Computes the complimentary CDF for the significant call proportion, R_m/m via asymptotic approximation. Included here mainly for pedagogic purposes.
cCDF.Rom(u, effect.size, n.sample, r.1, alpha, delta, groups = 2, N.tests,
type = c("paired", "balanced", "unbalanced"), grpj.per.grp1 = NULL,
FDP.control.method = "BHFDR", control = list(tol = 1e-08,
max.iter = c(1000, 20), distopt = 1, CS = list(NULL),
sim.level = 2, low.power.stop = TRUE, FDP.meth.thresh = FDP.cntl.mth.thrsh.def,
verb = FALSE))A sorted vector of values on the interval [0, 1] for which the cCDF of R_m/m should be computed.
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. Required for calculation of power
The proportion of simultaneous tests that are non-centrally located
The false discovery rate (in the BH case) or the upper bound on the probability that the FDP exceeds delta (Romano case)
If the "FDP.control.method" is set to 'Romano' or 'BHCLT', then the user can set the exceedance thresh-hold for the FDP tail probability control \(P\{ FDP > \delta \} < \alpha\). The default value is \(\alpha\).
The number of experimental groups to compare. Must be integral and >=1. The default value is 2.
The number of simultaneous hypothesis tests.
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.
A character string specifying how the false discovery proportion (FDP) is to be controlled. You may specify the whole word or any shortened uniquely identifying truncation. "BHFDR": the usual BH-FDR "BHCLT": use asymptotic approximation to the distribution of the FDP to find a smaller FDR which guarantees probability less than alpha that the FDP exceeds alpha. "Romano": use Romano's method which guarantees probability less than alpha that the FDP exceeds alpha. "Auto": in 'FixedPoint' mode, the program will use its own wisdom to determine which choice above to make. The order of conservatism is Romano > BHCLT > BHFDR, but BHFDR offers only expected control while the other two guarantee bounds on the excedance probabilty. If the distribution of the FDP is nearly degenerate, then BHFDR is the best option. Otherwise, if it can be reliably used, BHCLT would be the best choice. The 'effective' denominator, gamma*N.tests, in the CLT determines when the approximation is good enough and the asymptotic standard error of the FDP determines when the distribution is dispersed enough to matter. Use "Auto" to run through these checks and determine the best. A return argument, 'Auto', displays the choice made. See output components and details. "both": in 'simulation' mode, compute statistics R and T under BHCLT and Romano (in addition to BHFDR). Corresponding statistics are denoted R.st, T.st corresponding to BHCLT control of the FDP, and R.R and T.R corresponding to Romano control of the FDP. If sim.level is set to 2, (default) the statistics R.st.ht and T.st.ht, which are the number rejected and number true positives under BHCLT where r_0 = 1-r_1, gamma, and alpha.star have been estimated from the P-value data and then alpha.star computed from these.
Optionally, a list with components with the following components: '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 the iterated function limit and 1000 for all others by default. 'distop', specifying the distribution family of the central and non-centrally located sub-populations. distopt=1 gives normal (2 groups), distop=2 gives t- (2 groups) and distopt=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 a list CS = list(rho=0.80,n.WC=50) for example. 'sim.level' sim level 2 (default) stipulates, when FDP.control.method is set to "BHCLT", or "both", R.st.ht and T.st.ht are computed in addition to R.st and T.st (see above). 'low.power.stop' in simulation option, will result in an error message if the power computed via FixedPoint method is too low, which result in no solution for the BHCLT option. Default setting is TRUE. Set to FALSE to over-ride this behavior. 'FDP.meth.thresh' fine-tunes the 'Auto' voodoo (see above). Leave this alone. 'verb' vebosity level.
An object of class cdf which contains components
The call which produced the result
A data frame with columns u and cCDF.Rom
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>
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.
Lehmann E. L., Romano J. P.. Generalizations of the familywise error rate. Ann. Stat.. 2005;33(3):1138<U+2013>1154.
Romano Joseph P., Shaikh Azeem M.. Stepup procedures for control of generalizations of the familywise error rate. Ann. Stat.. 2006;34(4):1850-1873.
# NOT RUN {
library(pwrFDR)
u <- seq(from=0,to=1,len=100000)
rslt <- cCDF.Rom(u=u, effect.size=0.9, n.sample=70, r.1=0.05, alpha=0.15, N.tests=1000,
FDP.control.method="Auto")
## plot the result
with(rslt$cCDF.Rom, plot(u, cCDF.Rom, type="s"))
## compute the mean and median as a check
DX <- function(x)c(x[1], diff(x))
.mean. <- with(rslt$cCDF.Rom, sum(cCDF.Rom*DX(u)))
.median. <- with(rslt$cCDF.Rom, u[max(which(cCDF.Rom>0.5))])
# }
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