pwrFDR (version 2.8.9)

CDF.Pval: CDF of pooled (H0 and HA) population p-values

Description

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).

Usage

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

Arguments

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

Value

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

Details

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.

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

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

See Also

CDF.Pval.HA

Examples

# NOT RUN {
  ## 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) 
# }