is based on equation (1) of Hart et al (2013). It
#> assumes a negative binomial model for RNA-seq read counts and
#> equal sample size per group.}
#> \examples{
# # Load the package
#> power.hart # show the power function
#> n.hart=2*(qnorm(0.975)+qnorm(0.9))^2*(1/20+0.6^2)/(log(2)^2) # Equation 6 of Hart et al
#> power.hart(n.hart,0.05,log(2),20,0.6) # Recapitulate 90% power
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.hart,
#> eff.size=rep(c(log(2),0),c(100,900)),
#> null.effect=0,mu=5,sig=1)
#> res
#>}
#> END power.hart
#> BEGIN power.li
#> \title{Compute Power for RNA-Seq Experiments Assuming Poisson Distribution}
#> \description{Use the formula of Li et al (2013) to compute power for comparing RNA-seq expression across two groups assuming the Poisson distribution.}
#> \arguments{
power.li=function(n, #> \item{n}{per-group sample size}
alpha, #> \item{alpha}{p-value threshold}
rho, #> \item{rho}{fold-change, usual null hypothesis is that rho=1}
mu0, #> \item{mu0}{average count in control group}
w=1, #> \item{w}{ratio of total number of }
type="w") #> \item{type}{type of test: "w" for Wald, "s" for score, "lw" for log-transformed Wald, "ls" for log-transformed score.}
#> }
{
z.alpha=qnorm(alpha/2)
if (!is.element(type,c("w","lw","s","ls")))
stop("type must be 'w', 's', 'ls', or 'lw'.")
if (any(w<=0)) stop("w must be >0.")
if (any(rho<=0)) stop("rho must be >0.")
if (type=="w")
{
mu.z=sqrt(n*mu0*(rho-1)^2/(1+rho/w))
res=pnorm(z.alpha,mu.z)+pnorm(-z.alpha,mu.z,lower.tail=F)
return(res)
}
if (type=="lw")
{
mu.z=sqrt(n*mu0*(log(rho)^2)/(1+1/(rho*w)))
res=pnorm(z.alpha,mu.z)+pnorm(-z.alpha,mu.z,lower.tail=F)
return(res)
}
if (type=="s")
{
z.cut=z.alpha*sqrt((1+w*rho)/(w+rho))
mu.z=sqrt(n*mu0*(rho-1)^2/(1+rho/w))
res=pnorm(z.cut,mu.z)+pnorm(-z.cut,mu.z,lower.tail=F)
return(res)
}
if (type=="ls")
{
z.cut=z.alpha*sqrt((2+w+1/w)/((1+w*rho)*(1+1/(w*rho))))
mu.z=sqrt(n*mu0*(log(rho)^2)/(1+1/(rho*w)))
res=pnorm(z.cut,mu.z)+pnorm(-z.cut,mu.z,lower.tail=F)
return(res)
}
}
#> \value{vector of power estimates for two-sided tests}
#> \references{C-I Li, P-F Su, Y Guo, and Y Shyr (2013).
#> Sample size calculation for differential expression analysis of
#> RNA-seq data under Poisson distribution.
#> Int J Comput Biol Drug Des 6(4). doi:10.1504/IJCBDD.2013.056830}
#> \details{This function computes the power for each of a series of two-sided
#> tests defined by the input parameters. The power is based on the
#> sample size formulas in equations 10-13 of Li et al (2013).
#> Also, note that the null.effect is set to 1 in the examples because
#> the usual null hypothesis is that the fold-change = 1.}
#> \examples{
# # Load the package
#> power.li # show the power function
#> power.li(88,0.05,1.25,5,0.5,"w") # recapitulate 80% power in Table 1 of Li et al (2013)
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.li,
#> eff.size=rep(c(1.5,1),c(100,900)),
#> null.effect=1,
#> mu0=5,w=1,type="w")
#> res
#>}
#> END power.li
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Defines functions fdr.sampsize print.fdr.sampsize fdr.power alpha.fdr n.fdr average.power afdr alpha.power pi.star power.cox power.onesampt power.twosampt power.oneway power.tcorr power.ranksum power.signtest power.hart power.li
Documented in afdr alpha.fdr alpha.power average.power fdr.power fdr.sampsize n.fdr pi.star power.cox power.hart power.li power.onesampt power.oneway power.ranksum power.signtest power.tcorr power.twosampt print.fdr.sampsize
#> BEGIN DESCRIPTION
#> Package: FDRsampsize
#> Type: Package
#> Title: Compute Sample Size that Meets Requirements for Average Power and FDR
#> Version: 1.0
#> Author: Stan Pounds <stanley.pounds@stjude.org>
#> Maintainer: Stan Pounds <stanley.pounds@stjude.org>
#> Depends: R (>= 2.15.1)
#> Imports: stats
#> Date: 2016-01-06
#> Description: Defines a collection of functions to compute average power and sample size for studies that use the false discovery rate as the final measure of statistical significance.
#> License: GPL-2
#> END DESCRIPTION
#############################################################
#> BEGIN NAMESPACE
#> export(afdr,alpha.fdr,alpha.power,average.power,fdr.power,fdr.sampsize,pi.star,power.cox,power.hart,power.li,power.onesampt,power.oneway,power.ranksum,power.signtest,power.tcorr,power.twosampt)
#> importFrom("stats","pf","pnorm","qf","qnorm")
#> S3method(print,fdr.sampsize)
#> END NAMESPACE
#############################################################
#> BEGIN FDRsampsize-package
#> \alias{FDRsampsize-package}
#> \alias{FDRsampsize}
#> \docType{package}
#> \title{An R package to Perform Power and Sample Size Calculations for Microarray Studies}
#> \description{ A general approach to performing power and sample size calculations for microarray
#> studies has been developed in the literature. However, the software associated with those articles
#> implements the approach only for studies that will perform the t-test or one-way ANOVA to compare
#> gene expression across two or more groups. Here, we describe a set of R routines that implement
#> the general method for power and sample size calculations for a wider variety of statistical tests.
#> These routines accept the name of a function that computes the power for the statistical test of
#> interest and thus have the flexibility to perform calculations for virtually any statistical test
#> with a known power formula.}
#> \details{
#> \tabular{ll}{
#> Package: \tab FDRsampsize\cr
#> Type: \tab Package\cr
#> Version: \tab 1.0\cr
#> Date: \tab 2016-01-06\cr
#> License: \tab GPL(>=2) \cr }
#> }
#> \author{Stan Pounds <stanley.pounds@stjude.org> }
#> \references{A Onar-Thomas, S Pounds. FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance, Manuscript 2016.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END FDRsampsize-package
######################################
#> BEGIN fdr.sampsize
#> \alias{print.fdr.sampsize}
#> \title{Determine sample size required to achieve a desired average power while controlling the FDR at a specified level.}
#> \description{Determines the sample size needed to achieve a desired average power while controlling the FDR at a specified level.}
#> \arguments{
fdr.sampsize=function(fdr, #> \item{fdr}{Desired FDR (scalar numeric)}
ave.pow, #> \item{ave.pow}{Desired average power (scalar numeric)}
pow.func, #> \item{pow.func}{Character string name of function to compute power; must accept n, alpha, and eff.size as its first three arguments. Other optional arguments may also be provided.}
eff.size, #> \item{eff.size}{Numeric vector of effect sizes; interally, this will be provided as the third argument of pow.func}
null.effect, #> \item{null.effect}{Scalar value of the effect size under the null hypothesis. This may be 0 or 1 for tests that respectively use differences or ratios for comparisons.}
max.n=500, #> \item{max.n}{Maximum n to consider}
min.n=5, #> \item{min.n}{Minimum n to consider}
tol=0.00001, #> \item{tol}{Tolerance for bisection calculations}
eps=0.00001, #> \item{eps}{Epsilon for numerical differentiation}
lam=0.95, #> \item{lam}{Lambda for computing anticipated pi0 estimate, see Storey 2002.}
...) #> \item{...}{additional arguments for pow.func} }
{
# Compute alpha needed to give desired power at maximum sample size
maxpow=alpha.power(ave.pow,
max.n,
pow.func,
eff.size,
null.effect,
tol,...)
# Determine the anticipated FDR with that sample size and alpha
minfdr=afdr(max.n,
maxpow$alpha,
pow.func,
eff.size,
lam,eps,...)
pi0=mean(eff.size==null.effect) # Get the true value of pi0
if (minfdr>fdr) # Indicates that max.n cannot achieve specified FDR level
{
pi.hat=pi.star(max.n,pow.func,eff.size,lam=lam,eps=eps,...)
act.fdr=pi0*maxpow$alpha/(pi0*maxpow$alpha+(1-pi0)*maxpow$ave.pow)
res=list(OK=F,
n=max.n,
alpha=maxpow$alpha,
fdr.hat=minfdr,
act.fdr=act.fdr,
ave.pow=maxpow$ave.pow,
act.pi=pi0,
pi.hat=pi.hat,
eff.size=eff.size)
class(res)="fdr.sampsize"
return(res) # Return what is achievable with maximum sample size
}
# Determine FDR and power for minimum sample size
minpow=alpha.power(ave.pow,min.n,pow.func,eff.size,null.effect,tol,...)
maxfdr=afdr(min.n,minpow$alpha,pow.func,eff.size,lam,eps,...)
if (maxfdr<fdr)
{
pi.hat=pi.star(min.n,pow.func,eff.size,lam=lam,eps=eps,...)
act.fdr=pi0*minpow$alpha/(pi0*minpow$alpha+(1-pi0)*minpow$ave.pow)
res=list(OK=T,
n=min.n,
alpha=minpow$alpha,
fdr.hat=maxfdr,
act.fdr=act.fdr,
ave.pow=minpow$ave.pow,
act.pi=pi0,
pi.hat=pi.hat,
eff.size=eff.size)
class(res)="fdr.sampsize"
return(res)
}
res0=n.fdr(ave.pow,fdr,pow.func,eff.size,null.effect,
lam=lam,eps=eps,n0=min.n,n1=max.n,tol=tol,...)
res=list(OK=T,
n=res0$n,
alpha=res0$alpha,
fdr.hat=res0$fdr.hat,
act.fdr=res0$act.fdr,
ave.pow=res0$ave.pow,
act.pi=res0$act.pi,
pi.hat=res0$pi.hat,
eff.size=eff.size)
class(res)="fdr.sampsize"
return(res)
}
#> \details{This function checks the technical conditions regarding whether the desired FDR can be
#> achieved by min.n or max.n before calling n.fdr. Thus, for most applications,
#> fdr.sampsize should be used instead of n.fdr.}
#> \value{\code{fdr.sampsiz}e returns an object of class 'FDRsampsize', which is a list with the following components:
#> \item{OK}{indicates if the requirement is met}
#> \item{n}{the computed sample size}
#> \item{alpha}{the p-value threshold that gives the desired FDR}
#> \item{fdr.hat}{anticipated value of the FDR estimate given n and effect size }
#> \item{act.fdr}{actual expected FDR given n and effect size}
#> \item{ave.pow}{average power}
#> \item{act.pi}{actual value of pi0, the proportion of tests with a true null hypothesis.}
#> \item{pi.hat}{expected value of the pi0 estimate}
#> \item{eff.size}{input effect size vector}
#> }
#> \references{A Onar-Thomas, S Pounds. "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> \examples{
# # Load the package
#> power.twosampt # show the power.cox function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.twosampt,
#> eff.size=rep(c(1,0),c(10,990)),
#> null.effect=0)
#> res
#> }
#> END fdr.sampsize
# BEGIN print.FDRsampsize
# \title{Print the result object returned by fdr.sampsize.}
# \description{Print an object of class 'FDRsampsize', which is returned by fdr.sampsize.}
# \arguments{
print.fdr.sampsize=function(x,...) # \item{x}{an FDRsampsize object, which is the result of \link{fdr.sampsize}}
# \item{...}{additional arguments for print}
# }
{
object=x
res=c(n=round(object$n,2),
alpha=object$alpha,
ave.pow=object$ave.pow,
fdr.hat=object$fdr.hat,
act.fdr=object$act.fdr)
print(res,...)
}
# \value{prints a vector with the sample size estimate,
# the p-value threshold alpha, the average power,
# the anticipated FDR estimate, and the
# anticipated actual FDR}
# END print.FDRsampsize
######################################
#> BEGIN fdr.power
#> \title{Compute the average power at a specific FDR control level}
#> \description{Compute the average power at a specific level of FDR control for a given effect size and sample size}
#> \arguments{
fdr.power=function(fdr, #> \item{fdr}{Desired FDR, scalar}
n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{name of R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector; will be provided as the third argument of pow.func}
null.effect, #> \item{null.effect}{value of effect size that corresponds to null hypothesis}
lam=0.95, #> \item{lam}{name of R function to compute statistical power}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
tol=0.0001, #> \item{tol}{tolerance for bisection solution to alpha}
...) #> \item{...}{additional agruments for the functions} }
{
a0=alpha.fdr(fdr,n,pow.func,eff.size,null.effect,lam,eps,tol,...)
if (!a0$OK) return(0)
res=average.power(n,a0$alpha,pow.func,eff.size,null.effect=null.effect,...)
return(res)
}
#> \value{average power (scalar) of the tests with a false null hypothesis}
#> \examples{
#> fdr.power(fdr=0.10,n=50,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#>}
#> \references{A Onar-Thomas, S Pounds "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2016.
#>
#> Gadbury GL, et al. (2004) Power and sample size estimation in high dimensional biology. Statistical Methods in Medical Research 13(4):325-38.
#>
#> Pounds S and Cheng C (2005) Sample size determination for the false discovery rate. Bioinformatics 21(23): 4263-71.}
#> END fdr.power
######################################
#> BEGIN alpha.fdr
#> \title{Find the fixed p-value threshold that controls the FDR at a specified level}
#> \description{Find the p-value threshold that satisfies an FDR requirement (if such a threshold exists)}
#> \arguments{
alpha.fdr=function(fdr, #> \item{fdr}{Desired FDR, scalar}
n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{an R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{value of effect size that corresponds to the null hypothesis}
lam=0.95, #> \item{lam}{the lambda parameter in computing the pi0 (proportion of tests with a true null) estimate of Storey (2002)}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
tol=0.0001, #> \item{tol}{tolerance for bisection solution to alpha}
...) #> \item{...}{additional agruments for the functions} }
{
pi.hat=pi.star(n,pow.func,eff.size,lam,eps,...)
pow.temp=pow.func(n,eps,eff.size,...)
pr.rej=mean(pow.temp)
pdf0=exp(log(pr.rej)-log(eps))
minpossfdr=pi.hat/pdf0
if (minpossfdr>fdr) return(list(alpha=0,fdr=minpossfdr,OK=F))
a0=0
a1=1
fdr0=minpossfdr
fdr1=mean(eff.size==null.effect)
k=ceiling(-log(tol)/log(2))
for (i in 1:k)
{
a.mid=(a0+a1)/2
fdr.mid=afdr(n,a.mid,pow.func,eff.size,lam=lam,eps=eps,...)
if (fdr.mid>fdr)
{
a1=a.mid
fdr1=fdr.mid
}
else
{
a0=a.mid
fdr0=fdr.mid
}
}
return(list(alpha=a0,fdr=fdr0,OK=T))
}
#> \value{a list with the following components:
#> \item{fdr}{the FDR at that alpha}
#> \item{alpha}{the determined alpha}
#> \item{OK}{indicates if the requirement is met}
#> }
#> \examples{
#> alpha.fdr(fdr=0.1,n=50,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#> }
#> \references{A Onar-Thomas, S Pounds "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END alpha.fdr
######################################
#> BEGIN n.fdr
#> \title{Find the sample size that meets desired FDR and power criteria}
#> \description{Find smallest sample size that meets requirements for average power and FDR}
#> \arguments{
n.fdr=function(ave.pow, #> \item{ave.pow}{required average power (scalar)}
fdr, #> \item{fdr}{required FDR (scalar)}
pow.func, #> \item{pow.func}{name of R function that computes statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{Value of effect size that indicates null}
lam=0.95, #> \item{lam}{p-value at which to evaluate ensemble PDF}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
n0=5, #> \item{n0}{smallest sample size to be considered for bisection}
n1=500, #> \item{n1}{maximum sample size to be considered for bisection}
tol=0.000001, #> \item{tol}{tolerance for solving for alpha in iterations}
...) #> \item{...}{additional agruments for the functions} }
{
a0=alpha.power(ave.pow,n0,pow.func,eff.size,null.effect,tol,...)
a1=alpha.power(ave.pow,n1,pow.func,eff.size,null.effect,tol,...)
afdr0=afdr(n0,a0$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
afdr1=afdr(n1,a0$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
#print(afdr0)
k=ceiling((log(n1-n0)-log(tol))/log(2))
i=0
cl.eq=(ceiling(n1)==ceiling(n0))
while((i<k)&(!cl.eq))
{
i=i+1
n.mid=(n1+n0)/2
a.mid=alpha.power(ave.pow,n.mid,pow.func,eff.size,null.effect,tol,...)
afdr.mid=afdr(n.mid,a.mid$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
if (afdr.mid>fdr)
{
n0=n.mid
afdr0=afdr.mid
}
else
{
n1=n.mid
afdr1=afdr.mid
}
cl.eq=(ceiling(n1)==ceiling(n0))
}
n=ceiling(n1)
a=alpha.power(ave.pow,n,pow.func,eff.size,null.effect,tol,...)
pow.res=average.power(n,a$alpha,pow.func,eff.size,null.effect=null.effect,...)
fdr.hat=afdr(n,a$alpha,pow.func,eff.size,lam=lam,eps=eps,...)
pi.hat=pi.star(n,pow.func,eff.size,lam=lam,eps=eps,...)
act.pi0=mean(eff.size==null.effect)
act.fdr=act.pi0*a$alpha/(act.pi0*a$alpha+(1-act.pi0)*pow.res)
res=list(n=n,
alpha=a$alpha,
fdr.hat=fdr.hat,
ave.pow=pow.res,
act.fdr=act.fdr,
pi.hat=pi.hat,
act.pi=act.pi0)
class(res)="FDRsampsize"
return(res)
}
#> \details{This performs the sample size calculation without checking whether the
#> minimum or maximum sample size satisfy the desired requirements. The fdr.sampsize
#> function checks these conditions and then calls n.fdr. Thus, many users will
#> may prefer to use the sampsize.fdr function instead of n.fdr.}
#> \value{a list with the following components:
#> \item{n}{a sample size estimate}
#> \item{alpha}{the p-value cut-off}
#> \item{fdr.hat}{an approximation to the expected value of the FDR estimate given n }
#> \item{ave.pow}{the average power}
#> \item{fdr.act}{the actual FDR given n}
#> \item{pi.hat}{expected value of the pi.hat estimator given n}
#> \item{act.pi}{actual pi0}
#> }
#> \references{A Onar-Thomas, S Pounds. "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#>
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END n.fdr
######################################
#> BEGIN average.power
#> \title{Compute average power for a given sample size}
#> \description{Compute average power for given sample size, effect size, and p-value threshold}
#> \arguments{
average.power=function(n, #> \item{n}{sample size}
alpha, #> \item{alpha}{p-value cut off (scalar)}
pow.func, #> \item{pow.func}{an R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{value of effect size that corresponds to null hypothesis}
...) #> \item{...}{additional agruments for the functions} }
{
pi=mean(eff.size==null.effect)
pr.rej=mean(pow.func(n,alpha,eff.size,...))
res=exp(log(pr.rej-pi*alpha)-log(1-pi))
return(res)
}
#> \value{average power (scalar)}
#> \examples{
#> average.power(n=50,alpha=0.01,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#> }
#> \references{
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#> Gadbury GL, et al. (2004) Power and sample size estimation in high dimensional biology. Statistical Methods in Medical Research 13(4):325-38.
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END average.power
######################################
#> BEGIN afdr
#> \title{Compute the anticipated FDR}
#> \description{Compute the anticipated FDR for given sample size, p-value threshold, and effect size.}
#> \arguments{
afdr=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value cut-off (scalar)}
pow.func, #> \item{pow.func}{an R function that computes statistical power}
eff.size, #> \item{eff.size}{effect size vector}
lam=0.95, #> \item{lam}{p-value at which to evaluate ensemble PDF (for pi.star)}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
...) #> \item{...}{additional agruments for the functions} }
{
pi.hat=pi.star(n,pow.func,eff.size,lam,eps,...)
pow.temp=pow.func(n,alpha,eff.size,...)
pr.rej=mean(pow.temp)
return(pi.hat*alpha/pr.rej)
}
#> \value{the aFDR}
#> \details{The aFDR is defined by Pounds and Cheng (2005) as the anticipated false discovery rate
#> incurred by performing all tests with p-value threshold alpha given the same size
#> effect size and power function.}
#> \examples{
#> afdr(n=50,alpha=0.01,pow.func=power.twosampt,eff.size=rep(c(1,0),c(100,900)))
#> }
#> \references{Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END afdr
######################################
#> BEGIN alpha.power
#> \title{Find the p-value threshold that gives a specified average power}
#> \description{Find p-value cut-off that yields desired average power given n and effect size}
#> \arguments{
alpha.power=function(ave.pow, #> \item{ave.pow}{desired average power (scalar)}
n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{name of R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
null.effect, #> \item{null.effect}{value of effect size that corresponds to null hypothesis}
tol=0.000001, #> \item{tol}{tolerance for bisection solution to alpha}
...) #> \item{...}{additional agruments for the functions} }
{
pow0=0
pow1=1
a0=0
a1=1
k=ceiling(-log(tol)/log(2))
for (i in 1:k) # Bisection loop
{
a.mid=(a0+a1)/2
pow.mid=average.power(n,a.mid,pow.func,eff.size,null.effect=null.effect,...)
#print(pow.mid)
if (pow.mid>ave.pow)
{
a1=a.mid
pow1=pow.mid
}
else
{
a0=a.mid
pow0=pow.mid
}
}
return(list(alpha=a1,ave.pow=pow1))
}
#> \value{a list with the following components:
#> \item{alpha}{desired value of alpha}
#> \item{ave.pow}{average power at that alpha}
#> }
#> \examples{
#> alpha.power(ave.pow=0.8,n=50,pow.func=power.twosampt,
#> eff.size=rep(0:1,c(900,100)),null.effect=0)
#> }
#> \references{A Onar-Thomas, S Pounds. "FDRsampsize: An R package to Perform Generalized Power and Sample Size Calculations for Planning Studies that use the False Discovery Rate to Measure Significance", Manuscript 2015.
#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.
#>
#> Jung, Sin-Ho. "Sample size for FDR-control in microarray data analysis." Bioinformatics 21.14 (2005): 3097-3104.}
#> END alpha.power
######################################
#> BEGIN pi.star
#> \title{Compute the anticipated null proportion estimate}
#> \description{Compute an approximation of the expected value of the null proportion estimate given the sample size and effect size.}
#> \arguments{
pi.star=function(n, #> \item{n}{sample size}
pow.func, #> \item{pow.func}{an R function to compute statistical power}
eff.size, #> \item{eff.size}{effect size vector}
lam=0.95, #> \item{lam}{p-value at which to numerically evaluate p-value pdf (scalar)}
eps=0.0001, #> \item{eps}{epsilon for numerical differentiation}
...) #> \item{...}{additional agruments for the functions} }
{
pow.lam=pow.func(n,lam,eff.size,...)
pow.lam.eps=pow.func(n,lam-eps,eff.size,...)
res=exp(log(mean(pow.lam)-
mean(pow.lam.eps))-
log(eps))
return(res)
}
#> \value{scalar value for approximated E(pi.hat)}
#> \references{#> Pounds, Stan, and Cheng Cheng. "Sample size determination for the false discovery rate." Bioinformatics 21.23 (2005): 4263-4271.}
#> END pi.star
######################################
#> BEGIN power.cox
#> \title{Compute the power of a single-predictor Cox regression model}
#> \description{Use the formula of Hseih and Lavori (2000) to compute the power of a single-predictor Cox model.}
#> \arguments{
power.cox=function(n, #> \item{n}{number of events (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
logHR, #> \item{logHR}{log hazard ratio (vector)}
v) #> \item{v}{variance of predictor variable (vector)}
#> }
{
pnorm(qnorm(alpha/2),sqrt(n*v)*logHR)+
1-pnorm(qnorm(1-alpha/2),sqrt(n*v)*logHR)
}
#> \value{vector of power estimates for two-sided test}
#> \references{
#> Hsieh, FY and Lavori, Philip W (2000) Sample-size calculations for the Cox proportional hazards regression model with nonbinary covariates. Controlled Clinical Trials 21(6):552-560.
#> }
#> \examples{
# # Load the package
#> power.cox # show the power.cox function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.cox,
#> eff.size=rep(c(log(2),0),c(100,900)),
#> null.effect=0,
#> v=1)
#> res
#>}
#> END power.cox
######################################
#> BEGIN power.onesampt
#> \title{Compute power of the one-sample t-test}
#> \description{Estimate power of the one-sample t-test;Uses classical power formula for one-sample t-test}
#> \arguments{
power.onesampt=function(n, #> \item{n}{number of events (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
delta, #> \item{delta}{difference of actual mean from null mean (vector)}
sigma=1) #> \item{sigma}{standard deviation (vector or scalar, default=1)}
#> }
{
1-pf(qf(1-alpha,1,n-1),1,n-1,ncp=n*(delta/sigma)^2)
}
#> \value{vector of power estimates for two-sided test}
#> \examples{
# # Load the package
#> power.onesampt # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.onesampt,
#> eff.size=rep(c(2,0),c(100,900)),
#> null.effect=0,
#> sigma=1)
#> res
#>}
#> END power.onesampt
######################################
#> BEGIN power.twosampt
#> \title{Compute power of the two-samples t-test}
#> \description{Estimate power of the two-samples t-test;Uses classical power formula for two-sample t-test;Assumes equal variance and sample size }
#> \arguments{
power.twosampt=function(n, #> \item{n}{per-group sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
delta, #> \item{delta}{difference between population means (vector)}
sigma=1) #> \item{sigma}{standard deviation (vector or scalar)}
#> }
{
1-pf(qf(1-alpha,1,2*(n-1)),1,2*(n-1),ncp=n*(delta/sigma)^2/2)
}
#> \value{vector of power estimates for two-sided test}
#> \details{For many applications, the null.effect is zero difference of means.}
#> \examples{
# # Load the package
#> power.twosampt # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.twosampt,
#> eff.size=rep(c(2,0),c(100,900)),
#> null.effect=0,
#> sigma=1)
#> res
#>}
#> END power.twosampt
######################################
#> BEGIN power.oneway
#> \title{Compute power of one-way ANOVA}
#> \description{Compute power of one-way ANOVA;Uses classical power formula for ANOVA;Assumes equal variance and sample size }
#> \arguments{
power.oneway=function(n, #> \item{n}{per-group sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
theta, #> \item{theta}{sum of ((group mean - overall mean)/stdev)^2 across all groups for each hypothesis test (vector)}
k=2) #> \item{k}{the number of groups to be compared, default k=2}
#> }
{
1-pf(qf(1-alpha,k-1,k*(n-1)),k-1,k*(n-1),ncp=n*theta)
}
#> \value{vector of power estimates for test of equal means}
#> \details{For many applications, the null effect is zero for the parameter theta described above.}
#> \examples{
# # Load the package
#> power.oneway # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.oneway,
#> eff.size=rep(c(2,0),c(100,900)),
#> null.effect=0,
#> k=3)
#> res
#>}
#> END power.oneway
######################################
#> BEGIN power.tcorr
#> \title{Compute Power of the t-test for non-zero correlation}
#> \description{Estimate power of t-test for non-zero correlation;Uses classical power formula for t-test}
#> \arguments{
power.tcorr=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
rho) #> \item{rho}{population correlation coefficient (vector)}
#> }
{
1-pf(qf(1-alpha,1,n-2),1,n-2,ncp=(n-2)*rho^2/(1-rho^2))
}
#> \value{vector of power estimates for two-sided tests}
#> \details{For many applications, the null.effect is rho=0.}
#> \examples{
# # Load the package
#> power.tcorr # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.tcorr,
#> eff.size=rep(c(0.3,0),c(100,900)),
#> null.effect=0)
#> res
#>}
#> END power.tcorr
######################################
#> BEGIN power.ranksum
#> \title{Compute power of the rank-sum test}
#> \description{Compute power of rank-sum test;Uses formula of Noether (JASA 1987)}
#> \arguments{
power.ranksum=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
p) #> \item{p}{Pr (Y>X), as in Noether (JASA 1987)}
#> }
{ mu0=0.5*n*n
mu1=p*n*n
delta=(mu1-mu0)
sig2=n*n*(2*n+1)/12
sig=sqrt(sig2)
z.reject=-abs(qnorm(alpha/2,0,sig))
pow=pnorm(z.reject,delta,sig)+pnorm(-z.reject,delta,sig,lower.tail=F)
return(pow)
}
#> \value{vector of power estimates for two-sided tests}
#> \details{In most applications, the null effect size will be designated by p = 0.5, which indicates that
# the probability that a randomly selected subject from group A will have a
# greater value of the variable than a randomly selected subject from group B.
#> Thus, in the example below, the argument null.effect=0.5 is specified in the call to fdr.sampsize.}
#> \examples{
# # Load the package
#> power.ranksum # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.ranksum,
#> eff.size=rep(c(0.8,0.5),c(100,900)),
#> null.effect=0.5)
#> res
#>}
#> \references{Noether, Gottfried E (1987) Sample size determination for some common nonparametric tests.
#> Journal of the American Statistical Association, 82:645-647.}
#> END power.ranksum
######################################
#> BEGIN power.signtest
#> \title{Compute power of the sign test}
#> \description{Use the Noether (1987) formula to compute the power of the sign test}
#> \arguments{
power.signtest=function(n, #> \item{n}{sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
p) #> \item{p}{Pr (Y>X), as in Noether (JASA 1987)}
#> }
{
mu0=0.5*n
mu1=p*n
sig0=sqrt(n*0.25)
sig1=sqrt(n*p*(1-p))
pow=pnorm(qnorm(alpha/2,mu0,sig0),mu1,sig1)+
pnorm(qnorm(1-alpha/2,mu0,sig0),mu1,sig1,lower.tail=F)
return(pow)
}
#> \value{vector of power estimates for two-sided tests}
#> \details{In most applications, the null effect size will be designated by p = 0.5 instead of p = 0.
#> Thus, in the call to fdr.sampsize, we specify null.effect=0.5 in the example below.}
#> \references{Noether, Gottfried E (1987) Sample size determination for some common nonparametric tests.
#> Journal of the American Statistical Association, 82:645-647.}
#> \examples{
# # Load the package
#> power.signtest # show the power function
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.signtest,
#> eff.size=rep(c(0.8,0.5),c(100,900)),
#> null.effect=0.5)
#> res
#>}
#> END power.signtest
#> BEGIN power.hart
#> \title{Compute Power for RNA-seq Experiments Assuming Negative Binomial Distribution.}
#> \description{Use the formula of Hart et al (2013) to compute power for comparing RNA-seq expression across two groups assuming a negative binomial distribution.}
#> \arguments{
power.hart=function(n, #> \item{n}{per-group sample size (scalar)}
alpha, #> \item{alpha}{p-value threshold (scalar)}
log.fc, #> \item{log.fc}{log fold-change (vector), usual null hypothesis is log.fc=0}
mu, #> \item{mu}{read depth per gene (vector, same length as log.fc)}
sig) #> \item{sig}{coefficient of variation (CV) per gene (vector, same length as log.fc)}
#> }
{
z.alpha=qnorm(alpha/2)
res=pnorm(z.alpha,sqrt(n*log.fc^2/(2*(1/mu+sig^2))))+
pnorm(-z.alpha,sqrt(n*log.fc^2/(2*(1/mu+sig^2))),lower.tail=F)
return(res)
}
#> \value{vector of power estimates for the set of two-sided tests}
#> \references{SN Hart, TM Therneau, Y Zhang, GA Poland, and J-P Kocher (2013).
#> Calculating Sample Size Estimates for RNA Sequencing Data.
#> Journal of Computational Biology 20: 970-978.}
#> \details{This function is based on equation (1) of Hart et al (2013). It
#> assumes a negative binomial model for RNA-seq read counts and
#> equal sample size per group.}
#> \examples{
# # Load the package
#> power.hart # show the power function
#> n.hart=2*(qnorm(0.975)+qnorm(0.9))^2*(1/20+0.6^2)/(log(2)^2) # Equation 6 of Hart et al
#> power.hart(n.hart,0.05,log(2),20,0.6) # Recapitulate 90% power
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.hart,
#> eff.size=rep(c(log(2),0),c(100,900)),
#> null.effect=0,mu=5,sig=1)
#> res
#>}
#> END power.hart
#> BEGIN power.li
#> \title{Compute Power for RNA-Seq Experiments Assuming Poisson Distribution}
#> \description{Use the formula of Li et al (2013) to compute power for comparing RNA-seq expression across two groups assuming the Poisson distribution.}
#> \arguments{
power.li=function(n, #> \item{n}{per-group sample size}
alpha, #> \item{alpha}{p-value threshold}
rho, #> \item{rho}{fold-change, usual null hypothesis is that rho=1}
mu0, #> \item{mu0}{average count in control group}
w=1, #> \item{w}{ratio of total number of }
type="w") #> \item{type}{type of test: "w" for Wald, "s" for score, "lw" for log-transformed Wald, "ls" for log-transformed score.}
#> }
{
z.alpha=qnorm(alpha/2)
if (!is.element(type,c("w","lw","s","ls")))
stop("type must be 'w', 's', 'ls', or 'lw'.")
if (any(w<=0)) stop("w must be >0.")
if (any(rho<=0)) stop("rho must be >0.")
if (type=="w")
{
mu.z=sqrt(n*mu0*(rho-1)^2/(1+rho/w))
res=pnorm(z.alpha,mu.z)+pnorm(-z.alpha,mu.z,lower.tail=F)
return(res)
}
if (type=="lw")
{
mu.z=sqrt(n*mu0*(log(rho)^2)/(1+1/(rho*w)))
res=pnorm(z.alpha,mu.z)+pnorm(-z.alpha,mu.z,lower.tail=F)
return(res)
}
if (type=="s")
{
z.cut=z.alpha*sqrt((1+w*rho)/(w+rho))
mu.z=sqrt(n*mu0*(rho-1)^2/(1+rho/w))
res=pnorm(z.cut,mu.z)+pnorm(-z.cut,mu.z,lower.tail=F)
return(res)
}
if (type=="ls")
{
z.cut=z.alpha*sqrt((2+w+1/w)/((1+w*rho)*(1+1/(w*rho))))
mu.z=sqrt(n*mu0*(log(rho)^2)/(1+1/(rho*w)))
res=pnorm(z.cut,mu.z)+pnorm(-z.cut,mu.z,lower.tail=F)
return(res)
}
}
#> \value{vector of power estimates for two-sided tests}
#> \references{C-I Li, P-F Su, Y Guo, and Y Shyr (2013).
#> Sample size calculation for differential expression analysis of
#> RNA-seq data under Poisson distribution.
#> Int J Comput Biol Drug Des 6(4). doi:10.1504/IJCBDD.2013.056830}
#> \details{This function computes the power for each of a series of two-sided
#> tests defined by the input parameters. The power is based on the
#> sample size formulas in equations 10-13 of Li et al (2013).
#> Also, note that the null.effect is set to 1 in the examples because
#> the usual null hypothesis is that the fold-change = 1.}
#> \examples{
# # Load the package
#> power.li # show the power function
#> power.li(88,0.05,1.25,5,0.5,"w") # recapitulate 80% power in Table 1 of Li et al (2013)
#> res=fdr.sampsize(fdr=0.1,
#> ave.pow=0.8,
#> pow.func=power.li,
#> eff.size=rep(c(1.5,1),c(100,900)),
#> null.effect=1,
#> mu0=5,w=1,type="w")
#> res
#>}
#> END power.li
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