{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiple Linear Regression\n",
    "\n",
    "Dataset: https://www.superdatascience.com/training/\n",
    "\n",
    "Author: Filipa C. S. Rodrigues (filipacsrodrigues@gmail.com)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Multiple Linear Regression:\n",
    "\n",
    "\\begin{align}\n",
    "{y}= b_0 + b_1x_1 + b_2x_2 + ... + b_nx_n\n",
    "\\end{align}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.\n",
      "  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline \n",
    "import pandas\n",
    "import csv\n",
    "import numpy as np\n",
    "import statsmodels.api as sm\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>R&amp;D Spend</th>\n",
       "      <th>Administration</th>\n",
       "      <th>Marketing Spend</th>\n",
       "      <th>State</th>\n",
       "      <th>Profit</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>165349.20</td>\n",
       "      <td>136897.80</td>\n",
       "      <td>471784.10</td>\n",
       "      <td>New York</td>\n",
       "      <td>192261.83</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>162597.70</td>\n",
       "      <td>151377.59</td>\n",
       "      <td>443898.53</td>\n",
       "      <td>California</td>\n",
       "      <td>191792.06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>153441.51</td>\n",
       "      <td>101145.55</td>\n",
       "      <td>407934.54</td>\n",
       "      <td>California</td>\n",
       "      <td>191050.39</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>144372.41</td>\n",
       "      <td>118671.85</td>\n",
       "      <td>383199.62</td>\n",
       "      <td>New York</td>\n",
       "      <td>182901.99</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>142107.34</td>\n",
       "      <td>91391.77</td>\n",
       "      <td>366168.42</td>\n",
       "      <td>California</td>\n",
       "      <td>166187.94</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   R&D Spend  Administration  Marketing Spend       State     Profit\n",
       "0  165349.20       136897.80        471784.10    New York  192261.83\n",
       "1  162597.70       151377.59        443898.53  California  191792.06\n",
       "2  153441.51       101145.55        407934.54  California  191050.39\n",
       "3  144372.41       118671.85        383199.62    New York  182901.99\n",
       "4  142107.34        91391.77        366168.42  California  166187.94"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pandas.DataFrame.from_csv('50-Startups.csv', index_col=None)\n",
    "df[:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Variables\n",
    "\n",
    "- Profit: dependent variable (y)\n",
    "- R&D Spend, Administration, Marketing Spend: numeric independent variables (x)\n",
    "- State: categorical independet variable (x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "df_y = df['Profit']\n",
    "df_x = df.drop(['Profit'], axis = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create dummy variable for the categorical variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>California</th>\n",
       "      <th>New York</th>\n",
       "      <th>R&amp;D Spend</th>\n",
       "      <th>Administration</th>\n",
       "      <th>Marketing Spend</th>\n",
       "      <th>State</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>165349.20</td>\n",
       "      <td>136897.80</td>\n",
       "      <td>471784.10</td>\n",
       "      <td>New York</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>162597.70</td>\n",
       "      <td>151377.59</td>\n",
       "      <td>443898.53</td>\n",
       "      <td>California</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>153441.51</td>\n",
       "      <td>101145.55</td>\n",
       "      <td>407934.54</td>\n",
       "      <td>California</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>144372.41</td>\n",
       "      <td>118671.85</td>\n",
       "      <td>383199.62</td>\n",
       "      <td>New York</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>142107.34</td>\n",
       "      <td>91391.77</td>\n",
       "      <td>366168.42</td>\n",
       "      <td>California</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   California  New York  R&D Spend  Administration  Marketing Spend  \\\n",
       "0           0         1  165349.20       136897.80        471784.10   \n",
       "1           1         0  162597.70       151377.59        443898.53   \n",
       "2           1         0  153441.51       101145.55        407934.54   \n",
       "3           0         1  144372.41       118671.85        383199.62   \n",
       "4           1         0  142107.34        91391.77        366168.42   \n",
       "\n",
       "        State  \n",
       "0    New York  \n",
       "1  California  \n",
       "2  California  \n",
       "3    New York  \n",
       "4  California  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dummy = pandas.get_dummies(df_x['State'])\n",
    "df_x = dummy.join(df_x)\n",
    "df_x[:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only one dummy variable should be used to avoid the \"dummy variable trap\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>New York</th>\n",
       "      <th>R&amp;D Spend</th>\n",
       "      <th>Administration</th>\n",
       "      <th>Marketing Spend</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>165349.20</td>\n",
       "      <td>136897.80</td>\n",
       "      <td>471784.10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0</td>\n",
       "      <td>162597.70</td>\n",
       "      <td>151377.59</td>\n",
       "      <td>443898.53</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
       "      <td>153441.51</td>\n",
       "      <td>101145.55</td>\n",
       "      <td>407934.54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1</td>\n",
       "      <td>144372.41</td>\n",
       "      <td>118671.85</td>\n",
       "      <td>383199.62</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0</td>\n",
       "      <td>142107.34</td>\n",
       "      <td>91391.77</td>\n",
       "      <td>366168.42</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   New York  R&D Spend  Administration  Marketing Spend\n",
       "0         1  165349.20       136897.80        471784.10\n",
       "1         0  162597.70       151377.59        443898.53\n",
       "2         0  153441.51       101145.55        407934.54\n",
       "3         1  144372.41       118671.85        383199.62\n",
       "4         0  142107.34        91391.77        366168.42"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_x = df_x.drop(['California', 'State'], axis =1)\n",
    "df_x[:5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Backward Elimination \n",
    "\n",
    "Use Backward Elimination to find the most important features.\n",
    "\n",
    "Steps:\n",
    "\n",
    "1. Select a significance level (SL) to keep the variables in the model.\n",
    "2. Fit the full model with all possible predictors.\n",
    "3. Consider the predictor with the higher p-value. If p-value > SL, go to step 4. Otherwise go to step 6.\n",
    "4. Remove the predictor.\n",
    "5. Fit model without this variable.\n",
    "6. The model is ready"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "   - Apply ordinary least squares (OLS) Regression\n",
    "   \n",
    "SL = 0.05"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add a constant \\begin{align} b_0 \\end{align} to the model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>const</th>\n",
       "      <th>New York</th>\n",
       "      <th>R&amp;D Spend</th>\n",
       "      <th>Administration</th>\n",
       "      <th>Marketing Spend</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>165349.2</td>\n",
       "      <td>136897.80</td>\n",
       "      <td>471784.10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>162597.7</td>\n",
       "      <td>151377.59</td>\n",
       "      <td>443898.53</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   const  New York  R&D Spend  Administration  Marketing Spend\n",
       "0      1         1   165349.2       136897.80        471784.10\n",
       "1      1         0   162597.7       151377.59        443898.53"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df_x = sm.add_constant(df_x)\n",
    "df_x[:2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a model with all variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                 Profit   R-squared:                       0.951\n",
      "Model:                            OLS   Adj. R-squared:                  0.947\n",
      "Method:                 Least Squares   F-statistic:                     218.4\n",
      "Date:                Thu, 22 Dec 2016   Prob (F-statistic):           7.53e-29\n",
      "Time:                        11:28:30   Log-Likelihood:                -525.25\n",
      "No. Observations:                  50   AIC:                             1060.\n",
      "Df Residuals:                      45   BIC:                             1070.\n",
      "Df Model:                           4                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "===================================================================================\n",
      "                      coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
      "-----------------------------------------------------------------------------------\n",
      "const            5.042e+04   6653.545      7.577      0.000       3.7e+04  6.38e+04\n",
      "New York        -1332.0930   2690.180     -0.495      0.623     -6750.393  4086.207\n",
      "R&D Spend           0.8080      0.046     17.662      0.000         0.716     0.900\n",
      "Administration     -0.0236      0.052     -0.455      0.651        -0.128     0.081\n",
      "Marketing Spend     0.0264      0.017      1.581      0.121        -0.007     0.060\n",
      "==============================================================================\n",
      "Omnibus:                       15.851   Durbin-Watson:                   1.306\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               25.631\n",
      "Skew:                          -0.951   Prob(JB):                     2.72e-06\n",
      "Kurtosis:                       5.947   Cond. No.                     1.41e+06\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 1.41e+06. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "model1 = sm.OLS(df_y, df_x).fit()\n",
    "print model1.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As one can observe, the variable with the largest p-value is the \"Administration\" -> p-value = 0.651 >> 0.05\n",
    "\n",
    "Let's see the relationship of this variable with the \"Profit\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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u/EfAXW79dLwxLfDu0zVAN996URz71Mgtxo0bl9U6Jk6cofCZgrqlVuEnvrLP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\nYP6m9Vx0zJjAhWrHjh0Z7b6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6F4uWb+QCNpnXMPKcadPu9wkUQGfWrJnJtGn3Z9EqwwgX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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f6fcad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = sm.graphics.plot_fit(model1, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It does not seem to exist a relationship between Profit and Administration. \n",
    "\n",
    "Let's discard this variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                 Profit   R-squared:                       0.951\n",
      "Model:                            OLS   Adj. R-squared:                  0.948\n",
      "Method:                 Least Squares   F-statistic:                     296.2\n",
      "Date:                Thu, 22 Dec 2016   Prob (F-statistic):           4.44e-30\n",
      "Time:                        11:28:34   Log-Likelihood:                -525.36\n",
      "No. Observations:                  50   AIC:                             1059.\n",
      "Df Residuals:                      46   BIC:                             1066.\n",
      "Df Model:                           3                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "===================================================================================\n",
      "                      coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
      "-----------------------------------------------------------------------------------\n",
      "const            4.772e+04   3018.340     15.811      0.000      4.16e+04  5.38e+04\n",
      "New York        -1484.6097   2646.166     -0.561      0.577     -6811.065  3841.846\n",
      "R&D Spend           0.8003      0.042     18.976      0.000         0.715     0.885\n",
      "Marketing Spend     0.0286      0.016      1.809      0.077        -0.003     0.060\n",
      "==============================================================================\n",
      "Omnibus:                       15.844   Durbin-Watson:                   1.291\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               25.957\n",
      "Skew:                          -0.943   Prob(JB):                     2.31e-06\n",
      "Kurtosis:                       5.984   Cond. No.                     6.73e+05\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 6.73e+05. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "model2 = sm.OLS(df_y, df_x.drop(['Administration'], axis = 1)).fit()\n",
    "print model2.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, the variable with the highest p-value is \"New York\" that has a p-value of 0.577, which is much grater than the significance level 0.05. \n",
    "\n",
    "Let's see the relationship of this variable with the Profit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EJPfsNp0LknmQ1MzuMLMlZjbPzPplOsZMqe9cmNkZZjY/vOaaWd9s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\nlHhERCSj/j+jZnIn87EGXgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f865390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = sm.graphics.plot_fit(model2, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Being New York (1) or California (0) does not seem to affect the Profit. So we can conclude that the variable \"New York\" does not bring any value to the model and it can be discarded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                 Profit   R-squared:                       0.950\n",
      "Model:                            OLS   Adj. R-squared:                  0.948\n",
      "Method:                 Least Squares   F-statistic:                     450.8\n",
      "Date:                Thu, 22 Dec 2016   Prob (F-statistic):           2.16e-31\n",
      "Time:                        11:28:44   Log-Likelihood:                -525.54\n",
      "No. Observations:                  50   AIC:                             1057.\n",
      "Df Residuals:                      47   BIC:                             1063.\n",
      "Df Model:                           2                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "===================================================================================\n",
      "                      coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
      "-----------------------------------------------------------------------------------\n",
      "const            4.698e+04   2689.933     17.464      0.000      4.16e+04  5.24e+04\n",
      "R&D Spend           0.7966      0.041     19.266      0.000         0.713     0.880\n",
      "Marketing Spend     0.0299      0.016      1.927      0.060        -0.001     0.061\n",
      "==============================================================================\n",
      "Omnibus:                       14.677   Durbin-Watson:                   1.257\n",
      "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               21.161\n",
      "Skew:                          -0.939   Prob(JB):                     2.54e-05\n",
      "Kurtosis:                       5.575   Cond. No.                     5.32e+05\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 5.32e+05. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "model3 = sm.OLS(df_y, df_x.drop(['Administration', 'New York'], axis = 1)).fit()\n",
    "print model3.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, the variable with the highest p-value is \"Marketing Spend\", but the value \"0.06\" is very close to the significance level \"0.05\".\n",
    "\n",
    "Let's see the relationship with \"Profit\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LC6/V8vLKRGcpKSxbtuyAtqurq9MZeXlaDjojL0/r6upUVXXChAlRy2M4gWPX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\nVfUlmneLOQuvkzDu79lu+Xv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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10f930a50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = sm.graphics.plot_fit(model3, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It seems to exist some kind of relationship between the two variables.\n",
    "\n",
    "Let's see what happens if this variable is not included in the model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                 Profit   R-squared:                       0.947\n",
      "Model:                            OLS   Adj. R-squared:                  0.945\n",
      "Method:                 Least Squares   F-statistic:                     849.8\n",
      "Date:                Thu, 22 Dec 2016   Prob (F-statistic):           3.50e-32\n",
      "Time:                        11:28:56   Log-Likelihood:                -527.44\n",
      "No. Observations:                  50   AIC:                             1059.\n",
      "Df Residuals:                      48   BIC:                             1063.\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
      "------------------------------------------------------------------------------\n",
      "const       4.903e+04   2537.897     19.320      0.000      4.39e+04  5.41e+04\n",
      "R&D Spend      0.8543      0.029     29.151      0.000         0.795     0.913\n",
      "==============================================================================\n",
      "Omnibus:                       13.727   Durbin-Watson:                   1.116\n",
      "Prob(Omnibus):                  0.001   Jarque-Bera (JB):               18.536\n",
      "Skew:                          -0.911   Prob(JB):                     9.44e-05\n",
      "Kurtosis:                       5.361   Cond. No.                     1.65e+05\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 1.65e+05. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "model4 = sm.OLS(df_y, df_x.drop(['Administration', 'New York', 'Marketing Spend'], axis = 1)).fit()\n",
    "print model4.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Adjusted R-squared in the model3, that contains the \"R&D Spend\" and the \"Marketing Spend\" variables is: 0.948\n",
    "\n",
    "The Adjusted R-squared in the model4, that contains only the \"R&D Spend\" variable is: 0.945\n",
    "\n",
    "Hence, when both variables are used, the value of the Adjusted R-squared is larger which means the model is better. \n",
    "So the \"Marketing Spend\" should be mantained in the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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