pval.ipynb.json

{
 "metadata": {
  "name": "pval"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "from scipy.stats import norm\nfrom statsmodels.sandbox.stats.multicomp import fdrcorrection0\n# I'm running IPython with --pylab=inline if you are not, uncomment the lines below\n#import numpy as np\n#from matplotlib import pyplot as plt",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Let's start by pulling 1,000,000 samples from a normal distribution. These are essentially $z$-values, as we're using a standard normal distribution with a $\\mu=0$ and $\\sigma^{2}=1$, so we will convert them straight to p-values.\nImagine, in this case, that every p-value (datapoint) is the reult of a hypothesis test."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "x = norm.sf(norm.rvs(size=1e6))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The mean p-value is, as expected, $0.5$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "\"Mean = {0:0.3f}\".format(np.mean(x))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": "'Mean = 0.500'"
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "And we appear to have a good sampling."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_ = plt.hist(x, bins=20, cumulative=True)\n_ = xlabel(r'$p\\leq x$')\n_ = ylabel('count')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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H8r///U9ERP7xj39Idna2ijbDor29EBE5f/683H777TJp0iTZtGmTgi5Dr719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\n",
       "text": "<matplotlib.figure.Figure at 0x1152f7d90>"
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Which, in turn, means that $\\sim5\\%$ of the pvalues will be $\\le0.05$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "\"Pvals <= 0.05: {0:0.2f}%\".format(100 * np.sum(x <= 0.05) / float(x.size))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": "'Pvals <=0.05: 5.01%'"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "\"So, out of our {0:d} samples, we have {1:d} false positives\".format(x.size, np.sum(x<=0.05))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": "'So, out of our 1000000 samples, we have 50122 false positives'"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "\"The *best* pvalue being {0:0.2e}\".format(np.min(x))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": "'The *best* pvalue being 5.46e-08'"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Seems pretty significant! This is despite it being randomly pulled from a normal distribution."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_ = plt.boxplot(np.log10(x), vert=False)\n_ = title(r'Boxplot of sample P-values ($log_{10}$ scale)')\n_ = xlabel(r'$\\log_{10}\\left(p-value\\right)$')\n_ = ylabel(r'$x$')",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ICgrCoUOH8MQTTwAAwsLCkJqaitDQUBw4cMBwSuh///sfPD098cMPP8DFxcVw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\nhjsRkYQY7kREEvo/yOO43rYugzEAAAAASUVORK5CYII=\n",
       "text": "<matplotlib.figure.Figure at 0x10fb87090>"
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "So now, lets see what happens if we apply the Benjamini-Hochberg FDR Correction with $\\alpha=0.05$"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rejected, corrected = fdrcorrection0(x, alpha=0.05)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "h2 = plt.hist(corrected)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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3VVRUyO/3y+/3a926dTanAAA4D6zC5LnnnpPjOPJ4PJKksrIyeb1effzxx7rhhhv04osv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\nJgAAa/8HPb+2b9lWFcsAAAAASUVORK5CYII=\n",
       "text": "<matplotlib.figure.Figure at 0x10e1c5350>"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "\"Smallest P value before FDR: {0:0.2e}\\nSmallest P value after FDR: {1:0.2e}\".format(np.min(x), np.min(corrected))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": "'Smallest P value before FDR: 5.46e-08\\nSmallest P value after FDR: 5.46e-02'"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "h2 = plt.boxplot(np.log10(corrected), vert=False)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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       "text": "<matplotlib.figure.Figure at 0x10fad0b90>"
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}