analyze_AB_test_results.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 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/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",
    "**As you work through this notebook, follow along in the classroom and answer the corresponding quiz questions associated with each question.** The labels for each classroom concept are provided for each question.  This will assure you are on the right track as you work through the project, and you can feel more confident in your final submission meeting the criteria.  As a final check, assure you meet all the criteria on the [RUBRIC](https://review.udacity.com/#!/projects/37e27304-ad47-4eb0-a1ab-8c12f60e43d0/rubric).\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": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "random.seed(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`1.` Now, read in the `ab_data.csv` data. Store it in `df`.  **Use your dataframe to answer the questions in Quiz 1 of the classroom.**\n",
    "\n",
    "a. Read in the dataset and take a look at the top few rows here:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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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": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('./data/raw_data/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": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "294478"
      ]
     },
     "execution_count": 3,
     "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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 4,
     "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": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11965919355605512"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.converted.mean()"
   ]
  },
  {
   "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": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3893"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 'treatment' but not 'new_page'\n",
    "treatment = df[(df['landing_page'] != 'new_page') & (df['group'] == 'treatment')].count()[0]\n",
    "# not 'treatment' but 'new_page'\n",
    "new_page = df[(df['landing_page'] == 'new_page') & (df['group'] != 'treatment')].count()[0]\n",
    "treatment + new_page # The number of times the `new_page` and `treatment` don't match."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.isnull().any(axis=1).any()"
   ]
  },
  {
   "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": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# rows to drop based on conditions\n",
    "drop = df[(df['landing_page'] != 'new_page') & (df['group'] == 'treatment') |\n",
    "  (df['landing_page'] == 'new_page') & (df['group'] != 'treatment')]\n",
    "# create new dataframe\n",
    "df2 = pd.DataFrame()\n",
    "# drop rows\n",
    "df.drop(drop.index, inplace=True)\n",
    "df2 = df.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "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.` 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": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 10,
     "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": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2893    773192\n",
       "Name: user_id, dtype: int64"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2[df2.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": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
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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>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",
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      ],
      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "df2[df2.duplicated('user_id')]"
   ]
  },
  {
   "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": {},
   "outputs": [],
   "source": [
    "df2.drop_duplicates('user_id', inplace=True)"
   ]
  },
  {
   "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": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cr = df2.converted.mean()\n",
    "cr"
   ]
  },
  {
   "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": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1203863045004612"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "control_cr = df2.query(\"group == 'control'\")['converted'].mean()\n",
    "control_cr"
   ]
  },
  {
   "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": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11880806551510564"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "experiment_cr = df2.query(\"group == 'treatment'\")['converted'].mean()\n",
    "experiment_cr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000619442226688"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "split = df2.query(\"landing_page == 'new_page'\").count()[0] / df2.shape[0]\n",
    "split"
   ]
  },
  {
   "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": [
    ">No, I do not think there is sufficient evidence to conclude that the new treatment pages leads to more conversions by simply considering the results from parts (a) through (d) above. There was a fair 50/50 split in receiving the treatment landing page and little variation demonstrated in the probability of converting between the treatment and control as well as compared to overall. In fact it appears you have a higher probability to convert in the control than the experiment but the difference does not seem meaningful."
   ]
  },
  {
   "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": [
    "$$H_0: p_{new} \\leq p_{old}$$\n",
    "\n",
    "$$H_1: p_{new} > p_{old}$$"
   ]
  },
  {
   "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": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new_cr = df2.converted.mean() # assume they are equal to the converted rate in ab_data.csv regardless of the page\n",
    "p_new_cr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **conversion rate** for $p_{old}$ under the null?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old_cr = df2.converted.mean() # assume they are equal to the converted rate in ab_data.csv regardless of the page\n",
    "p_old_cr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$, the number of individuals in the treatment group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new = df2.query(\"group == 'treatment'\").count()[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": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old = df2.query(\"group == 'control'\").count()[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": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 0, 0, ..., 0, 0, 0])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_page_converted = np.random.binomial(2, p_new_cr, n_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": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 0, ..., 0, 0, 1])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "old_page_converted = np.random.binomial(2, p_old_cr, n_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": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001234612705261151"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_page_converted.mean() - old_page_converted.mean()"
   ]
  },
  {
   "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": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Simulating null hypothesis ie no diff between treatment & control\n",
    "p_diffs = []\n",
    "new_page_converted = np.random.binomial(n_new, p_new_cr, 10000)/n_new\n",
    "old_page_converted = np.random.binomial(n_old, p_old_cr, 10000)/n_old\n",
    "p_diffs = new_page_converted - old_page_converted\n",
    "\n",
    "lb, ub = np.percentile(p_diffs, 2.5), np.percentile(p_diffs, 97.5) # 95% confidence interval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0015782389853555567"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Calculating actual difference observed in ab_data.csv\n",
    "obs_new = df2.query(\"group == 'treatment'\")['converted'].mean()\n",
    "obs_old = df2.query(\"group == 'control'\")['converted'].mean()\n",
    "obs_diff = obs_new - obs_old\n",
    "obs_diff"
   ]
  },
  {
   "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": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(p_diffs);\n",
    "plt.axvline(x=obs_diff, color='red');\n",
    "plt.axvline(x=lb, color='grey'); # lower bound 95%\n",
    "plt.axvline(x=ub, color='grey'); # upper bound\n",
    "\n",
    "# The plot looks normally distributed as expected"
   ]
  },
  {
   "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": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.901"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(p_diffs > obs_diff).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": [
    ">What was just computed in part j is the p-value or probability in this case of obtaining differences by bootstrapping that are greater or equal to the observed difference when simulated under the null hypothesis. We then compared those simulated expected differences to the observed difference and determined that the observed difference seems to have come from this null distribution since 90% of the simulated difference is greater then the observed so we fail to reject the null hypothesis that the treatment conversion rate is less than or equal to control conversion rate. "
   ]
  },
  {
   "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": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "convert_old = df2.query(\"landing_page == 'old_page'\")['converted'].sum()\n",
    "convert_new = df2.query(\"landing_page == 'new_page'\")['converted'].sum()\n",
    "n_old = df2.query(\"group == 'control'\").count()[0]\n",
    "n_new = df2.query(\"group == 'treatment'\").count()[0]"
   ]
  },
  {
   "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": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.3109241984234394, 0.9050583127590245)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zscore, pval = sm.stats.proportions_ztest([convert_new, convert_old], [n_new, n_old], alternative='larger')\n",
    "zscore, pval"
   ]
  },
  {
   "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": [
    ">So the p-value represents the proportion of standardize scores that we expect to be greater under the null and this matches the proportion seen in j & k. The negative zscore hints at a negative difference between the treatment and control conversion rate or that indeed the control is performing better than the treatment which also matches the data in j & k. This demonstrates how you can get similar results with bootstrapping that you can with traditional statistical inference."
   ]
  },
  {
   "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": [
    ">Logistic regression"
   ]
  },
  {
   "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": 31,
   "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": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept'] = 1\n",
    "df2['ab_page'] = pd.get_dummies(df2['group'])['treatment'] # dummy vars\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": 32,
   "metadata": {},
   "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": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "        <td>Model:</td>              <td>Logit</td>      <td>Pseudo R-squared:</td>    <td>0.000</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <td>Dependent Variable:</td>     <td>converted</td>          <td>AIC:</td>        <td>212780.3502</td>\n",
       "</tr>\n",
       "<tr>\n",
       "         <td>Date:</td>        <td>2020-07-27 21:07</td>       <td>BIC:</td>        <td>212801.5095</td>\n",
       "</tr>\n",
       "<tr>\n",
       "   <td>No. Observations:</td>       <td>290584</td>       <td>Log-Likelihood:</td>  <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "       <td>Df Model:</td>              <td>1</td>            <td>LL-Null:</td>      <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "     <td>Df Residuals:</td>         <td>290582</td>        <td>LLR p-value:</td>      <td>0.18988</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "      <td>Converged:</td>           <td>1.0000</td>           <td>Scale:</td>         <td>1.0000</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "    <td>No. Iterations:</td>        <td>6.0000</td>              <td></td>               <td></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>0.0081</td>  <td>-246.6690</td> <td>0.0000</td> <td>-2.0046</td> <td>-1.9730</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>-0.0150</td>  <td>0.0114</td>   <td>-1.3109</td>  <td>0.1899</td> <td>-0.0374</td> <td>0.0074</td> \n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary2.Summary'>\n",
       "\"\"\"\n",
       "                          Results: Logit\n",
       "==================================================================\n",
       "Model:              Logit            Pseudo R-squared: 0.000      \n",
       "Dependent Variable: converted        AIC:              212780.3502\n",
       "Date:               2020-07-27 21:07 BIC:              212801.5095\n",
       "No. Observations:   290584           Log-Likelihood:   -1.0639e+05\n",
       "Df Model:           1                LL-Null:          -1.0639e+05\n",
       "Df Residuals:       290582           LLR p-value:      0.18988    \n",
       "Converged:          1.0000           Scale:            1.0000     \n",
       "No. Iterations:     6.0000                                        \n",
       "-------------------------------------------------------------------\n",
       "              Coef.   Std.Err.      z      P>|z|    [0.025   0.975]\n",
       "-------------------------------------------------------------------\n",
       "intercept    -1.9888    0.0081  -246.6690  0.0000  -2.0046  -1.9730\n",
       "ab_page      -0.0150    0.0114    -1.3109  0.1899  -0.0374   0.0074\n",
       "==================================================================\n",
       "\n",
       "\"\"\""
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary2() #had to use 'results.summary2' as 'results.summary produces error'"
   ]
  },
  {
   "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**?<br><br>  **Hint**: What are the null and alternative hypotheses associated with your regression model, and how do they compare to the null and alternative hypotheses in **Part II**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">The p-value associated with ab_page is 0.1899. That differs from the value in Part II because that was testing a different hypothesis of..\n",
    "\n",
    "$$H_0: p_{new} \\leq p_{old}$$\n",
    "\n",
    "$$H_1: p_{new} > p_{old}$$\n",
    ">Where as this is testing..\n",
    "$$H_0: p_{new} - p_{old} = 0$$\n",
    "\n",
    "$$H_1: p_{new} - p_{old} \\neq 0$$"
   ]
  },
  {
   "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": [
    ">It is a good idea to consider other factors to add into your regression model as they may help strengthen your prediction and performace of the model. Things like transforming, recoding and other attempts at feature engineering can help capture trends int he data that can improve overall performance. But there are disadvantages suchs as multicollinearity or when variables have latent or hidden relationships which can distort the models output as independence between the variables is preferred. Additionally, you can also overfit the model as well by considering too many predicitive variables."
   ]
  },
  {
   "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": 34,
   "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>country</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>834778</td>\n",
       "      <td>UK</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>928468</td>\n",
       "      <td>US</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>822059</td>\n",
       "      <td>UK</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>711597</td>\n",
       "      <td>UK</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>710616</td>\n",
       "      <td>UK</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id country\n",
       "0   834778      UK\n",
       "1   928468      US\n",
       "2   822059      UK\n",
       "3   711597      UK\n",
       "4   710616      UK"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "countries = pd.read_csv('./countries.csv')\n",
    "countries.head()"
   ]
  },
  {
   "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>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>country</th>\n",
       "      <th>CA</th>\n",
       "      <th>UK</th>\n",
       "      <th>US</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>US</td>\n",
       "      <td>0</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>US</td>\n",
       "      <td>0</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>US</td>\n",
       "      <td>0</td>\n",
       "      <td>0</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",
       "      <td>US</td>\n",
       "      <td>0</td>\n",
       "      <td>0</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",
       "      <td>US</td>\n",
       "      <td>0</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 country  CA  UK  US  \n",
       "0          1        0      US   0   0   1  \n",
       "1          1        0      US   0   0   1  \n",
       "2          1        1      US   0   0   1  \n",
       "3          1        1      US   0   0   1  \n",
       "4          1        0      US   0   0   1  "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# join the two dataframes on `user_id`\n",
    "df2 = df2.join(countries.set_index('user_id'), on='user_id')\n",
    "# create dummy variables\n",
    "df2[['CA','UK','US']] = pd.get_dummies(df2['country'])\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "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",
       "<tr>\n",
       "        <td>Model:</td>              <td>Logit</td>      <td>Pseudo R-squared:</td>    <td>0.000</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <td>Dependent Variable:</td>     <td>converted</td>          <td>AIC:</td>        <td>212781.1253</td>\n",
       "</tr>\n",
       "<tr>\n",
       "         <td>Date:</td>        <td>2020-07-27 21:07</td>       <td>BIC:</td>        <td>212823.4439</td>\n",
       "</tr>\n",
       "<tr>\n",
       "   <td>No. Observations:</td>       <td>290584</td>       <td>Log-Likelihood:</td>  <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "       <td>Df Model:</td>              <td>3</td>            <td>LL-Null:</td>      <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "     <td>Df Residuals:</td>         <td>290580</td>        <td>LLR p-value:</td>      <td>0.17599</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "      <td>Converged:</td>           <td>1.0000</td>           <td>Scale:</td>         <td>1.0000</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "    <td>No. Iterations:</td>        <td>6.0000</td>              <td></td>               <td></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>-1.9794</td>  <td>0.0127</td>  <td>-155.4145</td> <td>0.0000</td> <td>-2.0044</td> <td>-1.9544</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>-0.0149</td>  <td>0.0114</td>   <td>-1.3069</td>  <td>0.1912</td> <td>-0.0374</td> <td>0.0075</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US</th>        <td>-0.0099</td>  <td>0.0133</td>   <td>-0.7433</td>  <td>0.4573</td> <td>-0.0359</td> <td>0.0162</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>CA</th>        <td>-0.0506</td>  <td>0.0284</td>   <td>-1.7835</td>  <td>0.0745</td> <td>-0.1063</td> <td>0.0050</td> \n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary2.Summary'>\n",
       "\"\"\"\n",
       "                          Results: Logit\n",
       "==================================================================\n",
       "Model:              Logit            Pseudo R-squared: 0.000      \n",
       "Dependent Variable: converted        AIC:              212781.1253\n",
       "Date:               2020-07-27 21:07 BIC:              212823.4439\n",
       "No. Observations:   290584           Log-Likelihood:   -1.0639e+05\n",
       "Df Model:           3                LL-Null:          -1.0639e+05\n",
       "Df Residuals:       290580           LLR p-value:      0.17599    \n",
       "Converged:          1.0000           Scale:            1.0000     \n",
       "No. Iterations:     6.0000                                        \n",
       "-------------------------------------------------------------------\n",
       "              Coef.   Std.Err.      z      P>|z|    [0.025   0.975]\n",
       "-------------------------------------------------------------------\n",
       "intercept    -1.9794    0.0127  -155.4145  0.0000  -2.0044  -1.9544\n",
       "ab_page      -0.0149    0.0114    -1.3069  0.1912  -0.0374   0.0075\n",
       "US           -0.0099    0.0133    -0.7433  0.4573  -0.0359   0.0162\n",
       "CA           -0.0506    0.0284    -1.7835  0.0745  -0.1063   0.0050\n",
       "==================================================================\n",
       "\n",
       "\"\"\""
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logit_mod = sm.Logit(df2['converted'], df2[['intercept', 'ab_page', 'US', 'CA']])\n",
    "results = logit_mod.fit()\n",
    "results.summary2()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">No it does not appear that country has an impact on predicting conversion due to the high p-values."
   ]
  },
  {
   "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": {},
   "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>country</th>\n",
       "      <th>CA</th>\n",
       "      <th>UK</th>\n",
       "      <th>US</th>\n",
       "      <th>CA_page</th>\n",
       "      <th>US_page</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
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       "    <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",
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       "      <td>0</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",
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       "      <td>1</td>\n",
       "      <td>0</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",
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       "      <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",
       "      <td>US</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</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 country  CA  UK  US  CA_page  US_page  \n",
       "0          1        0      US   0   0   1        0        0  \n",
       "1          1        0      US   0   0   1        0        0  \n",
       "2          1        1      US   0   0   1        0        1  \n",
       "3          1        1      US   0   0   1        0        1  \n",
       "4          1        0      US   0   0   1        0        0  "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# creating the new columns to show interaction\n",
    "df2['CA_page'] = df2['ab_page'] * df2['CA']\n",
    "df2['US_page'] = df2['ab_page'] * df2['US']\n",
    "df2.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",
       "<tr>\n",
       "        <td>Model:</td>              <td>Logit</td>      <td>Pseudo R-squared:</td>    <td>0.000</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <td>Dependent Variable:</td>     <td>converted</td>          <td>AIC:</td>        <td>212777.1060</td>\n",
       "</tr>\n",
       "<tr>\n",
       "         <td>Date:</td>        <td>2020-07-27 21:07</td>       <td>BIC:</td>        <td>212808.8450</td>\n",
       "</tr>\n",
       "<tr>\n",
       "   <td>No. Observations:</td>       <td>290584</td>       <td>Log-Likelihood:</td>  <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "       <td>Df Model:</td>              <td>2</td>            <td>LL-Null:</td>      <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "     <td>Df Residuals:</td>         <td>290581</td>        <td>LLR p-value:</td>     <td>0.030766</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "      <td>Converged:</td>           <td>1.0000</td>           <td>Scale:</td>         <td>1.0000</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "    <td>No. Iterations:</td>        <td>6.0000</td>              <td></td>               <td></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>-1.9873</td>  <td>0.0072</td>  <td>-275.5728</td> <td>0.0000</td> <td>-2.0014</td> <td>-1.9732</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US_page</th>   <td>-0.0197</td>  <td>0.0121</td>   <td>-1.6337</td>  <td>0.1023</td> <td>-0.0434</td> <td>0.0039</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>CA_page</th>   <td>-0.0842</td>  <td>0.0378</td>   <td>-2.2251</td>  <td>0.0261</td> <td>-0.1583</td> <td>-0.0100</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary2.Summary'>\n",
       "\"\"\"\n",
       "                          Results: Logit\n",
       "==================================================================\n",
       "Model:              Logit            Pseudo R-squared: 0.000      \n",
       "Dependent Variable: converted        AIC:              212777.1060\n",
       "Date:               2020-07-27 21:07 BIC:              212808.8450\n",
       "No. Observations:   290584           Log-Likelihood:   -1.0639e+05\n",
       "Df Model:           2                LL-Null:          -1.0639e+05\n",
       "Df Residuals:       290581           LLR p-value:      0.030766   \n",
       "Converged:          1.0000           Scale:            1.0000     \n",
       "No. Iterations:     6.0000                                        \n",
       "-------------------------------------------------------------------\n",
       "              Coef.   Std.Err.      z      P>|z|    [0.025   0.975]\n",
       "-------------------------------------------------------------------\n",
       "intercept    -1.9873    0.0072  -275.5728  0.0000  -2.0014  -1.9732\n",
       "US_page      -0.0197    0.0121    -1.6337  0.1023  -0.0434   0.0039\n",
       "CA_page      -0.0842    0.0378    -2.2251  0.0261  -0.1583  -0.0100\n",
       "==================================================================\n",
       "\n",
       "\"\"\""
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logit_mod = sm.Logit(df2['converted'], df2[['intercept', 'US_page', 'CA_page']])\n",
    "results = logit_mod.fit()\n",
    "results.summary2()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ">There does not appear to be any there significant effects on predicting conversion when looking at the interactions between page and country.\n",
    "\n",
    "**Conclusion**\n",
    "\n",
    ">In conclusion, we have statistical evidence both via bootstrapping/sampling, z-test, and logistic regression which point at that the treatment page does not improve conversions or increase the likelihood of one. You could explore further and design other features from the data set to see if anything is found to have a positive relationship in predicting conversions and then use those design elements or factors in the designing of a new treatment. You could also identify those things in the treatment that seems to have a negative impact on performance hence treatment not performing no better than control. Once you have identified those things that you feel are impacting treatment negatively you could explore and see if those things are also impacting other tests in a negative fashion. This way you could build up knowledge on what to avoid when designing treatments or new pages. You could also explore other treatments or this dataset further to see if this story holds true when looking at meaningful or important segments for your organization such a mobile vs desktop or foreign vs domestic."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='conclusions'></a>\n",
    "## Finishing Up\n",
    "\n",
    "> Congratulations!  You have reached the end of the A/B Test Results project!  You should be very proud of all you have accomplished!\n",
    "\n",
    "\n",
    "## Directions to Submit\n",
    "\n",
    "> Before you submit your project, you need to create a .html or .pdf version of this notebook in the workspace here. To do that, run the code cell below. If it worked correctly, you should get a return code of 0, and you should see the generated .html file in the workspace directory (click on the orange Jupyter icon in the upper left).\n",
    "\n",
    "> Alternatively, you can download this report as .html via the **File** > **Download as** submenu, and then manually upload it into the workspace directory by clicking on the orange Jupyter icon in the upper left, then using the Upload button.\n",
    "\n",
    "> Once you've done this, you can submit your project by clicking on the \"Submit Project\" button in the lower right here. This will create and submit a zip file with this .ipynb doc and the .html or .pdf version you created. Congratulations!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 1,
     "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",
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    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
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