Analyze_ab_test_results_notebook.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyze A/B Test Results\n",
    "\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/B tests are very commonly performed by data analysts and data scientists.  It is important that you get some practice working with the difficulties of these \n",
    "\n",
    "For this project, you will be working to understand the results of an A/B test run by an e-commerce website.  Your goal is to work through this notebook to help the company understand if they should implement the new page, keep the old page, or perhaps run the experiment longer to make their decision.\n",
    "\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": 201,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "#setting the seed to get the same answers on quizzes\n",
    "random.seed(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`1.` **Quiz 1**\n",
    "\n",
    "a. Exploring the dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<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",
       "  </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"
      ]
     },
     "execution_count": 202,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('ab_data.csv')\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the cell below to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "294478"
      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. The number of unique users in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.user_id.nunique()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "d. The proportion of users converted."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11.965919355605511"
      ]
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.converted.mean()*100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "e. The number of times the `new_page` and `treatment` don't match."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3893"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.query('group==\"treatment\" and landing_page != \"new_page\" or group==\"control\" and landing_page==\"new_page\"').count()['user_id']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {},
   "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": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.isnull().sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` For the rows where **treatment** does not match with **new_page** or **control** does not match with **old_page**, we cannot be sure if this row truly received the new or old page.  Use **Quiz 2** in the classroom to figure out how we should handle these rows.  \n",
    "\n",
    "a. Now use the answer to the quiz to create a new dataset that meets the specifications from the quiz.  Store your new dataframe in **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "Int64Index: 290585 entries, 0 to 294477\n",
      "Data columns (total 5 columns):\n",
      "user_id         290585 non-null int64\n",
      "timestamp       290585 non-null object\n",
      "group           290585 non-null object\n",
      "landing_page    290585 non-null object\n",
      "converted       290585 non-null int64\n",
      "dtypes: int64(2), object(3)\n",
      "memory usage: 13.3+ MB\n"
     ]
    }
   ],
   "source": [
    "df.drop(df.query(\"group == 'treatment' and landing_page == 'old_page'\").index, inplace=True)\n",
    "df.drop(df.query(\"group == 'control' and landing_page == 'new_page'\").index, inplace=True)\n",
    "df.info()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [],
   "source": [
    "df.to_csv('ab_data2.csv', index=False)\n",
    "df2 = pd.read_csv('ab_data2.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 210,
     "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.` Use **df2** and the cells below to answer questions for **Quiz3** in the classroom."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. How many unique **user_id**s are in **df2**?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.user_id.nunique()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "b. There is one **user_id** repeated in **df2**.  What is it?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1876    773192\n",
       "2862    773192\n",
       "Name: user_id, dtype: int64"
      ]
     },
     "execution_count": 212,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2.user_id.duplicated(keep=False)].user_id"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is the row information for the repeat **user_id**? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
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       "    }\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>1876</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>2862</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",
       "1876   773192  2017-01-09 05:37:58.781806  treatment     new_page          0\n",
       "2862   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 213,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2.user_id.duplicated(keep=False)]"
   ]
  },
  {
   "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": 214,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "Int64Index: 290584 entries, 0 to 290584\n",
      "Data columns (total 5 columns):\n",
      "user_id         290584 non-null int64\n",
      "timestamp       290584 non-null object\n",
      "group           290584 non-null object\n",
      "landing_page    290584 non-null object\n",
      "converted       290584 non-null int64\n",
      "dtypes: int64(2), object(3)\n",
      "memory usage: 13.3+ MB\n"
     ]
    }
   ],
   "source": [
    "df2.drop_duplicates('user_id', inplace=True)\n",
    "df2.info()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`4.` Use **df2** in the cells below to answer the quiz questions related to **Quiz 4** in the classroom.\n",
    "\n",
    "a. What is the probability of an individual converting regardless of the page they receive?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 215,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.converted.mean()"
   ]
  },
  {
   "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": 216,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1203863045004612"
      ]
     },
     "execution_count": 216,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"group == 'control'\").converted.mean()"
   ]
  },
  {
   "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": 217,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11880806551510564"
      ]
     },
     "execution_count": 217,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"group == 'treatment'\").converted.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000619442226688"
      ]
     },
     "execution_count": 218,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query(\"landing_page == 'new_page'\").shape[0] / df2.landing_page.shape[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Consider your results from parts (a) through (d) above, and explain below whether you think there is sufficient evidence to conclude that the new treatment page leads to more conversions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer:**\n",
    "\n",
    "The number of people who converted from either group is almost identical which was about 12% of each group. So, there isn't strong evidence to infer that those who explore either page would neccessarily lead to more conversions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='ab_test'></a>\n",
    "### Part II - A/B Test\n",
    "\n",
    "Notice that because of the time stamp associated with each event, you could technically run a hypothesis test continuously as each observation was observed.  \n",
    "\n",
    "However, then the hard question is do you stop as soon as one page is considered significantly better than another or does it need to happen consistently for a certain amount of time?  How long do you run to render a decision that neither page is better than another?  \n",
    "\n",
    "These questions are the difficult parts associated with A/B tests in general.  \n",
    "\n",
    "\n",
    "`1.` For now, consider you need to make the decision just based on all the data provided.  If you want to assume that the old page is better unless the new page proves to be definitely better at a Type I error rate of 5%, what should your null and alternative hypotheses be?  You can state your hypothesis in terms of words or in terms of **$p_{old}$** and **$p_{new}$**, which are the converted rates for the old and new pages."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_0: p_{old} - p_{new} \\ge 0$\n",
    "\n",
    "$H_1: p_{old} - p_{new} < 0$\n",
    "\n",
    "where $p_{old}$ and $p_{new}$ are the converted rates for the old and new pages respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` Assume 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, assume they are equal to the **converted** rate in **ab_data.csv** regardless of the page. <br><br>\n",
    "\n",
    "Use a sample size for each page equal to the ones in **ab_data.csv**.  <br><br>\n",
    "\n",
    "Perform the sampling distribution for the difference in **converted** between the two pages over 10,000 iterations of calculating an estimate from the null.  <br><br>\n",
    "\n",
    "Use the cells below to provide the necessary parts of this simulation.  If this doesn't make complete sense right now, don't worry - you are going to work through the problems below to complete this problem.  You can use **Quiz 5** in the classroom to make sure you are on the right track.<br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a. What is the **conversion rate** for $p_{new}$ under the null? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 219,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new = df2.converted.mean()\n",
    "p_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **conversion rate** for $p_{old}$ under the null? <br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old = df2.converted.mean()\n",
    "p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$, the number of individuals in the treatment group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new = df2.query(\"group == 'treatment'\").shape[0]\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$, the number of individuals in the control group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old = df2.query(\"group == 'control'\").shape[0]\n",
    "n_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Simulate $n_{new}$ transactions with a conversion 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": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1, 0, ..., 1, 0, 1])"
      ]
     },
     "execution_count": 223,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_page_converted = np.random.choice([0, 1], size = n_new, p = [p_new, 1 - p_new])\n",
    "new_page_converted"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Simulate $n_{old}$ transactions with a conversion 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": 224,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1, 1, ..., 1, 1, 1])"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "old_page_converted = np.random.choice([0, 1], size = n_old, p = [p_old, 1 - p_old])\n",
    "old_page_converted"
   ]
  },
  {
   "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": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00049075837790923948"
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_diff = new_page_converted.mean() - old_page_converted.mean()\n",
    "p_diff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">An error occurs when calculating difference b/w $p_{new}$ - $p_{old}$ due to mismatch in sizes; so, difference in means is calculated instead."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Create 10,000 $p_{new}$ - $p_{old}$ values using the same simulation process you used in parts (a) through (g) above. Store all 10,000 values in a NumPy array called **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_diffs = []\n",
    "\n",
    "for _ in range(10000):\n",
    "    new_page_converted = np.random.choice([0, 1], size = n_new, p = [p_new, 1 - p_new]).mean()\n",
    "    old_page_converted = np.random.choice([0, 1], size = n_old, p = [p_old, 1 - p_old]).mean()\n",
    "    p_diffs.append(new_page_converted - old_page_converted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "i. Plot a histogram of the **p_diffs**.  Does this plot look like what you expected?  Use the matching problem in the classroom to assure you fully understand what was computed here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f425ea66e10>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_diffs);\n",
    "plt.ylabel('# of Simulations')\n",
    "plt.xlabel('p_diffs')\n",
    "plt.title('Plot of 10,000 Simulated p_diffs');"
   ]
  },
  {
   "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": 228,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  1.68148192e-03,   1.80901683e-04,   2.15641747e-03, ...,\n",
       "        -1.50526650e-03,   5.03735595e-05,  -2.06931466e-03])"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#converting p_diffs into a numpy array\n",
    "p_diffs = np.array(p_diffs)\n",
    "p_diffs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0015782389853555567"
      ]
     },
     "execution_count": 229,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#actual difference observed in the csv dataset\n",
    "act_diffs = df2.query('group == \"treatment\"').converted.mean() - df2.query('group == \"control\"').converted.mean()\n",
    "act_diffs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.90510000000000002"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#proportion of p_diffs greater than act_diff\n",
    "(p_diffs > act_diffs).mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "k. Please explain using the vocabulary you've learned in this course 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": [
    "**Initially, we calculated the p-value; the probability of getting our statistic or a more extreme value if the null is true.\n",
    "Greater p-value means that the statistic is more likely to come from our null hypothesis. In this case 0.9 value of p is much bigger than 0.05 alpha. So, there is no statistical evidence to reject the null hypothesis which states that old pages are of the same or slightly better in conversion than the new 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": 231,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(17489, 17264, 145274, 145310)"
      ]
     },
     "execution_count": 231,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "convert_old = df2.query('group == \"control\"').converted.sum()\n",
    "convert_new = df2.query('group == \"treatment\"').converted.sum()\n",
    "n_old = df2.query(\"landing_page == 'old_page'\").shape[0]\n",
    "n_new = df2.query(\"landing_page == 'new_page'\").shape[0]\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](https://docs.w3cub.com/statsmodels/generated/statsmodels.stats.proportion.proportions_ztest/) is a helpful link on using the built in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.3109241984234394, 0.90505831275902449)"
      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_score, p_value = sm.stats.proportions_ztest([convert_old, convert_new], [n_old, n_new], alternative='smaller')\n",
    "z_score, p_value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.90505831275902449"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "norm.cdf(z_score) #significance of z-score."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.959963984540054"
      ]
     },
     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#threshold value at 95% confidence interval.\n",
    "norm.ppf(1-(0.05/2))"
   ]
  },
  {
   "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": [
    "**Since z-score value of 1.31 does not exceed the threshold value at 95% confidence interval (1.96); there is no statistical evidence to reject the null hypothesis. Moreover, p-value obtained is similar as compared to the result obtained from our previous findings in j. and k. which also fails to reject the null hypothesis as it provides evidence of a higher probability of the null hypothesis**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='regression'></a>\n",
    "### Part III - A regression approach\n",
    "\n",
    "`1.` In this final part, you will see that the result you achieved in the A/B test in Part II above can also be achieved by performing regression.<br><br> \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": [
    "**Since, we are considering rows with either conversions or no conversions which predicts a probability between 0 and 1, Logistic Regression may 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. However, you first need to create in df2 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": 235,
   "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>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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </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  \n",
       "0          1        0  \n",
       "1          1        0  \n",
       "2          1        1  \n",
       "3          1        1  \n",
       "4          1        0  "
      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept'] = 1\n",
    "df2[['ab_page2', 'ab_page']] = pd.get_dummies(df2['group'])\n",
    "df2 = df2.drop('ab_page2', axis = 1)\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Use **statsmodels** to instantiate your regression model on the two columns you created in part b., then 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": 236,
   "metadata": {},
   "outputs": [],
   "source": [
    "log_mod = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']])"
   ]
  },
  {
   "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": 237,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366118\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>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>Fri, 04 Sep 2020</td> <th>  Pseudo R-squ.:     </th>  <td>8.077e-06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>06:15:30</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:                Fri, 04 Sep 2020   Pseudo R-squ.:               8.077e-06\n",
       "Time:                        06:15:30   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": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#I got an error when using summary(), below two lines are for that error\n",
    "#not specifically related to project\n",
    "from scipy import stats\n",
    "stats.chisqprob = lambda chisq, df: stats.chi2.sf(chisq, df)\n",
    "\n",
    "results = log_mod.fit()\n",
    "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 the **Part II**?<br><br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The p-value associated with ab_page was 0.19 which was significantly lower than the one in Part II which was roughly 0.9. Significant difference occurred because the null and alternative hypothesis differed in each exercise.**\n",
    "\n",
    "**<u>Part II Case:</u>**\n",
    "$$H_0: p_{old} - p_{new} \\ge 0$$\n",
    "\n",
    "\n",
    "$$H_1: p_{old} - p_{new} < 0$$\n",
    "\n",
    "**<u>Logistic Regression Case:</u>**\n",
    "$$H_0: p_{old} = p_{new}$$\n",
    "\n",
    "\n",
    "$$H_1: p_{old} \\ne p_{new}$$\n",
    "\n",
    "**$p_{old}$ and $p_{new}$ are the converted rates for the old and new pages respectively.**\n",
    "\n",
    "**Because the latter relies solely on two possible outcomes, it may be the reason that it yields a lower probability in the null hypothesis than that in the case in Part II of the earlier exercise.**"
   ]
  },
  {
   "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": [
    "**<u>Advantages:</u>**<br>\n",
    "\n",
    "- Allows for a more sophisticated model to distinguish other factors which may contribute to the outcome.\n",
    "- Can be used to identify outliers.<br>\n",
    "\n",
    "**<u>Disdvantages:</u>**<br>\n",
    "\n",
    "- May deliver inaccurate results because of correlated errors.<br>"
   ]
  },
  {
   "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 in. You will need to read in the **countries.csv** dataset and merge together your datasets on the appropriate rows.  [Here](https://pandas.pydata.org/pandas-docs/stable/generated/pandas.DataFrame.join.html) are the docs for joining tables. \n",
    "\n",
    "Does it appear that country had an impact on conversion?  Don't forget to create dummy variables for these country columns - **Hint: You will need two columns for the three dummy variables.** Provide the statistical output as well as a written response to answer this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
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       "\n",
       "    .dataframe tbody tr th {\n",
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       "    }\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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </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  \n",
       "user_id                                 \n",
       "834778           0          1        0  \n",
       "928468           0          1        1  \n",
       "822059           1          1        1  \n",
       "711597           0          1        0  \n",
       "710616           0          1        1  "
      ]
     },
     "execution_count": 238,
     "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": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['UK', 'US', 'CA'], dtype=object)"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new.country.unique()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "  <thead>\n",
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       "      <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>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",
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       "      <th></th>\n",
       "    </tr>\n",
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       "  <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",
       "    </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>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>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",
       "    </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>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  UK  US  \n",
       "user_id                                         \n",
       "834778           0          1        0   1   0  \n",
       "928468           0          1        1   0   1  \n",
       "822059           1          1        1   1   0  \n",
       "711597           0          1        0   1   0  \n",
       "710616           0          1        1   1   0  "
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new[['UK', 'US']] = pd.get_dummies(df_new['country'])[['UK','US']]\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "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>Fri, 04 Sep 2020</td> <th>  Pseudo R-squ.:     </th>  <td>1.521e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>06:15:31</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:                Fri, 04 Sep 2020   Pseudo R-squ.:               1.521e-05\n",
       "Time:                        06:15:31   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": 241,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept', 'UK', 'US']])\n",
    "results = log_mod.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Inference:** p-value for both countries yields a value larger than 0.05. That's why there is no statistical evidence on country's significant impact on conversion."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Combining country and ab_page."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366113\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>290580</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     3</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Fri, 04 Sep 2020</td> <th>  Pseudo R-squ.:     </th>  <td>2.323e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>06:15:31</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>0.1760</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>   -2.0300</td> <td>    0.027</td> <td>  -76.249</td> <td> 0.000</td> <td>   -2.082</td> <td>   -1.978</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK</th>        <td>    0.0506</td> <td>    0.028</td> <td>    1.784</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>    1.516</td> <td> 0.130</td> <td>   -0.012</td> <td>    0.093</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>   -0.0149</td> <td>    0.011</td> <td>   -1.307</td> <td> 0.191</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:                   290580\n",
       "Method:                           MLE   Df Model:                            3\n",
       "Date:                Fri, 04 Sep 2020   Pseudo R-squ.:               2.323e-05\n",
       "Time:                        06:15:31   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1760\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -2.0300      0.027    -76.249      0.000      -2.082      -1.978\n",
       "UK             0.0506      0.028      1.784      0.074      -0.005       0.106\n",
       "US             0.0408      0.027      1.516      0.130      -0.012       0.093\n",
       "ab_page       -0.0149      0.011     -1.307      0.191      -0.037       0.007\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept', 'UK', 'US', 'ab_page']])\n",
    "results = log_mod.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Inference:** Even after adding <i>ab_page</i> we can't see any statistical evidence to show an impact on the conversion because p-values were all exceeding 0.05."
   ]
  },
  {
   "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": 243,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366117\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>Fri, 04 Sep 2020</td> <th>  Pseudo R-squ.:     </th>  <td>1.082e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>06:15:32</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>0.3164</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>   -1.9926</td> <td>    0.008</td> <td> -252.910</td> <td> 0.000</td> <td>   -2.008</td> <td>   -1.977</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK_page</th>   <td>    0.0112</td> <td>    0.018</td> <td>    0.626</td> <td> 0.532</td> <td>   -0.024</td> <td>    0.046</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US_page</th>   <td>   -0.0144</td> <td>    0.012</td> <td>   -1.155</td> <td> 0.248</td> <td>   -0.039</td> <td>    0.010</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:                Fri, 04 Sep 2020   Pseudo R-squ.:               1.082e-05\n",
       "Time:                        06:15:32   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.3164\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -1.9926      0.008   -252.910      0.000      -2.008      -1.977\n",
       "UK_page        0.0112      0.018      0.626      0.532      -0.024       0.046\n",
       "US_page       -0.0144      0.012     -1.155      0.248      -0.039       0.010\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new['UK_page'] = df_new['UK'] * df_new['ab_page']\n",
    "df_new['US_page'] = df_new['US'] * df_new['ab_page']\n",
    "\n",
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept', 'UK_page', 'US_page']])\n",
    "results = log_mod.fit()\n",
    "results.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Introducing interaction terms yields no difference. There's still no statisitcal evidence to reconsider our previous inference."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion: ##\n",
    "\n",
    ">From the above results there is no significant impact of the countries and new page on conversion rates. The company shouldn't implement this new page as there is no conclusive evidence that it is better than the old page."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from subprocess import call\n",
    "call(['python', '-m', 'nbconvert', 'Analyze_ab_test_results_notebook.ipynb'])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}