# FDR Controlling Procedure for Hierarchically Structured Hypotheses

### Description

This function implements the Hierarchical FDR controlling procedure developed Benjamini and Yekutieli. The procedure incorporates structural information about the hierarchical relationships between hypotheses in order to increase power and interpretability of a testing procedure while controlling the False Discovery Rate at a prespecified level. It is applicable whenever we there is a natural hierarchical structure between the hypotheses being tested before the data analysis begins. For example, the method has been used before in Clinical Studies, where nodes deeper in the tree correspond to secondary or tertiary endpoints. It has also been used in QTL analysis, where we first make statements about regions of chromosomes being associated with specific brain activity and then focus on more and more detailed subsets of chromosomes.

### Usage

1 | ```
hFDR.adjust(unadjp, tree.el, alpha = 0.05)
``` |

### Arguments

`unadjp` |
A vector of raw p-values resulting from an
experiment. The names of this vector should be contained
in the edgelist parameterizing the hierarchical structure between
hypothesis, inputted as |

`tree.el` |
The edgelist parameterizing the hierarchical structure between hypotheses. The edges must be stored so that each edge is a row of a two column matrix, where the first column gives the parent and the second gives the child. |

`alpha` |
The level of FDR control within families of the tree. Note that this is NOT necessarily the level of FDR control within the entire tree. Refer to the paper by Yekutieli and Benjamini for bounds on various FDR criterion. |

### Details

The FDR controlling procedure is described in more detail in the paper by Yekutieli and Benjamini 2009. The idea is to control for multiple testing error within families of hypotheses, and only test a descendant family of hypotheses if the associated parent hypotheses was deemed significant in the higher level. The families of hypotheses are taken to be the children of any particular node, and error is controlled within these families using the Benjamini-Hochberg procedure. Different bounds can be proven for the FDR when considered along whole tree, within a single level, and tips. In particular, the whole tree FDR is typically controlled at three times the FDR control within individual families, and this result holds for arbitrary hypotheses tests and configurations of trees.

### Value

An object of class `hypothesesTree`

.

### Author(s)

Kris Sankaran

### References

Yekutieli, D. Hierarchical false discovery rate-controlling methodology. Journal of the American Statistical Association, 103(481):309-316, 2008.

Benjamini, Y, and Yekutieli, D. Hierarchical fdr testing of trees of hypotheses. 2002.

Sankaran, K and Holmes, S. structSSI: Simultaneous and Selective Inference for Grouped or Hierarchically Structured Data. Journal of Statistical Software, 59(13), 1-21. 2014. http://jstatsoft.org/v59/i13/

### See Also

`hypothesesTree-class`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
library('igraph')
library('ape')
alternative.indices <- sample(1:49, 30)
unadj.p.values <- vector("numeric", length = 49)
unadj.p.values[alternative.indices] <- runif(30, 0, 0.01)
unadj.p.values[-alternative.indices] <- runif(19, 0, 1)
unadj.p.values[c(1:5)] <- runif(5, 0, 0.01)
names(unadj.p.values) <- paste("Hyp ", c(1:49))
tree <- as.igraph(rtree(25))
V(tree)$name <- names(unadj.p.values)
tree.el <- get.edgelist(tree)
hyp.tree <- hFDR.adjust(unadj.p.values, tree.el, 0.05)
## We can visualize the difference between the unadjusted and the
## adjusted trees.
plot(hyp.tree, adjust = FALSE)
plot(hyp.tree, adjust = TRUE)
``` |