L2_counterfactual.ipynb

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    "$$\n",
    "\\newcommand{P}[1]{\\mathrm{P}\\left( #1 \\right)}\n",
    "\\newcommand{Pc}[2]{\\mathrm{P}\\left( #1 \\mid #2 \\right)}\n",
    "\\newcommand{Ec}[2]{\\mathrm{E}\\left( #1 \\mid #2 \\right)}\n",
    "\\newcommand{In}[2]{ #1 \\perp\\!\\!\\!\\perp #2}\n",
    "\\newcommand{Cin}[3]{ #1 \\perp\\!\\!\\!\\perp #2 \\, | \\, #3}\n",
    "\\newcommand{do}[1]{\\mathrm{do}\\left( #1 \\right)}\n",
    "\\newcommand{\\ci}{\\perp\\!\\!\\!\\perp}\n",
    "$$"
   ]
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   "source": [
    "# Counterfactual Models and Causal Effects\n",
    "\n",
    "Day 3 Module 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Potential Outcome Framework\n",
    "\n",
    "As discussed in the previous secsion, for a study with a binary treatment variable $A \\in \\{0, 1\\}$, we can introduce two new random variables to the system: $Y^{0}$ and $Y^{1}$, under the counterfactual outcome framework. We imagine that these variables exist, and can be treated as any other random variable - the only difference is that they are never directly observed. \n",
    "\n",
    "* This shifts the problem from one about how distributions change under the intervention, to one about data drawn i.i.d. from some underlying distribution with [missing values](https://en.wikipedia.org/wiki/Missing_data). "
   ]
  },
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   "cell_type": "markdown",
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    "| index $i$ | $X_i$ | $A_i$ | $Y_i$ | $Y_i^0$ | $Y_i^1$ |\n",
    "| --- | --- | --- | --- | --- | --- | \n",
    "| 1 | 0 | 0 | 4 | ? | ? |\n",
    "| 2 | 0 | 0 | 3 | ? | ? |\n",
    "| 3 | 0 | 0 | 5 | ? | ? |\n",
    "| 4 | 0 | 1 | 8 | ? | ? |\n",
    "| 5 | 1 | 0 | 2 | ? | ? |\n",
    "| 6 | 1 | 0 | 4 | ? | ? |\n",
    "| 7 | 1 | 1 | 8 | ? | ? |\n",
    "| 8 | 1 | 1 | 6 | ? | ? |"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "#### Identification Assumptions (for the missing counterfactuals)\n",
    "\n",
    "*  Consistency: $Y_i^a = Y_i$ if $A_i=a$.\n",
    "\n",
    "| index $i$ | $X_i$ | $A_i$ | $Y_i$ | $Y_i^0$ | $Y_i^1$ |\n",
    "| --- | --- | --- | --- | --- | --- | \n",
    "| 1 | 0 | 0 | 4 | <font color='red'>4</font> | ? |\n",
    "| 2 | 0 | 0 | 3 | <font color='red'>7</font> | ? |\n",
    "| 3 | 0 | 0 | 5 | <font color='red'>2</font> | ? |\n",
    "| 4 | 0 | 1 | 8 | ? | <font color='red'>8</font> |\n",
    "| 5 | 1 | 0 | 2 | <font color='red'>2</font> | ? |\n",
    "| 6 | 1 | 0 | 4 | <font color='red'>4</font> | ? |\n",
    "| 7 | 1 | 1 | 8 | ? | <font color='red'>8</font> |\n",
    "| 8 | 1 | 1 | 6 | ? | <font color='red'>6</font> |\n",
    "\n",
    "*  Positivity: $0 < \\Pc {A_i=1} {X_{1,i},X_{2,i},\\cdots,X_{p,i}} < 1$, for all values of $X_{1,i},X_{2,i}, \\cdots, X_{p,i}$ that occur in the study population.\n",
    "    - In the example, $\\Pc{A_i=1}{X_i=0}=0.25$ and $\\Pc{A_i=1}{X_i=1}=0.5$, both arms exist on each level of $X$.\n",
    "\n",
    "*  Conditional Exchangeability (No Unmeasured Confounding Assumption, NUCA): $\\{Y_i^0, Y_i^1\\} \\ci A_i \\vert \\{X_{1,i},X_{2,i}, \\cdots, X_{p,i}\\}$\n",
    "    - $P(A_i, Y_i^{0}, Y_i^{1}| X_i) = P(A_i|X_i)P(Y_i^{0}, Y_i^{1}|X_i)$ or $E[Y_i^{1}|A_i=1, X_i] = E[Y_i^{1}| X_i] = E[Y_i^{1}|A_i=1, X_i]$, at given level of $X_i$, we may use information from one treatment arm to impute for the other arm.\n",
    "    - If we want to estimate the ATE using observational data, we need to use other information we have about the samples - specifically we need to **assume** that we have enough additional information to completely explain the choice of treatment each subject.\n",
    "    - This means that the observed treatment a sample receives, $A$, is completely explained by $X$. This is sometimes called the [\"ignorability\" assumption](https://en.wikipedia.org/wiki/Ignorability)."
   ]
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   "cell_type": "markdown",
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   "source": [
    "\n",
    "# Data Example: Causal Effect\n",
    "\n",
    "Often we do not care about the full distribution of the counterfactuals, and it is of more interest to estimate the [Average Treatment Effect (ATE)](https://en.wikipedia.org/wiki/Average_treatment_effect):\n",
    "\n",
    "$\\theta = E[Y^{1} - Y^{0}]$\n",
    "\n",
    "If we just try and estimate this quantity from the observational distribution, we get (the association):\n",
    "\n",
    "$\\alpha = E[Y|X=1] - E[Y|X=0]$\n",
    "\n",
    "If we model $Y$ against $A$, we get $\\alpha$, if we model $Y^a$ we get $\\theta$ (you may have missing data in this model)\n",
    "\n",
    "* When randomization occurs, $\\theta = \\alpha$\n",
    "* In observational studies, the two quantities are not equal in general.\n",
    "\n",
    "**This can be illustrated from the following data example:**\n",
    "\n",
    "Suppose we are interested in studying the effect of smoking on lung function. We collect information from $n=500$ subjects and record whether or not the person is a current and/or former smoker ($A=1$ for smoker, $A=0$ for non-smoker) at the begining of the study, and their lung function score $Y$ (can be considered as a continuous measure) at the end of study, and we also record a confounding variable $X$ (e.g. age or measure of certain biomarker), which affects both the smoking status and the lung function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "from sklearn.preprocessing import PolynomialFeatures\n",
    "\n",
    "import numpy as np\n",
    "np.warnings.filterwarnings('ignore')\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "sns.set_style(\"whitegrid\")\n",
    "sns.set_palette(\"colorblind\")\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the following code to generate a hypothetical dataset from such a study.\n",
    "\n",
    "In the simulation, for subject $i$ ($i=1,\\cdots,500$), we generate \n",
    "\n",
    "* the confounder $X_i \\sim N(0,1)$;\n",
    "* binary treatment indicator $A$ from $\\text{logit}[\\Pc{A_i=1}{X_i}]=1 + 2 X_i$;\n",
    "* the (true) counterfactuals: \n",
    "    - $Y^0_i = 2 X_i + \\epsilon_i$\n",
    "    - $Y^1_i = 0.5 + 2 X_i + \\epsilon_i$\n",
    "    - where $\\epsilon_i \\sim N(0,0.3^2)$\n",
    "* the observed outcome $Y_i = Y^1_i \\times A_i + Y^0_i \\times (1 - A_i)$"
   ]
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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>A</th>\n",
       "      <th>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>Y0</th>\n",
       "      <th>Y1</th>\n",
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       "  </thead>\n",
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       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>-0.852792</td>\n",
       "      <td>-0.733534</td>\n",
       "      <td>-1.233534</td>\n",
       "      <td>-0.733534</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1</td>\n",
       "      <td>0.509614</td>\n",
       "      <td>1.495843</td>\n",
       "      <td>0.995843</td>\n",
       "      <td>1.495843</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
       "      <td>-0.372404</td>\n",
       "      <td>-0.632358</td>\n",
       "      <td>-0.632358</td>\n",
       "      <td>-0.132358</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1</td>\n",
       "      <td>0.352165</td>\n",
       "      <td>1.102453</td>\n",
       "      <td>0.602453</td>\n",
       "      <td>1.102453</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0</td>\n",
       "      <td>-0.844309</td>\n",
       "      <td>-1.681160</td>\n",
       "      <td>-1.681160</td>\n",
       "      <td>-1.181160</td>\n",
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       "   A         X         Y        Y0        Y1\n",
       "0  1 -0.852792 -0.733534 -1.233534 -0.733534\n",
       "1  1  0.509614  1.495843  0.995843  1.495843\n",
       "2  0 -0.372404 -0.632358 -0.632358 -0.132358\n",
       "3  1  0.352165  1.102453  0.602453  1.102453\n",
       "4  0 -0.844309 -1.681160 -1.681160 -1.181160"
      ]
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   "source": [
    "\"\"\"\n",
    "Data Example 1: smoking\n",
    "  \n",
    "\"\"\"\n",
    "n_samples = 500\n",
    "X = np.random.normal(size=n_samples)\n",
    "p_a = np.exp(1 + 2 * X)/(1 + np.exp(1 + 2 * X))\n",
    "A = np.random.binomial(n=1, p=p_a, size=n_samples)\n",
    "eY = 0.3 * np.random.normal(size=n_samples)\n",
    "Y0 = 2 * X + eY\n",
    "Y1 = Y0 + 0.5\n",
    "Y = np.where(A == 0, Y0, Y1)\n",
    "\n",
    "example_smk = pd.DataFrame({\"X\":X, \"A\":A, \"Y\":Y, \"Y0\":Y0, \"Y1\":Y1})\n",
    "example_smk.head()"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Can we conclude that smoking led to worse lung function outcome? We may estimate, $\\alpha$, the different in mean of the two treatment groups:"
   ]
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   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "estimated_effect: 2.754 ; standard_error: 0.278\n"
     ]
    }
   ],
   "source": [
    "\"\"\"\n",
    "Estimate the difference in means between two groups.\n",
    "  \n",
    "\"\"\"\n",
    "ctrl_grp = example_smk[example_smk.A == 0]\n",
    "trt_grp = example_smk[example_smk.A == 1]\n",
    " \n",
    "alpha = trt_grp.Y.mean() - ctrl_grp.Y.mean()\n",
    "alpha_err = 1.96 * np.sqrt(\n",
    "        trt_grp.Y.var() / trt_grp.shape[0] + \n",
    "        ctrl_grp.Y.var() / ctrl_grp.shape[0])\n",
    "  \n",
    "print(\"estimated_effect: {:.3f}\".format(alpha), \"; standard_error: {:.3f}\".format(alpha_err))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yet, the link we are seeing, is a correlation, but it does not neccessarily provide us the causal link. For this simulated dataset, we actually know both counterfactuals, so we can calculate ATE $\\theta$ as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "true ATE: 0.500\n"
     ]
    }
   ],
   "source": [
    "\"\"\"\n",
    "Estimate the causal treatment effect.\n",
    "  \n",
    "\"\"\"\n",
    "theta = example_smk.Y1.mean() - example_smk.Y0.mean()\n",
    "print(\"true ATE: {:.3f}\".format(theta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, we can see generally $\\theta \\neq \\alpha$\n",
    "\n",
    "As mentioned earlier, two related quantities for causal treatment effect that we can also consider are:\n",
    "\n",
    "* The average causal affect in the treated (ATT): $\\text{ATT} = E[Y_i^1 − Y_i^0\\,\\vert\\, A_i=1]$\n",
    "\n",
    "* The average causal affect in the untreated (ATU, or ATC as 'in controls'): $\\text{ATU} = E[Y_i^1 − Y_i^0\\,\\vert\\, A_i=0]$\n",
    "\n",
    "One way to interpret ATU is as a measure of the effect of treating only samples which wouldn't naturally be treated, and vice versa for ATT. Depending on the goal of your study, they may be more natural measures of what you care about."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The counterfactual model\n",
    "\n",
    "However, in observational studies, we will not observe the counterfactuals as in this simulation example, counterfactual models provide tools to estimate the counterfactuals, as well as the causal quantities that related to them. For example, we can build estimators: \n",
    " \n",
    " - $\\hat{Y}^{0}(X) = E[Y|X, A=0]$\n",
    " - $\\hat{Y}^{1}(X) = E[Y|X, A=1]$. \n",
    " \n",
    "If we can model these two quantities, we can estimate the ATE as:\n",
    "\n",
    "$\\hat\\theta = \\frac{1}{N}\\sum_{i=1}^N [\\hat{Y}^{1}(x_{i}) - \\hat{Y}^{0}(x_{i})]$\n",
    "\n",
    "The success of this approach depends on how well we can model the potential outcomes.\n",
    "\n",
    "Counterfactual models:\n",
    "\n",
    "* Postulates a joint distribution over outcomes in experimental and observational settings\n",
    "\n",
    "* Typically, experimental outcomes are ‘primary’, of which observational outcomes are deterministic functions\n",
    "\n",
    "* Rich language allowing many quantities of interest to be formulated, e.g. ETT, NDE\n",
    "\n",
    "* ‘Reduces’ Causation to Missing Data; all outcomes ‘observable’ a priori\n",
    "    Allows precise characterization of identification assumptions as conditional independence"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Modeling the Counterfactual\n",
    "\n",
    "Here we will discuss some of the causal inference tools for modeling the counterfactuals:\n",
    "\n",
    "* regression based method\n",
    "\n",
    "* matching based method\n",
    "\n",
    "* weighting based method\n",
    "\n",
    "Next, we will discuss each method and see them in action in our data example:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Regression based method\n",
    "\n",
    "If we are able to build a good regression model for the counterfactual outcomes (based on the observational data):\n",
    "* $\\hat{Y}^{0}(X) = f_0(X)$\n",
    "* $\\hat{Y}^{1}(X) = f_1(X)$\n",
    "\n",
    "then we can impute using these models, for example\n",
    "\n",
    "| index $i$ | $X_i$ | $A_i$ | $Y_i$ | $Y_i^0$ (regression) | $Y_i^1$ (regression) |\n",
    "| --- | --- | --- | --- | --- | --- | \n",
    "| 1 | 0 | 0 | 4 | <font color='red'>4</font> | <font color='green'>$f_1(0)$</font> |\n",
    "| 2 | 0 | 0 | 3 | <font color='red'>7</font> | <font color='green'>$f_1(0)$</font> |\n",
    "| 3 | 0 | 0 | 5 | <font color='red'>2</font> | <font color='green'>$f_1(0)$</font> |\n",
    "| 4 | 0 | 1 | 8 | <font color='green'>$f_0(0)$</font> | <font color='red'>8</font> |\n",
    "| 5 | 1 | 0 | 2 | <font color='red'>2</font> | <font color='green'>$f_1(1)$</font> |\n",
    "| 6 | 1 | 0 | 4 | <font color='red'>4</font> | <font color='green'>$f_1(1)$</font> |\n",
    "| 7 | 1 | 1 | 8 | <font color='green'>$f_0(1)$</font> | <font color='red'>8</font> |\n",
    "| 8 | 1 | 1 | 6 | <font color='green'>$f_0(1)$</font> | <font color='red'>6</font> |\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our data example, we can first look at the data. First, scatter plot on the observed relationship of $X$ and $Y$"
   ]
  },
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1cf3a45ac18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "example_smk.plot.scatter(x=\"X\", y=\"Y\", c=\"A\", cmap=\"rainbow\", colorbar=False);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we can look at how $Y$ is distributed between the two exposure groups:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1cf3a829358>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1cf3a8187f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.kdeplot(example_smk.loc[lambda df: df.A == 0].Y, label=\"non-smoker\")\n",
    "sns.kdeplot(example_smk.loc[lambda df: df.A == 1].Y, label=\"smoker\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we look at the distribution of the independent variable, $X$, it is clear that there is also differences between the two groups. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x1cf3a770278>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1cf3a45a8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sns.kdeplot(example_smk.loc[lambda df: df.A == 0].X, label=\"non-smoker\")\n",
    "sns.kdeplot(example_smk.loc[lambda df: df.A == 1].X, label=\"smoker\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need some way to disentangle the effect of $A$ on $Y$ and the effect of $X$ on $Y$.\n",
    "\n",
    "The simplest type of model we can use is a linear model. Specifically we could assume \n",
    "\n",
    "$Y^{0}_i = \\beta_{0} + \\beta_{1} X + \\epsilon_i$\n",
    "\n",
    "$Y^{1}_i = Y^{0}_i + \\theta$\n",
    "\n",
    "If these models are correctly specified, we can fit the following model to the observational data\n",
    "\n",
    "$Y_i = \\beta_0 + \\beta_1 X_i + \\theta A_i + \\epsilon_i$\n",
    "\n",
    "gives us an estimate of the ATE.\n",
    "\n",
    "The [`causalinference`](https://pypi.org/project/CausalInference/) package gives us a simple interface to do this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Treatment Effect Estimates: OLS\n",
      "\n",
      "                     Est.       S.e.          z      P>|z|      [95% Conf. int.]\n",
      "--------------------------------------------------------------------------------\n",
      "           ATE      0.484      0.033     14.866      0.000      0.420      0.547\n",
      "\n"
     ]
    }
   ],
   "source": [
    "from causalinference import CausalModel\n",
    "\n",
    "cm = CausalModel(\n",
    "    Y=example_smk.Y.values, \n",
    "    D=example_smk.A.values, \n",
    "    X=example_smk.X.values)\n",
    "\n",
    "cm.est_via_ols(adj=1)\n",
    "print(cm.estimates)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The regression based method yields a good estimate of ATE, which suggested that this is a good way to go.\n",
    "\n",
    "##### Does regression based method always work? Hint: what assumptions you have here? \n",
    "\n",
    "* Are the common assumptions we need for estimating the missing counterfactuals enough?\n",
    "* Wait... What are those common assumptions?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The role of correct model specification\n",
    "\n",
    "The regression based method works only when \"you have a good model\". For example, for a subject who was actually assigned to the treatment arm:\n",
    "\n",
    "* What is missing and need to be imputed?\n",
    "\n",
    "* What you suggested the missing part be $\\hat{Y}^{0}(X) = f_0(X)$\n",
    "\n",
    "* Is it good? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>A</th>\n",
       "      <th>X</th>\n",
       "      <th>Y</th>\n",
       "      <th>Y0</th>\n",
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       "      <td>2.416946</td>\n",
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       "      <td>2.416946</td>\n",
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       "      <td>-0.392514</td>\n",
       "      <td>-0.392514</td>\n",
       "      <td>0.107486</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1</td>\n",
       "      <td>0.256707</td>\n",
       "      <td>0.511234</td>\n",
       "      <td>0.011234</td>\n",
       "      <td>0.511234</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0</td>\n",
       "      <td>-1.683440</td>\n",
       "      <td>5.894063</td>\n",
       "      <td>5.894063</td>\n",
       "      <td>6.394063</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   A         X         Y        Y0        Y1\n",
       "0  1 -0.980555  2.416946  1.916946  2.416946\n",
       "1  0  0.523839  0.039191  0.039191  0.539191\n",
       "2  0  0.143830 -0.392514 -0.392514  0.107486\n",
       "3  1  0.256707  0.511234  0.011234  0.511234\n",
       "4  0 -1.683440  5.894063  5.894063  6.394063"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "modified version of smoking example\n",
    "  \n",
    "\"\"\"\n",
    "n_samples = 500\n",
    "X = np.random.normal(size=n_samples)\n",
    "p_a = np.exp(1 + 2 * X)/(1 + np.exp(1 + 2 * X))\n",
    "A = np.random.binomial(n=1, p=p_a, size=n_samples)\n",
    "eY = 0.3 * np.random.normal(size=n_samples)\n",
    "Y0 = 2 * X * X + eY\n",
    "Y1 = Y0 + 0.5\n",
    "Y = np.where(A == 0, Y0, Y1)\n",
    "\n",
    "eg_smk2 = pd.DataFrame({\"X\":X, \"A\":A, \"Y\":Y, \"Y0\":Y0, \"Y1\":Y1})\n",
    "eg_smk2.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Treatment Effect Estimates: OLS\n",
      "\n",
      "                     Est.       S.e.          z      P>|z|      [95% Conf. int.]\n",
      "--------------------------------------------------------------------------------\n",
      "           ATE     -0.358      0.318     -1.128      0.259     -0.981      0.264\n",
      "\n"
     ]
    }
   ],
   "source": [
    "cm = CausalModel(\n",
    "    Y=eg_smk2.Y.values, \n",
    "    D=eg_smk2.A.values, \n",
    "    X=eg_smk2.X.values)\n",
    "\n",
    "cm.est_via_ols(adj=1)\n",
    "print(cm.estimates)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Seems like it is far away from the truth, the treatment if harmful! \n",
    "\n",
    "However, the reality is that you would never know which is the true model, because [\"all models are wrong but some are useful\"](https://en.wikipedia.org/wiki/All_models_are_wrong). \n",
    "\n",
    "Next, we will discuss some alternative approaches that can work free of this \"unrealistic\" assumption."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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