3.6 - Inverse Probability Weighing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"width: 100%; overflow: hidden;\">\n",
    "    <div style=\"width: 150px; float: left;\"> <img src=\"https://raw.githubusercontent.com/DataForScience/Networks/master/data/D4Sci_logo_ball.png\" alt=\"Data For Science, Inc\" align=\"left\" border=\"0\" width=150px> </div>\n",
    "    <div style=\"float: left; margin-left: 10px;\"> <h1>Causal Inference In Statistics - A Primer</h1>\n",
    "        <h1>3.6 Inverse Probability Weighing</h1>\n",
    "        <p>Bruno Gonçalves<br/>\n",
    "        <a href=\"http://www.data4sci.com/\">www.data4sci.com</a><br/>\n",
    "            @bgoncalves, @data4sci</p></div>\n",
    "    <div style=\"float: right; margin-right:10px;\"> <p><a href=\"https://amzn.to/3gsFlkO\" target=_blank><img src='data/causality.jpeg' width='100px'>\n",
    "        <!--Amazon Affiliate Link--></a></p></div>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "from pprint import pprint\n",
    "\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "from CausalModel import CausalModel\n",
    "\n",
    "import watermark\n",
    "\n",
    "%load_ext watermark\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by print out the versions of the libraries we're using for future reference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.8.5\n",
      "IPython version      : 7.19.0\n",
      "\n",
      "Compiler    : Clang 10.0.0 \n",
      "OS          : Darwin\n",
      "Release     : 20.2.0\n",
      "Machine     : x86_64\n",
      "Processor   : i386\n",
      "CPU cores   : 16\n",
      "Architecture: 64bit\n",
      "\n",
      "Git hash: 6ba5a323994ecc2dbeee8be27219dbf5207f8dfb\n",
      "\n",
      "numpy     : 1.19.2\n",
      "matplotlib: 3.3.2\n",
      "watermark : 2.1.0\n",
      "pandas    : 1.1.3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%watermark -n -v -m -g -iv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load default figure style"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.style.use('./d4sci.mplstyle')\n",
    "colors = plt.rcParams['axes.prop_cycle'].by_key()['color']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We load the DAG from Fig 1.10 for ease of reference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = CausalModel('dags/Primer.Fig.1.10.dot')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 660x660 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1, figsize=(2.2,2.2))\n",
    "G.plot(ax=ax)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our example, the data for the joint probability $P\\left(X, Y, Z\\right)$ is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "PXYZ = pd.DataFrame({\n",
    "    'X':    ['Yes',  'Yes',    'Yes',  'Yes',    'No',   'No',     'No',   'No'],\n",
    "    'Y':    ['Yes',  'Yes',    'No',   'No',     'Yes',  'Yes',    'No',   'No'],\n",
    "    'Z':    ['Male', 'Female', 'Male', 'Female', 'Male', 'Female', 'Male', 'Female'],\n",
    "    'Prob': [0.116,   0.274,    0.009,   0.101,    0.334,  0.079,    0.051,  0.036]\n",
    "})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>Z</th>\n",
       "      <th>Prob</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Yes</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.116</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Yes</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.274</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Yes</td>\n",
       "      <td>No</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.009</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Yes</td>\n",
       "      <td>No</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.101</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>No</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.334</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>No</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.079</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>No</td>\n",
       "      <td>No</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.051</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>No</td>\n",
       "      <td>No</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.036</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     X    Y       Z   Prob\n",
       "0  Yes  Yes    Male  0.116\n",
       "1  Yes  Yes  Female  0.274\n",
       "2  Yes   No    Male  0.009\n",
       "3  Yes   No  Female  0.101\n",
       "4   No  Yes    Male  0.334\n",
       "5   No  Yes  Female  0.079\n",
       "6   No   No    Male  0.051\n",
       "7   No   No  Female  0.036"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PXYZ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We check that it all sums up to one"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PXYZ['Prob'].sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The conditional probability $P\\left(Y, Z | X\\right)$ for X=Yes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "PYZ_X = PXYZ[PXYZ['X'] == 'Yes'].copy()\n",
    "PYZ_X['Prob'] /= PYZ_X['Prob'].sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>Z</th>\n",
       "      <th>Prob</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Yes</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.232</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Yes</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.548</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Yes</td>\n",
       "      <td>No</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.018</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Yes</td>\n",
       "      <td>No</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.202</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     X    Y       Z   Prob\n",
       "0  Yes  Yes    Male  0.232\n",
       "1  Yes  Yes  Female  0.548\n",
       "2  Yes   No    Male  0.018\n",
       "3  Yes   No  Female  0.202"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PYZ_X.round(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These numbers still differ slightly from the corrected in the [online Errata](http://bayes.cs.ucla.edu/PRIMER/pearl-etal-2016-primer-errata-pages-july2020.pdf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we note that these values are just the corresponding values of $P\\left(X, Y, Z\\right)$ divided by $P\\left(X=\\mathrm{Yes}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "PX = PXYZ[['X', 'Prob']].groupby('X', as_index=False).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>X</th>\n",
       "      <th>Prob</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>No</td>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Yes</td>\n",
       "      <td>0.5</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     X  Prob\n",
       "0   No   0.5\n",
       "1  Yes   0.5"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Please note that the calculations in earlier editions of the book are wrong. For instance, in Page 74 we have:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.501"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.116+0.274+0.01+0.101"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "since they didn't correct the 0.01 to 0.009."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The gender distribution is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "PZ = PXYZ[['Z', 'Prob']].groupby('Z').sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Prob</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Z</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Female</th>\n",
       "      <td>0.49</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Male</th>\n",
       "      <td>0.51</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        Prob\n",
       "Z           \n",
       "Female  0.49\n",
       "Male    0.51"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "PZ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the conditional probability $P\\left(\\mathrm{Drug}|\\mathrm{Gender}\\right)$ is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "PX_Z = PXYZ[['X', 'Z', 'Prob']].groupby(['X', 'Z'], as_index=False).sum()\n",
    "PX_Z['Prob'] = PX_Z.apply(lambda x: x['Prob']/PZ.loc[x.Z], axis=1) #Normalization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "PX_Z.set_index(['X', 'Z'], inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>Prob</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>X</th>\n",
       "      <th>Z</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th rowspan=\"2\" valign=\"top\">No</th>\n",
       "      <th>Female</th>\n",
       "      <td>0.234694</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Male</th>\n",
       "      <td>0.754902</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th rowspan=\"2\" valign=\"top\">Yes</th>\n",
       "      <th>Female</th>\n",
       "      <td>0.765306</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Male</th>\n",
       "      <td>0.245098</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "                Prob\n",
       "X   Z               \n",
       "No  Female  0.234694\n",
       "    Male    0.754902\n",
       "Yes Female  0.765306\n",
       "    Male    0.245098"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PX_Z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The lack of correction for 0.009 subsists even in $P\\left(Y|Male\\right)$ which is why we obtain 0.245 instead of 0.247"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results of the intervention, as calculated the inverse probability weighing are then:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "PYdoXZ = PXYZ.copy()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "PYdoXZ['Prob'] = PYdoXZ.apply(lambda x: x.Prob/PX_Z.loc[x.X, x.Z], axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>Z</th>\n",
       "      <th>Prob</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Yes</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.473</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Yes</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.358</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Yes</td>\n",
       "      <td>No</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Yes</td>\n",
       "      <td>No</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.132</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>No</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.442</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>No</td>\n",
       "      <td>Yes</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.337</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>No</td>\n",
       "      <td>No</td>\n",
       "      <td>Male</td>\n",
       "      <td>0.068</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>No</td>\n",
       "      <td>No</td>\n",
       "      <td>Female</td>\n",
       "      <td>0.153</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     X    Y       Z   Prob\n",
       "0  Yes  Yes    Male  0.473\n",
       "1  Yes  Yes  Female  0.358\n",
       "2  Yes   No    Male  0.037\n",
       "3  Yes   No  Female  0.132\n",
       "4   No  Yes    Male  0.442\n",
       "5   No  Yes  Female  0.337\n",
       "6   No   No    Male  0.068\n",
       "7   No   No  Female  0.153"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PYdoXZ.round(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The small differences we observe with respect to the corrected version in the [online Errata](http://bayes.cs.ucla.edu/PRIMER/pearl-etal-2016-primer-errata-pages-july2020.pdf) are due to the lack of correction from 0.01 to 0.009 as noted above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And finally, the intervention values are:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "PYdoX = PYdoXZ[['X', 'Y', 'Prob']].groupby(['Y', 'X'], as_index=False).sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>No</td>\n",
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       "      <td>0.221</td>\n",
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       "      <th>1</th>\n",
       "      <td>No</td>\n",
       "      <td>Yes</td>\n",
       "      <td>0.169</td>\n",
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      "text/plain": [
       "     Y    X   Prob\n",
       "0   No   No  0.221\n",
       "1   No  Yes  0.169\n",
       "2  Yes   No  0.779\n",
       "3  Yes  Yes  0.831"
      ]
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     "execution_count": 23,
     "metadata": {},
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    }
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   "source": [
    "PYdoX.round(3)"
   ]
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