pingouin.multicomp

pingouin.multicomp(pvals, alpha=0.05, method='holm')[source]

P-values correction for multiple comparisons.

Parameters
pvalsarray_like

Uncorrected p-values.

alphafloat

Significance level.

methodstring

Method used for testing and adjustment of p-values. Can be either the full name or initial letters. Available methods are

'bonf' : one-step Bonferroni correction
'sidak' : one-step Sidak correction
'holm' : step-down method using Bonferroni adjustments
'fdr_bh' : Benjamini/Hochberg FDR correction
'fdr_by' : Benjamini/Yekutieli FDR correction
'none' : pass-through option (no correction applied)
Returns
rejectarray, boolean

True for hypothesis that can be rejected for given alpha.

pvals_correctedarray

P-values corrected for multiple testing.

Notes

This function is similar to the p.adjust R function.

The correction methods include the Bonferroni correction (bonf) in which the p-values are multiplied by the number of comparisons. Less conservative methods are also included such as Sidak (1967) (sidak), Holm (1979) (holm), Benjamini & Hochberg (1995) (fdr_bh), and Benjamini & Yekutieli (2001) (fdr_by), respectively.

The first three methods are designed to give strong control of the family-wise error rate. Note that the Holm’s method is usually preferred. The fdr_bh and fdr_by methods control the false discovery rate, i.e. the expected proportion of false discoveries amongst the rejected hypotheses. The false discovery rate is a less stringent condition than the family-wise error rate, so these methods are more powerful than the others.

The Bonferroni adjusted p-values are defined as:

\[\widetilde {p}_{{(i)}}= n \cdot p_{{(i)}}\]

where \(n\) is the number of finite p-values (i.e. excluding NaN).

The Sidak adjusted p-values are defined as:

\[\widetilde {p}_{{(i)}}= 1 - (1 - p_{{(i)}})^{n}\]

The Holm adjusted p-values are the running maximum of the sorted p-values divided by the corresponding increasing alpha level:

\[\widetilde {p}_{{(i)}}=\max _{{j\leq i}}\left\{(n-j+1)p_{{(j)}} \right\}_{{1}}\]

The Benjamini–Hochberg procedure (BH step-up procedure) controls the false discovery rate (FDR) at level \(\alpha\). It works as follows:

1. For a given \(\alpha\), find the largest \(k\) such that \(P_{(k)}\leq \frac {k}{n}\alpha.\)

2. Reject the null hypothesis for all \(H_{(i)}\) for \(i = 1, \ldots, k\).

The BH procedure is valid when the \(n\) tests are independent, and also in various scenarios of dependence, but is not universally valid.

The Benjamini–Yekutieli procedure (BY) controls the FDR under arbitrary dependence assumptions. This refinement modifies the threshold and finds the largest \(k\) such that:

\[P_{(k)} \leq \frac{k}{n \cdot c(n)} \alpha\]

References

  • Bonferroni, C. E. (1935). Il calcolo delle assicurazioni su gruppi di teste. Studi in onore del professore salvatore ortu carboni, 13-60.

  • Šidák, Z. K. (1967). “Rectangular Confidence Regions for the Means of Multivariate Normal Distributions”. Journal of the American Statistical Association. 62 (318): 626–633.

  • Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65–70.

  • 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.

  • Benjamini, Y., and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics, 29, 1165–1188.

Examples

FDR correction of an array of p-values

>>> import pingouin as pg
>>> pvals = [.50, .003, .32, .054, .0003]
>>> reject, pvals_corr = pg.multicomp(pvals, method='fdr_bh')
>>> print(reject, pvals_corr)
[False  True False False  True] [0.5    0.0075 0.4    0.09   0.0015]

Holm correction with missing values

>>> import numpy as np
>>> pvals[2] = np.nan
>>> reject, pvals_corr = pg.multicomp(pvals, method='holm')
>>> print(reject, pvals_corr)
[False  True False False  True] [0.5    0.009     nan 0.108  0.0012]