chi-square-of-distribution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculating the chi-square for a distribution and a histogram.\n",
    "\n",
    "We will do it two ways. With and without normalized histograms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline --no-import-all\n",
    "from scipy.stats import norm #will use this for plotting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### make some data from a N(0,1) distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_bins = 50\n",
    "x = norm.rvs(size=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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7Wr2jdTLWO+Z+c1UtAVTVJeDmpvwW4MKy/S42ZdpEvZOp363jkubFuKZCris5FhcXX9vu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\ncJekFjLcJamFDHdJaqH/D+kom+9uebc0AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dc5ea10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "counts, bins, patches = plt.hist(x, bins=n_bins,normed=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Get Bin centers\n",
    "\n",
    "Get the location for the bin centers. Here I do it step by step so you can see what is going on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-3.75518813, -3.60372897, -3.45226981, -3.30081065, -3.1493515 ,\n",
       "       -2.99789234, -2.84643318, -2.69497402, -2.54351486, -2.3920557 ,\n",
       "       -2.24059654, -2.08913738, -1.93767822, -1.78621906, -1.6347599 ,\n",
       "       -1.48330074, -1.33184158, -1.18038242, -1.02892326, -0.87746411,\n",
       "       -0.72600495, -0.57454579, -0.42308663, -0.27162747, -0.12016831,\n",
       "        0.03129085,  0.18275001,  0.33420917,  0.48566833,  0.63712749,\n",
       "        0.78858665,  0.94004581,  1.09150497,  1.24296413,  1.39442328,\n",
       "        1.54588244,  1.6973416 ,  1.84880076,  2.00025992,  2.15171908,\n",
       "        2.30317824,  2.4546374 ,  2.60609656,  2.75755572,  2.90901488,\n",
       "        3.06047404,  3.2119332 ,  3.36339236,  3.51485152,  3.66631068,\n",
       "        3.81776983])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# all bins edges\n",
    "bins"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-3.60372897, -3.45226981, -3.30081065, -3.1493515 , -2.99789234,\n",
       "       -2.84643318, -2.69497402, -2.54351486, -2.3920557 , -2.24059654,\n",
       "       -2.08913738, -1.93767822, -1.78621906, -1.6347599 , -1.48330074,\n",
       "       -1.33184158, -1.18038242, -1.02892326, -0.87746411, -0.72600495,\n",
       "       -0.57454579, -0.42308663, -0.27162747, -0.12016831,  0.03129085,\n",
       "        0.18275001,  0.33420917,  0.48566833,  0.63712749,  0.78858665,\n",
       "        0.94004581,  1.09150497,  1.24296413,  1.39442328,  1.54588244,\n",
       "        1.6973416 ,  1.84880076,  2.00025992,  2.15171908,  2.30317824,\n",
       "        2.4546374 ,  2.60609656,  2.75755572,  2.90901488,  3.06047404,\n",
       "        3.2119332 ,  3.36339236,  3.51485152,  3.66631068,  3.81776983])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "right_edges = bins[1:]\n",
    "right_edges"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-3.75518813, -3.60372897, -3.45226981, -3.30081065, -3.1493515 ,\n",
       "       -2.99789234, -2.84643318, -2.69497402, -2.54351486, -2.3920557 ,\n",
       "       -2.24059654, -2.08913738, -1.93767822, -1.78621906, -1.6347599 ,\n",
       "       -1.48330074, -1.33184158, -1.18038242, -1.02892326, -0.87746411,\n",
       "       -0.72600495, -0.57454579, -0.42308663, -0.27162747, -0.12016831,\n",
       "        0.03129085,  0.18275001,  0.33420917,  0.48566833,  0.63712749,\n",
       "        0.78858665,  0.94004581,  1.09150497,  1.24296413,  1.39442328,\n",
       "        1.54588244,  1.6973416 ,  1.84880076,  2.00025992,  2.15171908,\n",
       "        2.30317824,  2.4546374 ,  2.60609656,  2.75755572,  2.90901488,\n",
       "        3.06047404,  3.2119332 ,  3.36339236,  3.51485152,  3.66631068])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "left_edges = bins[:-1]\n",
    "left_edges"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-3.67945855, -3.52799939, -3.37654023, -3.22508108, -3.07362192,\n",
       "       -2.92216276, -2.7707036 , -2.61924444, -2.46778528, -2.31632612,\n",
       "       -2.16486696, -2.0134078 , -1.86194864, -1.71048948, -1.55903032,\n",
       "       -1.40757116, -1.256112  , -1.10465284, -0.95319368, -0.80173453,\n",
       "       -0.65027537, -0.49881621, -0.34735705, -0.19589789, -0.04443873,\n",
       "        0.10702043,  0.25847959,  0.40993875,  0.56139791,  0.71285707,\n",
       "        0.86431623,  1.01577539,  1.16723455,  1.31869371,  1.47015286,\n",
       "        1.62161202,  1.77307118,  1.92453034,  2.0759895 ,  2.22744866,\n",
       "        2.37890782,  2.53036698,  2.68182614,  2.8332853 ,  2.98474446,\n",
       "        3.13620362,  3.28766278,  3.43912194,  3.5905811 ,  3.74204025])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bin_centers = (right_edges+left_edges)/2\n",
    "bin_centers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "bin_width = bins[1]-bins[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Raw histogram counts\n",
    "\n",
    "To go from the pdf to the expected counts in a bin, you need to include a normalization constant."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "normalization = x.size*(bins[1]-bins[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "prediction = normalization*norm.pdf(bin_centers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1101f51d0>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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FZjY9vN/azGabWZ6ZzTKzVuWOHWZmy81sqZn1SUTgkga2bYOnnwbUJFO5pljbtswoKoLi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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x110423410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "counts, bins, patches = plt.hist(x, bins=n_bins,normed=False)\n",
    "plt.plot(bin_centers,prediction, c='r', lw=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(9998.4828255329667, 10000.0)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(prediction), np.sum(counts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1135c0b10>"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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kIIDLXlzG2fuiufBQY8q/tyXzOpPERN+4JXRISDDt2jW2O0a+FPsif2DJF7R4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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1130d82d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# make a quick plot of (counts-prediction)/sqrt(prediction)\n",
    "# it should look like N(0,1) for chi-square formula to be appropriate\n",
    "temp = plt.hist((counts-prediction)/np.sqrt(prediction), bins=7, normed=True, alpha=0.2)\n",
    "plt.plot(temp[1], norm.pdf(temp[1]))\n",
    "plt.plot(temp[1], norm.pdf(temp[1]), label='N(0,1)')\n",
    "plt.legend()\n",
    "plt.xlabel('(counts-prediction)/sqrt(prediction)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chisquare"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.stats import chisquare"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import chi2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "45.498862093830645"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# right way\n",
    "np.sum((counts-prediction)**2/prediction)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(45.498862093830645, 0.61586732424588986)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# right way\n",
    "chisq_obs, pvalue = chisquare(counts, prediction)\n",
    "chisq_obs, pvalue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can calculate the p-value for having a chi-square value this large.\n",
    "Importantly we need to have the number of degrees of freedom be the number of bins."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.65436627278078441"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#p-value\n",
    "chi2.sf(chisq_obs,counts.size)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Note \n",
    "A p-value of 0 or 1 is very unlikely. You should expect to sit in the middle of the distribution and have a p-value of ~0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normalized Histogram (wrong way)\n",
    "\n",
    "Here we make a normed histogram, which is equivalent to dividing by the normalization constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "prediction_normed = norm.pdf(bin_centers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1124c78d0>]"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Ia7ZzNvCZfT49x7puTM0FhlTy619Dmg2pTDYHaMTT/AaAP7oci6kZa9Gb0HJz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\nA8JxpmCbAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1121e2b50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "counts_normed, bins, patches = plt.hist(x, bins=n_bins,normed=True)\n",
    "plt.plot(bin_centers,prediction_normed, c='r', lw=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(6.6014382152927498, 6.6024399206195126)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Note, they don't add up to 1., they add up to (roughly) 1/bin_width\n",
    "np.sum(prediction_normed), np.sum(counts_normed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.6024399206195179"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1/bin_width"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1135c0ed0>"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ev3WmpFB6307K7NhIWa8vgbI7NhJyYA+na0WlK/nbXwPRJFWtlevqnwMHlnHj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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1148e3bd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# make a quick plot of (counts-prediction)/sqrt(prediction)\n",
    "# it should look like N(0,1) for chi-square formula to be appropriate\n",
    "temp = plt.hist((counts_normed-prediction_normed)/np.sqrt(prediction_normed), bins=7, normed=True, alpha=0.2)\n",
    "plt.plot(temp[1], norm.pdf(temp[1]), label='N(0,1) -- needed for chi-square')\n",
    "plt.plot(temp[1], norm.pdf(temp[1], 0,1./np.sqrt(normalization)), label='N(0,1/sqrt(norm))')\n",
    "plt.legend()\n",
    "plt.xlabel('(counts_normed-prediction_normed)/sqrt(prediction_normed)')\n",
    "plt.title('note, the scale is way off!!!', color='r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.030040350343106814"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#wrong way\n",
    "np.sum((counts_normed-prediction_normed)**2/prediction_normed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.030040350343106814, 1.0)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#wrong way\n",
    "chisq_obs_wrong, pvalue_wrong = chisquare(counts_normed, prediction_normed)\n",
    "chisq_obs_wrong, pvalue_wrong"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(45.498862093830645, 0.61586732424588986)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## compare with the right way\n",
    "chisq_obs, pvalue"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#p-value\n",
    "chi2.sf(chisq_obs_wrong,counts.size)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Note \n",
    "A p-value of 0 or 1 is very unlikely. You should expect to sit in the middle of the distribution and have a p-value of ~0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}