9-3_statsmodels.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Classical Statistics Modeling in Python with statsmodels\n",
    "\n",
    "Most popular package for modeling is scikit learn. Scikit-learn is not the best for a 'traditional' stats approach.\n",
    "\n",
    "The statsmodels library is designed for traditional statistics models like linear regression and logistic regression.\n",
    "\n",
    "Be sure to visit and read the documentation on the StatsModels website: https://www.statsmodels.org/stable/index.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# statsmodels.api uses numpy array notation\n",
    "# statsmodels.formula.api use formula notation (similar to R's formula notation)\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A minimal OLS example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Four pairs of points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([1,2,3,4])\n",
    "y = np.array([2,6,4,8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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.scatter(x,y, marker = '.')\n",
    "plt.xlim(0,5)\n",
    "plt.ylim(0,10)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "   x  y\n",
      "0  1  2\n",
      "1  2  6\n",
      "2  3  4\n",
      "3  4  8\n"
     ]
    }
   ],
   "source": [
    "# make a dataframe of our data\n",
    "d = pd.DataFrame({'x':x, 'y':y})\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Seaborn lmplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<seaborn.axisgrid.FacetGrid at 0x19c36025408>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.lmplot(x = 'x', y = 'y', data = d, ci = False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.5"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2909944487358056"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(x, ddof = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.581988897471611"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(y, ddof = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 0.8],\n",
       "       [0.8, 1. ]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.corrcoef(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6000000000000003"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# manual calculation of slope: b1 = r * sy / sx\n",
    "0.8 * np.std(y, ddof = 1) / np.std(x, ddof = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# manual calculation of intercept: b0 = ybar - b1 * xbar\n",
    "5 - 1.6 * 2.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## formula notation with statsmodels\n",
    "use statsmodels.formula.api (often imported as smf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# data is in a dataframe\n",
    "model = smf.ols('y ~ x', data = d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<statsmodels.regression.linear_model.OLS object at 0x0000019C3620A988>\n"
     ]
    }
   ],
   "source": [
    "print(model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# estimation of coefficients is not done until you call fit() on the model\n",
    "results = model.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<statsmodels.regression.linear_model.RegressionResultsWrapper object at 0x0000019C362149C8>\n"
     ]
    }
   ],
   "source": [
    "print(results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   R-squared:                       0.640\n",
      "Model:                            OLS   Adj. R-squared:                  0.460\n",
      "Method:                 Least Squares   F-statistic:                     3.556\n",
      "Date:                Fri, 04 Mar 2022   Prob (F-statistic):              0.200\n",
      "Time:                        09:44:49   Log-Likelihood:                -6.8513\n",
      "No. Observations:                   4   AIC:                             17.70\n",
      "Df Residuals:                       2   BIC:                             16.48\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "Intercept      1.0000      2.324      0.430      0.709      -8.998      10.998\n",
      "x              1.6000      0.849      1.886      0.200      -2.051       5.251\n",
      "==============================================================================\n",
      "Omnibus:                          nan   Durbin-Watson:                   3.400\n",
      "Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.308\n",
      "Skew:                          -0.000   Prob(JB):                        0.857\n",
      "Kurtosis:                       1.640   Cond. No.                         7.47\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\miles\\anaconda3\\lib\\site-packages\\statsmodels\\stats\\stattools.py:75: ValueWarning: omni_normtest is not valid with less than 8 observations; 4 samples were given.\n",
      "  \"samples were given.\" % int(n), ValueWarning)\n"
     ]
    }
   ],
   "source": [
    "print(results.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the abline_plot function for plotting the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "sm.graphics.abline_plot(model_results = results)\n",
    "plt.scatter(d.x, d.y)\n",
    "\n",
    "plt.xlim(0,5)\n",
    "plt.ylim(0,10)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generating an anova table"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "           df  sum_sq  mean_sq         F  PR(>F)\n",
      "x         1.0    12.8     12.8  3.555556     0.2\n",
      "Residual  2.0     7.2      3.6       NaN     NaN\n"
     ]
    }
   ],
   "source": [
    "print(sm.stats.anova_lm(results))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Making predictions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    4.2\n",
       "dtype: float64"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results.predict({'x' : 2})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## numpy array notation\n",
    "similar to sklearn's notation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3 4]\n"
     ]
    }
   ],
   "source": [
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = sm.add_constant(x)  # need to add a constant for the intercept term.\n",
    "# because we are using the numpy notation, we use sm rather than smf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 1.]\n",
      " [1. 2.]\n",
      " [1. 3.]\n",
      " [1. 4.]]\n"
     ]
    }
   ],
   "source": [
    "print(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$y_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$\n",
    "\n",
    "$$\\mathbf{\\hat{Y}} =\\mathbf{X} \\boldsymbol{\\beta}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 6, 4, 8])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "model2 = sm.OLS(y, X)  # OLS is capitalized in the numpy notation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "results2 = model2.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                      y   R-squared:                       0.640\n",
      "Model:                            OLS   Adj. R-squared:                  0.460\n",
      "Method:                 Least Squares   F-statistic:                     3.556\n",
      "Date:                Fri, 04 Mar 2022   Prob (F-statistic):              0.200\n",
      "Time:                        09:44:49   Log-Likelihood:                -6.8513\n",
      "No. Observations:                   4   AIC:                             17.70\n",
      "Df Residuals:                       2   BIC:                             16.48\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          1.0000      2.324      0.430      0.709      -8.998      10.998\n",
      "x1             1.6000      0.849      1.886      0.200      -2.051       5.251\n",
      "==============================================================================\n",
      "Omnibus:                          nan   Durbin-Watson:                   3.400\n",
      "Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.308\n",
      "Skew:                          -0.000   Prob(JB):                        0.857\n",
      "Kurtosis:                       1.640   Cond. No.                         7.47\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\miles\\anaconda3\\lib\\site-packages\\statsmodels\\stats\\stattools.py:75: ValueWarning: omni_normtest is not valid with less than 8 observations; 4 samples were given.\n",
      "  \"samples were given.\" % int(n), ValueWarning)\n"
     ]
    }
   ],
   "source": [
    "print(results2.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OLS solution:\n",
    "\n",
    "$$(X^TX)^{-1}X^TY$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1., 1.],\n",
       "       [1., 2.],\n",
       "       [1., 3.],\n",
       "       [1., 4.]])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1. , 1.6])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.inv(X.T @ X) @ (X.T @ y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple Linear Regression Example with Olympic data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "# winning time of the mens 100 meter dash\n",
    "olympic = pd.read_csv(\"https://raw.githubusercontent.com/sdrogers/fcmlcode/master/R/data/olympics/male100.csv\", names = [\"year\",\"time\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>year</th>\n",
       "      <th>time</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1896</td>\n",
       "      <td>12.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1900</td>\n",
       "      <td>11.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1904</td>\n",
       "      <td>11.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1906</td>\n",
       "      <td>11.20</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1908</td>\n",
       "      <td>10.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1912</td>\n",
       "      <td>10.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1920</td>\n",
       "      <td>10.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>1924</td>\n",
       "      <td>10.60</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>1928</td>\n",
       "      <td>10.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>1932</td>\n",
       "      <td>10.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1936</td>\n",
       "      <td>10.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>1948</td>\n",
       "      <td>10.30</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>1952</td>\n",
       "      <td>10.40</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1956</td>\n",
       "      <td>10.50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>1960</td>\n",
       "      <td>10.20</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>1964</td>\n",
       "      <td>10.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>1968</td>\n",
       "      <td>9.95</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>1972</td>\n",
       "      <td>10.14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>1976</td>\n",
       "      <td>10.06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>1980</td>\n",
       "      <td>10.25</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>1984</td>\n",
       "      <td>9.99</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>1988</td>\n",
       "      <td>9.92</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>1992</td>\n",
       "      <td>9.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>1996</td>\n",
       "      <td>9.84</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>2000</td>\n",
       "      <td>9.87</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>2004</td>\n",
       "      <td>9.85</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>2008</td>\n",
       "      <td>9.69</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>2012</td>\n",
       "      <td>9.63</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    year   time\n",
       "0   1896  12.00\n",
       "1   1900  11.00\n",
       "2   1904  11.00\n",
       "3   1906  11.20\n",
       "4   1908  10.80\n",
       "5   1912  10.80\n",
       "6   1920  10.80\n",
       "7   1924  10.60\n",
       "8   1928  10.80\n",
       "9   1932  10.30\n",
       "10  1936  10.30\n",
       "11  1948  10.30\n",
       "12  1952  10.40\n",
       "13  1956  10.50\n",
       "14  1960  10.20\n",
       "15  1964  10.00\n",
       "16  1968   9.95\n",
       "17  1972  10.14\n",
       "18  1976  10.06\n",
       "19  1980  10.25\n",
       "20  1984   9.99\n",
       "21  1988   9.92\n",
       "22  1992   9.96\n",
       "23  1996   9.84\n",
       "24  2000   9.87\n",
       "25  2004   9.85\n",
       "26  2008   9.69\n",
       "27  2012   9.63"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "olympic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "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.scatter('year', 'time', data = olympic)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### With formula api"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                   time   R-squared:                       0.826\n",
      "Model:                            OLS   Adj. R-squared:                  0.819\n",
      "Method:                 Least Squares   F-statistic:                     123.1\n",
      "Date:                Fri, 04 Mar 2022   Prob (F-statistic):           2.34e-11\n",
      "Time:                        09:44:50   Log-Likelihood:                 2.6222\n",
      "No. Observations:                  28   AIC:                            -1.244\n",
      "Df Residuals:                      26   BIC:                             1.420\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "Intercept     36.3091      2.339     15.523      0.000      31.501      41.117\n",
      "year          -0.0133      0.001    -11.095      0.000      -0.016      -0.011\n",
      "==============================================================================\n",
      "Omnibus:                       25.913   Durbin-Watson:                   1.281\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               56.627\n",
      "Skew:                           1.827   Prob(JB):                     5.05e-13\n",
      "Kurtosis:                       8.932   Cond. No.                     1.06e+05\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 1.06e+05. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "olympic_model = smf.ols('time ~ year', data = olympic)\n",
    "results = olympic_model.fit()\n",
    "print(results.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MpMIhLPX7uGteCeMKVFFTpDMU9DIgnK5v4qXNVYQiUd7dd5Q0g/lTRhP0+7hpepEqaop0oEdBb2ZPALcB1c65mfG2kcB/ABOB/cBfOeeOJdh3P3ASaAaa2utEWwp6+eDIacoiUZ6JRKmsOUNBbiZ3zC4mGPAxq6QgZaZ2BuKnKPGmngb9NcAp4Netgv4R4Khz7mEzexAY4Zz7WoJ99wMB59zhrnRYQS9nNbc4/vj+YULhKKu2fkh9UwtTi4bGyi7MLaEwP7vf+jbQzouIt/V46sbMJgIvtAr6ncC1zrkqMxsHvO6cm5pgv/0o6KWX1NQ18vzGSkKRKBvLj5ORZlw3bQxBv4/rpo0hs4/LLqTatQsyuCVjHX2Rc64KIB72Y9rZzgG/NzMH/NI591gHnbwfuB9g/Pjx3eyWeFlBbiafvmICn75iArsPniQUibJ8XQWrtx2kMD+LRXNKWBrwMW3ssD7pT6pduyDSnmQvXL7KOVcZ/0Ow2sx2OOfWJtow/kfgMYgd0Se5XzLATSkaytcXXswDC6aydtchQuEoT769n8f/sI9ZJQUEAz7umF3M8LzkVdRM1WsXRNrq7mfdg/EpG+LfqxNt5JyrjH+vBp4FLuvm+4kklJmexg0XF/GL/+rnnWU38O3bptPU4vj2yq1c9v01fOGpdby+s5rmlt4/dnhgwVRy26wEys1M54EFH5nFFOlX3T2ifw64B3g4/n1l2w3MbAiQ5pw7GX/8CeC73e2oyPmMys/m3vmTuHf+JLZU1FAWiVXUfHFTFWOH5XDXvBKW+n1MHt07FTXPnnDVqhtJdZ1ZdfM0cC1QCBwEHgJWAP8JjAcOAEHn3FEzKwYed84tNLPJxI7iIfYH5Snn3Pc70ymdjJXeUt/UzJrt1YTC5byx6xAtDgITRhAM+Lj1kmLys1V2QbxBF0yJAAdPnGH5ugpCkXL2HjpNbmY6t8waS9BfyuWTRpKWZloXLwOWgl6kFecc6w4cpyxSzvMbqzhV30TpyFxmFBfw2o5q6pta/ryt1sXLQKGgF2lHXUMzL2+tIhSO8sf3jyTcRuviZSDoKOh1Y08Z1HKz0lk818dTn7+i3W0qjteRigdEIp2loBeJK+lg/fuNP36D//P6+xw8caYPeyTSOxT0InGJ1sXnZKRx96WljMjL4kcv7+DKH67hv/3qXV7aXEV9U3M7rySSWrS2TCTufOvi9x46RVm87MLf/mYdw/MyY2UX/D5mlhT0Z9dFOqSTsSJd1NzieHP3IUKRKKu3HqShuYWLxw0j6I9V1Bw5JHllF0Tao1U30mNdXV+eSuvRk9mX47UNPLexklA4yuaKGjLTjRumFREM+Pj4RaPJ6OOKmjJ4KeilR7padz2V6rT3ZV92fHiCUDjKivUVHDndwOih2dw1t4RgwMeFY4b26nuJtKWglx7pat31VKrT3h99aWxu4dUd1ZRFory2o5qmFsfs0uEE/T5un11MQW5mUt5XBrdk1KOXQaSrdddTqU57f/QlMz2NBTPGsmDGWA6drGflhgpC4SjfXLGF772wjQUzxhIM+LjqgkLS0lLjlojibQp6Oa+u1l1PpTrt/d2X0UOz+dzVk7lv/iQ2V9QQCkdZuaGC5zZWUlyQwxK/j6V+HxNGDemT/vSFVDo/IzE6UyTn1dW666lUpz1V+mJmXOIbzvcWzeTdb9zIP39yLhcWDeXnr+3h44++zl/98m1C4XJO1zf1ab9629lzIhXH63DEripetnwzK9ZX9HfXBjXN0UunaNVNclTV1MUqaobL2X+klrysdG6dNY5goJRLJ47AbGBN7aTS+ZnBRidjRVKcc47wB8cIhct5cVMVpxuamTgqj6V+H3fN8w2Y2xNOevBFEiWKAfsevrWvuzOo6GSsSAro6JOFmXHpxJFcOnEk37ljBi9t/pBQuJx/+P0u/nH1LuZfWEgwUMonpheR02YqKpX09zkRSUxH9CJ9oLvr+Q8cqaVsXZRnIlEqjtcxLCeD22cXEwyUMttXkHJTO6l0DcVgo6kbkX7W07nrlhbH23uPEAqX87stH1Lf1MJFRfks9ftYPNfH6KHZyeh2t6TyOREvU9CL9LPenLs+caaRFzZWEYqUs/7AcdLTjOumjmapv5Trp40hK0OL6QYjzdGL9LPenLselpPJpy4fz6cuH8+e6pOE4hU1X9lezcghWSyaEyu7cPG4Yb3RdfEAHdGL9IFkz103NbewdvchQuEor2w/SGOzY2bJMIL+Uu6cU8zwvHMramp6xXs0dSOSAvoqXI+ebvhz2YVtVSfISk/jpulFLA34uGbKaJ7fWKkTph6koBcZpLZW/qXswrHaRoqGZXO6vplTCa7A1UVNA5vm6EUGqRnFBcy4o4BlC6fx6vZqQpEor+6oTrhtfxSdk76h0/Mig0B2Rjq3zBrHE5+9lLHDchJuM2pIFi0tqfcJX3pOQS8yyDx4y7SPFHoDOHy6gWsefY2frN5F+dHafuiZJIumbkQGmUQ3Qf/SDVPIzkwjFI7ys1d3809rdnPl5FEEAz5umTmO3KzULbsg56eTsSJyjuixWpavq6AsEuXA0VqyM9JITzNqG5opLsjhqzdP0+qcFNTRyVhN3YjIOXwj8vifN0zh9a9cyxevu5CmZkdtQ2wpZmXNGb4S2siTf9zfv52ULlHQi0hCaWnGs+sraG7zqb+pxfHQc1v57K/e5cVNVdQ3NbfzCpIqNEcvIu3qaMnljqqTfOGpdRTkZnLnnGKC/lJmlgxLuYqaoqAXkQ60V6OnZHgua796HX/Yc5hQuJzfvlfOr9/+gGljh8YrapYwKj91KmoOduc9GWtmTwC3AdXOuZnxtpHAfwATgf3AXznnjiXY92bgn4B04HHn3MOd6ZROxoqkRj2aztboqalt5LmNFYQiUTZFa8hIM66fNoZgoJRrp44mM12zxMnWoxIIZnYNcAr4daugfwQ46px72MweBEY4577WZr90YBdwExAF3gM+6Zzbdr4OK+hlsEulG3h09Q/Ozg9PUhYp59n1FRw+1UBhfjaL58ZulnJR0dAev74k1uNaN2Y2EXihVdDvBK51zlWZ2Tjgdefc1Db7XAl8xzm3IP7zMgDn3A/P934KehnsvHCT7cbmFl7feYhQuJxXd1TT1OKY7StgaaCUOy4ppiAvM6X+oA10yah1U+ScqwKIh/2YBNuUAOWtfo4Cl3fQyfuB+wHGjx/fzW6JeEN7J0EHUj2azHjVzJumF3H4VD0r1sfW5n9rxRa+98I2FswYyx/3HD4n5AHqGpt5dNXOXgl6fVqISebJ2ESn3tv9+OCcewx4DGJH9MnqlMhA0NUblaR6oBXmZ/O5qydz3/xJbKk4QShSzsoNldTUNSbcvjf+oLX9tFBxvI5lyzcDpNR/m77Q3TMkB+NTNsS/JyqHFwVKW/3sAyq7+X4ig8oDC6Z+pB5NbmY6DyyY+pFtzwZaxfE6HH8JtBXrK/qot51nZszyFfDdO2fyp6/fwIi8zITbjS1IXHitKx5dtbPdTwuDTXeD/jngnvjje4CVCbZ5D5hiZpPMLAu4O76fiJzHorkl/PCuWZQMz8WIzc23N289UAMtJzOdh26fkbDA2pHTDXwltJE/7T1Cd8u0eGH6q7ecd+rGzJ4GrgUKzSwKPAQ8DPynmd0HHACC8W2LiS2jXOicazKzLwKriC2vfMI5tzU5wxDxnkVzSzo1xTCQA61tgbVxBTks9fs4eKKeFzZVUhaJMmFUHkvn+bjL7+O9fUc7PUXVm/fpHehU1ExkgPPCCp1Eahua+N3mDymLRHl77xEMMIPWJfM7WqEz2Fb0qKiZiId1ZT5/IMnLymCJ38fT91/Bm1+9jvzsDNreF6WusZlHXt6RcP+uTH95nUogiAxwierLp9qqm54qHZmX8D63EKuo+cs33mfxvBLGDD33JG5np7+8TlM3IjIgtDdFlZWeRkNzC+lpxrUXjSYY8HH9tCKyMpI7YZFqS1p1c3ARGfAeWDC13Tn3mSUFlEWiLF8XZc2OakYOyfpzRc3pxcN6vS9dXaPf338UdEQvIgPG+QKzqbmFN3cfJhQp55Vt1TQ0tzCjeBhBv48755QwYkhWt1+7ta6cAO+rk8I9rnXT1xT0ItJTx043sHJDrKLm1soTZKWnceP0MQT9pVw9pZCMVhU1uxrGkx58MeFl/gbse/jWc9r6alWUpm5EZNAZMSSLz141ic9eNYltlX8pu/DS5g8ZMzSbxfNKCPpLuXBMfocXnSUK+q6s0U+F6xy0vFJEPG968TAeun0G7yy7gV982s8lvgIef3MfN/74DRb/y1sJQxvaD+OuLGlt7wKtvrxwS0f0IjJoZGWkcfPMsdw8cyzVJ8+wYn0FoXC03e3bC+OuLGlt7yRyX17noDl6ERnUnHP8bM1ufvbqHppbXZGVkWYsu2Ua9109ucfv0RerbnQyVkTkPFasr+CRl3dQWXOG7Iw0GppacMAVk0cS9Jdyy6yx5GWl7iSIgl5EpIsqjtexPBKlbF2UD47Ukp+dwa2zxhEM+PBPGIFZoltu9B8FvYhINznneHffUUKRKC9trqK2oZnJhUNY4vexZJ6vV2rn9wYFvYhILzhd38SLm6soi0R5d99R0gzmTxlN0O/jpulF5CSord9XFPQiIr1s/+HTPLMuyjORKJU1ZyjIzeSO2cUEAz5mlRT0+dSOgl5EJEmaWxx/fP8woXCUl7d+SENTC1OLhhIM+Fg0t4TC/Ow+6YeCXkSkD9TUNfL8xkpCkSgby4+TkWZcN20MQb+P66aNITO9c9eodmc5poJeRKSP7Tp4Ml5Rs4LDp+opzM9i0ZwSgoFSpo4d2u5+3S2CpqAXEeknjc0tvLHzEKFIOWu2V9PU4rjEV0DQ7+OO2SUU5GWes313i6CpqJmISD/JTE/jxulF3Di9iCOn6lmxoZJQuJxvrdzK917cziemFxEMlDL/wkLS0ywpRdAU9CIifWRUfjb3zZ/EvVdNZGvlCULhclZurOSFTVWMHZbDXfNKGDM0m4Mn6z+yb0+KoCnoRUT6mJkxs6SAmSUFfP3Wi1mzvZpQuJxfvPE+LQ7SjHNuhN5REbSzJ26zxl7ob+/9FPQiIv0oOyOdhbPGsXDWOA6eOMPydRX86q19VMeP6nMz07l3/kTumF38kX0TnbhNRCdjRURSjHOOdQeOUxYp5/mNVZyqb6J0ZC5L55WyxF+Cb0QecO6J26onv0x91e6EV2npiF5EJMWYGf4JI/BPGMG3b5vBy1urCIWj/OSVXfx0zS4+dsEogv7Sdm+Y0paCXkQkheVmpbN4ro/Fc32UH63lmXVRyiJRvvwfGzBIeO/atnQrQRGRAaJ0ZB5fvvEi1j5wHU99/nL8E0Z0aj8FvYjIAJOWZnzsgkLK/uZjPHzXLIbnZna4vU7Gioh4QEdXxuqIXkTE4xT0IiIep6AXEfG4HgW9mX3JzLaY2VYz+3KC5681sxoz2xD/+nZP3k9ERLqu2+vozWwm8HngMqABeNnMXnTO7W6z6ZvOudt60EcREemBnhzRXwy845yrdc41AW8Ai3unWyIi0lt6EvRbgGvMbJSZ5QELgdIE211pZhvN7HdmNqO9FzOz+80sbGbhQ4cO9aBbIiLSWrenbpxz283sR8Bq4BSwEWhqs9k6YIJz7pSZLQRWAFPaeb3HgMcgto6+u/0SEZFz9doFU2b2AyDqnPuXDrbZDwScc4fP81qHgA86+daFQIev5xEap3cMhjGCxtnXJjjnRid6okdFzcxsjHOu2szGA3cBV7Z5fixw0DnnzOwyYlNFR873uu11tp0+hNu7GsxLNE7vGAxjBI0zlfS0euUzZjYKaAS+4Jw7ZmZ/DeCc+wWwFPgbM2sC6oC7XSrWXBAR8bAeBb1z7uoEbb9o9fjnwM978h4iItIzXrgy9rH+7kAf0Ti9YzCMETTOlJGS1StFRKT3eOGIXkREOqCgFxHxuJQLejN7wsyqzWxLq7bZZva2mW02s+fNbFir55aZ2R4z22lmC1q1++Pb7zGzn5lZwruj95eujNPMbjKzSLw9YmbXt9rHM+Ns9fx4MztlZl9p1eapcZrZJfHntsafz4m3p+w4u/g7m2lmT8bbt5vZslb7pOwYAcys1Mxei/d7q5l9Kd4+0sxWm9nu+PcRrfZJ7RxyzqXUF3ANMA/Y0qrtPeDj8cf3At+LP55O7IrcbGAS8D6QHn/uXWLr+g34HXBLf4+tB+OcCxTHH88EKlrt45lxtnr+GSAEfMWL4yS22m0TMDv+86iB8HvbxTF+Cvht/HEesB+YmOpjjPdvHDAv/ngosCueNY8AD8bbHwR+FH+c8jmUckf0zrm1wNE2zVOBtfHHq4El8cd3EvtlqnfO7QP2AJeZ2ThgmHPubRf7r/1rYFHSO98FXRmnc269c64y3r4VyDGzbK+NE8DMFgF7iY3zbJvXxvkJYJNzbmN83yPOueZUH2cXx+iAIWaWAeQSq3B7ItXHCOCcq3LOrYs/PglsB0qI5c2T8c2e5C/9TvkcSrmgb8cW4I744yB/KZ5WApS32i4abyuJP27bnuraG2drS4D1zrl6PDZOMxsCfA34+zbbe2qcwEWAM7NVZrbOzL4abx+I42xvjGXAaaAKOAD8g3PuKANsjGY2kdgn6j8BRc65Koj9MQDGxDdL+RwaKEF/L/AFM4sQ+yjVEG9PNN/lOmhPde2NEwCLVf/8EfDfzzYleI2BPM6/B37inDvVZnuvjTMDmA/8l/j3xWZ2AwNznO2N8TKgGSgmNp3xv8xsMgNojGaWT2wa8cvOuRMdbZqgLaVyqKclEPqEc24HsY+7mNlFwK3xp6Kce9TrAyrj7b4E7Smtg3FiZj7gWeAzzrn3481eG+flwFIzewQYDrSY2Rli/9i8NM4o8IaLF/czs5eIzX3/OwNsnB2M8VPAy865RqDazN4CAsCbDIAxmlkmsd+73zjnlsebD5rZOOdcVXxapjrenvI5NCCO6M1sTPx7GvBN4GyZheeAu+Pz1ZOIlUB+N/6x6qSZXRE/y/0ZYGU/dL1L2hunmQ0HXgSWOefeOru918bpnLvaOTfROTcR+CnwA+fcz702TmAVcImZ5cXnsD8ObBuI4+xgjAeA6y1mCHAFsGMgjDHer/8LbHfO/bjVU88B98Qf38Nf+p36OdQfZ4A7+gKeJjav10jsL+J9wJeInfneBTxM/Ire+PbfIHaWeyetzmgTO3rYEn/u5633SYWvroyT2D+g08CGVl9jvDbONvt9h3NX3XhqnMCniZ1w3gI8MhDG2cXf2XxiK6e2AtuABwbCGOP9m09simVTq39vC4mtjloD7I5/H9lqn5TOIZVAEBHxuAExdSMiIt2noBcR8TgFvYiIxynoRUQ8TkEvIuJxCnoREY9T0IuIeNz/Bzld4U0F6IkgAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sm.graphics.abline_plot(model_results = results)\n",
    "plt.scatter(olympic.year, olympic.time)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### With statsmodels api"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                   time   R-squared:                       0.826\n",
      "Model:                            OLS   Adj. R-squared:                  0.819\n",
      "Method:                 Least Squares   F-statistic:                     123.1\n",
      "Date:                Fri, 04 Mar 2022   Prob (F-statistic):           2.34e-11\n",
      "Time:                        09:44:50   Log-Likelihood:                 2.6222\n",
      "No. Observations:                  28   AIC:                            -1.244\n",
      "Df Residuals:                      26   BIC:                             1.420\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const         36.3091      2.339     15.523      0.000      31.501      41.117\n",
      "year          -0.0133      0.001    -11.095      0.000      -0.016      -0.011\n",
      "==============================================================================\n",
      "Omnibus:                       25.913   Durbin-Watson:                   1.281\n",
      "Prob(Omnibus):                  0.000   Jarque-Bera (JB):               56.627\n",
      "Skew:                           1.827   Prob(JB):                     5.05e-13\n",
      "Kurtosis:                       8.932   Cond. No.                     1.06e+05\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 1.06e+05. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "array_year = sm.add_constant(olympic.year)\n",
    "mod = sm.OLS(olympic.time, array_year)\n",
    "res = mod.fit()\n",
    "print(res.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic Regression Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "# fictional dataset showing potential relationship between age and congestive heart disease.\n",
    "chd = pd.read_csv(\"chdage.csv\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the variable CHD is recorded as 0 or 1.\n",
    "# pd.Categorical makes the data categorical rather than integer\n",
    "chd.CHD = pd.Categorical(chd.CHD)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "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>AGE</th>\n",
       "      <th>CHD</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>20</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>23</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>24</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>25</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>25</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>63</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>64</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>64</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>65</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>69</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>100 rows × 2 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    AGE CHD\n",
       "0    20   0\n",
       "1    23   0\n",
       "2    24   0\n",
       "3    25   1\n",
       "4    25   0\n",
       "..  ...  ..\n",
       "95   63   1\n",
       "96   64   1\n",
       "97   64   0\n",
       "98   65   1\n",
       "99   69   1\n",
       "\n",
       "[100 rows x 2 columns]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "AGE       int64\n",
       "CHD    category\n",
       "dtype: object"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# change in data type is reflected here.\n",
    "chd.dtypes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x19c3c453f48>"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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.scatter(chd.AGE, chd.CHD, alpha = 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='AGE', ylabel='CHD'>"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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CuWxiT4b6xr3DDPzg6F7065TIlP4dmTm8dj9O6Z/BzOFZdEqK4dbj+xMZ7uru0zGBa6f2AeA3pw4m0bdfU+MiayZ4bz6uX01IRkWE8fMZA0mMieT8I3vUXGFjjBuqG9UjjSN7deC8Md2pHlof2yuN88Z0Izk2kp/PyCHKF/Ld0+L40bH9Gt3mBysnK4krJvWs2a/DuqVw6fieIa2jxcbgAYwx2cDrXhyDr7Yyt4Ck2Ig6ZxSHw5odBcREhtd7ZhsK1lqWbysgIzH6gMf3t+eXsKeojAGdA8fF80vK2bi7iP6dEw/4qoYVufmkxkXRKegsfMPuQqosNSHZlP2lFazfWUjfTgkBV6tUVFaxIreArqmxpMQFno2u2l5AQnQEWUGfWjbvKaKkvKrOZY15hWVs3VtMTmZSwK32xWWVrNmxnz4dE4iNqq27qsqyPDefzkkxdb6nZc2O/URHhNU5s922r5iCkgr6dUrkQJRWVLIqdz89M+JrDnJQu1/TE6PomBi4bdfvKiTMUGdCfUd+CbsLyxjQObFmotFayysLtvDV+jxOGZbFBL+D0xfrdvPqgi2M6pHKGSO7BmyTFbn5pMRG1Tlh2Li7iIqqKnplBG7bXftL2ZFfysDMxIBJzsJSd/9A8H6trLIs35ZPl5RYUoM+ZazeXkBcdESdT6Nb9hZTXFZBn46B23ZPYRlb9hYzMDOpxb76IHdfCfuKywMmlw9GY2PwCngRaZZf/XcpT/pN5P7+jCGcc0R3Xvl6Mz96oXYi97wx3bnn9CGt0cR2oVUmWY0xzwGfA/2NMZuNMZe3VF0icngVlVXw7FcbA8r+Nmc9AI/7/q/2wrxN5JcETibK4dFi18Fba89rqXWLSOsyGMKMAWpHACJ8QxjBQxlhBvS9jq2j1SdZRaTtiY1yk6bVjHH3NwABk+0AF4/LJvEw3B8idbXoGPzB0hi8SNvy6ZpdfLM1nwl90gPu5F6ZW8Cc1Tvp3zmxybuN5dA0Ngav74MXkWab0Cc94OqZav07Jzb7qhAJHQ3RiIh4lAJeRMSjFPAiIh6lgBcR8SgFvIiIRyngRUQ8SgEvIuJRCngREY9SwIuIeJQCXkTEoxTwIiIepYAXEfEoBbyIiEcp4EVEPEoBLyLiUQp4ERGPUsCLiHiUAl5ExKMU8CIiHqWAFxHxKAW8iIhHKeBFRDxKAS8i4lEKeBERj1LAi4h4lAJeRMSjFPAiIh6lgBcR8SgFvIiIRyngRUQ8SgEvIuJRCngREY9SwIuIeJQCXkTEoxTwIiIepYAXEfEoBbyIiEdFNPUEY0wEcAIwwFe0HHjbWlvRkg0TEZFD0+gZvDEmC1gG3AxkAV2AW4FlvsdEROQ7qqkz+N8Cj1hrH/AvNMZcD9wDXNxC7RIRkUPUVMCPtdZeElxorf2TMWZlyzRJRERCoalJ1uJGHisKZUNERCS0mjqDTzbGnF5PuQGSWqA9IiISIk0F/EfAyQ089nGI2yIiIiHUaMBbay89XA0REZHQajTgjTE/auxxa+19oW2OiIiESlOTrIl+/24JWk5s2aaJiMihaGqI5s7qn40xp/ovi4jId9vBfBeNbbFWiIhIyOnLxkREPKqpSdYl1J659zHGLPZ/3Fo7tKUaJiIih6ap6+BPBzoBm4LKewBbW6RFIiISEk0N0dwP5FtrN/j/w31Nwf0t3zwREWmupgI+21q7OLjQWjsPyG6RFomISEg0FfAxjTwWG8qGiIhIaDUV8HONMVcEFxpjLgfmt0yTREQkFJqaZL0R+I8x5gJqA300EAWc1oLtEhGRQ9TUnazbgfHGmCnAYF/xG9baD1q8ZSIickia/KPbANba2cDsFm6LiIiEkO5kFRHxKAW8iIhHKeBFRDxKAS8i4lEKeBERj1LAi4h4lAJeRMSjFPAiIh6lgBcR8SgFvIiIRyngRUQ8SgEvIuJRCngREY9SwIuIeJQCXkTEoxTwIiIepYAXEfEoBbyIiEcp4EVEPEoBLyLiUQp4ERGPUsCLiHiUAl5ExKMU8CIiHqWAFxHxKAW8iIhHKeBFRDwqorUbIM2Utw4W/Rui4mD4hRDfIfR1bF0I37wKyV1h2PmuLhFpMxTwbdHutfDo0VBW4Jbn/R2u+jy0AbzuQ3j6dLCVbnnpK3Dpm6Fbv4i0OA3RtEUL/lUb7gB7voWVIQ7fr/5WG+4AGz6FbYtDW4eItCgFfFsUEV1PWcxhqKOeMhH5zlLAt0UjL4aEzrXLWSOg3/GhrWP8dRAZX7uccypk9A9tHSLSohTwLWXzPPj2U6iqOrDnF2yHVe/A/p1NPzcpE676DMbfAFN/CZe9A+GRTb8ubx2sfg9K9weWF+6GVe9C/rbasqwR8MNPYcyVcNL9cOY/ah+rqoINn8Gmr+rWkbsE1s6GirLA8r0bXR3FewPLi/e68r2bmm5/U3asgDWzoLwksDx/q6ujKC+wvLTAbY+89YHllRWw7iM3yXyo9u90de/fcWDPr6qCbz+BzfMPve7WVLLPt183tnZL2jVNsoZaZQU8ezasneWWM4fBxa9DTFLDr1n2H3jlSqgsg/BoOOsfMOCkhp9fvAeePBl2LHPLW+bBOc9AWCPH6zn3way7AAuxqXDRq5A13P0SvvA9qCiGsAg49REYerabyP3nSVDgC/28tXD83VBW6Ore4gugnkfBha+4A8yr18DCf7nylB5w2duQlOXG89+6DWwVRCXCBS9Aj/EuyJ49B8r2gwmHE/8AR1x+ABu5Hm/dDl8+4n5OzIJL34C0XrDwOXjtWqiqgIhYOPdf0OcY2PI1PH0alOwFDBx7J0y4AQp3wT9OgF2r3LoGnQZn/bN5bfrmNXj5ct9+jYIzHoecmQ0/v2Sf2+a5S9xy3+PgvOchLLx59beWjV/Av85080QmDKb/Ho68srVb1S7pDD7UVr9TG+4A2xbBwmcbf807P3chAFBZCu/+ovHnz3+yNtzBTbCu+6Dh5xfvgQ/vAWzQMvDeL124gwvBd34G1sIn99WGO8DnD7nJ3MX/rg13gPUfw4rXff38V2353g3wxcPubPr9O124g/uln/Vr9/Osu1y4g5vQff9OqChtvO/1yVtXG+4ABVvhkwfc2fC7P3f9AtfP9+5wP8/+rS/ccdtl9m9dwH71t9pwB3fw3fDZwbcJXN01+7XMLTdmwb9qwx1g9buw5v3m1d2aZv269iIAWwWz7oTy4tZtUzulgA+1wnqGWAqb+Hge/Jr61tHU440N7ZTk1wZNzfN31L+u4jwXiHXWZ91QTn317N/ZcHl5UeAVP1C7PYLrLs1vXhAU7qqnbCdUlbuDWb11B+2TihI3ZFPvtj3A4ZWm2tXU8Ft99TS37tYUvG3L9kNZUeu0pZ1TwIda/xMhJrl2OTwKBp/Z+GuGnRu0fF7jzx9ylhtOqRab1vgka2oP6D4+sGz4+fXXPeQsN9wyPKgNGQPcuPzg0wOv2IlKhIEzoOckSO7m9wIDw86BuDToNz1wXdX9C+5n/xMhNqXhfjSkyyhI7xdUx7nuqp/BZzRQ9/mB5dmT3A1dQ892wwrV4ju6IZ3mCN62wds02JAzIcxvLiUmBfqf0Ly6W1Pwfu17fMvciCdNMtballu5MdOBB4Fw4HFr7e8ae/7o0aPtvHnzDr6i8hIXpI2NQR+osqL6bxgqK4LIWDAmsLyi1IWt/zjpjhXwxSNuSOCIK6DbEY3XWVEGXz0KGz53QXnEFRDuF+DlxS5U/eve+AXMfQKi4mHctZDep/E6ivfC53+BnStg4CkuyACqKt161n0I3cbA2KtqL4dc/joseQGSu8OE6yGhoyvf8jV89Zjr99iroVOOK9+7CT7/sztTHXEh9Jnmykv3uyGe3MXuQDTiItcXa+HrJ91EZ+ZwGHe1609Nv0vcwSZ4DLqy3P3vP7FcsB0++xPs2wxDz4EBJ9au44uH3aR37ykw+vLa98mi593wUsYAd9VQ9YF5/RyY/083VzHuGkjr6Vd3hRt2iIgKbFNVlRtei4wNbOdXj7nJ9h7j4cgfBLa5vNjNufi/bzd9BXMfd+sZew1k+B24qip9cwlBl6ta69bV0Ps2FDfAVVW6/kQewOW41rrhplVvQ+ch7v0ZnVD7eEP7tcG6q9wn0AOpu7kqK9wwYRu8FNgYM99aO7q+x1psktUYEw48BBwLbAbmGmNes9Z+E7JKyorgv9e42+njOsBxd7uzxubYMh/+80M3/tr1CDjjCXfmW7DdTZR9OwdSs+GUv7gQriiD1290Y9LRiTDtlzD6MreulW/C0pfdmzI2zZ1hNnbw2b3Ghc32pbBvE/Se5n6xi/LglSvcOGxSV5hxP/Q7zv2yLXnR9TsiBjr0gfRrG+/f1q/dVxvs2+gCt/dUiE934+qLnoWtC9xYdu+pkDnUDVcseg5WvAEJnaDrKDfhaC0se8WNTZtwd9ZeHfDrPoQlL7mDSVgEZE90vzC5S2Dx8279RXmuf8ldIH+LmwTd9AXsWO4mbLsf6QLgtWvd3bOxqXDcr2s/ccy+Bz77swvZI6+EY+9y5Rs+gcUvQNEu91jPo1yo7Fzhtu2ulbA/1x100nq5g9DCZ2H9R27+IHsi9JrsQmzxv922jUpwl4aOucLV8emf4OM/uOGcUZe4ycOwMFj5NrzxI3e1Tt9j4fTHXLt3r3XbMHeJu5qkzzToONBtn1eudGPsiZkw4z53pl5V5bbfslfdCUtqz9qAn/u4b2x7v/tkMOMBF5LrPoLXrnNzHj2Pcu/bhI7uyqCXL3fv6/T+cPqj7hNYcyx8zs0fFO9x74GZfwk8kAUr2Ob6veFT2L7MtavHeHcy9Np1ro8xyW5ie+T3Gq/7m//Cm7fB/u3uwoPT/up+30Lp84fgw9+7E7IRF8KJf2x7E9sNaLEzeGPMOOAOa+3xvuWfAFhr72noNQd9Bv/RvTD77trlsEi4aRkkdjq4xloLfxntgrZa3+Pgghfh5SvcWWy1hM6ujrmPw9s/9luJgeu/duPUTwR9pD/z73WHCvw9NtkFbLVuY+Hyd+CNm1091WKS4UcrXPi8elXgOn4wxwVzfSpK4b6BULS7tmzERe4X9cmT3URptY45cPXnbsLzk/tqyyNi4EfL3YTjvy8IXP8lb7irZh4cFnj367F3ubO3B4e5A1e1nJlw9lPw/AXuDLpaSne4fhF8+oCbmKtmwuGmpbBrNTx1SmDd5z4H3cfC/w1wZ9DVJt0C034BDx3pQr5arynwvVfh1ath4TO15fEZcNM3sOApt939XTvPBetjkwPLT3vMhc59A938QbUxP4AT74XHj4HNc2vLu4yCKz4IvOIH3DDXzcvdgeKV7wfWccUHEJ3s3p/4/a6e+Ed3kLkvJ3DMe9h5LgSfPj1wsj+9H1zr15YDlb8NHhhcO1ENMPUXcNQtDb/mxUvcCUC1pK5w42L3Scp/otmEwQ2L3H6vT/Ee+L+BtRcBAEy40R0YQiV3Cfx1YmDZzIdc0LcRjZ3Bt+QYfBfA/+Lmzb6yAMaYK40x84wx83buPIBrwP0FX6dcVe7Ogg9WaUFguPuve1tQHftz3RlKcDnW3cpfp7yedjb1ePU6/EMf3FUee9bXv7766q22d1NguAfUsSiwfMc37tNJcN0VJbBzZd1ycO3JXRIY7tXlRbsDw726HNyZc0A7N7pf6uC+2ErIXVp/H7ctdG32D/fq8vLiwHCvLoe6/SjcCfmbG9i2ixre5nnrAsM9oI6g11QvB9ddVuDO9hvattsWEhDu1eX5W+pOaDb0vt21qnkTnduXBoZ7fesOFtyP/M1u+wZvD1sVeNVQsJ2rAsP9QOo+WPXt11Dc//Ad0ZIBb+opq/NxwVr7mLV2tLV2dEZGxsHV0HNS4HJkvDtLOlgxSW4MOGDdR7n/s4PqSOvlJuOCy8Oj3JlkjwnU6Xr1uhqSHXQG0VDdCZ3dmVhwv024r94GpPUMmgAFsn11BK+r21g3vhxcHpPsPiEEl2NcWbcxdb8uoedRbrggY0BQuW8dwf3rmOMm44LLI2Kh6+i65dV1ZA5zZ7nB5ZGx0GV03fL66k7pASnZdfdVWITbr9kTAydfq9eRMcBNxAaX+/czuO7g8vgMN3RTZ7+GuXq7jwucfK1eV3J3N2wYUN7Atu0yqnlj8V1GQWTQ63oe3fhrgutO7w+Jnev2Lzwauo5peD2dh7iJ5sbWfah6jHe/P/6a+n1tQ1oy4DcD/qnSFdga0hrGXOluTkno7H7Jz3u2eVdhgLu5qNcUN2Y+6HR30w24j4PDzndj/D0muhuKjHFjwkff7m6q6TgIzn7avYk75bgx2A593S/g8b9147KNOe1Rd6VJbBoMmAGn/NmVT74dRl3qAqDbka5/4ZEw8GQ45g4X2un93Q00HXo3vP6wcDj3GXclTVy6G56Z+jP32IwH3KRrbBr0ORbO+JsrH3+9m0CN7whZI90NN1Hxbpz6hD+4QEzr7T7OZg5z4/nnPAOdhrhx5Uk3u69UADcc0/Motw2HnuO2CcD0e9xVO3EdXGic/ZQrH305TLzJ7dfOQ1zb49Kgy0g3B5LWy9V/4h9dAEYnum2TNcLNF4y9xv0DOPMJN+Yfm+aGhk7yDTsd8yv3MTwu3W2Xc303ig09G6b8DJK6uAPO2U+5A3p6Xzj9b+4Am9zNDT8NONEdDM973gVVfIZr+1G3ujpmPuy7MigN+p0Apz7syo+6FY74vnt+1yPc6yOi3Tj8cb9x75v0fq6+jP5uvuLsp1x7ErNg8k9cO8PC3DbvMcH1Y/iFMO1Xro6T7nNfLxGb5vp/xhONvwcbEpfmtk3noW5/TLixdq6pIcffDUPPdfs1e1Ltfh3xPTd0lpjp3ifnPgMJjZzURcW5bdNllHsfHnmV+30PpQ693Xskvb8bSpr2K8g5penXtREtOQYfAawCpgFbgLnA+dbaZQ29ptlX0bRHS1/xXUUT58K0+9jQ1/HFX2HpSy7gJv808IqOtqIoDz74jRs26DnJHZT1vfbiIa1yFY21tsIYcy3wDu4yyb83Fu5yENZ/DC9d6rc8B65f4L6jJlTm/b12EnnzXHf53vUL614e+F33ypWw5j3389av3RUsp/ypVZskcri06I1O1to3rbX9rLW9rbV3N/0KOSArgr77vaI49Le0L389cDl/S/2TgN9l5SV1t8uK1+t/rogH6U7WtiitV92yDk3c6HSwgsf0wyLcfQFtSUS0G17yl9bIXIWIxyjg26KRF9XePm/C3GRzj3GhrWPSLW6CE9zVDsfe5SaR2xJj3CRy9ZUYiZlwQqM3U4t4Sot+VcHB0iTrQchb5yZBY5Lc1QUt9V0fO1e5K2Ti0lpm/YdDebG7szO974F9b75IG9Iqk6zSgoL/6PaSF0P/R7ertcUrZ4JFxtZ+nYJIO6IhmrbocPzRbRFp8xTwbdHh+KPbItLmKeDbosPxR7dFpM3TGHxblJQJ13zhvso3Ms59o6EmD0UkiAK+rYpNbVNfaSoih5+GaEREPEoBLyLiUQp4ERGPUsCLiHiUAl5ExKMU8CIiHqWAFxHxKAW8iIhHKeBFRDxKAS8i4lEKeBERj1LAi4h4lAJeRMSjFPAiIh6lgBcR8SgFvIiIRyngRUQ8SgEvIuJRCngREY9SwIuIeJQCXkTEoxTwIiIepYAXEfEoBbyIiEcp4EVEPEoBLyLiUQp4ERGPUsCLiHiUAl5ExKMU8CIiHqWAFxHxKAW8iIhHKeBFRDxKAS8i4lEKeBERj1LAi4h4lAJeRMSjFPAiIh6lgBcR8SgFvIiIRxlrbWu3oYYxZiewoZkvTwd2hbA5bYX63b6o3+3LgfS7h7U2o74HvlMBfyiMMfOstaNbux2Hm/rdvqjf7cuh9ltDNCIiHqWAFxHxKC8F/GOt3YBWon63L+p3+3JI/fbMGLyIiATy0hm8iIj4UcCLiHhUmwt4Y0w3Y8xsY8xyY8wyY8wNvvI0Y8x7xpjVvv9TW7utoWSMiTHGfGWMWeTr952+ck/3u5oxJtwYs8AY87pvub30+1tjzBJjzEJjzDxfmef7boxJMca8ZIxZ4ftdH+f1fhtj+vv2c/W/fGPMjYfS7zYX8EAFcLO1diAwFrjGGJMD3A7Mstb2BWb5lr2kFJhqrR0GDAemG2PG4v1+V7sBWO633F76DTDFWjvc73ro9tD3B4G3rbUDgGG4fe/pfltrV/r283BgFFAE/IdD6be1tk3/A/4LHAusBDJ9ZZnAytZuWwv2OQ74GjiyPfQb6Op7Y08FXveVeb7fvr59C6QHlXm670ASsB7fRSDtpd9BfT0O+PRQ+90Wz+BrGGOygRHAl0Ana+02AN//HVuxaS3CN0yxENgBvGetbRf9Bh4AbgOq/MraQ78BLPCuMWa+MeZKX5nX+94L2An8wzcs97gxJh7v99vfucBzvp+b3e82G/DGmATgZeBGa21+a7fncLDWVlr38a0rMMYYM7iVm9TijDEzgB3W2vmt3ZZWMsFaOxI4ATcceVRrN+gwiABGAo9Ya0cAhXhsOKYxxpgo4BTgxUNdV5sMeGNMJC7cn7HWvuIr3m6MyfQ9nok7y/Uka+1e4ENgOt7v9wTgFGPMt8DzwFRjzL/wfr8BsNZu9f2/AzceOwbv930zsNn3CRXgJVzge73f1U4AvrbWbvctN7vfbS7gjTEGeAJYbq29z++h14CLfT9fjBub9wxjTIYxJsX3cyxwDLACj/fbWvsTa21Xa2027mPrB9baC/F4vwGMMfHGmMTqn3HjskvxeN+ttbnAJmNMf1/RNOAbPN5vP+dROzwDh9DvNncnqzFmIjAHWELtmOxPcePwLwDdgY3AWdbavFZpZAswxgwFngTCcQfmF6y1dxljOuDhfvszxkwGbrHWzmgP/TbG9MKdtYMbtnjWWnt3O+n7cOBxIApYB1yK732Pt/sdB2wCellr9/nKmr2/21zAi4jIgWlzQzQiInJgFPAiIh6lgBcR8SgFvIiIRyngRUQ8SgEvAhhjTjPGWGPMAL+yMcaYD33f4ve1MeYNY8wQ32N3GGO2BH37X0qrdUCkHrpMUgQwxryA+yKnWdbaO4wxnXD3Vpxvrf3M95yJuC/+etUYcwew31r7x1ZrtEgTdAYv7Z7ve40mAJfj7pYFuBZ4sjrcAay1n1hrXz38LRRpHgW8CJyK++7xVUCeMWYkMAj3lcyNuclveGZ2SzdS5GAp4EXcd3887/v5ed9yAGPMl76/LPSgX/H91vcHGqy1Uw5HQ0UORkRrN0CkNfm+52MqMNgYY3Hf9WNx3/szEt8XO1lrjzTGnAnMaK22ihwsncFLe3cm8JS1toe1Ntta2w3314TeBS4xxoz3e25cq7RQpJl0Bi/t3XnA74LKXgbOB84Bfm+M6YL7Du5dwF1+z7vJGHOh3/Kp1tpvW7CtIgdFl0mKiHiUhmhERDxKAS8i4lEKeBERj1LAi4h4lAJeRMSjFPAiIh6lgBcR8aj/B/sxkCNBRngvAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# seaborn swarm plot shows the density of values\n",
    "sns.swarmplot(x = \"AGE\", y = \"CHD\", data = chd, orient = \"horizontal\", order = [1,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<AxesSubplot:xlabel='AGE', ylabel='CHD'>"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "# shows potential logistic regression curve\n",
    "sns.regplot(x=\"AGE\", y=\"CHD\", data=chd, logistic=True, ci=None)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$y_i = 1 / \\exp( - (\\beta_0 + \\beta_1 x_i + \\epsilon_i) )$$\n",
    "\n",
    "$$\\mathbf{\\hat{Y}} =1 / \\exp( -\\mathbf{X} \\boldsymbol{\\beta})$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.536765\n",
      "         Iterations 6\n"
     ]
    }
   ],
   "source": [
    "array_age = sm.add_constant(chd.AGE)\n",
    "log_reg = sm.Logit(chd.CHD, array_age).fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                           Logit Regression Results                           \n",
      "==============================================================================\n",
      "Dep. Variable:                    CHD   No. Observations:                  100\n",
      "Model:                          Logit   Df Residuals:                       98\n",
      "Method:                           MLE   Df Model:                            1\n",
      "Date:                Fri, 04 Mar 2022   Pseudo R-squ.:                  0.2145\n",
      "Time:                        09:44:50   Log-Likelihood:                -53.677\n",
      "converged:                       True   LL-Null:                       -68.331\n",
      "Covariance Type:            nonrobust   LLR p-value:                 6.168e-08\n",
      "==============================================================================\n",
      "                 coef    std err          z      P>|z|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const         -5.3095      1.134     -4.683      0.000      -7.531      -3.088\n",
      "AGE            0.1109      0.024      4.610      0.000       0.064       0.158\n",
      "==============================================================================\n"
     ]
    }
   ],
   "source": [
    "print(log_reg.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "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.scatter(chd.AGE, chd.CHD, alpha = 0.5)\n",
    "plt.plot(chd.AGE, log_reg.predict(), color = \"orange\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}