Statistics project.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Проект по математической статистике\n",
    "### Хальман Михаил. Задание Siblings\n",
    "\n",
    "Задание состоит в том, чтобы по одномерной выборке посчитать среднее значение (медиану), вместе с соответствующими доверительными интервалами, а затем проверить результат на отложенных данных."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Описание статистических методов, использованных в решении:\n",
    "\n",
    "Общий метод проверки гипотез состоит в том, чтобы посчитать по выборке статистику $T(X)$ и сравнить её значение с квантилью распределения статистики в зависимости от уровня значимости.\n",
    "\n",
    "### Student's t-test\n",
    "\n",
    "https://en.wikipedia.org/wiki/Student%27s_t-test\n",
    "\n",
    "Семейство статистических тестов, для проверки равенства средних, в которых статистика имеет распределение Стьюдента.\n",
    "\n",
    "Выделяют:\n",
    "#### Одновыборочный тест о проверке равенства среднего заданной константе $\\mu_0$\n",
    "\n",
    "**Тестовая статистика:**\n",
    "$$ t = \\frac{\\bar{X} - \\mu_0}{s / \\sqrt{n}} $$\n",
    "\n",
    "В предположениях нормальности выборки $X$ статистика имеет распределение Стьюдента c $n-1$ степенями свободы. Гипотеза $H_0$ о равенстве отвергается, если значение статистики $|t| > u_{1-\\alpha/2}$, где $u_q$ - $q$-квантиль распределения стьюдента, а $\\alpha$ --- уровень значимости.\n",
    "\n",
    "#### Двувыборочный тест о проверке равенства средних для двух выборок.\n",
    "\n",
    "**Тестовая статистика:**\n",
    "$$ t = \\frac{\\bar{X} - \\mu_0}{\\tilde{s}}$$\n",
    "\n",
    "Где $\\tilde{s}$ определённым образом определяется в зависимости от того, отличаются или совпадают размеры или дисперсии выборок.\n",
    "\n",
    "В предположениях нормальности выборки $X$ статистика имеет распределение Стьюдента. Гипотеза $H_0$ о равенстве отвергается, если значение статистики $|t| > u_{1-\\alpha/2}$, где $u_q$ - $q$-квантиль распределения стьюдента, а $\\alpha$ --- уровень значимости.\n",
    "\n",
    "\n",
    "### Mann Whitney U test\n",
    "\n",
    "https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test\n",
    "\n",
    "Непараметрический аналог двувыборочного t-теста. Отвергает гипотезу о том, что две выборки пришли из одного распределения. Главное отличие состоит в том, что тест не предполагает нормальности выборок. \n",
    "\n",
    "Тестовая статистика считается опредёлнным суммированием рангов (порядковых индексов в отсортированной выборке, полученной смешением двух выборок). \n",
    "\n",
    "### Shapiro-Wilk test\n",
    "\n",
    "https://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test\n",
    "\n",
    "Тест, отвергающий нормальность выборки.\n",
    "\n",
    "### Chi-squared test\n",
    "\n",
    "https://en.wikipedia.org/wiki/Chi-squared_test\n",
    "\n",
    "Тест, отвергающий принадлежность выборки определённому семейству распределений. Тестовая статистика считает отклонения эмпирического распределения от теоретического и имеет распределение Хи-квадрат."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Уровень значимости:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ALPHA_P_VALUE = 0.05"
   ]
  },
  {
   "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\n",
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import scipy.stats as st\n",
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = pd.read_csv('siblings.csv')"
   ]
  },
  {
   "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>Siblings</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Siblings\n",
       "0         2\n",
       "1         4\n",
       "2         5\n",
       "3         4\n",
       "4         3"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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>Siblings</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>220.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>2.018182</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>1.685186</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>2.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>3.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>11.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Siblings\n",
       "count  220.000000\n",
       "mean     2.018182\n",
       "std      1.685186\n",
       "min      0.000000\n",
       "25%      1.000000\n",
       "50%      2.000000\n",
       "75%      3.000000\n",
       "max     11.000000"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "siblings = np.array(data.Siblings)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average number of sibling: 2.018182;\n",
      "Skewed geometric mean (sqrt(prod(1 + x_i)) - 1): 1.670798 \n",
      "Median: 2.000000\n"
     ]
    }
   ],
   "source": [
    "print 'Average number of sibling: %f;\\nSkewed geometric mean (sqrt(prod(1 + x_i)) - 1): %f \\nMedian: %f' % \\\n",
    "    (np.mean(siblings),\n",
    "     np.exp(np.mean(np.log(1 + siblings))) - 1,\n",
    "     np.median(siblings))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Т.к. данные дискретные, тест Шапиро-Уилка не применим [3]. **Для проверки нормальности будем применять тест хи-квадрат.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not Gaussian (Chi-squared test p-value 2.990120e-27)\n",
      "\n",
      "    \n",
      "    Average number of sibling: 2.018182\n",
      "    95% confidence interval for mean: (1.796007, 2.240357)\n",
      "    Median: 2.000000\n",
      "    95% confidence interval for median: (1.721545, 2.278455) (Assumes normality)\n",
      "    \n"
     ]
    },
    {
     "data": {
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RULTHT4JY7FyDVLuNIAor4krSUlY6N49UcKgzg/mOLEmSJEnqa5hbrfwoItZn5g8j4nnA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\nVSwC/FWRljft1W4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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10856f3d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def do_plots(data, callback=None):\n",
    "    #sm.qqplot(data)\n",
    "    if callback is None:\n",
    "        callback = lambda x:x\n",
    "    n = data.size\n",
    "    sigma = np.std(data) / np.sqrt(n)\n",
    "    avg = np.mean(data)\n",
    "    med = np.median(data)\n",
    "    q = st.norm.ppf(1 - ALPHA_P_VALUE / 2)\n",
    "    MAGIC_CONST = np.sqrt(np.pi / 2) * q\n",
    "\n",
    "    for normality_test_function, name in [(st.normaltest, 'Chi-squared')]: #, (st.shapiro, 'Shapiro-Wilk')]:\n",
    "        _, gaussian_p_value = normality_test_function(data)\n",
    "        if gaussian_p_value < ALPHA_P_VALUE:\n",
    "            print 'Not Gaussian (%s test p-value %e)' % (name, gaussian_p_value)\n",
    "        else:\n",
    "            print 'Gaussian (%s test p-value %e)' % (name, gaussian_p_value)\n",
    "\n",
    "    print '''\n",
    "    \n",
    "    Average number of sibling: %f\n",
    "    %d%% confidence interval for mean: (%f, %f)\n",
    "    Median: %f\n",
    "    %d%% confidence interval for median: (%f, %f) (Assumes normality)\n",
    "    ''' % \\\n",
    "    (callback(avg),\n",
    "     int((1 - ALPHA_P_VALUE) * 100),\n",
    "     callback(avg - sigma * q),\n",
    "     callback(avg + sigma * q),\n",
    "     callback(med),\n",
    "     int((1 - ALPHA_P_VALUE) * 100),\n",
    "     callback(med - sigma * MAGIC_CONST),\n",
    "     callback(med + sigma * MAGIC_CONST))\n",
    "    \n",
    "    figure(figsize=(16, 6))\n",
    "    subplot(121)\n",
    "    _ = boxplot(data)\n",
    "    subplot(122)\n",
    "    _ = hist(data, bins=np.unique(data), normed=True)\n",
    "    axvline(avg, c='r', linewidth=2, label='Mean')\n",
    "    axvline(med, c='g', linewidth=2, label='Median')\n",
    "    axvline(avg - sigma * q, c='r', ls='--')\n",
    "    axvline(avg + sigma * q, c='r', ls='--')\n",
    "    axvline(med - sigma * MAGIC_CONST, c='g', ls='--')\n",
    "    axvline(med + sigma * MAGIC_CONST, c='g', ls='--')\n",
    "    #axvline(np.exp(np.mean(np.log(1 + siblings))) - 1, c='g', linewidth=2, label='Skewed geometric mean')\n",
    "    legend()\n",
    "do_plots(data=siblings)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6813512865431575"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(np.concatenate((siblings, siblings)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sqrt-преобразование"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not Gaussian (Chi-squared test p-value 8.439340e-03)\n",
      "\n",
      "    \n",
      "    Average number of sibling: 1.676146\n",
      "    95% confidence interval for mean: (1.482013, 1.882223)\n",
      "    Median: 2.000000\n",
      "    95% confidence interval for median: (1.735427, 2.283335) (Assumes normality)\n",
      "    \n"
     ]
    },
    {
     "data": {
      "image/png": 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4noQ/VMYoJtGO4j6j2kjf7R9rc70Ukb7bS1f6blZ503cLQ/oukMecpe9meOGT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\n/H+9NNquII6r+wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10852eb50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sqrt_siblings = np.sqrt(siblings)\n",
    "do_plots(data=sqrt_siblings, callback=lambda x: x ** 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Логарифмическое преобразование:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gaussian (Chi-squared test p-value 1.244265e-01)\n",
      "\n",
      "    \n",
      "    Average number of sibling: 1.670798\n",
      "    95% confidence interval for mean: (1.505508, 1.846993)\n",
      "    Median: 2.000000\n",
      "    95% confidence interval for median: (1.769157, 2.250086) (Assumes normality)\n",
      "    \n"
     ]
    },
    {
     "data": {
      "image/png": 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vtyEeKZ3FxCOlOzXvkdLBGNpr6Egp3cy7Ni8q6AgAAAB2TFMKAABAbzSlAAAA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\nw3MgFgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10852eed0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "log_siblings = log(1 + siblings)\n",
    "do_plots(log_siblings, callback=lambda x:np.exp(x) - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Тест хи-квадрат не отвергает гипотезу норимальности для логарифмического преобразования. Поэтому мы можем применить t-test для проверки равенства медиан на новых данных."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean: 2.018182\n",
      "Median: 2.000000\n"
     ]
    }
   ],
   "source": [
    "mean_siblings = np.mean(siblings)\n",
    "print 'Mean: %f' % mean_siblings\n",
    "median_siblings = np.median(siblings)\n",
    "print 'Median: %f' % median_siblings"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Считаем теперь отложенные данные:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "hold_out_data = pd.read_csv('siblingsexercise2.csv')\n",
    "ho_siblings = np.array(hold_out_data.Siblings)\n",
    "log_ho_siblings = np.log(1 + ho_siblings)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Проверка нормальности, а также статистики"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gaussian (Chi-squared test p-value 1.659853e-01)\n",
      "\n",
      "    \n",
      "    Average number of sibling: 3.196970\n",
      "    95% confidence interval for mean: (2.710420, 3.683519)\n",
      "    Median: 3.000000\n",
      "    95% confidence interval for median: (2.390200, 3.609800) (Assumes normality)\n",
      "    \n"
     ]
    },
    {
     "data": {
      "image/png": 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IcNnzqDLW5uFkW1INnpds4uUZzQ61uP0y5ncRqLxu1maOWAIDQiI6AAAA6orGEgCAAbO9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\nACShsQQAAAAAJPn/oaEPrm+6IOIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10852e590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "do_plots(ho_siblings)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gaussian (Chi-squared test p-value 1.335344e-01)\n",
      "\n",
      "    \n",
      "    Average number of sibling: 2.681164\n",
      "    95% confidence interval for mean: (2.230948, 3.194115)\n",
      "    Median: 3.000000\n",
      "    95% confidence interval for median: (2.396669, 3.710497) (Assumes normality)\n",
      "    \n"
     ]
    },
    {
     "data": {
      "image/png": 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OScslLZa00szOmqLfLZK+KokrygCA0iIpBQCgWpZI2unuu9z9NUkbJV3ao9+/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\nH9v01THT7aYjAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109df6050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "do_plots(log_ho_siblings, callback=lambda x:np.exp(x) - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Можно видеть, что:\n",
    "1. Отложенная выборка \"проходит\" хи-квадрат проверку нормальности ещё до применения логарифмирования. (Хи-квадрат тест не отвергает нормальность)\n",
    "2. Выборочные медиана и среднее смещены (значительно больше), чем на первой выборке."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Непараметрический ранговый тест об одинаковой распредённости"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mann-Whitney U-test\n",
      "\tNot equal\n",
      "\tp-value=1.159785e-06\n"
     ]
    }
   ],
   "source": [
    "_, p_value = st.mannwhitneyu(siblings, ho_siblings)\n",
    "print 'Mann-Whitney U-test'\n",
    "if p_value < ALPHA_P_VALUE:\n",
    "    print '\\tNot equal'\n",
    "else: \n",
    "    print '\\tEqual'\n",
    "print '\\tp-value=%e' % p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ранговый тест показывает, отвергает одинаковую распределённость выборок"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Проверим гипотезу о равенстве медиан"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1-выборочный тест для отложенной выборки"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not equal\n",
      "p-value= 3.301715e-03\n"
     ]
    }
   ],
   "source": [
    "_, p_value = st.ttest_1samp(log_ho_siblings, np.log(1 + median_siblings)) \n",
    "if p_value < ALPHA_P_VALUE:\n",
    "    print 'Not equal'\n",
    "else: \n",
    "    print 'Equal'\n",
    "print 'p-value= %e' % p_value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not equal\n",
      "p-value= 1.022438e-05\n"
     ]
    }
   ],
   "source": [
    "_, p_value = st.ttest_1samp(ho_siblings, median_siblings)\n",
    "if p_value < ALPHA_P_VALUE:\n",
    "    print 'Not equal'\n",
    "else: \n",
    "    print 'Equal'\n",
    "print 'p-value= %e' % p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Двух-выборочный тест"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not equal\n",
      "p-value= 3.328580e-06\n"
     ]
    }
   ],
   "source": [
    "_, p_value = st.ttest_ind(siblings, ho_siblings)\n",
    "if p_value < ALPHA_P_VALUE:\n",
    "    print 'Not equal'\n",
    "else: \n",
    "    print 'Equal'\n",
    "print 'p-value= %e' % p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Для средних:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not equal\n",
      "p-value= 4.280463e-03\n"
     ]
    }
   ],
   "source": [
    "_, p_value = st.ttest_1samp(log_ho_siblings, np.log(1 + mean_siblings)) \n",
    "if p_value < ALPHA_P_VALUE:\n",
    "    print 'Not equal'\n",
    "else: \n",
    "    print 'Equal'\n",
    "print 'p-value= %e' % p_value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Not equal\n",
      "p-value= 1.334698e-05\n"
     ]
    }
   ],
   "source": [
    "_, p_value = st.ttest_1samp(ho_siblings, mean_siblings)\n",
    "if p_value < ALPHA_P_VALUE:\n",
    "    print 'Not equal'\n",
    "else: \n",
    "    print 'Equal'\n",
    "print 'p-value= %e' % p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Box-plot-ы для убедительности :)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a2eb410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "boxplot(x=(siblings, ho_siblings), widths=0.5)\n",
    "xticks([1, 2], ('Train', 'Test'))\n",
    "grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Выводы:\n",
    "1. Все тесты (как параметрические, так и непараметрические) отвергают одинаковую распределённость выборок. Существует статистически значимое различие в данных собранных в первую и вторую таблицы. Как средние, так и медианы различны\n",
    "2. 95-процентные доверительные интервалы, основанные на гипотезе нормальности:\n",
    " * Для первой выборки:\n",
    "     - Среднее: 2.02 (1.797, 2.24)\n",
    "     - Медиана: 2.00 (1.77, 2.25)\n",
    " * Для второй выборки:\n",
    "     - Среднее: 3.2 (2.71, 3.68)\n",
    "     - Медиана: 3.0 (2.39, 3.6)\n",
    "3. Выводы, полученные при использовании статистических тестов соотносятся с результатом, полученным простым сравнением доверительных интервалов."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Ссылки\n",
    "1. Как посчитать распределение выборочной медианы.\n",
    "    * https://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/162/Handouts/MedianThm04.pdf\n",
    "2. Использованные тесты:\n",
    "    * https://en.wikipedia.org/wiki/Student%27s_t-test\n",
    "    * https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test\n",
    "    * https://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test\n",
    "    * https://en.wikipedia.org/wiki/Chi-squared_test\n",
    "3. Проверка нормальности дискретных данных \n",
    "    * http://blog.minitab.com/blog/quality-data-analysis-and-statistics/assumptions-and-normality"
   ]
  },
  {
   "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": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}