CH4/templates/09_stat.non_parametric_tests_1sample.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Непараметрические криетрии"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  Критерий | Одновыборочный | Двухвыборочный | Двухвыборочный (связанные выборки)  \n",
    "  ------------- | -------------|\n",
    "  **Знаков**  | $\\times$ | | $\\times$ \n",
    "  **Ранговый** | $\\times$  | $\\times$ | $\\times$  \n",
    "  **Перестановочный** | $\\times$  | $\\times$ | $\\times$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mirrors as potential environmental enrichment for individually housed laboratory mice"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(Sherwin, 2004): 16 лабораторных мышей были помещены в двухкомнатные клетки, в одной из комнат висело зеркало. С целью установить, есть ли у мышей какие-то предпочтения насчет зеркал, измерялась доля времени, которое каждая мышь проводила в каждой из своих двух клеток."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import itertools\n",
    "\n",
    "from scipy import stats\n",
    "from statsmodels.stats.descriptivestats import sign_test\n",
    "from statsmodels.stats.weightstats import zconfint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Загрузка данных"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mouses_data = pd.read_csv('mirror_mouses.txt', header = None)\n",
    "mouses_data.columns = ['proportion_of_time']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>proportion_of_time</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.348471</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.640620</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.549818</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.537454</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.400444</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.426292</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.417156</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.477977</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.450657</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.476602</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.488677</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.484741</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.481614</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.489406</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.467365</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.467330</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    proportion_of_time\n",
       "0             0.348471\n",
       "1             0.640620\n",
       "2             0.549818\n",
       "3             0.537454\n",
       "4             0.400444\n",
       "5             0.426292\n",
       "6             0.417156\n",
       "7             0.477977\n",
       "8             0.450657\n",
       "9             0.476602\n",
       "10            0.488677\n",
       "11            0.484741\n",
       "12            0.481614\n",
       "13            0.489406\n",
       "14            0.467365\n",
       "15            0.467330"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mouses_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>proportion_of_time</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>16.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>0.475289</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.066065</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>0.348471</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>0.444566</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>0.477289</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>0.488859</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>0.640620</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       proportion_of_time\n",
       "count           16.000000\n",
       "mean             0.475289\n",
       "std              0.066065\n",
       "min              0.348471\n",
       "25%              0.444566\n",
       "50%              0.477289\n",
       "75%              0.488859\n",
       "max              0.640620"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mouses_data.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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2f3Pl5bhW0i/afoek/5X0jKRfXmvb9Y7ZpTb5Je2QdLNXflXBkqTrI+L2DZb9mE3W23ZR\n2dfKUJJfA4+9VJz/rbZ/S9IRrZw7b1tr27Hk10jGPyIetH2bpPskfU/StRGxT5LGMP4nym/7pWo4\n/nxoCAAS4LZxAJAAZQ4ACVDmAJAAZQ4ACVDmAJAAZQ4ACVDmAJAAZQ4ACfw/BJG1k9NhySUAAAAA\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f04cba05290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.hist(mouses_data.proportion_of_time)\n",
    "pylab.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Одновыборочные критерии"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "95% confidence interval for the mean time: [0.442918, 0.507660]\n"
     ]
    }
   ],
   "source": [
    "print '95%% confidence interval for the mean time: [%f, %f]' % zconfint(mouses_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Критерий знаков"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_0\\colon$ медиана доли времени, проведенного в клетке с зеркалом, равна 0.5\n",
    "\n",
    "$H_1\\colon$ медиана доли времени, проведенного в клетке с зеркалом, не равна 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M: -5, p-value: 0.021271\n"
     ]
    }
   ],
   "source": [
    "print \"M: %d, p-value: %f\" % sign_test(mouses_data, 0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Критерий знаковых рангов Вилкоксона"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "WilcoxonResult(statistic=35.0, pvalue=0.087935607142362426)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m0 = 0.5\n",
    "stats.wilcoxon(mouses_data.proportion_of_time - m0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Перестановочный критерий"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "$H_0\\colon$ среднее равно 0.5\n",
    "\n",
    "$H_1\\colon$ среднее не равно 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def permutation_t_stat_1sample(sample, mean):\n",
    "    t_stat = sum(map(lambda x: x - mean, sample))\n",
    "    return t_stat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.39537769999999972"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "permutation_t_stat_1sample(mouses_data.proportion_of_time, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def permutation_zero_distr_1sample(sample, mean, max_permutations = None):\n",
    "    centered_sample = map(lambda x: x - mean, sample)\n",
    "    if max_permutations:\n",
    "        signs_array = set([tuple(x) for x in 2 * np.random.randint(2, size = (max_permutations, \n",
    "                                                                              len(sample))) - 1 ])\n",
    "    else:\n",
    "        signs_array =  itertools.product([-1, 1], repeat = len(sample))\n",
    "    distr = [sum(centered_sample * np.array(signs)) for signs in signs_array]\n",
    "    return distr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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gHUPNr+rKWuVntWvXrteXe70evV5vnFWQpBWn3+/T7/cn0lfG+ZLxJA8D36+q3x4q2w28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\nb3BmalUIAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f04db113310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.hist(permutation_zero_distr_1sample(mouses_data.proportion_of_time, 0.5), bins = 15)\n",
    "pylab.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def permutation_test(sample, mean, max_permutations = None, alternative = 'two-sided'):\n",
    "    if alternative not in ('two-sided', 'less', 'greater'):\n",
    "        raise ValueError(\"alternative not recognized\\n\"\n",
    "                         \"should be 'two-sided', 'less' or 'greater'\")\n",
    "    \n",
    "    t_stat = permutation_t_stat_1sample(sample, mean)\n",
    "    \n",
    "    zero_distr = permutation_zero_distr_1sample(sample, mean, max_permutations)\n",
    "    \n",
    "    if alternative == 'two-sided':\n",
    "        return sum([1. if abs(x) >= abs(t_stat) else 0. for x in zero_distr]) / len(zero_distr)\n",
    "    \n",
    "    if alternative == 'less':\n",
    "        return sum([1. if x <= t_stat else 0. for x in zero_distr]) / len(zero_distr)\n",
    "\n",
    "    if alternative == 'greater':\n",
    "        return sum([1. if x >= t_stat else 0. for x in zero_distr]) / len(zero_distr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p-value: 0.157776\n"
     ]
    }
   ],
   "source": [
    "print \"p-value: %f\" % permutation_test(mouses_data.proportion_of_time, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p-value: 0.157923\n"
     ]
    }
   ],
   "source": [
    "print \"p-value: %f\" % permutation_test(mouses_data.proportion_of_time, 0.5, 10000)"
   ]
  },
  {
   "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": "2.7.13"
  }
 },
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}