# Simultaneous critical values for t-tests in very high dimensions

### Description

Implements the method developed by Cao and Kosorok (2011) for the significance analysis of thousands of features in high-dimensional biological studies. It is an asymptotically valid data-driven procedure to find critical values for rejection regions controlling the k-familywise error rate, false discovery rate, and the tail probability of false discovery proportion.

### Usage

1 2 |

### Arguments

`dataSet1 ` |
data.frame or matrix containing the dataset for subset 1 for the two-sample t-test. |

`dataSet2 ` |
data.frame or matrix containing the dataset for subset 2 for the two-sample t-test. |

`gammas ` |
vector of significance levels at which feature significance is to be determined. |

`compare ` |
one of ("ST", "BH", "Both", "None"). In addition to the Cao-Kosorok method, obtain feature significance indicators using the Storey-Tibshirani method (ST) (Storey and Tibshirani, 2003), the Benjamini-Hochberg method (BH), (Benjamini andHochberg, 1995), "both" the ST and the BH methods, or do not consider alternative methods (none). |

`cSequence` |
A vector specifying the values of c to be considered in estimating the proportion of alternative hypotheses. If no vector is provided, a default of seq(0.01,6,0.01) is used. See Section 2.3 of Cao and Kosorok (2011) for more information. |

`tSequence` |
A vector specifying the search space for the critical t value. If no vector is provided, a default of seq(0.01,6,0.01) is used. |

### Details

The Storey-Tibshirani (2003), ST, method implemented in highTtest is adapted from the implementation written by Alan Dabney and John D. Storey and available from

http://www.bioconductor.org/packages/release/bioc/html/qvalue.html.

The comparison capability is included only for convenience and reproducibility of the original manuscript. For a complete analysis based on the ST method, the user is referred to the qvalue package available through the bioconductor archive.

The following methods retrieve individual results from a highTtest object, x:

`BH(x)`

:
Retrieves a matrix of logical values. The
rows correspond to features, the columns to levels
of significance. Matrix elements are TRUE if feature
was determined to be significant by the Benjamini-Hochberg
(1995) method.

`CK(x)`

:
Retrieves a matrix of logical values. The
rows correspond to features, the columns to levels
of significance. Matrix elements are TRUE if feature
was determined to be significant by the Cao-Kosorok
(2011) method.

`pi_alt(x)`

: Retrieves the
estimated proportion of alternative hypotheses
obtained by the Cao-Kosorok (2011) method.

`pvalue(x)`

: Retrieves the
vector of p-values calculated using the
two-sample t-statistic.

`ST(x)`

:
Retrieves a matrix of logical values. The
rows correspond to features, the columns to levels
of significance. Matrix elements are TRUE if feature
was determined to be significant by the Storey-Tibshirani
(2003) method.

A simple x-y plot comparing the number of significant features as a function of the level significance level can be generated using

`plot(x,...)`

: Generates a plot
of the number of significant features as a function of the
level of significance as calculated for each method (CK,BH, and/or
ST). Additional plot controls can be passed through the ellipsis.

When comparisons to the ST and BH methods are requested, Venn diagrams can be generated using provided that package colorfulVennPlot is installed.

`vennD(x, gamma, ...)`

: Generates
two- and three-dimensional Venn diagrams comparing the
features selected by each method. Implements methods of
package colorfulVennPlot. In addition to the `highTtest`

object, the level of significance, `gamma`

, must
also be provided. Most control argument of the
colorfulVennPlot package can be passed through the ellipsis.

### Value

Returns an object of class `highTtest`

.

### Author(s)

Authors: Hongyuan Cao, Michael R. Kosorok, and Shannon T. Holloway <sthollow@ncsu.edu> Maintainer: Shannon T. Holloway <sthollow@ncsu.edu>

### References

Cao, H. and Kosorok, M. R. (2011). Simultaneous critical values for t-tests in very high dimensions. Bernoulli, 17, 347–394. PMCID: PMC3092179.

Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society: Series B, 57, 289–300.

Storey, J. and Tibshirani, R. (2003). Statistical significance for genomewide studies. Proceedings of the National Academy of Sciences, USA, 100, 9440–9445.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
set.seed(123)
x1 <- matrix(c(runif(500),runif(500,0.25,1)),nrow=100)
obj <- highTtest(dataSet1=x1[,1:5],
dataSet2=x1[,6:10],
gammas=seq(0.1,1,0.1),
tSequence=seq(0.001,3,0.001))
#Print number of significant features identified in each method
colSums(CK(obj))
colSums(ST(obj))
colSums(BH(obj))
#Plot the number of significant features identified in each method
plot(obj, main="Example plot")
ltry <- try(library(colorfulVennPlot),silent=TRUE)
if( !is(ltry,"try-error") ) vennD(obj, 0.8, Title="Example vennD")
#Proportion of alternative hypotheses
pi_alt(obj)
#p-values
pvalue(obj)
``` |