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added M/Z bias sims
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‎Chapter16.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Chapter 16: Difficulties of Unconfoundedness in Observational Studies for Causal Effects"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import pandas as pd\n",
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"import scipy as sp\n",
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"import statsmodels.api as sm\n",
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"import statsmodels.formula.api as smf\n",
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"# viz\n",
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"import matplotlib\n",
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"import matplotlib.pyplot as plt\n",
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"import seaborn as sns\n",
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"font = {'family' : 'IBM Plex Sans Condensed',\n",
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" 'weight' : 'normal',\n",
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" 'size' : 10}\n",
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"plt.rc('font', **font)\n",
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"plt.rcParams['figure.figsize'] = (10, 10)\n",
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"%matplotlib inline\n",
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"\n",
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"from utils import *\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<div>\n",
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"<style scoped>\n",
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" .dataframe tbody tr th:only-of-type {\n",
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" vertical-align: middle;\n",
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" }\n",
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"\n",
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" .dataframe tbody tr th {\n",
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" vertical-align: top;\n",
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" }\n",
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"\n",
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" .dataframe thead th {\n",
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" text-align: right;\n",
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" }\n",
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"</style>\n",
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"<table border=\"1\" class=\"dataframe\">\n",
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" <thead>\n",
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" <tr style=\"text-align: right;\">\n",
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" <th></th>\n",
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" <th>U1</th>\n",
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" <th>U2</th>\n",
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" <th>X</th>\n",
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" <th>Y</th>\n",
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" </tr>\n",
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" </thead>\n",
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" <tbody>\n",
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" <tr>\n",
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" <th>0</th>\n",
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" <td>0.868786</td>\n",
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" <td>3.271211</td>\n",
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" <td>2.870715</td>\n",
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" <td>2.776858</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>1</th>\n",
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" <td>0.102776</td>\n",
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" <td>-1.424613</td>\n",
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" <td>-0.343647</td>\n",
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" <td>-2.323463</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>2</th>\n",
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" <td>-0.473300</td>\n",
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" <td>-0.808196</td>\n",
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" <td>-2.437951</td>\n",
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" <td>-0.117681</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>3</th>\n",
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" <td>-0.524105</td>\n",
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" <td>-0.641949</td>\n",
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" <td>-0.149231</td>\n",
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" <td>-0.537228</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>4</th>\n",
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" <td>-0.183823</td>\n",
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" <td>0.540470</td>\n",
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" <td>-0.029903</td>\n",
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" <td>1.374849</td>\n",
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" </tr>\n",
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" </tbody>\n",
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"</table>\n",
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"</div>"
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],
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"text/plain": [
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" U1 U2 X Y\n",
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"0 0.868786 3.271211 2.870715 2.776858\n",
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"1 0.102776 -1.424613 -0.343647 -2.323463\n",
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"2 -0.473300 -0.808196 -2.437951 -0.117681\n",
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"3 -0.524105 -0.641949 -0.149231 -0.537228\n",
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"4 -0.183823 0.540470 -0.029903 1.374849"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"n = int(1e6)\n",
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"df = simulate(\n",
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" U1 = lambda: np.random.normal(size = n),\n",
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" U2 = lambda: np.random.normal(size = n),\n",
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" X = lambda U1, U2: U1 + U2 + np.random.normal(size=n),\n",
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" Y = lambda U2: U2 + np.random.normal(size=n),\n",
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" )\n",
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"\n",
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"df.head()\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## M-bias"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### continuous treatment\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(-0.0005873841517624718, -0.19992316827000112)"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"\n",
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"df['Z'] = df.U1 + np.random.normal(size=n)\n",
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"\n",
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"smf.ols(\"Y ~ Z\", df).fit().params[1], smf.ols(\"Y ~ Z + X\", df).fit().params[1]\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### binary treatment"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0.0010055221102859783, -0.4154833890606614)"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"df['Z'] = df.Z >= 0\n",
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"\n",
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"smf.ols(\"Y ~ Z\", df).fit().params[1], smf.ols(\"Y ~ Z + X\", df).fit().params[1]\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Z-bias"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"n = int(1e6)\n",
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"df = simulate(\n",
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" U = lambda: np.random.normal(size = n),\n",
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" X = lambda: np.random.normal(size = n),\n",
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" Z = lambda X, U: X + U + np.random.normal(size=n),\n",
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" Y = lambda U: U + np.random.normal(size=n),\n",
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" )\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0.33315108130802534, 0.4997989461297992)"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"smf.ols(\"Y ~ Z\", df).fit().params[1], smf.ols(\"Y ~ Z + X\", df).fit().params[1]\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Adjusted comparison is more biased."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### stronger association"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0.16699612964603475, 0.4998217107196198)"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"df['Z'] = 2 * df.X + df.U + np.random.normal(size=n)\n",
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"smf.ols(\"Y ~ Z\", df).fit().params[1], smf.ols(\"Y ~ Z + X\", df).fit().params[1]\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0.00990024072283937, 0.500804991941852)"
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]
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},
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"df['Z'] = 10 * df.X + df.U + np.random.normal(size=n)\n",
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"smf.ols(\"Y ~ Z\", df).fit().params[1], smf.ols(\"Y ~ Z + X\", df).fit().params[1]\n"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "metrics",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

‎utils.py

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import numpy as np
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import pandas as pd
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def simulate(**kwargs):
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values = {}
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for k,v in kwargs.items():
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inputs = {arg: values[arg] for arg in v.__code__.co_varnames}
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values[k] = v(**inputs)
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return pd.DataFrame(values)

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