all iv chapters

Author: apoorvalal

Chapter22.ipynb

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+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Chapter 22: Disentangling Mixture Distributions and Instrumental Variable Inequalities"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# n_z,d,d1  under monotonicity\n",
+    "def IVbinary(n111, n110, n101, n100, n011, n010, n001, n000):\n",
+    "    n_tr = n111 + n110 + n101 + n100\n",
+    "    n_co = n011 + n010 + n001 + n000\n",
+    "    n = n_tr + n_co\n",
+    "\n",
+    "    # proportions of latent strata\n",
+    "    pi_n = (n101 + n100)/ n_tr # p(d = 0 | z = 1) never taker\n",
+    "    pi_a = (n011 + n010)/ n_co # p(d = 1 | z = 0) always taker\n",
+    "    pi_c = 1 - pi_n - pi_a # by monotonicity, the only remaining group are compliers\n",
+    "\n",
+    "    # four observed means of the outcomes (Z = z, D = d)\n",
+    "    mean_y_11 = n111 / (n111 + n110)\n",
+    "    mean_y_10 = n101 / (n101 + n100)\n",
+    "    mean_y_01 = n011 / (n011 + n010)\n",
+    "    mean_y_00 = n001 / (n001 + n000)\n",
+    "    # all four means are positive for binary outcomes\n",
+    "\n",
+    "    # means of the outcome of two strata\n",
+    "    mu_n1, mu_a0  = mean_y_10, mean_y_01\n",
+    "    # exclusion implies 0 and 1 are same for always takers and never takers\n",
+    "    mu_n0, mu_a1 = mu_n1, mu_a0\n",
+    "    # stratum (Z=1, D = 1) is a mixture of c, a\n",
+    "    mu_c1 = ((pi_c + pi_a) * mean_y_11 - pi_a * mu_a1) / pi_c\n",
+    "    # stratum (Z=0, D = 0) is a mixture of c, n\n",
+    "    mu_c0 = ((pi_c + pi_n) * mean_y_00 - pi_n * mu_n0) / pi_c\n",
+    "    # identifiable quantities\n",
+    "    return {\n",
+    "        'pi_c' : pi_c,\n",
+    "        'pi_n' : pi_n,\n",
+    "        'pi_a' : pi_a,\n",
+    "        'mu_n1': mu_n1,\n",
+    "        'mu_n0': mu_n0,\n",
+    "        'mu_a1': mu_a1,\n",
+    "        'mu_a0': mu_a0,\n",
+    "        'mu_c1': mu_c1,\n",
+    "        'mu_c0': mu_c0,\n",
+    "        'tau_c': mu_c1 - mu_c0,\n",
+    "    }\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{'pi_c': 0.44305817033089756,\n",
+       " 'pi_n': 0.4247104247104247,\n",
+       " 'pi_a': 0.1322314049586777,\n",
+       " 'mu_n1': 0.6181818181818182,\n",
+       " 'mu_n0': 0.6181818181818182,\n",
+       " 'mu_a1': 0.75,\n",
+       " 'mu_a0': 0.75,\n",
+       " 'mu_c1': 0.7086064097947424,\n",
+       " 'mu_c0': 0.6292041771696075,\n",
+       " 'tau_c': 0.0794022326251349}"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "## Investigators et al.(2014) data\n",
+    "(investigators_analysis := IVbinary(\n",
+    "  n111 = 107,\n",
+    "  n110 = 42,\n",
+    "  n101 = 68,\n",
+    "  n100 = 42,\n",
+    "  n011 = 24,\n",
+    "  n010 = 8,\n",
+    "  n001 = 131,\n",
+    "  n000 = 79\n",
+    "))\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{'pi_c': 0.11839971280558428,\n",
+       " 'pi_n': 0.6922554347826086,\n",
+       " 'pi_a': 0.18934485241180707,\n",
+       " 'mu_n1': 0.08243375858684986,\n",
+       " 'mu_n0': 0.08243375858684986,\n",
+       " 'mu_a1': 0.11406844106463879,\n",
+       " 'mu_a0': 0.11406844106463879,\n",
+       " 'mu_c1': -0.004548064490810916,\n",
+       " 'mu_c0': 0.12000941833518534,\n",
+       " 'tau_c': -0.12455748282599625}"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(flu_analysis := IVbinary(\n",
+    "  n111 = 31,\n",
+    "  n110 = 422,\n",
+    "  n101 = 84,\n",
+    "  n100 = 935,\n",
+    "  n011 = 30,\n",
+    "  n010 = 233,\n",
+    "  n001 = 99,\n",
+    "  n000 = 1027\n",
+    "))\n"
+   ]
+  }
+ ],
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