Analyze_ab_test_results_notebook.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyze A/B Test Results\n",
    "\n",
    "You may either submit your notebook through the workspace here, or you may work from your local machine and submit through the next page.  Either way assure that your code passes the project [RUBRIC](https://review.udacity.com/#!/projects/37e27304-ad47-4eb0-a1ab-8c12f60e43d0/rubric).  **Please save regularly.**\n",
    "\n",
    "This project will assure you have mastered the subjects covered in the statistics lessons.  The hope is to have this project be as comprehensive of these topics as possible.  Good luck!\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",
    "#We are setting the seed to assure you get the same answers on quizzes as we set up\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": {
      "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",
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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>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>804228</td>\n",
       "      <td>2017-01-12 08:01:45.159739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>661590</td>\n",
       "      <td>2017-01-11 16:55:06.154213</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>853541</td>\n",
       "      <td>2017-01-08 18:28:03.143765</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>936923</td>\n",
       "      <td>2017-01-10 15:20:49.083499</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>679687</td>\n",
       "      <td>2017-01-19 03:26:46.940749</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>719014</td>\n",
       "      <td>2017-01-17 01:48:29.539573</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>817355</td>\n",
       "      <td>2017-01-04 17:58:08.979471</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>839785</td>\n",
       "      <td>2017-01-15 18:11:06.610965</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   user_id                   timestamp      group landing_page  converted\n",
       "0   851104  2017-01-21 22:11:48.556739    control     old_page          0\n",
       "1   804228  2017-01-12 08:01:45.159739    control     old_page          0\n",
       "2   661590  2017-01-11 16:55:06.154213  treatment     new_page          0\n",
       "3   853541  2017-01-08 18:28:03.143765  treatment     new_page          0\n",
       "4   864975  2017-01-21 01:52:26.210827    control     old_page          1\n",
       "5   936923  2017-01-10 15:20:49.083499    control     old_page          0\n",
       "6   679687  2017-01-19 03:26:46.940749  treatment     new_page          1\n",
       "7   719014  2017-01-17 01:48:29.539573    control     old_page          0\n",
       "8   817355  2017-01-04 17:58:08.979471  treatment     new_page          1\n",
       "9   839785  2017-01-15 18:11:06.610965  treatment     new_page          1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv('ab_data.csv')\n",
    "df.head(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the cell below to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(294478, 5)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape"
   ]
  },
  {
   "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": [
       "user_id         290584\n",
       "timestamp       294478\n",
       "group                2\n",
       "landing_page         2\n",
       "converted            2\n",
       "dtype: int64"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.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": [
    "len(df[df['converted'] == 1])/len(df)"
   ]
  },
  {
   "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": [
       "user_id         3893\n",
       "timestamp       3893\n",
       "group           3893\n",
       "landing_page    3893\n",
       "converted       3893\n",
       "dtype: int64"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.query('group == \"control\" and landing_page == \"new_page\"').count() + df.query('group == \"treatment\" and landing_page == \"old_page\"').count()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Do any of the rows have missing values?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'pandas.core.frame.DataFrame'>\n",
      "RangeIndex: 294478 entries, 0 to 294477\n",
      "Data columns (total 5 columns):\n",
      "user_id         294478 non-null int64\n",
      "timestamp       294478 non-null object\n",
      "group           294478 non-null object\n",
      "landing_page    294478 non-null object\n",
      "converted       294478 non-null int64\n",
      "dtypes: int64(2), object(3)\n",
      "memory usage: 11.2+ MB\n"
     ]
    }
   ],
   "source": [
    "df.info()"
   ]
  },
  {
   "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": [
    {
     "data": {
      "text/plain": [
       "(290585, 5)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "index_names = df[((df['group'] == 'treatment') == (df['landing_page'] == 'new_page')) == False].index \n",
    "df2 = df.drop(index_names)\n",
    "df2.shape"
   ]
  },
  {
   "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['user_id'].duplicated() == True]['user_id']"
   ]
  },
  {
   "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": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1899</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-09 05:37:58.781806</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2893</th>\n",
       "      <td>773192</td>\n",
       "      <td>2017-01-14 02:55:59.590927</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      user_id                   timestamp      group landing_page  converted\n",
       "1899   773192  2017-01-09 05:37:58.781806  treatment     new_page          0\n",
       "2893   773192  2017-01-14 02:55:59.590927  treatment     new_page          0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.query('user_id == 773192')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. Remove **one** of the rows with a duplicate **user_id**, but keep your dataframe as **df2**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2.drop(2893, inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(290584, 5)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.shape"
   ]
  },
  {
   "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": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P_converting = len(df2[df2['converted'] == 1])/len(df2)\n",
    "P_converting"
   ]
  },
  {
   "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": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1203863045004612"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_control = df2[df2['group'] == 'control']\n",
    "P_converting_G_Control = len(df_control[df_control['converted'] == 1])/len(df_control)\n",
    "P_converting_G_Control"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Given that an individual was in the `treatment` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11880806551510564"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_treatment = df2[df2['group'] == 'treatment']\n",
    "P_converting_G_Treatment = len(df_treatment[df_treatment['converted'] == 1])/len(df_treatment)\n",
    "P_converting_G_Treatment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000619442226688"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(df2[df2['group'] == 'treatment'])/len(df2)"
   ]
  },
  {
   "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": [
    "### Our results looked like this:\n",
    "* Probability of converting regardless of the page Users received = 11.96%\n",
    "* Probability of converting given the user in control = 12.03%\n",
    "* Probability of converting given the user in Treatment = 11.88%\n",
    "* Probability of receiving the new page / Control vs Treatment = 50.01% \n",
    "\n",
    "### Analysis:\n",
    "* Our division of the control vs treatment is almost similar with 50.01% of the users will receive the new treatment, which is good.\n",
    "<br><br>\n",
    "* As for the converting rate, there is not that much of a change between the new page and the old page, though the old page has a 0.15% converting rate higher than the new page.\n",
    "<br><br>\n",
    "* So, from what we have now using probabilities of Conversion we don't have sufficient evidence that the new treatment will lead to more conversions.\n",
    "\n",
    "### What could we check later?\n",
    "* maybe looking at the duration of the test will have some information, maybe its a small duration so we need more time, we will test later and will find out"
   ]
  },
  {
   "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": [
    "### Our Hypothesis:\n",
    "\n",
    "* **$H_{0}$** : **$p_{old}$** >= **$p_{new}$**\n",
    "* **$H_{1}$** : **$p_{old}$** <  **$p_{new}$**"
   ]
  },
  {
   "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": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pnew = df2.converted.mean()\n",
    "pnew"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **conversion rate** for $p_{old}$ under the null? <br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pold = df2.converted.mean()\n",
    "pold"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$, the number of individuals in the treatment group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nnew = df2[df2['group'] == 'treatment'].user_id.count()\n",
    "nnew"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$, the number of individuals in the control group?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nold = df2[df2['group'] == 'control'].user_id.count()\n",
    "nold"
   ]
  },
  {
   "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": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_page_converted = np.random.choice([0, 1], size=nnew, p=[1-pnew, pnew])"
   ]
  },
  {
   "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": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "old_page_converted = np.random.choice([0, 1], size=nold, p=[1-pold, pold])"
   ]
  },
  {
   "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": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0010550333489407343"
      ]
     },
     "execution_count": 25,
     "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": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_diffs = []\n",
    "for _ in range(10000):\n",
    "    new_page_converted = np.random.choice([0, 1], size=nnew, p=[1-pnew, pnew])   \n",
    "    old_page_converted = np.random.choice([0, 1], size=nold, p=[1-pold, pold])\n",
    "    p_diffs.append(new_page_converted.mean() - old_page_converted.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "i. Plot a histogram of the **p_diffs**.  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": [
       "<matplotlib.figure.Figure at 0x7fc2284b72e8>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(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": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x7fc1fee62c88>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc1fee62c18>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "obs_diffs = df2[df2['group'] == 'treatment'].converted.mean() - df2[df2['group'] == 'control'].converted.mean()\n",
    "plt.hist(p_diffs);\n",
    "plt.axvline(obs_diffs, color='red')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.90649999999999997"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P_val =  (p_diffs > obs_diffs).mean()\n",
    "P_val"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.3069270907770565"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_score = (obs_diffs - np.array(p_diffs).mean())/np.array(p_diffs).std()\n",
    "z_score"
   ]
  },
  {
   "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": [
    "### Explaination\n",
    "* we started this section with our assumption of the null, we assumed all convention values are equal.\n",
    "* then we simulated this null hypothesis, using our assumptions of equal values and the sample size from our original data, and we made this sample distribution of the difference in convention means for this null. \n",
    "###### (kinda similter to calculating the stander deviation of the sample and then calculate its normal distribution on the null hypotheses)\n",
    "<br>\n",
    "* after that we calculated the difference on our original data and compared it with the null to see is it true, can we accept this null hypothesis or we should reject it, and we found that a P-value = 0.9 is more than enough to accept this null and therefore we concluded that there is no need for the new page."
   ]
  },
  {
   "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": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/conda/lib/python3.6/site-packages/statsmodels/compat/pandas.py:56: FutureWarning: The pandas.core.datetools module is deprecated and will be removed in a future version. Please use the pandas.tseries module instead.\n",
      "  from pandas.core import datetools\n"
     ]
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "n_convert_old = df2[(df2['group'] == 'control') & (df2['converted'] == 1)].user_id.count()\n",
    "n_convert_new = df2[(df2['group'] == 'treatment') & (df2['converted'] == 1)].user_id.count()\n",
    "n_old = df2[df2['group'] == 'control'].user_id.count()\n",
    "n_new = df2[df2['group'] == 'treatment'].user_id.count()"
   ]
  },
  {
   "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": "markdown",
   "metadata": {},
   "source": [
    "Ref: https://sonalake.com/latest/hypothesis-testing-of-proportion-based-samples/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.90505831275902449"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sample_success_a, sample_size_a = (n_convert_old, n_old)\n",
    "sample_success_b, sample_size_b = (n_convert_new, n_new)\n",
    "\n",
    "successes = np.array([sample_success_b, sample_success_a])\n",
    "samples = np.array([sample_size_b, sample_size_a])\n",
    "\n",
    "z_stat, p_value = sm.stats.proportions_ztest(count=successes, nobs=samples, alternative='larger')\n",
    "p_value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.3109241984234394"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_stat"
   ]
  },
  {
   "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": [
    "* It concludes the same as our previous results, that this null is accepted and we don't need the new page.\n",
    "* the z score means that the sample mean is 1.31 standard deviation less than the mean of the null.\n",
    "* and the p-value is a score that indicates if we should reject the null or not."
   ]
  },
  {
   "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": 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>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": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept'] = 1\n",
    "df2['ab_page'] = pd.get_dummies(df['group'])['treatment']\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": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366118\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "model = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']])\n",
    "results = model.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": "markdown",
   "metadata": {},
   "source": [
    "###### The code in the cell below is to fix a bug happens when calling summary function on the logistic model.\n",
    "###### ref: https://github.com/statsmodels/statsmodels/issues/3931"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import stats\n",
    "stats.chisqprob = lambda chisq, df: stats.chi2.sf(chisq, df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290582</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     1</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Sun, 03 Jan 2021</td> <th>  Pseudo R-squ.:     </th>  <td>8.077e-06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>12:07:55</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:                Sun, 03 Jan 2021   Pseudo R-squ.:               8.077e-06\n",
       "Time:                        12:07:55   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": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "intercept    0.136863\n",
       "ab_page      0.985123\n",
       "dtype: float64"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(results.params) "
   ]
  },
  {
   "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 here is for a null hypotisys of:\n",
    "* **$H_{0}$** : **$p_{old}$** - **$p_{new}$** = 0\n",
    "* **$H_{1}$** : **$p_{old}$** - **$p_{new}$** ≠ 0\n",
    "<br>\n",
    "\n",
    "which can be calculated using two tailed p-value calculation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.18609999999999999"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(p_diffs < obs_diffs).mean() + (p_diffs > 0 + (0 - obs_diffs)).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "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>294473</th>\n",
       "      <td>751197</td>\n",
       "      <td>2017-01-03 22:28:38.630509</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>294474</th>\n",
       "      <td>945152</td>\n",
       "      <td>2017-01-12 00:51:57.078372</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>294475</th>\n",
       "      <td>734608</td>\n",
       "      <td>2017-01-22 11:45:03.439544</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>294476</th>\n",
       "      <td>697314</td>\n",
       "      <td>2017-01-15 01:20:28.957438</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>294477</th>\n",
       "      <td>715931</td>\n",
       "      <td>2017-01-16 12:40:24.467417</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": [
       "        user_id                   timestamp      group landing_page  \\\n",
       "294473   751197  2017-01-03 22:28:38.630509    control     old_page   \n",
       "294474   945152  2017-01-12 00:51:57.078372    control     old_page   \n",
       "294475   734608  2017-01-22 11:45:03.439544    control     old_page   \n",
       "294476   697314  2017-01-15 01:20:28.957438    control     old_page   \n",
       "294477   715931  2017-01-16 12:40:24.467417  treatment     new_page   \n",
       "\n",
       "        converted  intercept  ab_page  \n",
       "294473          0          1        0  \n",
       "294474          0          1        0  \n",
       "294475          0          1        0  \n",
       "294476          0          1        0  \n",
       "294477          0          1        1  "
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.tail()"
   ]
  },
  {
   "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": [
    "### why it's a good idea to consider other factors:\n",
    "\n",
    "In our ab testing, we are discussing if the new treatment works better, so far we used one variable of this test, which is whether the user landed on the new page or the old page, with well-choosen distribution with nearly 50% of users are in the control group versus the treatment group, but many other factors might affect the user decision of converting or not:\n",
    "\n",
    "* An old user who just exploring the new treatment, may not make any conversion.\n",
    "* The location of a user may be a cultural factor for acting to the new treatment.\n",
    "* Maybe our unbiased 50% control group has another biased factor in selecting them like gender or the time those users used our service.\n",
    "* The duration of the test may also affect the experiment, had we do it enough?\n",
    "\n",
    "from all of that we are not just rejecting the new treatment, we can know what we can do better or change that will increase our conversion in future updates.\n",
    "<hr> \n",
    "\n",
    "### disadvantages of adding additional terms to our regression:\n",
    "\n",
    "##### By adding multiple factors to our regression model we are now using multiple regression, which can have its problems:\n",
    "<br>\n",
    "like maybe a new factor adds outliers to our model that will hurt it.\n",
    "\n",
    "* we should check for each factor and try to clean it as much as possible\n",
    "\n",
    "multicollinearity can happen between factors, and it means that one independent factor may be correlated with one other independent factor, and it's bad for our analysis on the model, we may found that it has a negative correlation in the model with a positive correlation in the real world\n",
    "* we can avoid that by detecting those correlations and don't use them, we can do that using scatter plots of VIFs measures"
   ]
  },
  {
   "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": 41,
   "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": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_countries = pd.read_csv('countries.csv')\n",
    "df_countries.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "        vertical-align: middle;\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",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>country</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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </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  \n",
       "0          1        0      US  \n",
       "1          1        0      US  \n",
       "2          1        1      US  \n",
       "3          1        1      US  \n",
       "4          1        0      US  "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df2.join(df_countries.set_index('user_id'), on='user_id')\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2 = df2.join(pd.get_dummies(df2['country']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "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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       "<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": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366113\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "model = sm.Logit(df2['converted'], df2[['intercept', 'ab_page', 'US', 'UK']])\n",
    "results = model.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>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>Sun, 03 Jan 2021</td> <th>  Pseudo R-squ.:     </th>  <td>2.323e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>12:07:56</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>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",
       "<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>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",
       "</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:                Sun, 03 Jan 2021   Pseudo R-squ.:               2.323e-05\n",
       "Time:                        12:07:56   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",
       "ab_page       -0.0149      0.011     -1.307      0.191      -0.037       0.007\n",
       "US             0.0408      0.027      1.516      0.130      -0.012       0.093\n",
       "UK             0.0506      0.028      1.784      0.074      -0.005       0.106\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "intercept    0.131332\n",
       "ab_page      0.985168\n",
       "US           1.041599\n",
       "UK           1.051944\n",
       "dtype: float64"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(results.params)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "looks like US and Uk have a very close bus also very small effect on the data"
   ]
  },
  {
   "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": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "df2['US_page'] = df2['US']*df2['ab_page']\n",
    "df2['UK_page'] = df2['UK']*df2['ab_page']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "</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>US_page</th>\n",
       "      <th>UK_page</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>851104</td>\n",
       "      <td>2017-01-21 22:11:48.556739</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>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",
       "    <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",
       "      <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",
       "      <td>US</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>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",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>864975</td>\n",
       "      <td>2017-01-21 01:52:26.210827</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>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  US_page  UK_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        1        0  \n",
       "3          1        1      US   0   0   1        1        0  \n",
       "4          1        0      US   0   0   1        0        0  "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366109\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "model = sm.Logit(df2['converted'], df2[['intercept', 'ab_page', 'US', 'UK','US_page','UK_page']])\n",
    "results = model.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>     <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>               <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290578</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>MLE</td>       <th>  Df Model:          </th>   <td>     5</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>          <td>Sun, 03 Jan 2021</td> <th>  Pseudo R-squ.:     </th>  <td>3.482e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>              <td>12:07:57</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>           <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th> </th>                      <td> </td>        <th>  LLR p-value:       </th>   <td>0.1920</td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>intercept</th> <td>   -2.0040</td> <td>    0.036</td> <td>  -55.008</td> <td> 0.000</td> <td>   -2.075</td> <td>   -1.933</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>   -0.0674</td> <td>    0.052</td> <td>   -1.297</td> <td> 0.195</td> <td>   -0.169</td> <td>    0.034</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US</th>        <td>    0.0175</td> <td>    0.038</td> <td>    0.465</td> <td> 0.642</td> <td>   -0.056</td> <td>    0.091</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK</th>        <td>    0.0118</td> <td>    0.040</td> <td>    0.296</td> <td> 0.767</td> <td>   -0.066</td> <td>    0.090</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US_page</th>   <td>    0.0469</td> <td>    0.054</td> <td>    0.872</td> <td> 0.383</td> <td>   -0.059</td> <td>    0.152</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>UK_page</th>   <td>    0.0783</td> <td>    0.057</td> <td>    1.378</td> <td> 0.168</td> <td>   -0.033</td> <td>    0.190</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290578\n",
       "Method:                           MLE   Df Model:                            5\n",
       "Date:                Sun, 03 Jan 2021   Pseudo R-squ.:               3.482e-05\n",
       "Time:                        12:07:57   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "                                        LLR p-value:                    0.1920\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -2.0040      0.036    -55.008      0.000      -2.075      -1.933\n",
       "ab_page       -0.0674      0.052     -1.297      0.195      -0.169       0.034\n",
       "US             0.0175      0.038      0.465      0.642      -0.056       0.091\n",
       "UK             0.0118      0.040      0.296      0.767      -0.066       0.090\n",
       "US_page        0.0469      0.054      0.872      0.383      -0.059       0.152\n",
       "UK_page        0.0783      0.057      1.378      0.168      -0.033       0.190\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "intercept    0.134794\n",
       "ab_page      0.934776\n",
       "US           1.017682\n",
       "UK           1.011854\n",
       "US_page      1.048001\n",
       "UK_page      1.081428\n",
       "dtype: float64"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.exp(results.params)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "the p-value of the UK and US increased but still there is no much change in convention rates overall"
   ]
  },
  {
   "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": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 55,
     "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"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
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   "pygments_lexer": "ipython3",
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