Ch20-CDF-Pval-ar-eq-u: Calculates the fixed point for the Romano procedure.
In pwrFDR: FDR Power

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

Calculates the fixed point for the Romano procedure, e.g. finds u which solves u = G( psi(u, d) a) where G is the common p-value CDF, and psi(u, d) = u d/(1 - (1-a) u). Essentially an internal function and included at the user level for pedagogic purposes.

1 2	CDF.Pval.ar.eq.u(effect.size, n.sample, r.1, alpha, delta, groups, type, grpj.per.grp1, control)

`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. Required for calculation of power
`r.1`	The proportion of simultaneous tests that are non-centrally located
`alpha`	The upper bound on the probability that the FDP exceeds delta.
`delta`	The exceedance thresh-hold for the FDP tail probability control method (BHCLT or Romano) P\{ FDP > δ \} < α. The default value is α.
`groups`	The number of experimental groups to compare. Must be integral and >=1. The 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: '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

`call`	The call which produced the result
`gamma`	The fixed point for the Romano method.

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>

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):1138–1154.

Romano Joseph P., Shaikh Azeem M.. Stepup procedures for control of generalizations of the familywise error rate. Ann. Stat.. 2006;34(4):1850-1873.

CDF.Pval.ua.eq.u

  ## An example showing that the Romano method is more conservative than the BHCLT method
  ## which is in turn more conservative than the BH-FDR method based upon ordering of the
  ## significant call proportions, R_m/m

  ## First find alpha.star for the BH-CLT method at level alpha=0.15
  a.st.BHCLT <-controlFDP(effect.size=0.8,r.1=0.05,N.tests=1000,n.sample=70,alpha=0.15)$alpha.star

  ## now find the significant call fraction under the BH-FDR method at level alpha=0.15
  gamma.BHFDR <- CDF.Pval.ua.eq.u(effect.size = 0.8, n.sample = 70, r.1 = 0.05, alpha=0.15)

  ## now find the significant call fraction under the Romano method at level alpha=0.15
  gamma.romano <- CDF.Pval.ar.eq.u(effect.size = 0.8, n.sample = 70, r.1 = 0.05, alpha=0.15)

  ## now find the significant call fraction under the BH-CLT method at level alpha=0.15
  gamma.BHCLT <- CDF.Pval.ua.eq.u(effect.size = 0.8, n.sample = 70, r.1 = 0.05, alpha=a.st.BHCLT)