CDF.Pval.HA {pwrFDR}

CDF of p-values for test statistics distribted under HA.

Description

Computes the CDF of p-values for test statistics distribted under HA.

Usage

CDF.Pval.HA(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 20 for function iteration and 1000 for all others 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)

Details

Computes the CDF of p-values for test statistics distribted under HA. 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 a P-value for a test statistic distributed under HA is

G_A(u) = Fc_A(Fc_0^-1(u))

The limiting true positive fraction is the infinite simultaneous tests average power,

lim_m T_m/M_m = average.power (a.s.),

which is used to approximate the average power for finite 'm', is G_1 at gamma alpha:

G_1( gamma alpha) = average.pwer

where alpha is the nominal FDR and gamma = lim_m R_m/m (a.s.) is the limiting positive call fraction.

Value

A list with components

call

The call which produced the result

u

The argument that was passed to the function

CDF.Pval.HA

The value of the CDF

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>

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

Examples

  ## First calculate an average power for a given set of parameters
  rslt.avgp <- pwrFDR(effect.size=0.79, n.sample=42, r.1=0.05, alpha=0.15)

  ## Now verify that G_A( gamma f ) = average.power

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

  GA.gma.alpha <- CDF.Pval.HA(u=gma*alpha, r.1=0.05, effect.size=0.79, n.sample=42)

  c(G.gm.alpha=GA.gma.alpha$CDF.Pval.HA$CDF.Pval.HA, average.power=rslt.avgp$average.power)