tr03-tailRankPower: Power of the tail-rank test

In TailRank: The Tail-Rank Statistic

Power of the tail-rank test

Description

Compute the significance level and the power of a tail-rank test.

Usage

tailRankPower(G, N1, N2, psi, phi, conf = 0.95,
              model=c("bb", "betabinom", "binomial"))
tailRankCutoff(G, N1, N2, psi, conf,
               model=c("bb", "betabinom", "binomial"),
               method=c('approx', 'exact'))
                           

Arguments

G	An integer; the number of genes being assessed as potnetial biomarkers. Statistically, the number of hypotheses being tested.
N1	An integer; the number of "train" or "healthy" samples used.
N2	An integer; the number of "test" or "cancer" samples used.
psi	A real number between 0 and 1; the desired specificity of the test.
phi	A real number between 0 and 1; the sensitivity that one would like to be able to detect, conditional on the specificity.
conf	A real number between 0 and 1; the confidence level of the results. Can be obtained by subtracting the family-wise Type I error from 1.
model	A character string that determines whether significance and power are computed based on a binomial or a beta-binomial (bb) model.
method	A character string; either "exact" or "approx". The deafult is to use a Bonferroni approximation.

Details

A power estimate for the tail-rank test can be obtained as follows. First, let X ~ Binom(N,p) denote a binomial random variable. Under the null hypotheis that cancer is not different from normal, we let p = 1 - ψ be the expected proportion of successes in a test of whether the value exceeds the psi-th quantile. Now let

α = P(X > x,| N, p)

be one such binomial measurement. When we make G independent binomial measurements, we take

conf = P(all\ G\ of\ the\ X's ≤ x | N, p).

(In our paper on the tail-rank statistic, we write everything in terms of γ = 1 - conf.) Then we have

conf = P(X ≤ x | N, p)^G = (1 - alpha)^G.

Using a Bonferroni-like approximation, we can take

conf ~= 1 - α*G.

Solving for α, we find that

α ~= (1-conf)/G.

So, the cutoff that ensures that in multiple experiments, each looking at G genes in N samples, we have confidence level conf (or significance level γ = 1 - conf) of no false positives is computed by the function tailRankCutoff.

The final point to note is that the quantiles are also defined in terms of q = 1 - α, so there are lots of disfiguring "1's" in the implementation.

Now we set M to be the significance cutoff using the procedure detailed above. A gene with sensitivity φ gets detected if the observed number of cases above the threshold is greater than or equal to M. The tailRankPower function implements formula (1.3) of our paper on the tail-rank test.

Value

tailRankCutoff returns an integer that is the maximum expected value of the tail rank statistic under the null hypothesis.

tailRankPower returns a real numbe between 0 and 1 that is the power of the tail-rank test to detect a marker with true sensitivity equal to phi.

Author(s)

Kevin R. Coombes <[e-mail]>

See Also

TailRankTest, tailRankPower, biomarkerPowerTable, matrixMean, toleranceBound

Examples

psi.0 <- 0.99
confide <- rev(c(0.8, 0.95, 0.99))
nh <- 20
ng <- c(100, 1000, 10000, 100000)
ns <- c(10, 20, 50, 100, 250, 500)
formal.cut <- array(0, c(length(ns), length(ng), length(confide)))
for (i in 1:length(ng)) {
  for (j in 1:length(ns)) {
    formal.cut[j, i, ] <- tailRankCutoff(ng[i], nh, ns[j], psi.0, confide)
  }
}
dimnames(formal.cut) <- list(ns, ng, confide)
formal.cut

phi <- seq(0.1, 0.7, by=0.1)
N <- c(10, 20, 50, 100, 250, 500)
pows <- matrix(0, ncol=length(phi), nrow=length(N))
for (ph in 1:length(phi)) {
  pows[, ph] <-  tailRankPower(10000, nh, N, 0.95, phi[ph], 0.9)
}
pows <- data.frame(pows)
dimnames(pows) <- list(as.character(N), as.character(round(100*phi)))
pows

TailRank documentation built on Jan. 13, 2023, 3:02 a.m.