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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).
CDF.Pval(u, effect.size, n.sample, r.1, groups=2, type="balanced",
grpj.per.grp1=1, control)Argument of the CDF. Result will be Pr( P_i <= u )
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.
The number of experimental replicates.
The proportion of all test statistics that are distributed under HA.
The number of experimental groups to compare. Default value is 2.
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.
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.
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
A list with components
The call which produced the result
The argument that was passed to the function
The value of the CDF
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.
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.
# 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)
# }
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