fwer2gfwer: Function to compute augmentation MTP adjusted p-values
In multtest: Resampling-based multiple hypothesis testing

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

Augmentation multiple testing procedures (AMTPs) to control the generalized family-wise error rate (gFWER), the tail probability of the proportion of false positives (TPPFP), and false discovery rate (FDR) based on any initial procudeure controlling the family-wise error rate (FWER). AMTPs are obtained by adding suitably chosen null hypotheses to the set of null hypotheses already rejected by an initial FWER-controlling MTP. A function for control of FDR given any TPPFP controlling procedure is also provided.

fwer2gfwer(adjp, k = 0)

fwer2tppfp(adjp, q = 0.05)

fwer2fdr(adjp, method = "both", alpha = 0.05)

`adjp`	Numeric vector of adjusted p-values from any FWER-controlling procedure.
`k`	Maximum number of false positives.
`q`	Maximum proportion of false positives.
`method`	Character string indicating which FDR controlling method should be used. The options are "conservative" for a conservative, general method, "restricted" for a less conservative, but restricted method, or "both" (default) for both.
`alpha`	Nominal level for an FDR controlling procedure (can be a vector of levels).

The gFWER and TPPFP functions control Type I error rates defined as tail probabilities for functions g(Vn,Rn) of the numbers of Type I errors (Vn) and rejected hypotheses (Rn). The gFWER and TPPFP correspond to the special cases g(Vn,Rn)=Vn (number of false positives) and g(Vn,Rn)=Vn/Rn (proportion of false positives among the rejected hypotheses), respectively.

Adjusted p-values for an AMTP are simply shifted versions of the adjusted p-values of the original FWER-controlling MTP. For control of gFWER (Pr(Vn>k)), for example, the first k adjusted p-values are set to zero and the remaining p-values are the adjusted p-values of the FWER-controlling MTP shifted by k. One can therefore build on the large pool of available FWER-controlling procedures, such as the single-step and step-down maxT and minP procedures.

Given a FWER-controlling MTP, the FDR can be conservatively controlled at level alpha by considering the corresponding TPPFP AMTP with q=alpha/2 at level alpha/2, so that Pr(Vn/Rn>alpha/2)<=alpha/2. A less conservative procedure (general=FALSE) is obtained by using an AMTP controlling the TPPFP with q=1-sqrt(1-alpha) at level 1-sqrt(1-alpha), so that Pr(Vn/Rn>1-sqrt(1-alpha))<=1-sqrt(1-alpha). The first, more general method can be used with any procedure that asymptotically controls FWER. The second, less conservative method requires the following additional assumptions: (i) the true alternatives are asymptotically always rejected by the FWER-controlling procedure, (ii) the limit of the FWER exists, and (iii) the FWER-controlling procedure provides exact asymptotic control. See http://www.bepress.com/sagmb/vol3/iss1/art15/ for more details. The method implemented in fwer2fdr for computing rejections simply uses the TPPFP AMTP fwer2tppfp with q=alpha/2 (or 1-sqrt(1-alpha)) and rejects each hypothesis for which the TPPFP adjusted p-value is less than or equal to alpha/2 (or 1-sqrt(1-alpha)). The adjusted p-values are based directly on the FWER adjusted p-values, so that very occasionally a hypothesis will have the indicator that it is rejected in the matrix of rejections, but the adjusted p-value will be slightly greater than the nominal level. The opposite might also occur occasionally.

For fwer2gfwer and fwer2tppfp, a numeric vector of AMTP adjusted p-values. For fwer2fdr, a list with two components: (i) a numeric vector (or a length(adjp) by 2 matrix if method="both") of adjusted p-values for each hypothesis, (ii) a length(adjp) by length(alpha) matrix (or length(adjp) by length(alpha) by 2 array if method="both") of indicators of whether each hypothesis is rejected at each value of the argument alpha.

Katherine S. Pollard with design contributions from Sandrine Dudoit and Mark J. van der Laan.

M.J. van der Laan, S. Dudoit, K.S. Pollard (2004), Augmentation Procedures for Control of the Generalized Family-Wise Error Rate and Tail Probabilities for the Proportion of False Positives, Statistical Applications in Genetics and Molecular Biology, 3(1). http://www.bepress.com/sagmb/vol3/iss1/art15/

M.J. van der Laan, S. Dudoit, K.S. Pollard (2004), Multiple Testing. Part II. Step-Down Procedures for Control of the Family-Wise Error Rate, Statistical Applications in Genetics and Molecular Biology, 3(1). http://www.bepress.com/sagmb/vol3/iss1/art14/

S. Dudoit, M.J. van der Laan, K.S. Pollard (2004), Multiple Testing. Part I. Single-Step Procedures for Control of General Type I Error Rates, Statistical Applications in Genetics and Molecular Biology, 3(1). http://www.bepress.com/sagmb/vol3/iss1/art13/

Katherine S. Pollard and Mark J. van der Laan, "Resampling-based Multiple Testing: Asymptotic Control of Type I Error and Applications to Gene Expression Data" (June 24, 2003). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 121. http://www.bepress.com/ucbbiostat/paper121

MTP, MTP-class, MTP-methods, mt.minP, mt.maxT

data<-matrix(rnorm(200),nr=20)
group<-c(rep(0,5),rep(1,5))
fwer.mtp<-MTP(X=data,Y=group)
fwer.adjp<-fwer.mtp@adjp
gfwer.adjp<-fwer2gfwer(adjp=fwer.adjp,k=c(1,5,10))
compare.gfwer<-cbind(fwer.adjp,gfwer.adjp)
mt.plot(adjp=compare.gfwer,teststat=fwer.mtp@statistic,proc=c("gFWER(0)","gFWER(1)","gFWER(5)","gFWER(10)"),col=1:4,lty=1:4)
title("Comparison of Single-step MaxT gFWER Controlling Methods")