# Estimation of Proportion of Null Hypotheses among p-values

### Description

This function makes three routines available for estimating the
true proportion of null hypotheses from a vector of unadjusted
p-values arising from a multiple testing experiment.
The specific methods are the Least Slope method (`lsl`

), the
Two Step Test method (`tst`

), and the Storey tail proportion of
p-values method (`storey`

). These methods are derived and
explained in the references given below.

### Usage

1 | ```
estimate.pi0(pvalues, method, alpha = 0.05, lambda = 0.5)
``` |

### Arguments

`pvalues` |
A vector of the unadjusted p-values resulting from a multiple testing experiment. |

`method` |
The technique used to estimate the proportion of true null
hypotheses. Valid arguments are |

`alpha` |
In the Two Step Test method, the level of the Benjamini-Hochberg procedure used to estimate the propotion of true null hypotheses. |

`lambda` |
In the Storey tail proportion of p-values method, the cutoff used to differentiate p-values. |

### Details

The Least Slope method uses the insight that, if we plot the sorted unadjusted p-values, then the p-values corresponding to null hypotheses will have steep slopes, as they are uniformly distributed between 0 and 1. If we find the position where the slope of the sorted p-values increases the most, we can fix that slope and extrapolate to the x-axis, and the position where the line intersects the x-axis is the proportion of p-values estimated to be null. The formal derivation is presented in the reference below.

Storey's method uses the idea that most of the p-values above some parameter lambda (usually set to 0.5) correspons to null hypotheses. The number of hypotheses in this interval can then be used to estimate the number of hypotheses overall which are null hypotheses. See the paper

The Two Step Test method uses the idea that the result of a multiple comparisons procedure gives an estimate for how many hypotheses are null. That is, if we perform the BH procedure on 100 hypotheses, and 10 of them are rejected, then a reasonable estimate of the proportion of null hypotheses among those 100 is pi0 = 0.9.

### Value

An estimate of the proportion of true null hypotheses from the result of the multiple testing experiment that the unadjusted p-values were extracted from.

### Author(s)

Kris Sankaran

### References

Banjamini, Y, Krieger, A.M., and Yekutieli, D. Adaptive linear step-up procedures that control the false discovery rate. Biometrica, 93(3):491, 2006.

Benjamini, Y, and Hochberg, Y. “On the adaptive control of the false discovery rate in multiple testing with independent statistics.” Journal of Educational and Behavioral Statistics, 25(1):60, 2000.

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/

Storey, J.D., Taylor, J.E., and Siegmund, D. Strong control, conservative point estimation, and simultaneous conservative consistency of false discovery rates: a unified approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology),66(1):187-205. 2004.

### Examples

1 2 3 4 5 6 7 | ```
true.p.1 <- runif(20, 0, 0.01)
null.p.1 <- runif(980, 0, 1)
p.val.1 <- c(true.p.1, null.p.1)
estimate.pi0(p.val.1, "lsl")
estimate.pi0(p.val.1, "storey", lambda = 0.2)
estimate.pi0(p.val.1, "tst")
``` |