howmany_dependent: Number of correct rejections, for dependent test statistics
In howmany: A lower bound for the number of correct rejections

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

Lower bounds for the number of correct rejections, for multiple tests of associations with dependent test statistics.

1 2	howmany_dependent(X, Y, alpha = 0.05, test = wilcox.test, alternative = "two.sided", n.permutation=round(20/alpha) )

`X`	a n*p matrix, where each of the p columns contains n observations
`Y`	a factor or numerical vector of length n, containing a binary class variable
`alpha`	the level, a scalar in [0,1]
`test`	the test to be used
`alternative`	an alternative for the test, supplied as an argument to function `test`
`n.permutation`	the number of permutations to use to determine the bounding function

For multiple tests of associations (with possibly dependent test statistics), a lower bound for the number of correct rejections is calculated, which is valid simultaneously for all possible number of rejections and under arbitrary dependence between test statistics. The bound is monotonically increasing with the number of made rejections.

The matrix X contains the observations, while Y contains binary class labels. For each hypothesis k=1,...p, a p-value is calculated internally according to supplied function test, with first argument X[Y==0,k], second argument X[Y==1,k], and additional argument alternative (if the class labels are not 0 and 1, they are converted accordingly). The object returned by test has to have a component p.value, containing (perhaps unsurprisingly) the p-value of the corresponding test.

The focus of the current implementation is on portability, not speed, and computations might take some time.

An object of class howmany, for which summary, plot, and print methods are available.

The lower bound for the number of correct rejections (as a function of the number of rejections) can be accessed with the function lowerbound.

Nicolai Meinshausen, nicolai@stat.berkeley.edu

N. Meinshausen and J. Rice (2006) "Estimating the proportion of false discoveries among a large number of independently tested hypotheses", Annals of Statistics 34(1), 373-393

N. Meinshausen (2006) "False discovery control for multiple tests of association under general dependence", Scandinavian Journal of Statistics 33(2), 227-237

N. Meinshausen and P. Buhlmann (2005) "Lower bounds for the number of false null hypotheses for multiple testing of associations", Biometrika 92(4), 893-907

lowerbound for extracting the number of correct rejections (as a function of the number of made rejections).

howmany for similar functionality for multiple tests of associations, where test statistics are independent.

## Warning: running example might take a
## few minutes of computing time...

## create observation matrix X 
## for p=200 hypotheses with n=40 observations
p <- 200
n <- 40
Indep <- matrix( rnorm(p*n) , ncol= p )  
C <- diag(p); C <- C+matrix( 0.01*rbinom(p^2,1,0.2) , ncol=p ) 
X <- Indep%*%C

## create binary class variables Y
Y <- c( rep(0,round(n/2)), rep(1,n-round(n/2)) )

## 100 false null hypotheses (random effects)
for (k in 1:100){  X[Y==1, k] <- X[Y==1, k] + rnorm(1) }



## compute object of class 'howmany' and print the result
(object <- howmany_dependent(X,Y))

## extract the lower bound
(lower <- lowerbound(object))

## plot the result
plot(object)

## for comparison: number of rejections with Bonferroni correction
(bonf <- sum(object$pvalues<0.05/p))