CDF.Pval.ua.eq.u {pwrFDR}

Function which solves the implicit equation u = G( u alpha)

Description

Function which solves the implicit equation u = G( u alpha) where G is the pooled P-value CDF and alpha is the FDR

Usage

  CDF.Pval.ua.eq.u(effect.size, n.sample, r.1, alpha, groups, type, 
                   grpj.per.grp1, control)
  

Arguments

effect.size

The per statistic effect size

n.sample

The per statistic sample size

r.1

The proportion of Statistics distributed according to the alternative distribution

alpha

The false discovery rate.

groups

Number of experimental groups from which the test statistic is calculated

type

A character string specifying, in the groups=2 case, whether the test is 'paired', 'balanced', or 'unbalanced' and in the case when groups >=3, whether the test is 'balanced' or 'unbalanced'. The default in all cases is 'balanced'. Left unspecified in the one sample (groups=1) case.

grpj.per.grp1

Required when type="unbalanced", specifies the group 0 to group 1 ratio in the two group case, and in the case of 3 or more groups, the group j to group 1 ratio, where group 1 is the group with the largest effect under the alternative hypothesis.

control

Optionally, a list with components with the following components: 'groups', used when distop=3 (F-dist), specifying number of groups. '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

A list with a single component,

gamma

The solution of the implicit equation u = G( u alpha), where G is the pooled P-value CDF. This represents the infinite tests limiting proportion of hypothesis tests that are called significant by the BH-FDR procedure at alpha.

Author(s)

Grant Izmirlian <izmirlian at nih dot gov>

References

Izmirlian G. (2020) Strong consistency and asymptotic normality for quantities related to the Benjamini-Hochberg false discovery rate procedure. Statistics and Probability Letters; 108713, <doi:10.1016/j.spl.2020.108713>.

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>

Jung S-H. (2005) Sample size for FDR-control in microarray data analysis. Bioinformatics; 21:3097-3104.

Liu P. and Hwang J-T. G. (2007) Quick calculation for sample size while controlling false discovery rate with application to microarray analysis. Bioinformatics; 23:739-746.

Examples

  ## An example showing that the Romano method is more conservative than the BHCLT method
  ## which is in turn more conservative than the BH-FDR method based upon ordering of the
  ## significant call proportions, R_m/m

  ## First find alpha.star for the BH-CLT method at level alpha=0.15
  a.st.BHCLT <-controlFDP(effect.size=0.8,r.1=0.05,N.tests=1000,n.sample=70,alpha=0.15)$alpha.star

  ## now find the significant call fraction under the BH-FDR method at level alpha=0.15
  gamma.BHFDR <- CDF.Pval.ua.eq.u(effect.size = 0.8, n.sample = 70, r.1 = 0.05, alpha=0.15)

  ## now find the significant call fraction under the Romano method at level alpha=0.15
  gamma.romano <- CDF.Pval.ar.eq.u(effect.size = 0.8, n.sample = 70, r.1 = 0.05, alpha=0.15)

  ## now find the significant call fraction under the BH-CLT method at level alpha=0.15
  gamma.BHCLT <- CDF.Pval.ua.eq.u(effect.size = 0.8, n.sample = 70, r.1 = 0.05, alpha=a.st.BHCLT)