Ch09-CDF-Pval: CDF of pooled (H0 and HA) population p-values
In pwrFDR: FDR Power

Description Usage Arguments Details Value Author(s) References See Also Examples

1 2	CDF.Pval(u, effect.size, n.sample, r.1, groups=2, type="balanced", grpj.per.grp1=1, control)

`u`	Argument of the CDF. Result will be Pr( P_i <= u )
`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.
`groups`	The number of experimental groups to compare. Default value is 2.
`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. '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

Computes the CDF of the pooled population p-values under the mixture model, e.g. the p-values are i.i.d. with CDF a mixture between a uniform (CDF in the null distributed population) and a concave function (CDF in the non-null distributed population). If Fc_0 is the cCDF of a test statistic under H0 and Fc_A is the cCDF of a test statistic under HA then the CDF of the P-values is

G(u) = (1-r) u + r Fc_A(Fc_0^-1(u))

The limiting positve call fraction, lim_m V_m/m = gamma (a.s.) is the solution to the equation

G( gamma alpha) = gamma

where alpha is the nominal FDR.

A list with components

`call`	The call which produced the result
`u`	The argument that was passed to the function
`CDF.Pval`	The value of the CDF

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

Genovese, C. and L. Wasserman. (2004) A stochastic process approach to false discovery control. Annals of Statistics. 32 (3), 1035-1061.

CDF.Pval.HA

  ## First calculate an average power for a given set of parameters
  rslt.avgp <- pwrFDR(effect.size=0.79, n.sample=46, r.1=2000/54675, alpha=0.15)

  ## Now verify that G( gamma alpha ) = gamma

  gma <- rslt.avgp$gamma
  alpha <- rslt.avgp$call$alpha

  G.gma.a <- CDF.Pval(u=gma*alpha, r.1=2000/54675, effect.size=0.79, n.sample=46)$CDF.Pval$CDF.Pval

  c(G.of.gamma.alpha=G.gma.a, gamma=gma)