fwer2gfwer {multtest} | R Documentation

## Function to compute augmentation MTP adjusted p-values

### Description

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.

### Usage

    fwer2gfwer(adjp, k = 0)

    fwer2tppfp(adjp, q = 0.05)

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

### Arguments

`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).

### Details

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.

### Value

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`.

### Author(s)

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

### References

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

### See Also

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

### Examples

    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")

[Package multtest version 1.22.0 Index]