Chapter27mediationAnalysis.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 27: Mediation Analysis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "statsmodels      : 0.14.0\n",
      "pandas           : 2.0.3\n",
      "matplotlib       : 3.7.2\n",
      "numpy            : 1.24.3\n",
      "matplotlib_inline: 0.1.6\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm\n",
    "\n",
    "from joblib import Parallel, delayed\n",
    "\n",
    "np.random.seed(42)\n",
    "\n",
    "%load_ext watermark\n",
    "%watermark --iversions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def BKmediation(Z, M, Y, X):\n",
    "    # mediator reg\n",
    "    medreg = sm.OLS(M, np.c_[sm.add_constant(Z), X]).fit(vcov_type=\"HC1\")\n",
    "    medZcoef, medZse = medreg.params[1], medreg.bse[1]\n",
    "    outreg = sm.OLS(Y, np.c_[sm.add_constant(Z), M, X]).fit(vcov_type=\"HC1\")\n",
    "    outZcoef, outZse = outreg.params[1], outreg.bse[1]\n",
    "    outMcoef, outMse = outreg.params[2], outreg.bse[2]\n",
    "    NDE, NIE = outZcoef, outMcoef * medZcoef\n",
    "    # sobel's variance\n",
    "    NDEse = outZse\n",
    "    NIEse = np.sqrt(outMse**2 * medZcoef**2 + medZse**2 * outMcoef**2)\n",
    "    return np.array([NDE, NDEse, NDE / NDEse, NIE, NIEse, NIE / NIEse])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulation(beta1, theta2, theta1):\n",
    "    n = 200\n",
    "    Z = np.random.binomial(1, 0.5, n)\n",
    "    X = np.random.normal(size=n)\n",
    "    M = beta1 * Z + X + np.random.normal(size=n)\n",
    "    Y = theta1 * Z + theta2 * M + X + np.random.normal(size=n)\n",
    "    return BKmediation(Z, M, Y, X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\beta_1 = \\theta_2 = \\theta_1 = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.16168798729204314 0.1592993872148648\n",
      "0.15508938768915878 0.1594246671589853\n"
     ]
    },
    {
     "data": {
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 600x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res = Parallel(n_jobs=-1)(delayed(simulation)(1, 1, 1) for _ in range(1000))\n",
    "res = np.vstack(res)\n",
    "print(res[:, 0].std(), res[:, 1].mean())\n",
    "print(res[:, 3].std(), res[:, 4].mean())\n",
    "\n",
    "f, ax = plt.subplots(1, 2, figsize=(6, 3))\n",
    "ax[0].hist(res[:, 0], bins=30, density=True)\n",
    "ax[0].vlines(1, 0, 1, color=\"red\")\n",
    "ax[0].set_title(\"NDE\")\n",
    "ax[1].hist(res[:, 3], bins=30, density=True)\n",
    "ax[1].vlines(\n",
    "    1,\n",
    "    0,\n",
    "    1,\n",
    "    color=\"red\",\n",
    ")\n",
    "ax[1].set_title(\"NIE\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\theta_2$ = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.16070039875959033 0.15894602890501858\n",
      "0.07139184520607268 0.07184486066517198\n"
     ]
    },
    {
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       "     </g>\n",
       "     <g id=\"text_22\">\n",
       "      <!-- 5 -->\n",
       "      <g transform=\"translate(199.358807 51.402604) scale(0.1 -0.1)\">\n",
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       "   </g>\n",
       "   <g id=\"patch_68\">\n",
       "    <path d=\"M 212.721307 188.638125 \n",
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       "   </g>\n",
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       "   <g id=\"patch_70\">\n",
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       "   <g id=\"patch_71\">\n",
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       "   </g>\n",
       "   <g id=\"text_23\">\n",
       "    <!-- NIE -->\n",
       "    <g transform=\"translate(278.762216 16.318125) scale(0.12 -0.12)\">\n",
       "     <defs>\n",
       "      <path id=\"DejaVuSans-49\" d=\"M 628 4666 \n",
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       "L 1259 0 \n",
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       "     </defs>\n",
       "     <use xlink:href=\"#DejaVuSans-4e\"/>\n",
       "     <use xlink:href=\"#DejaVuSans-49\" x=\"74.804688\"/>\n",
       "     <use xlink:href=\"#DejaVuSans-45\" x=\"104.296875\"/>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p0df3cfbfd2\">\n",
       "   <rect x=\"30.103125\" y=\"22.318125\" width=\"152.181818\" height=\"166.32\"/>\n",
       "  </clipPath>\n",
       "  <clipPath id=\"p317dbb18b6\">\n",
       "   <rect x=\"212.721307\" y=\"22.318125\" width=\"152.181818\" height=\"166.32\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 600x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res = Parallel(n_jobs=-1)(delayed(simulation)(1, 0, 1) for _ in range(1000))\n",
    "res = np.vstack(res)\n",
    "print(res[:, 0].std(), res[:, 1].mean())\n",
    "print(res[:, 3].std(), res[:, 4].mean())\n",
    "\n",
    "f, ax = plt.subplots(1, 2, figsize=(6, 3))\n",
    "ax[0].hist(res[:, 0], bins=30, density=True)\n",
    "ax[0].vlines(1, 0, 3, color=\"red\")\n",
    "ax[0].set_title(\"NDE\")\n",
    "ax[1].hist(res[:, 3], bins=30, density=True)\n",
    "ax[1].vlines(\n",
    "    0,\n",
    "    0,\n",
    "    3,\n",
    "    color=\"red\",\n",
    ")\n",
    "ax[1].set_title(\"NIE\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\beta_1 = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.13398443810512395 0.1421209837224335\n",
      "0.010519116540859337 0.012717706993520813\n"
     ]
    },
    {
     "data": {
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 600x300 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "res = Parallel(n_jobs=-1)(delayed(simulation)(0, 0, 1) for _ in range(1000))\n",
    "res = np.vstack(res)\n",
    "print(res[:, 0].std(), res[:, 1].mean())\n",
    "print(res[:, 3].std(), res[:, 4].mean())\n",
    "\n",
    "f, ax = plt.subplots(1, 2, figsize=(6, 3))\n",
    "ax[0].hist(res[:, 0], bins=30, density=True)\n",
    "ax[0].vlines(1, 0, 3, color=\"red\")\n",
    "ax[0].set_title(\"NDE\")\n",
    "ax[1].hist(res[:, 3], bins=30, density=True)\n",
    "ax[1].vlines(\n",
    "    0,\n",
    "    0,\n",
    "    3,\n",
    "    color=\"red\",\n",
    ")\n",
    "ax[1].set_title(\"NIE\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import formulaic as fm\n",
    "\n",
    "jobsdata = pd.read_csv(\"jobsdata.csv\")\n",
    "X = (\n",
    "    fm.Formula(\n",
    "        \"\"\"\n",
    "                ~ econ_hard + depress1 + sex + age + occp + marital +\n",
    "                nonwhite + educ + income -1\n",
    "                \"\"\"\n",
    "    )\n",
    "    .get_model_matrix(data=jobsdata)\n",
    "    .values\n",
    ")\n",
    "Z, M, Y = jobsdata[[\"treat\", \"job_seek\", \"depress2\"]].values.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "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>coef</th>\n",
       "      <th>se</th>\n",
       "      <th>t</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>NDE</th>\n",
       "      <td>-0.036789</td>\n",
       "      <td>0.040794</td>\n",
       "      <td>-0.901813</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NIE</th>\n",
       "      <td>-0.013733</td>\n",
       "      <td>0.009008</td>\n",
       "      <td>-1.524646</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         coef        se         t\n",
       "NDE -0.036789  0.040794 -0.901813\n",
       "NIE -0.013733  0.009008 -1.524646"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = BKmediation(Z, M, Y, X)\n",
    "pd.DataFrame([res[0:3], res[3:6]], index=[\"NDE\", \"NIE\"], columns=[\"coef\", \"se\", \"t\"])"
   ]
  }
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