A function which computes the asymptotic variance of the true positive fraction, S_m/M_m in the BH-FDR procedure on m=N.tests simultaneous tests.
var.rtm.SoM(x, groups, effect.size, n.sample, r.1, FDR, N.tests, control)Calls to this function can be made either specifying the single
argument, x, which is an object of class "pwr" returned from the
function, pwrFDR or by a full call specification, by
specifying all the arguments required by the function pwrFDR
(see below)
The number of experimental groups to compare. Default value is 2.
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.
Number of simultaneous tests.
Optionally, a list with components with the following components: 'groups', used when distop=3 (F-dist), specifying number of groups. 'version', used only in the 'JL' method, choice 0 gives the 'JL' version as published, whereas choice 1 replaces the FDR with r.0*FDR resulting in the infinite simultaneous tests limiting average power, which is the 'Iz' version, but this is redundant because you can specify the 'Iz' method to use this option. '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
Returns a value of class vvv, containing components
The computed asymptotic variance
The average power
The expected proportion of significant calls
The 'q-value', which is the value of the criterion on the scale of the statistic (t of given number of 2 n - 2 degrees of freedom) which can be used as a per test criterion resulting in the equivalent Benjamini-Hochberg procedure
The call which produced the result
The true positive fraction, S_m/M_m, i.e., is the proportion of all non-centrally located statistical tests that are declared significant by the Benjamini-Hochberg procedure. It is shown, in the cited publication, to be root-m consistent and asymptotically normal i.e.
m^1/2 ( S_m/M_m - average.power ) --D--> N(0, sigma^2)
which is to say that the distribution of S_m/M_m is spiked about its mean, the average power, and the width dies off as 1/m^1/2. This noraml approximation is of interest in its own right and is also useful for powering multiple testing experiments on a more conservative operating characteristic than the average power, E[ S_m / M_m ]. For example, we can power the experiment on the lower 10th percentile of the distribution of S_m/M_m, which is approximated as
average.power + qnorm(0.10)*(vS/N.tests)^0.5
While the width of this distribution is negligible for micro-array studies, e.g. when N.tests=54675, it is non-negligible for as many as 200 simultaneous tests, where the average power could be 80% but the lower quantile of the empirical average power could be as low as 50%.
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
# NOT RUN {
## call using result of pwrFDR
rslt.Iz <- pwrFDR(effect.size=0.79, n.sample=46, r.1=2000/54675, FDR=0.15)
vS <- var.rtm.SoM(rslt.Iz)
## call via argument list specification
vS <- var.rtm.SoM(effect.size=0.79, n.sample=46, r.1=2000/54675, FDR=0.15)
# }
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