causality_and_discretization.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/user/prj/chisq-other/.venv/lib/python3.12/site-packages/tqdm/auto.py:21: TqdmExperimentalWarning: Using `tqdm.autonotebook.tqdm` in notebook mode. Use `tqdm.tqdm` instead to force console mode (e.g. in jupyter console)\n",
      "  from .autonotebook import tqdm as notebook_tqdm\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from statsmodels.formula.api import ols\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "from tqdm.auto import tqdm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following causal model:\n",
    "\n",
    "```\n",
    "      P\n",
    "     / \\\n",
    "    /   \\\n",
    "   v     v\n",
    "   R     Q  \n",
    "```\n",
    "\n",
    "This $R$ and $Q$ are independent conditionally on $P$. Assume that we don't know that and want to estimate the effect of $Q$ on $R$ with linear regression, controlling on $P$.\n",
    "\n",
    "$$Q = \\beta_R R + \\beta_P P$$\n",
    "\n",
    "For correct model, we expect non-significant coefficient for $\\beta_R$.\n",
    "\n",
    "Though, what happens if instead of $P$ we observe either discretized or non-linearly transformed version of $P$, called $P'$? Let's check."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f7f5f55597fc43a78f52ce393bed8d5c",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/1000 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "false_positive_rate=0.109\n"
     ]
    }
   ],
   "source": [
    "sample_size = 1000\n",
    "n_iterations = 1000\n",
    "q_pvalues = []\n",
    "\n",
    "for _ in tqdm(range(n_iterations)):\n",
    "    p = np.random.uniform(size=sample_size)\n",
    "    r = 2 * p + np.random.normal(size=sample_size)\n",
    "    q = 3 * p + np.random.normal(size=sample_size)\n",
    "\n",
    "    p_prime = np.floor(p * 5) / 5\n",
    "\n",
    "    fit = ols(\"r ~ p + q\", data={\"r\": r, \"p\": p_prime, \"q\": q}).fit()\n",
    "    q_pvalues.append(fit.pvalues[\"q\"])\n",
    "\n",
    "false_positive_rate = np.mean(np.array(q_pvalues) < 0.05)\n",
    "print(f\"{false_positive_rate=:0.3f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([180., 115., 110.,  95.,  88.,  85.,  79.,  89.,  86.,  73.]),\n",
       " array([1.09230406e-05, 9.99918217e-02, 1.99972720e-01, 2.99953619e-01,\n",
       "        3.99934518e-01, 4.99915417e-01, 5.99896315e-01, 6.99877214e-01,\n",
       "        7.99858113e-01, 8.99839011e-01, 9.99819910e-01]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(q_pvalues)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": ".venv",
   "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": "3.12.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}