CH4/templates/03_stat.two_proporions_conf_int_upd.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Доверительные интервалы для двух долей "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import scipy\n",
    "from statsmodels.stats.weightstats import *\n",
    "from statsmodels.stats.proportion import proportion_confint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Загрузка данных"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = pd.read_csv('banner_click_stat.txt', header = None, sep = '\\t')\n",
    "data.columns = ['banner_a', 'banner_b']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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>banner_a</th>\n",
       "      <th>banner_b</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   banner_a  banner_b\n",
       "0         0         0\n",
       "1         1         1\n",
       "2         0         0\n",
       "3         0         0\n",
       "4         0         0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.head()"
   ]
  },
  {
   "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>banner_a</th>\n",
       "      <th>banner_b</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>count</th>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1000.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mean</th>\n",
       "      <td>0.037000</td>\n",
       "      <td>0.053000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>std</th>\n",
       "      <td>0.188856</td>\n",
       "      <td>0.224146</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>min</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25%</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50%</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>75%</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>max</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          banner_a     banner_b\n",
       "count  1000.000000  1000.000000\n",
       "mean      0.037000     0.053000\n",
       "std       0.188856     0.224146\n",
       "min       0.000000     0.000000\n",
       "25%       0.000000     0.000000\n",
       "50%       0.000000     0.000000\n",
       "75%       0.000000     0.000000\n",
       "max       1.000000     1.000000"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Интервальные оценки долей"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\frac1{ 1 + \\frac{z^2}{n} } \\left( \\hat{p} + \\frac{z^2}{2n} \\pm z \\sqrt{ \\frac{ \\hat{p}\\left(1-\\hat{p}\\right)}{n} + \\frac{z^2}{4n^2} } \\right), \\;\\; z \\equiv z_{1-\\frac{\\alpha}{2}}$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "conf_interval_banner_a = proportion_confint(sum(data.banner_a), \n",
    "                                            data.shape[0],\n",
    "                                            method = 'wilson')\n",
    "conf_interval_banner_b = proportion_confint(sum(data.banner_b), \n",
    "                                            data.shape[0],\n",
    "                                            method = 'wilson')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "interval for banner a [0.026961, 0.050582]\n",
      "interval for banner b [0.040747, 0.068675]\n"
     ]
    }
   ],
   "source": [
    "print 'interval for banner a [%f, %f]' % conf_interval_banner_a\n",
    "print 'interval for banner b [%f, %f]' % conf_interval_banner_b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Как их сравнить?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Доверительный интервал для разности долей (независимые выборки)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "   | $X_1$ | $X_2$  \n",
    "  ------------- | -------------|\n",
    "  1  | a | b \n",
    "  0  | c | d \n",
    "  $\\sum$ | $n_1$| $n_2$\n",
    "  \n",
    "$$ \\hat{p}_1 = \\frac{a}{n_1}$$\n",
    "\n",
    "$$ \\hat{p}_2 = \\frac{b}{n_2}$$\n",
    "\n",
    "\n",
    "$$\\text{Доверительный интервал для }p_1 - p_2\\colon \\;\\; \\hat{p}_1 - \\hat{p}_2 \\pm z_{1-\\frac{\\alpha}{2}}\\sqrt{\\frac{\\hat{p}_1(1 - \\hat{p}_1)}{n_1} + \\frac{\\hat{p}_2(1 - \\hat{p}_2)}{n_2}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def proportions_confint_diff_ind(sample1, sample2, alpha = 0.05):    \n",
    "    z = scipy.stats.norm.ppf(1 - alpha / 2.)   \n",
    "    p1 = float(sum(sample1)) / len(sample1)\n",
    "    p2 = float(sum(sample2)) / len(sample2)\n",
    "    \n",
    "    left_boundary = (p1 - p2) - z * np.sqrt(p1 * (1 - p1)/ len(sample1) + p2 * (1 - p2)/ len(sample2))\n",
    "    right_boundary = (p1 - p2) + z * np.sqrt(p1 * (1 - p1)/ len(sample1) + p2 * (1 - p2)/ len(sample2))\n",
    "    \n",
    "    return (left_boundary, right_boundary)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "confidence interval: [-0.034157, 0.002157]\n"
     ]
    }
   ],
   "source": [
    "print \"confidence interval: [%f, %f]\" % proportions_confint_diff_ind(data.banner_a, data.banner_b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Доверительный интервал для разности долей (связанные выборки)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "  $X_1$ \\ $X_2$ | 1| 0 | $\\sum$\n",
    "  ------------- | -------------|\n",
    "  1  | e | f | e + f\n",
    "  0  | g | h | g + h\n",
    "  $\\sum$ | e + g| f + h | n  \n",
    "  \n",
    "$$ \\hat{p}_1 = \\frac{e + f}{n}$$\n",
    "\n",
    "$$ \\hat{p}_2 = \\frac{e + g}{n}$$\n",
    "\n",
    "$$ \\hat{p}_1 - \\hat{p}_2 = \\frac{f - g}{n}$$\n",
    "\n",
    "\n",
    "$$\\text{Доверительный интервал для }p_1 - p_2\\colon \\;\\;  \\frac{f - g}{n} \\pm z_{1-\\frac{\\alpha}{2}}\\sqrt{\\frac{f + g}{n^2} - \\frac{(f - g)^2}{n^3}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def proportions_confint_diff_rel(sample1, sample2, alpha = 0.05):\n",
    "    z = scipy.stats.norm.ppf(1 - alpha / 2.)\n",
    "    sample = zip(sample1, sample2)\n",
    "    n = len(sample)\n",
    "        \n",
    "    f = sum([1 if (x[0] == 1 and x[1] == 0) else 0 for x in sample])\n",
    "    g = sum([1 if (x[0] == 0 and x[1] == 1) else 0 for x in sample])\n",
    "    \n",
    "    left_boundary = float(f - g) / n  - z * np.sqrt(float((f + g)) / n**2 - float((f - g)**2) / n**3)\n",
    "    right_boundary = float(f - g) / n  + z * np.sqrt(float((f + g)) / n**2 - float((f - g)**2) / n**3)\n",
    "    return (left_boundary, right_boundary)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'data' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-2-24f2ee11d972>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[1;32mprint\u001b[0m \u001b[1;34m\"confidence interval: [%f, %f]\"\u001b[0m \u001b[1;33m%\u001b[0m \u001b[0mproportions_confint_diff_rel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mbanner_a\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mbanner_b\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m: name 'data' is not defined"
     ]
    }
   ],
   "source": [
    "print \"confidence interval: [%f, %f]\" % proportions_confint_diff_rel(data.banner_a, data.banner_b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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