Ch19-CDF-Pval-ua-eq-u: Function which solves the implicit equation u = G( u alpha)
In pwrFDR: FDR Power

Description Usage Arguments Value Author(s) References Examples

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

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

`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

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.

Grant Izmirlian <izmirlian at nih dot gov>

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 λ-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.

  ## 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)