pwrFDR (version 1.95)

cCDF.ToJ: Complementary CDF for the False Discovery Fraction

Description

Computes the complementary CDF for the False Discovery Fraction, T_m/J_m, via approximation based upon the asymptotic distribution.

Usage

cCDF.ToJ(lambda, x=NULL, groups, effect.size, n.sample, r.1, FDR, N.tests, 
           control)

Arguments

lambda

Argument of the complementary CDF. Result will be Pr( S_m/M_m > lambda )

x

In the abbreviated call sequence, the user only needs to specify lambda, N.tests, and x, an object of class pwr returned by the function pwrFDR.

groups

The number of experimental groups to compare. Default value is 2.

effect.size

The effect size (mean over standard deviation) for test statistics having non-zero means. Assumed to be a constant (in magnitude) over non-zero mean test statistics.

n.sample

The number of experimental replicates.

r.1

The proportion of all test statistics that are distributed under HA.

FDR

The false discovery rate.

N.tests

The number of simultaneous hypothesis tests.

control

Optionally, a list with components with the following components: 'groups', used when distop=3 (F-dist), specifying number of groups. 'version', used only in the 'JL' method, choice 0 gives the 'JL' version as published, whereas choice 1 replaces the FDR with r.0*FDR resulting in the infinite simultaneous tests limiting average power, which is the 'Iz' version, but this is redundant because you can specify the 'Iz' method to use this option. 'tol' is a convergence criterion used in iterative methods which is set to 1e-8 by default 'max.iter' is an iteration limit, set to 1000 by default 'distop', specifying the distribution family of the central and non-centrally located sub-populations. =1 gives normal (2 groups) =2 gives t- (2 groups) and =3 gives F- (2+ groups) 'CS', correlation structure, for use only with 'method="simulation"' which will simulate m simulatenous tests with correlations 'rho' in blocks of size 'n.WC'. Specify as list CS = list(rho=0.80, n.WC=50) for example

Value

An object of class "vvv" which is a list having components

cCDF.ToJ

The result

average.power

The average power at the supplied arguments

c.g

The per test threshold that is equivalent to the BH-FDR, on the test statistic scale

gamma

The limiting proportion of tests that were called significant

objective

Result of optimization producing the average power, should be close to zero.

err.III

The probability mass on side of the oppositely signed alternative in two sided tests

sigma.rtm.ToJ

The square root of the asymptotic variance of the root-m scaled false discovery fraction, m^0.5 * T_m/J_m

call

The call which produced the result

Details

The complementary CDF for the False Discovery Fraction, T_m/J_m, is approximated using its asymptotic distribution. Since

m^0.5 ( T_m/J_m - (1 - r.1) FDR ) --D--> N(0, tau^2)

then

P( T_m/J_m > lambda)

~ 1 - Phi( m^0.5 (lambda - (1 - r.1) FDR)/tau )

The approximation is reasonable as long as m tau^2 is large enough A formula for the asymptotic variance is given in the cited manuscript. There is a user level function, var.rtm.ToJ, which computes the asymptotic variance.

References

Izmirlian G. (2017) Average Power and \(\lambda\)-power in Multiple Testing Scenarios when the Benjamini-Hochberg False Discovery Rate Procedure is Used. arXiv:1801.03989

See Also

controlFDF cCDF.SoM

Examples

# NOT RUN {
## Example 1: Explicit call 
   ccdf <- cCDF.ToJ(lambda=(1-2000/54675)*0.15, effect.size=0.79, n.sample=46, r.1=2000/54675, 
                    FDR=0.15, N.tests=1000)
   ccdf

## Example 2: Abbreviated call using result of pwrFDR
   rslt.avgp <- pwrFDR(effect.size=0.79, n.sample=46, r.1=2000/54675, FDR=0.15)
   ccdf <- cCDF.ToJ(lambda=(1-2000/54675)*0.15, x=rslt.avgp, N.tests=1000)
   ccdf
# }