# P-value combination using the inverse normal method

### Description

Combines one sided p-values using the inverse normal method.

### Usage

1 | ```
invnorm(indpval, nrep, BHth = 0.05)
``` |

### Arguments

`indpval` |
List of vectors of one sided p-values to be combined. |

`nrep` |
Vector of numbers of replicates used in each study to calculate the previous one-sided p-values. |

`BHth` |
Benjamini Hochberg threshold. By default, the False Discovery Rate is controlled at 5%. |

### Details

For each gene *g*, let

*N_g = ∑_{s=1}^S ω_s Φ^{-1}(1-p_{gs}),*

where *p_{gs}* corresponds to the raw *p*-value obtained for gene *g* in a differential
analysis for study *s* (assumed to be uniformly distributed under the null hypothesis), *Φ* the
cumulative distribution function of the standard normal distribution, and *ω_s* a set of weights.
We define the weights *ω_s* as in Marot and Mayer (2009):

*ω_s = √{\frac{∑_c R_{cs}}{∑_\ell ∑_c R_{c\ell}}},*

where *∑_c R_{cs}* is the total number of biological replicates in study *s*. This allows
studies with large numbers of biological replicates to be attributed a larger weight than smaller studies.

Under the null hypothesis, the test statistic *N_g* follows a N(0,1) distribution. A unilateral
test on the righthand tail of the distribution may then be performed, and classical procedures for the
correction of multiple testing, such as that of Benjamini and Hochberg (1995), may subsequently be applied to
the obtained *p*-values to control the false discovery rate at a desired level *α*.

### Value

`DEindices ` |
Indices of differentially expressed genes at the chosen Benjamini Hochberg threshold. |

`TestStatistic ` |
Vector with test statistics for differential expression in the meta-analysis. |

`rawpval ` |
Vector with raw p-values for differential expression in the meta-analysis. |

`adjpval ` |
Vector with adjusted p-values for differential expression in the meta-analysis. |

### Note

This function resembles the function `directpvalcombi`

in the *metaMA* R package; there is, however, one
important difference in the calculation of *p*-values. In particular, for microarray data, it is typically
advised to separate under- and over-expressed genes prior to the meta-analysis. In the case of RNA-seq data,
differential analyses from individual studies typically make use of negative binomial models and exact tests,
which lead to one-sided, rather than two-sided, p-values. As such, we suggest performing a meta-analysis over
the full set of genes, followed by an a posteriori check, and if necessary filter, of genes with conflicting
results (over vs. under expression) among studies.

### References

Y. Benjamini and Y. Hochberg (1995). Controlling the false discovery rate: a pratical and powerful approach
to multiple testing. *JRSS B* (57): 289-300.

Hedges, L. and Olkin, I. (1985). Statistical Methods for Meta-Analysis. Academic Press.

Marot, G. and Mayer, C.-D. (2009). Sequential analysis for microarray data based on sensitivity and meta-analysis.
*SAGMB* 8(1): 1-33.

A. Rau, G. Marot and F. Jaffrezic (2014). Differential meta-analysis of RNA-seq data. *BMC Bioinformatics* **15**:91

### See Also

`metaRNASeq`

### Examples

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