U Qv fHDã@sPddlZddlmZdgZddd„Zddd „Zdd d „Zdd d „Zddd„Z dS)éN)ÚSeriesÚ multicompçš™™™™™©?Úfdr_bhc Cs.| ¡dkst‚t |¡}|j}| ¡}t |¡ ¡}t |¡}||}| ¡}|j |}t  d|d¡t |ƒ} | ¡dkr t dt  d|d¡¡} | | } |d|…| } tj   | ddd…¡ddd…} t | dd¡} t | t |tj¡¡} | | |¡} tjddt | |¡} W5QRX| | fS) u P-values FDR correction with Benjamini/Hochberg and Benjamini/Yekutieli procedure. This covers Benjamini/Hochberg for independent or positively correlated and Benjamini/Yekutieli for general or negatively correlated tests. Parameters ---------- pvals : array_like Array of p-values of the individual tests. alpha : float Error rate (= alpha level). method : str FDR correction methods. Can be 'fdr_bh' or 'fdr_by'. Returns ------- reject : array, bool True if a hypothesis is rejected, False if not pval_corrected : array P-values adjusted for multiple hypothesis testing using the BH or BY correction. See also -------- bonf : Bonferroni correction holm : Holm-Bonferroni correction Notes ----- From Wikipedia: The **Benjamini–Hochberg** procedure (BH step-up procedure) controls the false discovery rate (FDR) at level :math:`\alpha`. It works as follows: 1. For a given :math:`\alpha`, find the largest :math:`k` such that :math:`P_{(k)}\leq \frac {k}{m}\alpha.` 2. Reject the null hypothesis (i.e., declare discoveries) for all :math:`H_{(i)}` for :math:`i = 1, \ldots, k`. The BH procedure is valid when the m 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 :math:`k` such that: .. math:: P_{(k)} \leq \frac{k}{m \cdot c(m)} \alpha References ---------- - 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. - https://en.wikipedia.org/wiki/False_discovery_rate 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', alpha=.05) >>> print(reject, pvals_corr) [False True False False True] [0.5 0.0075 0.4 0.09 0.0015] )rÚfdr_byérçð?NéÿÿÿÿÚignore©Úinvalid)ÚlowerÚAssertionErrorÚnpÚasarrayÚshapeÚravelÚisnanÚsumÚargsortÚsizeÚarangeÚfloatZminimumÚ accumulateÚclipÚappendÚfullÚnanÚreshapeÚerrstateÚless)ÚpvalsÚalphaÚmethodÚ shape_initÚnum_nanÚ pvals_sortindÚ pvals_sortedÚ sortrevindÚntestsZ ecdffactorÚcmÚ pvals_corrÚpvals_correctedÚreject©r.ú6/tmp/pip-unpacked-wheel-2te3nxqf/pingouin/multicomp.pyÚfdrs*J     r0c Csdt |¡}t |¡ ¡}|t|jƒ|}t |dd¡}tjddt ||¡}W5QRX||fS)açP-values correction with Bonferroni method. Parameters ---------- pvals : array_like Array of p-values of the individual tests. alpha : float Error rate (= alpha level). Returns ------- reject : array, bool True if a hypothesis is rejected, False if not pval_corrected : array P-values adjusted for multiple hypothesis testing using the Bonferroni procedure (= multiplied by the number of tests). See also -------- holm : Holm-Bonferroni correction fdr : Benjamini/Hochberg and Benjamini/Yekutieli FDR correction Notes ----- From Wikipedia: Statistical hypothesis testing is based on rejecting the null hypothesis if the likelihood of the observed data under the null hypotheses is low. If multiple hypotheses are tested, the chance of a rare event increases, and therefore, the likelihood of incorrectly rejecting a null hypothesis (i.e., making a Type I error) increases. The Bonferroni correction compensates for that increase by testing each individual hypothesis :math:`p_i` at a significance level of :math:`p_i = \alpha / n` where :math:`\alpha` is the desired overall alpha level and :math:`n` is the number of hypotheses. For example, if a trial is testing :math:`n=20` hypotheses with a desired :math:`\alpha=0.05`, then the Bonferroni correction would test each individual hypothesis at :math:`\alpha=0.05/20=0.0025``. The Bonferroni adjusted p-values are defined as: .. math:: \widetilde {p}_{{(i)}}= n \cdot p_{{(i)}} The Bonferroni correction tends to be a bit too conservative. Note that NaN values are not taken into account in the p-values correction. References ---------- - Bonferroni, C. E. (1935). Il calcolo delle assicurazioni su gruppi di teste. Studi in onore del professore salvatore ortu carboni, 13-60. - https://en.wikipedia.org/wiki/Bonferroni_correction Examples -------- >>> import pingouin as pg >>> pvals = [.50, .003, .32, .054, .0003] >>> reject, pvals_corr = pg.multicomp(pvals, method='bonf', alpha=.05) >>> print(reject, pvals_corr) [False True False False True] [1. 0.015 1. 0.27 0.0015] Nrr r ) rrrrrrrrr )r!r"r%r,r-r.r.r/Úbonfzs@ r1c CsÎt |¡}|j}| ¡}t |¡ ¡}t |¡}||}| ¡}|j|}|d|…t |dd¡}tj   |¡}t  |dd¡}t  |t  |tj¡¡}|| |¡} tjddt | |¡} W5QRX| | fS)u^P-values correction with Holm method. Parameters ---------- pvals : array_like Array of p-values of the individual tests. alpha : float Error rate (= alpha level). Returns ------- reject : array, bool True if a hypothesis is rejected, False if not pvals_corrected : array P-values adjusted for multiple hypothesis testing using the Holm procedure. See also -------- bonf : Bonferroni correction fdr : Benjamini/Hochberg and Benjamini/Yekutieli FDR correction Notes ----- From Wikipedia: In statistics, the Holm–Bonferroni method (also called the Holm method) is used to counteract the problem of multiple comparisons. It is intended to control the family-wise error rate and offers a simple test uniformly more powerful than the Bonferroni correction. The Holm adjusted p-values are the running maximum of the sorted p-values divided by the corresponding increasing alpha level: .. math:: \frac{\alpha}{n}, \frac{\alpha}{n-1}, ..., \frac{\alpha}{1} where :math:`n` is the number of test. The full mathematical formula is: .. math:: \widetilde {p}_{{(i)}}=\max _{{j\leq i}}\left\{(n-j+1)p_{{(j)}} \right\}_{{1}} Note that NaN values are not taken into account in the p-values correction. References ---------- - Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, 65-70. - https://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni_method Examples -------- >>> import pingouin as pg >>> pvals = [.50, .003, .32, .054, .0003] >>> reject, pvals_corr = pg.multicomp(pvals, method='holm', alpha=.05) >>> print(reject, pvals_corr) [False True False False True] [0.64 0.012 0.64 0.162 0.0015] Nrr rr r )rrrrrrrrrÚmaximumrrrrrrrr ) r!r"r$r%r&r'r(r)r+r,r-r.r.r/ÚholmÃs A    r3c Cstt |¡}t |¡ ¡}t|jƒ|}dt d||¡}t |dd¡}tjddt  ||¡}W5QRX||fS)u‚P-values correction with Sidak method. Parameters ---------- pvals : array_like Array of p-values of the individual tests. alpha : float Error rate (= alpha level). Returns ------- reject : array, bool True if a hypothesis is rejected, False if not pval_corrected : array P-values adjusted for multiple hypothesis testing using the Sidak procedure. See also -------- bonf, holm, fdr, multicomp Notes ----- The Sidak adjusted p-values are defined as: .. math:: \widetilde {p}_{{(i)}}= 1 - (1 - p_{{(i)}})^{n} The Sidak correction is slightly more liberal than the Bonferroni correction. Note that NaN values are not taken into account in the p-values correction. References ---------- - Å idák, Z. K. (1967). "Rectangular Confidence Regions for the Means of Multivariate Normal Distributions". Journal of the American Statistical Association. 62 (318): 626–633. - https://en.wikipedia.org/wiki/%C5%A0id%C3%A1k_correction Examples -------- >>> import numpy as np >>> import pingouin as pg >>> pvals = [.50, .003, .32, .054, .0003] >>> reject, pvals_corr = pg.multicomp(pvals, method='sidak', alpha=.05) >>> print(reject, np.round(pvals_corr, 4)) [False True False False True] [0.9688 0.0149 0.8546 0.2424 0.0015] rrNr r ) rrrrrrÚpowerrrr )r!r"r%r)r,r-r.r.r/Úsidaks3 r5c CsJt|ttjtfƒstdƒ‚t|tƒs,tdƒ‚t|tƒs>tdƒ‚d|krRdks\ntdƒ‚t |¡}|  ¡dkr„t ||d\}}n¾|  ¡d kr¢t ||d\}}n |  ¡d krÀt ||d\}}n‚|  ¡d kràt ||d d \}}nb|  ¡dkrt ||dd \}}n@|  ¡dkr:|}tjddt ||¡}W5QRXntdƒ‚||fS)uÊP-values correction for multiple comparisons. Parameters ---------- pvals : array_like Uncorrected p-values. alpha : float Significance level. method : string 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 ------- reject : array, boolean True for hypothesis that can be rejected for given alpha. pvals_corrected : array 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** [1]_ adjusted p-values are defined as: .. math:: \widetilde {p}_{{(i)}}= n \cdot p_{{(i)}} where :math:`n` is the number of *finite* p-values (i.e. excluding NaN). The **Sidak** [2]_ adjusted p-values are defined as: .. math:: \widetilde {p}_{{(i)}}= 1 - (1 - p_{{(i)}})^{n} The **Holm** [3]_ adjusted p-values are the running maximum of the sorted p-values divided by the corresponding increasing alpha level: .. math:: \widetilde {p}_{{(i)}}=\max _{{j\leq i}}\left\{(n-j+1)p_{{(j)}} \right\}_{{1}} The **Benjamini–Hochberg** procedure (BH step-up procedure, [4]_) controls the false discovery rate (FDR) at level :math:`\alpha`. It works as follows: 1. For a given :math:`\alpha`, find the largest :math:`k` such that :math:`P_{(k)}\leq \frac {k}{n}\alpha.` 2. Reject the null hypothesis for all :math:`H_{(i)}` for :math:`i = 1, \ldots, k`. The BH procedure is valid when the :math:`n` tests are independent, and also in various scenarios of dependence, but is not universally valid. The **Benjamini–Yekutieli** procedure (BY, [5]_) controls the FDR under arbitrary dependence assumptions. This refinement modifies the threshold and finds the largest :math:`k` such that: .. math:: P_{(k)} \leq \frac{k}{n \cdot c(n)} \alpha References ---------- .. [1] Bonferroni, C. E. (1935). Il calcolo delle assicurazioni su gruppi di teste. Studi in onore del professore salvatore ortu carboni, 13-60. .. [2] Å idák, Z. K. (1967). "Rectangular Confidence Regions for the Means of Multivariate Normal Distributions". Journal of the American Statistical Association. 62 (318): 626–633. .. [3] Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65–70. .. [4] 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. .. [5] 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] zpvals must be list or arrayzalpha must be a float.zmethod must be a string.rrzalpha must be between 0 and 1.)Úbr1Z bonferroni)r")Úhr3)Úsr5)r0rZbhr)r"r#)rZbyrÚnoner r z)Multiple comparison method not recognized)Ú isinstanceÚlistrZndarrayrrrÚstrrr r1r3r5r0rr Ú ValueError)r!r"r#r-r,r.r.r/r^s*~     )rr)r)r)r)rr3) ZnumpyrZpandasrÚ__all__r0r1r3r5rr.r.r.r/Ús  l I Y B