Project_Analyze_ab_test_results_Oleksandra_Laurent.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analyze A/B Test Results\n",
    "\n",
    "\n",
    "\n",
    "## Table of Contents\n",
    "- [Introduction](#intro)\n",
    "- [Part I - Probability](#probability)\n",
    "- [Part II - A/B Test](#ab_test)\n",
    "- [Part III - Regression](#regression)\n",
    "\n",
    "\n",
    "<a id='intro'></a>\n",
    "### Introduction\n",
    "\n",
    "A/B tests are very commonly performed by data analysts and data scientists. \n",
    "\n",
    "For this project, we will be working to understand the results of an A/B test run by an e-commerce website.  Our 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",
    "<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`.\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('ab_data.csv')\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Use the below cell to find the number of rows in the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "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": [
       "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 line up."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3893"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Filtering new_page rows and calculating amount of rows there group doesn't equal treatment\n",
    "new_page=df[df['landing_page']=='new_page']\n",
    "a=sum(new_page['group']!='treatment')\n",
    "\n",
    "#Filtering treatment rows and calculating amount of rows there landing_page doesn't equal new_page\n",
    "treatment=df[df['group']=='treatment']\n",
    "b=sum(treatment['landing_page']!='new_page')\n",
    "\n",
    "a+b"
   ]
  },
  {
   "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",
      " #   Column        Non-Null Count   Dtype \n",
      "---  ------        --------------   ----- \n",
      " 0   user_id       294478 non-null  int64 \n",
      " 1   timestamp     294478 non-null  object\n",
      " 2   group         294478 non-null  object\n",
      " 3   landing_page  294478 non-null  object\n",
      " 4   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": [
    "> As we can see there is no rows with missing values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`2.` For the rows where **treatment** is not aligned with **new_page** or **control** is not aligned with **old_page**, we cannot be sure if this row truly received the new or old page.  \n",
    "> We will remove rows where the landing_page and group columns don't match."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Select the rows where values don't match\n",
    "drop1=df[((df['group']=='treatment') & (df['landing_page']!='new_page'))]\n",
    "drop2=df[((df['group']!='treatment')&(df['landing_page']=='new_page'))]\n",
    "\n",
    "#droping those rows\n",
    "df=df.drop(drop1.index)\n",
    "df2=df.drop(drop2.index)"
   ]
  },
  {
   "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()].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",
       "        vertical-align: middle;\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>user_id</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>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"
    }
   ],
   "source": [
    "df2[df2['user_id'].duplicated()]"
   ]
  },
  {
   "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": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2.drop_duplicates(subset='user_id', inplace=True)\n",
    "sum(df2.duplicated())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "290584"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(df2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`4.` Use **df2** in the below cells to answer the questions.\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": [
    "df2['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. Given that an individual was in the `control` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1203863045004612"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "control=df2[df2['group']=='control']\n",
    "control['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Given that an individual was in the `treatment` group, what is the probability they converted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11880806551510564"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "treatment=df2[df2['group']=='treatment']\n",
    "treatment['converted'].mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is the probability that an individual received the new page?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000619442226688"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(df2[df2['landing_page']=='new_page'])/len(df2['landing_page'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4999380557773312"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#probability that an individual received old page\n",
    "len(df2[df2['landing_page']=='old_page'])/len(df2['landing_page'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Consider your results from a. through d. above, and explain below whether you think there is sufficient evidence to say that the new treatment page leads to more conversions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> The probability of an individual in the **treatment** group converting is **0.120** and probability of an individual in the **control** group converting is **0.118**. As we can see numbers are almost the same. There is no sufficient evidence to say that new treatment page leads to more conversions. Also we can see that probabilities of receiving old and new pages are almost the same which says that these events are well balanced.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='ab_test'></a>\n",
    "### Part II - A/B Test\n",
    "\n",
    "Notice that because of the time stamp associated with each event, you could technically run a hypothesis test continuously as each observation was observed.  \n",
    "\n",
    "However, then the hard question is do you stop as soon as one page is considered significantly better than another or does it need to happen consistently for a certain amount of time?  How long do you run to render a decision that neither page is better than another?  \n",
    "\n",
    "These questions are the difficult parts associated with A/B tests in general.  \n",
    "\n",
    "\n",
    "`1.` For now, consider you need to make the decision just based on all the data provided.  If you want to assume that the old page is better unless the new page proves to be definitely better at a Type I error rate of 5%, what should your null and alternative hypotheses be?  You can state your hypothesis in terms of words or in terms of **$p_{old}$** and **$p_{new}$**, which are the converted rates for the old and new pages."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$H_0: p_{old} - p_{new} \\geq 0$$\n",
    "\n",
    "\n",
    "$$H_1: p_{new} - p_{old} > 0$$\n",
    "\n",
    "$$α=0.05 $$\n",
    "\n",
    "\n"
   ]
  },
  {
   "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 **convert rate** for $p_{new}$ under the null? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_new=df2['converted'].mean()\n",
    "p_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b. What is the **convert rate** for $p_{old}$ under the null? <br><br>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.11959708724499628"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_old=df2['converted'].mean()\n",
    "p_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. What is $n_{new}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145310"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_new=len(df2[df2['landing_page']=='new_page'])\n",
    "n_new"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "d. What is $n_{old}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "145274"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n_old=len(df2[df2['landing_page']=='old_page'])\n",
    "n_old"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. Simulate $n_{new}$ transactions with a convert rate of $p_{new}$ under the null.  Store these $n_{new}$ 1's and 0's in **new_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "new_page_converted = np.random.binomial(1, p_new, n_new)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "f. Simulate $n_{old}$ transactions with a convert rate of $p_{old}$ under the null.  Store these $n_{old}$ 1's and 0's in **old_page_converted**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "old_page_converted= np.random.binomial(1, p_old, n_old)"
   ]
  },
  {
   "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": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.000986291702403011"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the observed difference\n",
    "obs_diff = new_page_converted.mean()- old_page_converted.mean() \n",
    "obs_diff"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "h. Simulate 10,000 $p_{new}$ - $p_{old}$ values using this same process similarly to the one you calculated in parts **a. through g.** above.  Store all 10,000 values in a numpy array called **p_diffs**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "p_diffs = []\n",
    "for i in range(10000):\n",
    "    new_page_converted = np.random.binomial(1, p_new, n_new)\n",
    "    old_page_converted = np.random.binomial(1, p_old, n_old)\n",
    "    diff= new_page_converted.mean() - old_page_converted.mean()\n",
    "    p_diffs.append(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": 28,
   "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.title('Simulated Differences in Conversion Rates for the Null Hypothesis', fontsize=12);"
   ]
  },
  {
   "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": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9091"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculation of actual difference and p-value\n",
    "act_diff=(df2[df2['group'] == \"treatment\"]['converted'].mean()) - (df2[df2['group'] == \"control\"]['converted'].mean())\n",
    "p_diffs=np.array(p_diffs)\n",
    "p= (p_diffs>act_diff).mean()\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "k. In words, explain what you just computed in part **j.**  What is this value called in scientific studies?  What does this value mean in terms of whether or not there is a difference between the new and old pages?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> First we calculated the actual difference: difference calculated between converted rates of new page and old page based on data ab_data.csv. Then we calculated p-value. The p-value is the probability of getting our statistic or a more extreme value if the null is true. \n",
    "In our case p-value is large, we have evidence that our statistic was likely to come from the null hypothesis. Therefore, we failed to reject the null hypothesis. The conversion rate for old page is better or equal to 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": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "convert_old = sum(df2[df2['landing_page']=='old_page'].converted)\n",
    "convert_new = sum(df2[df2['landing_page']=='new_page'].converted)\n",
    "n_old = len(df2[df2['landing_page']=='old_page'])\n",
    "n_new = len(df2[df2['landing_page']=='new_page'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "m. Now use `stats.proportions_ztest` to compute your test statistic and p-value.  [Here](http://knowledgetack.com/python/statsmodels/proportions_ztest/) is a helpful link on using the built in."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-1.3109241984234394, 0.9050583127590245)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z_score, p_value = sm.stats.proportions_ztest([convert_new, convert_old], [n_new, n_old], alternative='larger')\n",
    "(z_score, p_value)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "n. What do the z-score and p-value you computed in the previous question mean for the conversion rates of the old and new pages?  Do they agree with the findings in parts **j.** and **k.**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> The p-value agrees with the previous finding in part j. It means that we failed to reject the null. The conversion rate for old page is better or equal to the new page."
   ]
  },
  {
   "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 acheived in the previous A/B test can also be acheived 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 since we have binary values."
   ]
  },
  {
   "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 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": 32,
   "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": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2['intercept']=1\n",
    "df2['ab_page'] = pd.get_dummies(df2.group)['treatment']\n",
    "df2.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "c. Use **statsmodels** to import your regression model.  Instantiate the model, and fit the model using the two columns you created in part **b.** to predict whether or not an individual converts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366118\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "log_mod = sm.Logit(df2['converted'], df2[['intercept', 'ab_page']])\n",
    "results = log_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": 34,
   "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>Mon, 20 Apr 2020</td> <th>  Pseudo R-squ.:     </th>  <td>8.077e-06</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                <td>22:45:29</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>Covariance Type:</th>     <td>nonrobust</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:                Mon, 20 Apr 2020   Pseudo R-squ.:               8.077e-06\n",
       "Time:                        22:45:29   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "Covariance Type:            nonrobust   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": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e. What is the p-value associated with **ab_page**? Why does it differ from the value you found in **Part II**?<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 the **Part II**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> p-value associated with ab_page is 0.19 which is lower than p-value calculated in Part II. Difference is due to Logistic Regression is based on a two-tailed test. But it's still greater then  $\\alpha$ and we still do not reject our null hypothesis. \n",
    "$$ H_0: P_{new} = P_{old} $$$$ H_1: P_{new} \\neq P_{old} $$"
   ]
  },
  {
   "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's good to add other factors to avoid  Simpson's paradox. We might consider adding factors like  the timestamps so it can bring light on which part of the day the individuals converted the most. Also adding additional information about the individuals like age, gender, nationality can reveal another way on looking at the data. Knowing if users are new/old to website can be useful factor as novelty effect can affect the rate of conversions. \n",
    "Adding too many terms can decrease true associations and lead to large standard errors and imprecise confidence intervals. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "g. Now along with testing if the conversion rate changes for different pages, also add an effect based on which country a user lives. You will need to read in the **countries.csv** dataset and merge together your datasets on the approporiate rows.  [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": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "countries_df = pd.read_csv('./countries.csv')\n",
    "df_new = countries_df.set_index('user_id').join(df2.set_index('user_id'), how='inner')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['UK', 'US', 'CA'], dtype=object)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#check which countries we have in the data set\n",
    "countries_df['country'].unique()"
   ]
  },
  {
   "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>country</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>CA</th>\n",
       "      <th>US</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>user_id</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>834778</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-14 23:08:43.304998</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>928468</th>\n",
       "      <td>US</td>\n",
       "      <td>2017-01-23 14:44:16.387854</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>822059</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 14:04:14.719771</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>711597</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-22 03:14:24.763511</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>710616</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 13:14:44.000513</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        country                   timestamp      group landing_page  \\\n",
       "user_id                                                               \n",
       "834778       UK  2017-01-14 23:08:43.304998    control     old_page   \n",
       "928468       US  2017-01-23 14:44:16.387854  treatment     new_page   \n",
       "822059       UK  2017-01-16 14:04:14.719771  treatment     new_page   \n",
       "711597       UK  2017-01-22 03:14:24.763511    control     old_page   \n",
       "710616       UK  2017-01-16 13:14:44.000513  treatment     new_page   \n",
       "\n",
       "         converted  intercept  ab_page  CA  US  \n",
       "user_id                                         \n",
       "834778           0          1        0   0   0  \n",
       "928468           0          1        1   0   1  \n",
       "822059           1          1        1   0   0  \n",
       "711597           0          1        0   0   0  \n",
       "710616           0          1        1   0   0  "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Create the necessary dummy variables\n",
    "df_new[['CA', 'US']] = pd.get_dummies(df_new['country'])[['CA','US' ]]\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366113\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>       <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                 <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290580</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>                 <td>MLE</td>       <th>  Df Model:          </th>   <td>     3</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>            <td>Mon, 20 Apr 2020</td> <th>  Pseudo R-squ.:     </th>  <td>2.323e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                <td>22:45:31</td>     <th>  Log-Likelihood:    </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>converged:</th>             <td>True</td>       <th>  LL-Null:           </th> <td>-1.0639e+05</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>     <td>nonrobust</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>   -1.9794</td> <td>    0.013</td> <td> -155.415</td> <td> 0.000</td> <td>   -2.004</td> <td>   -1.954</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>CA</th>        <td>   -0.0506</td> <td>    0.028</td> <td>   -1.784</td> <td> 0.074</td> <td>   -0.106</td> <td>    0.005</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>US</th>        <td>   -0.0099</td> <td>    0.013</td> <td>   -0.743</td> <td> 0.457</td> <td>   -0.036</td> <td>    0.016</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ab_page</th>   <td>   -0.0149</td> <td>    0.011</td> <td>   -1.307</td> <td> 0.191</td> <td>   -0.037</td> <td>    0.007</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290580\n",
       "Method:                           MLE   Df Model:                            3\n",
       "Date:                Mon, 20 Apr 2020   Pseudo R-squ.:               2.323e-05\n",
       "Time:                        22:45:31   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "Covariance Type:            nonrobust   LLR p-value:                    0.1760\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -1.9794      0.013   -155.415      0.000      -2.004      -1.954\n",
       "CA            -0.0506      0.028     -1.784      0.074      -0.106       0.005\n",
       "US            -0.0099      0.013     -0.743      0.457      -0.036       0.016\n",
       "ab_page       -0.0149      0.011     -1.307      0.191      -0.037       0.007\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_new['intercept'] = 1\n",
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept','CA', 'US', 'ab_page']])\n",
    "result = log_mod.fit()\n",
    "result.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> p-value for countries are above alpha value. So again we fail to reject the null hypothesis. So it doesn't appear that countries have an impact on conversion."
   ]
  },
  {
   "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": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country</th>\n",
       "      <th>timestamp</th>\n",
       "      <th>group</th>\n",
       "      <th>landing_page</th>\n",
       "      <th>converted</th>\n",
       "      <th>intercept</th>\n",
       "      <th>ab_page</th>\n",
       "      <th>CA</th>\n",
       "      <th>US</th>\n",
       "      <th>ca_page</th>\n",
       "      <th>us_page</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>user_id</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>834778</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-14 23:08:43.304998</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
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       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>928468</th>\n",
       "      <td>US</td>\n",
       "      <td>2017-01-23 14:44:16.387854</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>822059</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 14:04:14.719771</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>711597</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-22 03:14:24.763511</td>\n",
       "      <td>control</td>\n",
       "      <td>old_page</td>\n",
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       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>710616</th>\n",
       "      <td>UK</td>\n",
       "      <td>2017-01-16 13:14:44.000513</td>\n",
       "      <td>treatment</td>\n",
       "      <td>new_page</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        country                   timestamp      group landing_page  \\\n",
       "user_id                                                               \n",
       "834778       UK  2017-01-14 23:08:43.304998    control     old_page   \n",
       "928468       US  2017-01-23 14:44:16.387854  treatment     new_page   \n",
       "822059       UK  2017-01-16 14:04:14.719771  treatment     new_page   \n",
       "711597       UK  2017-01-22 03:14:24.763511    control     old_page   \n",
       "710616       UK  2017-01-16 13:14:44.000513  treatment     new_page   \n",
       "\n",
       "         converted  intercept  ab_page  CA  US  ca_page  us_page  \n",
       "user_id                                                           \n",
       "834778           0          1        0   0   0        0        0  \n",
       "928468           0          1        1   0   1        0        1  \n",
       "822059           1          1        1   0   0        0        0  \n",
       "711597           0          1        0   0   0        0        0  \n",
       "710616           0          1        1   0   0        0        0  "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#creating additional columns\n",
    "df_new['ca_page'] = df_new['CA'] * df_new['ab_page']\n",
    "df_new['us_page'] = df_new['US'] * df_new['ab_page']\n",
    "df_new.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.366109\n",
      "         Iterations 6\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Logit Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>       <td>converted</td>    <th>  No. Observations:  </th>   <td>290584</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                 <td>Logit</td>      <th>  Df Residuals:      </th>   <td>290581</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>                 <td>MLE</td>       <th>  Df Model:          </th>   <td>     2</td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>            <td>Mon, 20 Apr 2020</td> <th>  Pseudo R-squ.:     </th>  <td>3.272e-05</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                <td>22:45:33</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>Covariance Type:</th>     <td>nonrobust</td>    <th>  LLR p-value:       </th>   <td>0.03077</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>    0.007</td> <td> -275.573</td> <td> 0.000</td> <td>   -2.001</td> <td>   -1.973</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>ca_page</th>   <td>   -0.0842</td> <td>    0.038</td> <td>   -2.225</td> <td> 0.026</td> <td>   -0.158</td> <td>   -0.010</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>us_page</th>   <td>   -0.0197</td> <td>    0.012</td> <td>   -1.634</td> <td> 0.102</td> <td>   -0.043</td> <td>    0.004</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                           Logit Regression Results                           \n",
       "==============================================================================\n",
       "Dep. Variable:              converted   No. Observations:               290584\n",
       "Model:                          Logit   Df Residuals:                   290581\n",
       "Method:                           MLE   Df Model:                            2\n",
       "Date:                Mon, 20 Apr 2020   Pseudo R-squ.:               3.272e-05\n",
       "Time:                        22:45:33   Log-Likelihood:            -1.0639e+05\n",
       "converged:                       True   LL-Null:                   -1.0639e+05\n",
       "Covariance Type:            nonrobust   LLR p-value:                   0.03077\n",
       "==============================================================================\n",
       "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
       "------------------------------------------------------------------------------\n",
       "intercept     -1.9873      0.007   -275.573      0.000      -2.001      -1.973\n",
       "ca_page       -0.0842      0.038     -2.225      0.026      -0.158      -0.010\n",
       "us_page       -0.0197      0.012     -1.634      0.102      -0.043       0.004\n",
       "==============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### Fit Your Linear Model And Obtain the Results\n",
    "df_new['intercept']=1\n",
    "log_mod = sm.Logit(df_new['converted'], df_new[['intercept', 'ca_page', 'us_page']])\n",
    "result = log_mod.fit()\n",
    "result.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> According to p-value interactions between page and US have no significant effects on conversion. However p-value for individuals from Canada is less than alpha therefor we can reject the null hypothesis. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='conclusions'></a>\n",
    "## Conclusions\n",
    "\n",
    "According to the analysis performed we found that the old page is better than the new page, therefore we fail to reject the null hypothesis. We do not have sufficient evidence to suggest that new page leads to more conversions than the old one.\n",
    "\n",
    "During our analysis we found that the probability of an individual in the treatment group converting is 0.120 and probability of an individual in the control group converting is 0.118 which led to conclusion that there is no sufficient evidence to say that new treatment page leads to more conversions.\n",
    "\n",
    "During A/B test we found that p-value is large. Therefore, we failed to reject the null hypothesis. The conversion rate for old page was again better or equal to the new page.\n",
    "\n",
    "Logistic regression gave the same result as A/B test. Also we added an effect based on which country a user lives. It gave the same results. We failed to reject the null hypothesis. So it doesn't appear that countries have an impact on conversion.\n",
    "Although there were one factor that resulted in in rejecting our null hypothesis. Interaction between new page and individuals from Canada resulted in small p-value suggesting our null not to be true.  \n",
    "\n",
    "So based on information gathered the old page is better than new one. The solution for the company based on these data is to keep the old page. However, I would suggest running the experiment longer with limitations stated below in mind.\n",
    "\n",
    "\n",
    "### Limitations\n",
    "There are limitations to this project, namely: duration of the experiment, novelty effects and limited amount of variables.\n",
    "\n",
    "\n",
    "\n",
    "### Resources used\n",
    "- https://docs.scipy.org/doc/scipy/reference/index.html\n",
    "- https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.where.html\n",
    "- https://stackoverflow.com/\n",
    "- https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5543767/\n",
    "- https://www.statsmodels.org/dev/generated/statsmodels.stats.proportion.proportions_ztest.html\n",
    "- https://www.noporkpies.com/blog/design/pros-and-cons-of-ab-testing/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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