Analyze_ab_test_results_notebook_final.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyze A/B Test Results\n",
    "\n",
    "From: https://github.com/ksatola/Analyze-A-B-Test-Results\n",
    "\n",
    "## Table of Contents\n",
    "- [Introduction](#intro)\n",
    "- [Part I - Probability](#probability)\n",
    "- [Part II - A/B Test](#ab_test)\n",
    "- [Part III - Regression](#regression)\n",
    "\n",
    "\n",
    "<a id='intro'></a>\n",
    "### Introduction\n",
    "\n",
    "A company has developed a new e-commerce web page in order to try and increase the number of users who \"convert,\" meaning the number of users who decide to pay for the company's product.\n",
    "\n",
    "The main goal of this project was to understand the results of an A/B test run by the website and provide statistical and practical interpretation on the test results.\n",
    "\n",
    "<a id='probability'></a>\n",
    "#### Part I - Probability\n",
    "\n",
    "To get started, let's import our libraries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`1.` Now, read in the `ab_data.csv` data.\n",
    "\n",
    "a. Read in the dataset and take a look at the top few rows here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>804228</td>\n",
       "      <td>2017-01-12 08:01:45.159739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>853541</td>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>936923</td>\n",
       "      <td>2017-01-10 15:20:49.083499</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>679687</td>\n",
       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>719014</td>\n",
       "      <td>2017-01-17 01:48:29.539573</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>817355</td>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>839785</td>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\n",
       "0   851104  2017-01-21 22:11:48.556739    control     old_page          0\n",
       "1   804228  2017-01-12 08:01:45.159739    control     old_page          0\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0\n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0\n",
       "4   864975  2017-01-21 01:52:26.210827    control     old_page          1\n",
       "5   936923  2017-01-10 15:20:49.083499    control     old_page          0\n",
       "6   679687  2017-01-19 03:26:46.940749  treatment     new_page          1\n",
       "7   719014  2017-01-17 01:48:29.539573    control     old_page          0\n",
       "8   817355  2017-01-04 17:58:08.979471  treatment     new_page          1\n",
       "9   839785  2017-01-15 18:11:06.610965  treatment     new_page          1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('ab_data.csv')\n",
    "df.head(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the below cell to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of rows in the dataset is 294478.\n"
     ]
    }
   ],
   "source": [
    "print('Number of rows in the dataset is {}.'.format(df.shape[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. The number of unique users in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of unique users in the dataset is 290584.\n"
     ]
    }
   ],
   "source": [
    "print('Number of unique users in the dataset is {}.'.format(df.user_id.nunique()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. The proportion of users converted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The proportion of users converted is 12%.\n"
     ]
    }
   ],
   "source": [
    "prop_users_converted = round(df.converted.mean() * 100)\n",
    "print('The proportion of users converted is {}%.'.format(prop_users_converted))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. The number of times the `new_page` and `treatment` don't line up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The number of times the new_page and treatment don't line up is 3893.\n"
     ]
    }
   ],
   "source": [
    "num_not_aligned = df[((df['group'] == 'treatment') == (df['landing_page'] == 'new_page')) == False].shape[0]\n",
    "print('The number of times the new_page and treatment don\\'t line up is {}.'.format(num_not_aligned))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "user_id         0\n",
       "timestamp       0\n",
       "group           0\n",
       "landing_page    0\n",
       "converted       0\n",
       "dtype: int64"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.isnull().sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** There are no missing values in the dataset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` For the rows where **treatment** is not aligned with **new_page** or **control** is not aligned with **old_page**, we cannot be sure if this row truly received the new or old page. We will remove these rows as we should use the rows where we can feel confident in the accuracy of data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "df2 = df.copy()\n",
    "df2 = df2[((df2['group'] == 'treatment') == (df2['landing_page'] == 'new_page')) != False]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Double Check all of the correct rows were removed - this should be 0\n",
    "df2[((df2['group'] == 'treatment') == (df2['landing_page'] == 'new_page')) == False].shape[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`3.` Remove duplicates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. How many unique **user_id**s are in **df2**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of unique users in the dataset is 290584.\n"
     ]
    }
   ],
   "source": [
    "print('Number of unique users in the dataset is {}.'.format(df2.user_id.nunique()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. There is one **user_id** repeated in **df2**.  What is it?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The repeated user_id in df2 is 773192.\n"
     ]
    }
   ],
   "source": [
    "print('The repeated user_id in df2 is {}.'.format(df2[df2.user_id.duplicated()].user_id.iloc[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is the row information for the repeat **user_id**? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1899</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-09 05:37:58.781806</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2893</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-14 02:55:59.590927</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "1899   773192  2017-01-09 05:37:58.781806  treatment     new_page          0\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2.user_id == 773192]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Remove **one** of the rows with a duplicate **user_id**, but keep your dataframe as **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1899</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-09 05:37:58.781806</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "1899   773192  2017-01-09 05:37:58.781806  treatment     new_page          0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.drop(2893, inplace=True)\n",
    "df2[df2.user_id == 773192]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`4.` Use **df2** in the below cells to answer some questions:\n",
    "\n",
    "a. What is the probability of an individual converting regardless of the page they receive?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The probability of an individual converting regardless of the page they receive is 0.11959708724499628.\n"
     ]
    }
   ],
   "source": [
    "conv_prob = df2.converted.mean()\n",
    "print('The probability of an individual converting regardless of the page they receive is {}.'.format(conv_prob))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Given that an individual was in the `control` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Given that an individual was in the control group, the probability they converted is 0.1203863045004612.\n"
     ]
    }
   ],
   "source": [
    "conv_prob_control = df2[df2.group == 'control'].converted.mean()\n",
    "print('Given that an individual was in the control group, the probability they converted is {}.'.format(conv_prob_control))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Given that an individual was in the `treatment` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Given that an individual was in the treatment group, the probability they converted is 0.11880806551510564.\n"
     ]
    }
   ],
   "source": [
    "conv_prob_treatment = df2[df2.group == 'treatment'].converted.mean()\n",
    "print('Given that an individual was in the treatment group, the probability they converted is {}.'.format(conv_prob_treatment))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The probability that an individual received the new page is 0.5000619442226688.\n"
     ]
    }
   ],
   "source": [
    "new_page_prob = (df2['landing_page'] == \"new_page\").mean()\n",
    "print('The probability that an individual received the new page is {}.'.format(new_page_prob))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Is there sufficient evidence to say that the new treatment page leads to more conversions?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** There is no sufficient evidence to say that the new treatment page leads to more conversions (the conversion probability for each page is 'almost' the same). Moreover, the probability of conversion of the control group is even slightly higher than the probability of conversion of the treatment group (conv_prob_control > conv_prob_treatment). The number of events leading to old and new pages is well ballanced (new_page_prob) for each page type (there is no bias existing potentially disturbing existence of the evidence)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='ab_test'></a>\n",
    "### Part II - A/B Test \n",
    "\n",
    "`1.` It is assumed that the old page is better unless the new page proves to be definitely better at a Type I error rate of 5%. The null and alternative hypotheses are (**$p_{old}$** and **$p_{new}$**, which are the converted rates for the old and new pages):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "H_0 : p_{new} - p_{old} \\leq 0 \\\\\n",
    "H_1 : p_{new} - p_{old} > 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` It is assumed under the null hypothesis, $p_{new}$ and $p_{old}$ both have \"true\" success rates equal to the **converted** success rate regardless of page - that is $p_{new}$ and $p_{old}$ are equal. Furthermore, it is assumed they are equal to the **converted** rate in **ab_data.csv** regardless of the page."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. What is the **convert rate** for $p_{new}$ under the null? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new = conv_prob\n",
    "p_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **convert rate** for $p_{old}$ under the null?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old = conv_prob\n",
    "p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new = df2[df2.landing_page == 'new_page'].shape[0]\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old = df2[df2.landing_page == 'old_page'].shape[0]\n",
    "n_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Simulate $n_{new}$ transactions with a convert rate of $p_{new}$ under the null.  Store these $n_{new}$ 1's and 0's in **new_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1190971027458537"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_page_converted = np.random.choice([0, 1], size=n_new, p=[(1 - p_new), p_new])\n",
    "new_page_converted.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Simulate $n_{old}$ transactions with a convert rate of $p_{old}$ under the null.  Store these $n_{old}$ 1's and 0's in **old_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.119539628563955"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "old_page_converted = np.random.choice([0, 1], size=n_old, p=[(1 - p_old), p_old])\n",
    "old_page_converted.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Find $p_{new}$ - $p_{old}$ for your simulated values from part (e) and (f)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0004425258181013081"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_page_converted.mean() - old_page_converted.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Simulate 10,000 $p_{new}$ - $p_{old}$ values using this same process similarly to the one you calculated in parts **a. through g.** above.  Store all 10,000 values in a numpy array called **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "p_diffs = []\n",
    "for _ in range(10000):\n",
    "    new_page_converted = np.random.choice([0, 1], size=n_new, p=[(1 - p_new), p_new])\n",
    "    old_page_converted = np.random.choice([0, 1], size=n_old, p=[(1 - p_old), p_old])\n",
    "    p_diffs.append(new_page_converted.mean() - old_page_converted.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "i. Plot a histogram of the **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "obs_diff = conv_prob_treatment - conv_prob_control\n",
    "p_diffs = np.asarray(p_diffs)\n",
    "plt.hist(p_diffs)\n",
    "plt.title(\"Simulated Differences in Conversion Rates for the Null Hypothesis\", fontsize=12)\n",
    "plt.xlabel(\"Probability Difference\", fontsize=12)\n",
    "plt.axvline(obs_diff, color='r');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "j. What proportion of the **p_diffs** are greater than the actual difference observed in **ab_data.csv**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8997"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(p_diffs >= obs_diff).mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "k. In words, explain what you just computed in part **j.**  What is this value called in scientific studies?  What does this value mean in terms of whether or not there is a difference between the new and old pages?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** In part j p-value is computed (in our case it is the probability of not observing any difference between convertion rates of control and treatment groups if the null hypothesis is true). In order to reject the null hypothesis, the p-value should be below our α level of 0.05 (agreed level of 5% chance of committing a Type I error if the null is true). In our case, the p-value calculated is 0.9, too high for us to be able to reject the null hypothesis. There is no sufficient evidence to say that there is convertion difference between the new and old pages."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "l. We could also use a built-in to achieve similar results.  Though using the built-in might be easier to code, the above portions are a walkthrough of the ideas that are critical to correctly thinking about statistical significance. Fill in the below to calculate the number of conversions for each page, as well as the number of individuals who received each page. Let `n_old` and `n_new` refer the the number of rows associated with the old page and new pages, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(17489, 17264, 145274, 145310)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "convert_old = df2.query('group == \"control\" & converted == 1')['converted'].count()\n",
    "convert_new = df2.query('group == \"treatment\" & converted == 1')['converted'].count()\n",
    "convert_old, convert_new, n_old, n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "m. Now use `stats.proportions_ztest` to compute your test statistic and p-value.  [Here](http://knowledgetack.com/python/statsmodels/proportions_ztest/) is a helpful link on using the built in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.3109241984234394, 0.9050583127590245)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_score, p_value = sm.stats.proportions_ztest([convert_new, convert_old], [n_new, n_old], alternative='larger')\n",
    "z_score, p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "n. What do the z-score and p-value you computed in the previous question mean for the conversion rates of the old and new pages?  Do they agree with the findings in parts **j.** and **k.**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** A z-score is the number of standard deviations from the mean a data point is (obs_diff). The z-score and p-value computed in the previous question agree with the findings in parts **j.** and **k** (p-value == 0.9)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='regression'></a>\n",
    "### Part III - A regression approach\n",
    "\n",
    "`1.` Confirm that the result achieved in the previous A/B test can also be acheived by performing regression.\n",
    "\n",
    "a. Since each row is either a conversion or no conversion, what type of regression should you be performing in this case?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** Because we are dealing with binary output, Logistic regression should be used."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. The goal is to use **statsmodels** to fit the regression model you specified in part **a.** to see if there is a significant difference in conversion based on which page a customer receives.  First we need to create a column for the intercept, and create a dummy variable column for which page each user received.  Add an **intercept** column, as well as an **ab_page** column, which is 1 when an individual receives the **treatment** and 0 if **control**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>old_page</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>804228</td>\n",
       "      <td>2017-01-12 08:01:45.159739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>853541</td>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted  \\\n",
       "0   851104  2017-01-21 22:11:48.556739    control     old_page          0   \n",
       "1   804228  2017-01-12 08:01:45.159739    control     old_page          0   \n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0   \n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0   \n",
       "4   864975  2017-01-21 01:52:26.210827    control     old_page          1   \n",
       "\n",
       "   intercept  ab_page  old_page  \n",
       "0          1        0         1  \n",
       "1          1        0         1  \n",
       "2          1        1         0  \n",
       "3          1        1         0  \n",
       "4          1        0         1  "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept'] = 1\n",
    "df2[['ab_page', 'old_page']] = pd.get_dummies(df2['landing_page'])\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Use **statsmodels** to import your regression model.  Instantiate the model, and fit the model using the two columns you created in part **b.** to predict whether or not an individual converts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366118\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "logit_mod = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']])\n",
    "results = logit_mod.fit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Provide the summary of your model below, and use it as necessary to answer the following questions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290582</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     1</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Thu, 20 Sep 2018</td> <th>  Pseudo R-squ.:     </th>  <td>8.077e-06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>22:19:45</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>           <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th> </th>                      <td> </td>        <th>  LLR p-value:       </th>   <td>0.1899</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th> <td>   -1.9888</td> <td>    0.008</td> <td> -246.669</td> <td> 0.000</td> <td>   -2.005</td> <td>   -1.973</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>   -0.0150</td> <td>    0.011</td> <td>   -1.311</td> <td> 0.190</td> <td>   -0.037</td> <td>    0.007</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290582\n",
       "Method:                           MLE   Df Model:                            1\n",
       "Date:                Thu, 20 Sep 2018   Pseudo R-squ.:               8.077e-06\n",
       "Time:                        22:19:45   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1899\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -1.9888      0.008   -246.669      0.000      -2.005      -1.973\n",
       "ab_page       -0.0150      0.011     -1.311      0.190      -0.037       0.007\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. What is the p-value associated with **ab_page**? Why does it differ from the value you found in **Part II**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** The p-value calculated here is 0.190. This is because the Logit Regression is based on a two-tailed test. 0.190 is still greater then 0.05 (our $\\alpha$), so we still cannot reject our null hypothesis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Now, you are considering other things that might influence whether or not an individual converts.  Discuss why it is a good idea to consider other factors to add into your regression model.  Are there any disadvantages to adding additional terms into your regression model?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** The convert rate may be related to some features of users like nationality, age, gender or specific cultural behavior. Adding additional information about users could reveal hidden value of the new version of the page for specific group of the users."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Now along with testing if the conversion rate changes for different pages, also add an effect based on which country a user lives. You will need to read in the **countries.csv** dataset and merge together your datasets on the approporiate rows. \n",
    "\n",
    "Does it appear that country had an impact on conversion?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>old_page</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>user_id</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>834778</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-14 23:08:43.304998</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>928468</th>\n",
       "      <td>US</td>\n",
       "      <td>2017-01-23 14:44:16.387854</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>822059</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 14:04:14.719771</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>711597</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-22 03:14:24.763511</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>710616</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 13:14:44.000513</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        country                   timestamp      group landing_page  \\\n",
       "user_id                                                               \n",
       "834778       UK  2017-01-14 23:08:43.304998    control     old_page   \n",
       "928468       US  2017-01-23 14:44:16.387854  treatment     new_page   \n",
       "822059       UK  2017-01-16 14:04:14.719771  treatment     new_page   \n",
       "711597       UK  2017-01-22 03:14:24.763511    control     old_page   \n",
       "710616       UK  2017-01-16 13:14:44.000513  treatment     new_page   \n",
       "\n",
       "         converted  intercept  ab_page  old_page  \n",
       "user_id                                           \n",
       "834778           0          1        0         1  \n",
       "928468           0          1        1         0  \n",
       "822059           1          1        1         0  \n",
       "711597           0          1        0         1  \n",
       "710616           0          1        1         0  "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "countries_df = pd.read_csv('./countries.csv')\n",
    "df_new = countries_df.set_index('user_id').join(df2.set_index('user_id'), how='inner')\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['UK', 'US', 'CA'], dtype=object)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Check what countries are in the new dataset\n",
    "df_new.country.unique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>old_page</th>\n",
       "      <th>ca</th>\n",
       "      <th>uk</th>\n",
       "      <th>us</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>user_id</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>834778</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-14 23:08:43.304998</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>928468</th>\n",
       "      <td>US</td>\n",
       "      <td>2017-01-23 14:44:16.387854</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>822059</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 14:04:14.719771</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>711597</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-22 03:14:24.763511</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>710616</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 13:14:44.000513</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        country                   timestamp      group landing_page  \\\n",
       "user_id                                                               \n",
       "834778       UK  2017-01-14 23:08:43.304998    control     old_page   \n",
       "928468       US  2017-01-23 14:44:16.387854  treatment     new_page   \n",
       "822059       UK  2017-01-16 14:04:14.719771  treatment     new_page   \n",
       "711597       UK  2017-01-22 03:14:24.763511    control     old_page   \n",
       "710616       UK  2017-01-16 13:14:44.000513  treatment     new_page   \n",
       "\n",
       "         converted  intercept  ab_page  old_page  ca  uk  us  \n",
       "user_id                                                       \n",
       "834778           0          1        0         1   0   1   0  \n",
       "928468           0          1        1         0   0   0   1  \n",
       "822059           1          1        1         0   0   1   0  \n",
       "711597           0          1        0         1   0   1   0  \n",
       "710616           0          1        1         0   0   1   0  "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Create the necessary dummy variables\n",
    "df_new[['ca', 'uk', 'us']] = pd.get_dummies(df_new['country'])\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366116\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290581</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     2</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Thu, 20 Sep 2018</td> <th>  Pseudo R-squ.:     </th>  <td>1.521e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>22:19:46</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>           <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th> </th>                      <td> </td>        <th>  LLR p-value:       </th>   <td>0.1984</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th> <td>   -2.0375</td> <td>    0.026</td> <td>  -78.364</td> <td> 0.000</td> <td>   -2.088</td> <td>   -1.987</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>uk</th>        <td>    0.0507</td> <td>    0.028</td> <td>    1.786</td> <td> 0.074</td> <td>   -0.005</td> <td>    0.106</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>us</th>        <td>    0.0408</td> <td>    0.027</td> <td>    1.518</td> <td> 0.129</td> <td>   -0.012</td> <td>    0.093</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290581\n",
       "Method:                           MLE   Df Model:                            2\n",
       "Date:                Thu, 20 Sep 2018   Pseudo R-squ.:               1.521e-05\n",
       "Time:                        22:19:46   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1984\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -2.0375      0.026    -78.364      0.000      -2.088      -1.987\n",
       "uk             0.0507      0.028      1.786      0.074      -0.005       0.106\n",
       "us             0.0408      0.027      1.518      0.129      -0.012       0.093\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Train the model\n",
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept', 'uk', 'us']])\n",
    "result = log_mod.fit()\n",
    "result.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** The p-values for the countries are above 0.05, so again we cannot reject the null hypothesis. Countries do not influence significantly differences in the conversion rates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Though you have now looked at the individual factors of country and page on conversion, we would now like to look at an interaction between page and country to see if there significant effects on conversion.  Create the necessary additional columns, and fit the new model.  \n",
    "\n",
    "Provide the summary results, and your conclusions based on the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>old_page</th>\n",
       "      <th>ca</th>\n",
       "      <th>uk</th>\n",
       "      <th>us</th>\n",
       "      <th>us_page</th>\n",
       "      <th>uk_page</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>user_id</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>834778</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-14 23:08:43.304998</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>928468</th>\n",
       "      <td>US</td>\n",
       "      <td>2017-01-23 14:44:16.387854</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>822059</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 14:04:14.719771</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>711597</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-22 03:14:24.763511</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>710616</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 13:14:44.000513</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        country                   timestamp      group landing_page  \\\n",
       "user_id                                                               \n",
       "834778       UK  2017-01-14 23:08:43.304998    control     old_page   \n",
       "928468       US  2017-01-23 14:44:16.387854  treatment     new_page   \n",
       "822059       UK  2017-01-16 14:04:14.719771  treatment     new_page   \n",
       "711597       UK  2017-01-22 03:14:24.763511    control     old_page   \n",
       "710616       UK  2017-01-16 13:14:44.000513  treatment     new_page   \n",
       "\n",
       "         converted  intercept  ab_page  old_page  ca  uk  us  us_page  uk_page  \n",
       "user_id                                                                         \n",
       "834778           0          1        0         1   0   1   0        0        0  \n",
       "928468           0          1        1         0   0   0   1        1        0  \n",
       "822059           1          1        1         0   0   1   0        0        1  \n",
       "711597           0          1        0         1   0   1   0        0        0  \n",
       "710616           0          1        1         0   0   1   0        0        1  "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create additional columns specifying what user/country converted\n",
    "df_new['us_page'] = df_new['us'] * df_new['ab_page']\n",
    "df_new['uk_page'] = df_new['uk'] * df_new['ab_page']\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366109\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290578</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     5</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Thu, 20 Sep 2018</td> <th>  Pseudo R-squ.:     </th>  <td>3.482e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>22:19:47</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>           <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th> </th>                      <td> </td>        <th>  LLR p-value:       </th>   <td>0.1920</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th> <td>   -2.0040</td> <td>    0.036</td> <td>  -55.008</td> <td> 0.000</td> <td>   -2.075</td> <td>   -1.933</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>   -0.0674</td> <td>    0.052</td> <td>   -1.297</td> <td> 0.195</td> <td>   -0.169</td> <td>    0.034</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>us</th>        <td>    0.0175</td> <td>    0.038</td> <td>    0.465</td> <td> 0.642</td> <td>   -0.056</td> <td>    0.091</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>uk</th>        <td>    0.0118</td> <td>    0.040</td> <td>    0.296</td> <td> 0.767</td> <td>   -0.066</td> <td>    0.090</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>us_page</th>   <td>    0.0469</td> <td>    0.054</td> <td>    0.872</td> <td> 0.383</td> <td>   -0.059</td> <td>    0.152</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>uk_page</th>   <td>    0.0783</td> <td>    0.057</td> <td>    1.378</td> <td> 0.168</td> <td>   -0.033</td> <td>    0.190</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290578\n",
       "Method:                           MLE   Df Model:                            5\n",
       "Date:                Thu, 20 Sep 2018   Pseudo R-squ.:               3.482e-05\n",
       "Time:                        22:19:47   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1920\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -2.0040      0.036    -55.008      0.000      -2.075      -1.933\n",
       "ab_page       -0.0674      0.052     -1.297      0.195      -0.169       0.034\n",
       "us             0.0175      0.038      0.465      0.642      -0.056       0.091\n",
       "uk             0.0118      0.040      0.296      0.767      -0.066       0.090\n",
       "us_page        0.0469      0.054      0.872      0.383      -0.059       0.152\n",
       "uk_page        0.0783      0.057      1.378      0.168      -0.033       0.190\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Fit Your Linear Model And Obtain the Results\n",
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept', 'ab_page', 'us', 'uk', 'us_page', 'uk_page']])\n",
    "result = log_mod.fit()\n",
    "result.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:** Looking at p-values for all variables (all higher than 0.05) we can say that interactions between page and country have no significant effects on conversion. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='conclusions'></a>\n",
    "## Conclusions\n",
    "\n",
    "The conclusion of the whole study and the general result is that we do not have sufficient evidence to suggest that the new page results in more conversions than the old one.\n",
    "\n",
    "Although there could be other variables identified regarding users which could result in rejecting our null hypothesis (and would require additional resources/money and time), the so-far analysis results would bias me toward suggesting to drop the new page as not bringing much value and focus on something else (maybe another brand new page with other features to test)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resources\n",
    "\n",
    "External sources used:\n",
    "- https://pandas.pydata.org/pandas-docs/stable/api.html#dataframe\n",
    "- http://knowledgetack.com/python/statsmodels/proportions_ztest/\n",
    "- https://www.statsmodels.org/dev/generated/statsmodels.stats.proportion.proportions_ztest.html\n",
    "- http://www.statisticshowto.com/probability-and-statistics/z-score/\n",
    "- http://blog.yhat.com/posts/logistic-regression-python-rodeo.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}