Analysis_07MAY2019_new.ipynb

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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Causal-model\" data-toc-modified-id=\"Causal-model-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Causal model</a></span><ul class=\"toc-item\"><li><span><a href=\"#Notes\" data-toc-modified-id=\"Notes-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Notes</a></span></li></ul></li><li><span><a href=\"#Synthetic-data\" data-toc-modified-id=\"Synthetic-data-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Synthetic data</a></span></li><li><span><a href=\"#Algorithms\" data-toc-modified-id=\"Algorithms-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Algorithms</a></span><ul class=\"toc-item\"><li><span><a href=\"#Contraction-algorithm\" data-toc-modified-id=\"Contraction-algorithm-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Contraction algorithm</a></span></li><li><span><a href=\"#Causal-algorithm\" data-toc-modified-id=\"Causal-algorithm-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Causal algorithm</a></span></li></ul></li><li><span><a href=\"#Performance-comparison\" data-toc-modified-id=\"Performance-comparison-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Performance comparison</a></span><ul class=\"toc-item\"><li><span><a href=\"#Predictive-models\" data-toc-modified-id=\"Predictive-models-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Predictive models</a></span></li><li><span><a href=\"#Visual-comparison\" data-toc-modified-id=\"Visual-comparison-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>Visual comparison</a></span></li></ul></li></ul></div>"
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    "##  Causal model\n",
    "\n",
    "Our model is defined by the probabilistic expression \n",
    "\n",
    "\\begin{equation} \\label{model_disc}\n",
    "P(Y=0 | \\text{do}(R=r)) = \\sum_x \\underbrace{P(Y=0|X=x, T=1)}_\\text{1} \n",
    "\\overbrace{P(T=1|R=r, X=x)}^\\text{2} \n",
    "\\underbrace{P(X=x)}_\\text{3}\n",
    "\\end{equation}\n",
    "\n",
    "which is equal to \n",
    "\n",
    "\\begin{equation}\\label{model_cont}\n",
    "P(Y=0 | \\text{do}(R=r)) = \\int_x P(Y=0|X=x, T=1)P(T=1|R=r, X=x)P(X=x)\n",
    "\\end{equation}\n",
    "\n",
    "for continuous $x$. Model as a graph (Z is a latent variable, and can be excluded from the expression with do-calculus by showing that $X$ is admissible for adjustment):\n",
    "\n",
    "![Model as picture](../figures/intervention_model.png \"Intervention model\")\n",
    "\n",
    "For predicting the probability of negative outcome the following should hold because by Pearl $P(Y=0 | \\text{do}(R=r), X=x) = P(Y=0 | R=r, X=x)$ when $X$ is an admissible set:\n",
    "\n",
    "\\begin{equation} \\label{model_pred}\n",
    "P(Y=0 | \\text{do}(R=r), X=x) = P(Y=0|X=x, T=1)P(T=1|R=r, X=x).\n",
    "\\end{equation}\n",
    "\n",
    "Still it should be noted that this prediction takes into account the probability of the individual to be given a positive decision ($T=1$), see second term in \\ref{model_pred}.\n",
    "\n",
    "----\n",
    "\n",
    "### Notes\n",
    "\n",
    "* Equations \\ref{model_disc} and \\ref{model_cont} describe the whole causal effect in the population (the causal effect of changing $r$ over all strata $X$).\n",
    "* Prediction should be possible with \\ref{model_pred}. Both terms can be learned from the data. NB: the probability $P(Y=0 | \\text{do}(R=r), X=x)$ is lowest when the individual $x$ is the most dangerous or the least dangerous. How could we infer/predict the counterfactual \"what is the probability of $Y=0$ if we were to let this individual go?\" has yet to be calculated.\n",
    "* Is the effect of R learned/estimated correctly if it is just plugged in to a predictive model (e.g. logistic regression)?\n",
    "* $P(Y=0 | do(R=0)) = 0$ only in this application. My predictive models say that when $r=0$ the probability $P(Y=0) \\approx 0.027$ which would be a natural estimate in another application/scenario (e.g. in medicine the probability of an adverse event when a stronger medicine is distributed to everyone. Then the probability will be close to zero but not exactly zero.)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Imports\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "from datetime import datetime\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as scs\n",
    "import scipy.integrate as si\n",
    "import seaborn as sns\n",
    "import numpy.random as npr\n",
    "from sklearn.preprocessing import OneHotEncoder\n",
    "from sklearn.linear_model import LogisticRegression\n",
    "from sklearn.ensemble import RandomForestClassifier\n",
    "\n",
    "# Settings\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "plt.rcParams.update({'font.size': 16})\n",
    "plt.rcParams.update({'figure.figsize': (14, 7)})\n",
    "\n",
    "# Suppress deprecation warnings.\n",
    "\n",
    "import warnings\n",
    "\n",
    "def fxn():\n",
    "    warnings.warn(\"deprecated\", DeprecationWarning)\n",
    "\n",
    "with warnings.catch_warnings():\n",
    "    warnings.simplefilter(\"ignore\")\n",
    "    fxn()"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## Synthetic data\n",
    "\n",
    "In the chunk below, we generate the synthetic data as described by Lakkaraju et al. The default values and definitions of $Y$ and $T$ values follow their description.\n",
    "\n",
    "**Parameters**\n",
    "\n",
    "* M = `nJudges_M`, number of judges\n",
    "* N = `nSubjects_N`, number of subjects assigned to each judge\n",
    "* betas $\\beta_i$ = `beta_i`, where $i \\in \\{X, Z, W\\}$ are coefficients for the respected variables\n",
    "\n",
    "**Columns of the data:**\n",
    "\n",
    "* `judgeID_J` = judge IDs as running numbering from 0 to `nJudges_M - 1`\n",
    "* R = `acceptanceRate_R`, acceptance rates\n",
    "* X = `X`, invidual's features observable to all (models and judges)\n",
    "* Z = `Z`, information observable for judges only\n",
    "* W = `W`, unobservable / inaccessible information\n",
    "* T = `decision_T`, bail-or-jail decisions where $T=0$ represents jail decision and $T=1$ bail decision.\n",
    "* Y = `result_Y`, result variable, if $Y=0$ person will or would recidivate and if $Y=1$ person will or would not commit a crime."
   ]
  },
  {
   "cell_type": "code",
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       "      <th></th>\n",
       "      <th>judgeID_J</th>\n",
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       "      <td>1.476504</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.521224</td>\n",
       "      <td>1.288442</td>\n",
       "      <td>-0.120573</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.633221</td>\n",
       "      <td>0</td>\n",
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       "      <th>3</th>\n",
       "      <td>99.0</td>\n",
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       "      <td>-0.020646</td>\n",
       "      <td>1.636132</td>\n",
       "      <td>-0.460398</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.596941</td>\n",
       "      <td>0</td>\n",
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       "      <th>4</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.473597</td>\n",
       "      <td>0.570864</td>\n",
       "      <td>-1.763391</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.549796</td>\n",
       "      <td>0</td>\n",
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       "      <td>99.0</td>\n",
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       "      <td>2.441796</td>\n",
       "      <td>-2.163222</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.496034</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>1.546339</td>\n",
       "      <td>1.173717</td>\n",
       "      <td>0.775945</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.423059</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>-0.005012</td>\n",
       "      <td>0.854531</td>\n",
       "      <td>0.772508</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.406603</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>1.124008</td>\n",
       "      <td>0.194219</td>\n",
       "      <td>-0.793756</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.363169</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.954372</td>\n",
       "      <td>2.227896</td>\n",
       "      <td>0.471742</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.351900</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.139221</td>\n",
       "      <td>-0.118334</td>\n",
       "      <td>-2.218456</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.328329</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>-0.894260</td>\n",
       "      <td>0.806683</td>\n",
       "      <td>-0.174320</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.328299</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.146317</td>\n",
       "      <td>0.090848</td>\n",
       "      <td>0.575790</td>\n",
       "      <td>0.0</td>\n",
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       "      <td>0</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>1.061130</td>\n",
       "      <td>0.311556</td>\n",
       "      <td>0.271090</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.304708</td>\n",
       "      <td>0</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.364795</td>\n",
       "      <td>1.340871</td>\n",
       "      <td>0.741578</td>\n",
       "      <td>0.0</td>\n",
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       "      <td>0.0</td>\n",
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       "      <td>0</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.098206</td>\n",
       "      <td>1.030595</td>\n",
       "      <td>1.369878</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.249515</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>2.009392</td>\n",
       "      <td>0.348242</td>\n",
       "      <td>-0.918357</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.242619</td>\n",
       "      <td>0</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>2.650009</td>\n",
       "      <td>0.548038</td>\n",
       "      <td>-0.734014</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.224764</td>\n",
       "      <td>0</td>\n",
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       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>1.104756</td>\n",
       "      <td>-1.249380</td>\n",
       "      <td>1.073779</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.207050</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>1.418292</td>\n",
       "      <td>0.664494</td>\n",
       "      <td>0.564188</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.188848</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>-0.895593</td>\n",
       "      <td>-0.054607</td>\n",
       "      <td>0.983115</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.185423</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.685227</td>\n",
       "      <td>1.374469</td>\n",
       "      <td>-0.796697</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.183456</td>\n",
       "      <td>0</td>\n",
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       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>-0.422198</td>\n",
       "      <td>0.394662</td>\n",
       "      <td>-2.208503</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.174958</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.956056</td>\n",
       "      <td>0.839686</td>\n",
       "      <td>-1.294539</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.170929</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.588787</td>\n",
       "      <td>1.398952</td>\n",
       "      <td>2.395113</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.165979</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>-1.344696</td>\n",
       "      <td>1.856761</td>\n",
       "      <td>0.909326</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.159617</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>-0.551863</td>\n",
       "      <td>-0.071659</td>\n",
       "      <td>1.297963</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.154631</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>0.735048</td>\n",
       "      <td>1.507983</td>\n",
       "      <td>-0.539594</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.146622</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>99.0</td>\n",
       "      <td>0.186313</td>\n",
       "      <td>2.294530</td>\n",
       "      <td>-1.381245</td>\n",
       "      <td>-0.660446</td>\n",
       "      <td>0.0</td>\n",
       "      <td>1.138679</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49970</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>0.711615</td>\n",
       "      <td>-2.915390</td>\n",
       "      <td>0.172405</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.110999</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49971</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.827231</td>\n",
       "      <td>-1.777676</td>\n",
       "      <td>0.876569</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.113757</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49972</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.200296</td>\n",
       "      <td>-1.591935</td>\n",
       "      <td>-2.098238</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.117259</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49973</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.236819</td>\n",
       "      <td>-0.646515</td>\n",
       "      <td>-0.073568</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.118655</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49974</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.067620</td>\n",
       "      <td>0.232980</td>\n",
       "      <td>0.700920</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.123346</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49975</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.767348</td>\n",
       "      <td>-1.909278</td>\n",
       "      <td>-0.040354</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.128207</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49976</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.937825</td>\n",
       "      <td>-1.963602</td>\n",
       "      <td>0.233229</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.128611</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49977</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>0.614167</td>\n",
       "      <td>-2.312697</td>\n",
       "      <td>0.246288</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.139810</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49978</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.245388</td>\n",
       "      <td>-0.787053</td>\n",
       "      <td>-1.123660</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.145423</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49979</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.514847</td>\n",
       "      <td>-1.908273</td>\n",
       "      <td>-0.349488</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.154516</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49980</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.269407</td>\n",
       "      <td>-0.587080</td>\n",
       "      <td>-1.341434</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.167074</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49981</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.089120</td>\n",
       "      <td>-0.700974</td>\n",
       "      <td>0.623632</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.179464</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49982</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>0.690144</td>\n",
       "      <td>-0.734551</td>\n",
       "      <td>0.779340</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-0.183530</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49983</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.875618</td>\n",
       "      <td>-0.732720</td>\n",
       "      <td>-0.642521</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.190108</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49984</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.401851</td>\n",
       "      <td>-0.333382</td>\n",
       "      <td>0.338832</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.195607</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49985</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.977555</td>\n",
       "      <td>-1.152621</td>\n",
       "      <td>-1.279561</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.219627</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49986</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-2.038125</td>\n",
       "      <td>-0.382141</td>\n",
       "      <td>-0.997567</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.224630</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49987</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.315269</td>\n",
       "      <td>0.453257</td>\n",
       "      <td>0.625528</td>\n",
       "      <td>0.0</td>\n",
       "      <td>-0.230150</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49988</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.506943</td>\n",
       "      <td>-0.522032</td>\n",
       "      <td>0.850031</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.231718</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49989</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.377669</td>\n",
       "      <td>-0.269111</td>\n",
       "      <td>-2.687594</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.243685</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49990</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.142379</td>\n",
       "      <td>-0.175690</td>\n",
       "      <td>0.513888</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.249886</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49991</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.607483</td>\n",
       "      <td>1.504558</td>\n",
       "      <td>0.239068</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.280709</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49992</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.818221</td>\n",
       "      <td>-1.289808</td>\n",
       "      <td>-1.831835</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.285505</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49993</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.599375</td>\n",
       "      <td>-0.171306</td>\n",
       "      <td>-0.084856</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.299202</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49994</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.469176</td>\n",
       "      <td>-1.982240</td>\n",
       "      <td>-1.780709</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.315370</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49995</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.006017</td>\n",
       "      <td>-0.388426</td>\n",
       "      <td>-1.401947</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.321114</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49996</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.219101</td>\n",
       "      <td>-1.348249</td>\n",
       "      <td>0.724337</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.330852</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49997</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-2.471645</td>\n",
       "      <td>-0.267131</td>\n",
       "      <td>-0.077586</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.357114</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49998</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-1.077745</td>\n",
       "      <td>-0.237860</td>\n",
       "      <td>-1.101762</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.372307</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>49999</th>\n",
       "      <td>0.0</td>\n",
       "      <td>0.399632</td>\n",
       "      <td>-0.964923</td>\n",
       "      <td>-0.476811</td>\n",
       "      <td>0.241492</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.434957</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>50000 rows × 8 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "       judgeID_J  acceptanceRate_R         X         Z         W  result_Y  \\\n",
       "0           99.0          0.186313  0.826274  2.105072  1.476504       0.0   \n",
       "1           99.0          0.186313 -0.028464  0.297879 -0.975877       0.0   \n",
       "2           99.0          0.186313  0.521224  1.288442 -0.120573       0.0   \n",
       "3           99.0          0.186313 -0.020646  1.636132 -0.460398       0.0   \n",
       "4           99.0          0.186313  0.473597  0.570864 -1.763391       0.0   \n",
       "5           99.0          0.186313  0.968785  2.441796 -2.163222       0.0   \n",
       "6           99.0          0.186313  1.546339  1.173717  0.775945       0.0   \n",
       "7           99.0          0.186313 -0.005012  0.854531  0.772508       0.0   \n",
       "8           99.0          0.186313  1.124008  0.194219 -0.793756       0.0   \n",
       "9           99.0          0.186313  0.954372  2.227896  0.471742       0.0   \n",
       "10          99.0          0.186313  0.139221 -0.118334 -2.218456       1.0   \n",
       "11          99.0          0.186313 -0.894260  0.806683 -0.174320       1.0   \n",
       "12          99.0          0.186313  0.146317  0.090848  0.575790       0.0   \n",
       "13          99.0          0.186313  1.061130  0.311556  0.271090       0.0   \n",
       "14          99.0          0.186313  0.364795  1.340871  0.741578       0.0   \n",
       "15          99.0          0.186313  1.359628 -0.331233  1.216847       0.0   \n",
       "16          99.0          0.186313  0.098206  1.030595  1.369878       0.0   \n",
       "17          99.0          0.186313  2.009392  0.348242 -0.918357       0.0   \n",
       "18          99.0          0.186313  2.650009  0.548038 -0.734014       0.0   \n",
       "19          99.0          0.186313  1.104756 -1.249380  1.073779       0.0   \n",
       "20          99.0          0.186313  1.418292  0.664494  0.564188       0.0   \n",
       "21          99.0          0.186313 -0.895593 -0.054607  0.983115       1.0   \n",
       "22          99.0          0.186313  0.685227  1.374469 -0.796697       0.0   \n",
       "23          99.0          0.186313 -0.422198  0.394662 -2.208503       1.0   \n",
       "24          99.0          0.186313  0.956056  0.839686 -1.294539       0.0   \n",
       "25          99.0          0.186313  0.588787  1.398952  2.395113       0.0   \n",
       "26          99.0          0.186313 -1.344696  1.856761  0.909326       0.0   \n",
       "27          99.0          0.186313 -0.551863 -0.071659  1.297963       1.0   \n",
       "28          99.0          0.186313  0.735048  1.507983 -0.539594       0.0   \n",
       "29          99.0          0.186313  2.294530 -1.381245 -0.660446       0.0   \n",
       "...          ...               ...       ...       ...       ...       ...   \n",
       "49970        0.0          0.399632  0.711615 -2.915390  0.172405       1.0   \n",
       "49971        0.0          0.399632 -0.827231 -1.777676  0.876569       1.0   \n",
       "49972        0.0          0.399632 -1.200296 -1.591935 -2.098238       1.0   \n",
       "49973        0.0          0.399632 -0.236819 -0.646515 -0.073568       1.0   \n",
       "49974        0.0          0.399632 -1.067620  0.232980  0.700920       1.0   \n",
       "49975        0.0          0.399632 -0.767348 -1.909278 -0.040354       1.0   \n",
       "49976        0.0          0.399632 -0.937825 -1.963602  0.233229       1.0   \n",
       "49977        0.0          0.399632  0.614167 -2.312697  0.246288       1.0   \n",
       "49978        0.0          0.399632 -0.245388 -0.787053 -1.123660       1.0   \n",
       "49979        0.0          0.399632 -1.514847 -1.908273 -0.349488       1.0   \n",
       "49980        0.0          0.399632 -0.269407 -0.587080 -1.341434       1.0   \n",
       "49981        0.0          0.399632 -0.089120 -0.700974  0.623632       1.0   \n",
       "49982        0.0          0.399632  0.690144 -0.734551  0.779340       0.0   \n",
       "49983        0.0          0.399632 -0.875618 -0.732720 -0.642521       1.0   \n",
       "49984        0.0          0.399632 -1.401851 -0.333382  0.338832       1.0   \n",
       "49985        0.0          0.399632 -0.977555 -1.152621 -1.279561       1.0   \n",
       "49986        0.0          0.399632 -2.038125 -0.382141 -0.997567       1.0   \n",
       "49987        0.0          0.399632 -0.315269  0.453257  0.625528       0.0   \n",
       "49988        0.0          0.399632 -0.506943 -0.522032  0.850031       1.0   \n",
       "49989        0.0          0.399632 -1.377669 -0.269111 -2.687594       1.0   \n",
       "49990        0.0          0.399632 -0.142379 -0.175690  0.513888       1.0   \n",
       "49991        0.0          0.399632 -1.607483  1.504558  0.239068       1.0   \n",
       "49992        0.0          0.399632 -0.818221 -1.289808 -1.831835       1.0   \n",
       "49993        0.0          0.399632 -0.599375 -0.171306 -0.084856       1.0   \n",
       "49994        0.0          0.399632 -0.469176 -1.982240 -1.780709       1.0   \n",
       "49995        0.0          0.399632 -1.006017 -0.388426 -1.401947       1.0   \n",
       "49996        0.0          0.399632 -0.219101 -1.348249  0.724337       1.0   \n",
       "49997        0.0          0.399632 -2.471645 -0.267131 -0.077586       1.0   \n",
       "49998        0.0          0.399632 -1.077745 -0.237860 -1.101762       1.0   \n",
       "49999        0.0          0.399632 -0.964923 -0.476811  0.241492       1.0   \n",
       "\n",
       "       probabilities_T  decision_T  \n",
       "0             1.692504           0  \n",
       "1             1.681097           0  \n",
       "2             1.633221           0  \n",
       "3             1.596941           0  \n",
       "4             1.549796           0  \n",
       "5             1.496034           0  \n",
       "6             1.423059           0  \n",
       "7             1.406603           0  \n",
       "8             1.363169           0  \n",
       "9             1.351900           0  \n",
       "10            1.328329           0  \n",
       "11            1.328299           0  \n",
       "12            1.326697           0  \n",
       "13            1.304708           0  \n",
       "14            1.283888           0  \n",
       "15            1.276119           0  \n",
       "16            1.249515           0  \n",
       "17            1.242619           0  \n",
       "18            1.224764           0  \n",
       "19            1.207050           0  \n",
       "20            1.188848           0  \n",
       "21            1.185423           0  \n",
       "22            1.183456           0  \n",
       "23            1.174958           0  \n",
       "24            1.170929           0  \n",
       "25            1.165979           0  \n",
       "26            1.159617           0  \n",
       "27            1.154631           0  \n",
       "28            1.146622           0  \n",
       "29            1.138679           0  \n",
       "...                ...         ...  \n",
       "49970        -0.110999           1  \n",
       "49971        -0.113757           1  \n",
       "49972        -0.117259           1  \n",
       "49973        -0.118655           1  \n",
       "49974        -0.123346           1  \n",
       "49975        -0.128207           1  \n",
       "49976        -0.128611           1  \n",
       "49977        -0.139810           1  \n",
       "49978        -0.145423           1  \n",
       "49979        -0.154516           1  \n",
       "49980        -0.167074           1  \n",
       "49981        -0.179464           1  \n",
       "49982        -0.183530           1  \n",
       "49983        -0.190108           1  \n",
       "49984        -0.195607           1  \n",
       "49985        -0.219627           1  \n",
       "49986        -0.224630           1  \n",
       "49987        -0.230150           1  \n",
       "49988        -0.231718           1  \n",
       "49989        -0.243685           1  \n",
       "49990        -0.249886           1  \n",
       "49991        -0.280709           1  \n",
       "49992        -0.285505           1  \n",
       "49993        -0.299202           1  \n",
       "49994        -0.315370           1  \n",
       "49995        -0.321114           1  \n",
       "49996        -0.330852           1  \n",
       "49997        -0.357114           1  \n",
       "49998        -0.372307           1  \n",
       "49999        -0.434957           1  \n",
       "\n",
       "[50000 rows x 8 columns]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set seed for reproducibility\n",
    "npr.seed(42)\n",
    "\n",
    "def sigmoid(x):\n",
    "    return 1 / (1 + np.exp(-x))\n",
    "\n",
    "def generateData(nJudges_M=100,\n",
    "                 nSubjects_N=500,\n",
    "                 beta_X=1.0,\n",
    "                 beta_Z=1.0,\n",
    "                 beta_W=0.2):\n",
    "\n",
    "    # Assign judge IDs as running numbering from 0 to nJudges_M - 1\n",
    "    judgeID_J = np.repeat(np.arange(0, nJudges_M, dtype=np.int32), nSubjects_N)\n",
    "\n",
    "    # Sample acceptance rates uniformly from a closed interval\n",
    "    # from 0.1 to 0.9 and round to tenth decimal place.\n",
    "    acceptance_rates = np.round(npr.uniform(.1, .9, nJudges_M), 10)\n",
    "\n",
    "    # Replicate the rates so they can be attached to the corresponding judge ID.\n",
    "    acceptanceRate_R = np.repeat(acceptance_rates, nSubjects_N)\n",
    "\n",
    "    # Sample the variables from standard Gaussian distributions.\n",
    "    X = npr.normal(size=nJudges_M * nSubjects_N)\n",
    "    Z = npr.normal(size=nJudges_M * nSubjects_N)\n",
    "    W = npr.normal(size=nJudges_M * nSubjects_N)\n",
    "\n",
    "    probabilities_Y = sigmoid(beta_X * X + beta_Z * Z + beta_W * W)\n",
    "\n",
    "    # 0 if P(Y = 0| X = x; Z = z; W = w) >= 0.5 , 1 otherwise\n",
    "    result_Y = 1 - probabilities_Y.round()\n",
    "    \n",
    "    # For the conditional probabilities of T we add noise ~ N(0, 0.1)\n",
    "    probabilities_T = sigmoid(beta_X * X + beta_Z * Z)\n",