{
 "cells": [
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Structural Causal Models and Parameters\n",
    "\n",
    "## Lab 03 for PH 290: Targeted Learning in Biomedical Big Data\n",
    "\n",
    "### Author: [Nima Hejazi](https://nimahejazi.org)\n",
    "\n",
    "### Date: 31 January 2018\n",
    "\n",
    "### Attribution: adapted from source materials by [David Benkeser](https://www.benkeserstatistics.com/)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "── Attaching packages ─────────────────────────────────────── tidyverse 1.2.1 ──\n",
      "✔ ggplot2 2.2.1.9000     ✔ purrr   0.2.4     \n",
      "✔ tibble  1.4.2          ✔ dplyr   0.7.4     \n",
      "✔ tidyr   0.7.2.9000     ✔ stringr 1.2.0.9000\n",
      "✔ readr   1.1.1          ✔ forcats 0.2.0     \n",
      "── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──\n",
      "✖ dplyr::filter() masks stats::filter()\n",
      "✖ dplyr::lag()    masks stats::lag()\n"
     ]
    }
   ],
   "source": [
    "options(repr.plot.width = 4, repr.plot.height = 3)  ## resizing plots\n",
    "options(scipen = 999)  ## has scientific notation ever annoyed you?\n",
    "library(tidyverse)\n",
    "library(ggsci)\n",
    "set.seed(76548208)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## I. Introduction\n",
    "\n",
    "In this lab, we review how to simulate from structural causal models (SCMs) and how to approximate causal parameters based on SCMs. We also illustrate how to compare causal parameters to observed data parameters to verify identifiability results discussed in class."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the data structure $X = (W_1,W_2,A,Y)$ and $U = (U_{W,1}, U_{W,2},U_{A},U_Y)$. The first exercise will show how to explicitly simulate data from a structural model. Our causal model stipulates that $(U,X) \\sim P_{U,X}$ for some $P_{U,X} \\in \\mathcal{M}^X$, a model that assumes that $U_A \\perp U_Y$, but otherwise makes no assumptions about the distribution of $(U,X)$. One (of many) such $P_{U,X}$ that is in this model is given by the error distributions \\begin{align*}\n",
    "U_{W,1} &\\sim \\mbox{Bernoulli}(1/2) \\\\\n",
    "U_{W,2} &\\sim \\mbox{Bernoulli}(1/2) \\\\\n",
    "U_A &\\sim \\mbox{Normal}(0,1) \\\\\n",
    "U_Y &\\sim \\mbox{Normal}(0,1) \\ ,\n",
    "\\end{align*}\n",
    "and structural equations \\begin{align*}\n",
    "f_{W,1}(U_{W,1}) &= U_{W,1} \\\\\n",
    "f_{W,2}(U_{W,2}) &= U_{W,2} \\\\\n",
    "f_A(W_1, W_2, U_A) &= I(\\mbox{expit}(W_1 - W_2 + U_A) > 0.5)\\\\\n",
    "f_Y(W_1, W_2, A, U_Y) &= -W_1 + W_2 + A - U_Y \\ . \n",
    "\\end{align*}\n",
    "\n",
    "Let's think about how we can explicity code this distribution in R. Let's first write a function to represent each structural equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# structural equation for W_1\n",
    "# takes as input a vector U_W1 and returns a vector evaluating\n",
    "# f_{W,1}(U_W1)\n",
    "f_W1 <- function(U_W1){\n",
    "    return(U_W1)\n",
    "}\n",
    "\n",
    "# structural equation for W_2\n",
    "# takes as input a vector U_W2 and returns a vector evaluating\n",
    "# f_{W,2}(U_W2)\n",
    "f_W2 <- function(U_W2){\n",
    "    return(U_W2)\n",
    "}\n",
    "\n",
    "# structural equation for A\n",
    "f_A <- function(W_1, W_2, U_A){\n",
    "    return(as.numeric(plogis(W_1 - W_2 + U_A) > 0.5))\n",
    "}\n",
    "\n",
    "# structural equation for Y\n",
    "f_Y <- function(W_1, W_2, A, U_Y){\n",
    "    return(-W_1 + W_2 + A - U_Y)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now define a function to generate an observation from this SCM."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>W_1</th><th scope=col>W_2</th><th scope=col>A</th><th scope=col>Y</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0         </td><td>0         </td><td>1         </td><td> 1.3935595</td></tr>\n",
       "\t<tr><td>1         </td><td>0         </td><td>0         </td><td>-0.1991701</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>0         </td><td>-0.7243135</td></tr>\n",
       "\t<tr><td>0         </td><td>0         </td><td>0         </td><td>-0.6034721</td></tr>\n",
       "\t<tr><td>0         </td><td>0         </td><td>1         </td><td> 2.7803916</td></tr>\n",
       "\t<tr><td>1         </td><td>0         </td><td>1         </td><td> 0.7458899</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " W\\_1 & W\\_2 & A & Y\\\\\n",
       "\\hline\n",
       "\t 0          & 0          & 1          &  1.3935595\\\\\n",
       "\t 1          & 0          & 0          & -0.1991701\\\\\n",
       "\t 1          & 1          & 0          & -0.7243135\\\\\n",
       "\t 0          & 0          & 0          & -0.6034721\\\\\n",
       "\t 0          & 0          & 1          &  2.7803916\\\\\n",
       "\t 1          & 0          & 1          &  0.7458899\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "W_1 | W_2 | A | Y | \n",
       "|---|---|---|---|---|---|\n",
       "| 0          | 0          | 1          |  1.3935595 | \n",
       "| 1          | 0          | 0          | -0.1991701 | \n",
       "| 1          | 1          | 0          | -0.7243135 | \n",
       "| 0          | 0          | 0          | -0.6034721 | \n",
       "| 0          | 0          | 1          |  2.7803916 | \n",
       "| 1          | 0          | 1          |  0.7458899 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  W_1 W_2 A Y         \n",
       "1 0   0   1  1.3935595\n",
       "2 1   0   0 -0.1991701\n",
       "3 1   1   0 -0.7243135\n",
       "4 0   0   0 -0.6034721\n",
       "5 0   0   1  2.7803916\n",
       "6 1   0   1  0.7458899"
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   "source": [
    "sim_obs_scm <- function(n) {\n",
    "    ##############################################\n",
    "    # function to draw n observations from an scm\n",
    "    # n = the number of observations to draw\n",
    "    # returns a data.frame with named columns\n",
    "    ##############################################\n",
    "\n",
    "    # first we draw the errors\n",
    "    ## draw U_{W,1}\n",
    "    U_W1 <- rbinom(n,1,0.5)\n",
    "    ## draw U_{W,2}\n",
    "    U_W2 <- rbinom(n,1,0.5)\n",
    "    ## draw U_A\n",
    "    U_A <- rnorm(n,0,1)\n",
    "    ## draw U_Y\n",
    "    U_Y <- rnorm(n,0,1)\n",
    "\n",
    "    # now we can evaluate the observations sequentially\n",
    "    ## evaluate W_1\n",
    "    W_1 <- f_W1(U_W1)\n",
    "    ## evaluate W_2\n",
    "    W_2 <- f_W2(U_W2)\n",
    "    ## evaluate A\n",
    "    A <- f_A(W_1 = W_1, W_2 = W_2, U_A = U_A)\n",
    "    ## evaluate Y\n",
    "    Y <- f_Y(W_1 = W_1, W_2 = W_2, A = A, U_Y = U_Y)\n",
    "\n",
    "    # return a tibble\n",
    "    out <- as_tibble(list(W_1 = W_1, W_2 = W_2, A = A, Y = Y))\n",
    "    return(out)\n",
    "}\n",
    "\n",
    "# try it out \n",
    "sim_test <- sim_obs_scm(n = 100)\n",
    "head(sim_test)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above function generates a set of $n$ observations that might be observed in pratice were this $P_{X,U}$ to be the true data generating distribution. Because we are simulating the data, we can also perform manipulations on this SCM, as we would like to (but may be unable to) do in practice. \n",
    "\n",
    "Let's now define a function that allows us to set the value of $A$ to either 0 or 1 and returns a data set of counterfactual observations, i.e., observations we would have seen if we were able to manipulate the SCM. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>W_1</th><th scope=col>W_2</th><th scope=col>A</th><th scope=col>Y_1</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0         </td><td>1         </td><td>1         </td><td> 1.3922722</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>1         </td><td> 1.8913405</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>1         </td><td>-0.3918730</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>1         </td><td> 1.7296904</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>1         </td><td> 2.0408356</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>1         </td><td>-0.4072731</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " W\\_1 & W\\_2 & A & Y\\_1\\\\\n",
       "\\hline\n",
       "\t 0          & 1          & 1          &  1.3922722\\\\\n",
       "\t 1          & 1          & 1          &  1.8913405\\\\\n",
       "\t 1          & 1          & 1          & -0.3918730\\\\\n",
       "\t 1          & 1          & 1          &  1.7296904\\\\\n",
       "\t 1          & 1          & 1          &  2.0408356\\\\\n",
       "\t 1          & 1          & 1          & -0.4072731\\\\\n",
       "\\end{tabular}\n"
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       "\n",
       "W_1 | W_2 | A | Y_1 | \n",
       "|---|---|---|---|---|---|\n",
       "| 0          | 1          | 1          |  1.3922722 | \n",
       "| 1          | 1          | 1          |  1.8913405 | \n",
       "| 1          | 1          | 1          | -0.3918730 | \n",
       "| 1          | 1          | 1          |  1.7296904 | \n",
       "| 1          | 1          | 1          |  2.0408356 | \n",
       "| 1          | 1          | 1          | -0.4072731 | \n",
       "\n",
       "\n"
      ],
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       "  W_1 W_2 A Y_1       \n",
       "1 0   1   1  1.3922722\n",
       "2 1   1   1  1.8913405\n",
       "3 1   1   1 -0.3918730\n",
       "4 1   1   1  1.7296904\n",
       "5 1   1   1  2.0408356\n",
       "6 1   1   1 -0.4072731"
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     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>W_1</th><th scope=col>W_2</th><th scope=col>A</th><th scope=col>Y_0</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1         </td><td>0         </td><td>0         </td><td>-0.3711431</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>0         </td><td>-0.8501863</td></tr>\n",
       "\t<tr><td>0         </td><td>0         </td><td>0         </td><td> 0.6210759</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>0         </td><td> 1.6312533</td></tr>\n",
       "\t<tr><td>0         </td><td>0         </td><td>0         </td><td>-1.5470715</td></tr>\n",
       "\t<tr><td>1         </td><td>1         </td><td>0         </td><td>-1.5102034</td></tr>\n",
       "</tbody>\n",
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      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " W\\_1 & W\\_2 & A & Y\\_0\\\\\n",
       "\\hline\n",
       "\t 1          & 0          & 0          & -0.3711431\\\\\n",
       "\t 1          & 1          & 0          & -0.8501863\\\\\n",
       "\t 0          & 0          & 0          &  0.6210759\\\\\n",
       "\t 1          & 1          & 0          &  1.6312533\\\\\n",
       "\t 0          & 0          & 0          & -1.5470715\\\\\n",
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       "\n",
       "W_1 | W_2 | A | Y_0 | \n",
       "|---|---|---|---|---|---|\n",
       "| 1          | 0          | 0          | -0.3711431 | \n",
       "| 1          | 1          | 0          | -0.8501863 | \n",
       "| 0          | 0          | 0          |  0.6210759 | \n",
       "| 1          | 1          | 0          |  1.6312533 | \n",
       "| 0          | 0          | 0          | -1.5470715 | \n",
       "| 1          | 1          | 0          | -1.5102034 | \n",
       "\n",
       "\n"
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       "  W_1 W_2 A Y_0       \n",
       "1 1   0   0 -0.3711431\n",
       "2 1   1   0 -0.8501863\n",
       "3 0   0   0  0.6210759\n",
       "4 1   1   0  1.6312533\n",
       "5 0   0   0 -1.5470715\n",
       "6 1   1   0 -1.5102034"
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   "source": [
    "sim_int_scm <- function(n, set_A = 1) {\n",
    "    #########################################################\n",
    "    # function that draws n observations from an SCM that is \n",
    "    #   intervened on to set A = set_A\n",
    "    # n = number of observations\n",
    "    # set_A = the value to set A equal to (0 or 1)\n",
    "    # returns a tibble of coutnerfactual observations\n",
    "    #########################################################\n",
    "\n",
    "    # first we draw the errors\n",
    "    ## draw U_{W,1}\n",
    "    U_W1 <- rbinom(n,1,0.5)\n",
    "    ## draw U_{W,2}\n",
    "    U_W2 <- rbinom(n,1,0.5)\n",
    "    ## draw U_A\n",
    "    U_A <- rnorm(n,0,1)\n",
    "    ## draw U_Y\n",
    "    U_Y <- rnorm(n,0,1)\n",
    "\n",
    "    # now we can evaluate the observations sequentially\n",
    "    ## evaluate W_1\n",
    "    W_1 <- f_W1(U_W1)\n",
    "    ## evaluate W_2\n",
    "    W_2 <- f_W2(U_W2)\n",
    "    # we are now setting A = 1 for everyone\n",
    "    A <- rep(set_A, n)\n",
    "    ## evaluate Y with the set values of A\n",
    "    Y <- f_Y(W_1 = W_1, W_2 = W_2, A = A, U_Y = U_Y)\n",
    "\n",
    "    # return a tibble object\n",
    "    out <- as_tibble(list(W_1 = W_1, W_2 = W_2, A = A, Y = Y) )\n",
    "    ## lets rename the Y column to reflect the intervention\n",
    "    colnames(out)[4] <- paste0(\"Y_\",set_A)\n",
    "    return(out)\n",
    "}\n",
    "\n",
    "# try it out\n",
    "sim_test_a_is_1 <- sim_int_scm(n = 100, set_A = 1)\n",
    "head(sim_test_a_is_1)\n",
    "\n",
    "# and with setA = 0\n",
    "sim_test_a_is_0 <- sim_int_scm(n = 100, set_A = 0)\n",
    "head(sim_test_a_is_0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's think about the distribution of the counterfactual random variables $Y_a, a = 0,1$ implied by an intervention on the SCM setting $A=a$. We can use a large sample to approximate this distribution and plot an approximation with a histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
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    {
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VmWiwRIgARIwNMEuAzJ09XDzJEACcRCwL68iltTxkKPflNNgBpwqokzPRIgARIg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\nWvr16ycjRoyQnJwcmTZ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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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UZ7qjEQgAAEykUABVwCst62H7v77rvd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\nBCAAAQhAoAoE/h9SwxPJ1KowBwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# simulate a big data set from he SCM on which we intervened to set A = 0\n",
    "big_int_0 <- sim_int_scm(n = 1e5, set_A = 0)\n",
    "\n",
    "# plot the distribution of Y_0\n",
    "p_y0 <- ggplot(big_int_0, aes(x = Y_0)) +\n",
    "  geom_histogram(aes(y = ..count..), fill = \"blue\", binwidth = 0.1, alpha = 0.8) +\n",
    "  theme_bw() +\n",
    "  xlab(\"Y_0\") +\n",
    "  ggtitle(\"Distribution of Y_0\")\n",
    "\n",
    "# simulate a big data set from the SCM on which we intervened to set A = 1\n",
    "big_int_1 <- sim_int_scm(n = 1e5, set_A = 1)\n",
    "\n",
    "# plot the distribution of Y_1\n",
    "p_y1 <- ggplot(big_int_1, aes(x = Y_1)) +\n",
    "  geom_histogram(aes(y = ..count..), fill = \"blue\", binwidth = 0.1, alpha = 0.8) +\n",
    "  theme_bw() +\n",
    "  xlab(\"Y_1\") +\n",
    "  ggtitle(\"Distribution of Y_1\")\n",
    "\n",
    "# approximate the (counterfactual) mean of Y_0\n",
    "cf_mean_y0 <- mean(big_int_0$Y_0)\n",
    "## let's add it to the histogram\n",
    "p_y0 <- p_y0 + geom_vline(xintercept = cf_mean_y0, colour = \"red\")\n",
    "p_y0\n",
    "\n",
    "# approximate the (counterfactual) mean of Y_1\n",
    "cf_mean_y1 <- mean(big_int_1$Y_1)\n",
    "## let's add it to the histogram\n",
    "p_y1 <- p_y1 + geom_vline(xintercept = cf_mean_y1, colour = \"red\")\n",
    "p_y1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider computing the average treatment effect under $P_{U,X}$, $$\n",
    "\\Psi_{cf}(P_{U,X}) = E_{P_{U,X}}\\{ f_Y(w_1, w_2, 1, u_Y) - E_{P_{U,X}}\\{f_Y(w_1, w_2, 0, u_Y) \\} \\ . \n",
    "$$\n",
    "This quantity can be approximated using our simulated values as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.00320668516922"
      ],
      "text/latex": [
       "1.00320668516922"
      ],
      "text/markdown": [
       "1.00320668516922"
      ],
      "text/plain": [
       "[1] 1.003207"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cf_mean_y1 - cf_mean_y0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's take a look at the distribution of the observed $Y$ implied by this SCM. In particular, let's consider the distribution of $Y$ conditional on $A=a$ for $a=0,1$. A conditional distribution is nothing more than the distribution of a variable in a subset of the population. Here that subset is defined by the observed treatment level."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.336417832929758"
      ],
      "text/latex": [
       "0.336417832929758"
      ],
      "text/markdown": [
       "0.336417832929758"
      ],
      "text/plain": [
       "[1] 0.3364178"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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WncsHaAhgRSEeAirFR0vnZTb/36CgoS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\nMvKoEbCfzxs1+Slv4RC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     "metadata": {},
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bqnhpHfG2+8MdXKzTPvtNNO7kFKc8hzowcvbqa+vj5uIoXmWooMqK7FyajhVH1q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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# simulate large data set\n",
    "big_obs <- sim_obs_scm(n = 1e5)\n",
    "\n",
    "# plot a histogram of the conditional distribution of Y given A = 0 \n",
    "py_a_is_0 <- big_obs %>%\n",
    "  dplyr::filter(A == 0) %>%\n",
    "  ggplot(., aes(x = Y)) +\n",
    "    geom_histogram(aes(y = ..count..), fill = \"blue\", binwidth = 0.1, alpha = 0.8) +\n",
    "    theme_bw() +\n",
    "    xlab(\"Y\") +\n",
    "    ggtitle(\"Conditional dist. of Y | A = 0\")\n",
    "\n",
    "# plot a histogram of the conditional distribution of Y given A = 1\n",
    "py_a_is_1 <- big_obs %>%\n",
    "  dplyr::filter(A == 1) %>%\n",
    "  ggplot(., aes(x = Y)) +\n",
    "    geom_histogram(aes(y = ..count..), fill = \"blue\", binwidth = 0.1, alpha = 0.8) +\n",
    "    theme_bw() +\n",
    "    xlab(\"Y\") +\n",
    "    ggtitle(\"Conditional dist. of Y | A = 1\")\n",
    "\n",
    "# approximate the conditional mean of Y | A = 0 \n",
    "cond_mean_Y0 <- big_obs %>%\n",
    "  dplyr::filter(A == 0) %>%\n",
    "  summarise(mean(Y)) %>%\n",
    "  as.numeric()\n",
    "## let's add it to the histogram\n",
    "py_a_is_0 <- py_a_is_0 + geom_vline(xintercept = cond_mean_Y0, colour = \"red\")\n",
    "py_a_is_0\n",
    "cond_mean_Y0\n",
    "\n",
    "# approximate the conditional mean of Y | A = 0 \n",
    "cond_mean_Y1 <- big_obs %>%\n",
    "  dplyr::filter(A == 1) %>%\n",
    "   summarise(mean(Y)) %>%\n",
    "   as.numeric()\n",
    "## let's add it to the histogram\n",
    "py_a_is_1 <- py_a_is_1 + geom_vline(xintercept = cond_mean_Y1, colour = \"red\")\n",
    "py_a_is_1\n",
    "cond_mean_Y1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we take the mean of the large (\"infinite\") data set we are approximating the parameter $$ \n",
    "E_P(Y \\mid A = a)\n",
    "$$\n",
    "with $$\n",
    "\\frac{1}{N_a}\\sum\\limits_{i : A_i = a} Y_i \\ ,\n",
    "$$\n",
    "where $N_a$ is the number of observations with $A = a$. If $n$ is chosen to be very large, this approximation should be very close to the true integral. \n",
    "\n",
    "Consider the parameter that I'll call the \"naive causal effect\":\n",
    "$$\n",
    "\\Psi_{naive}(P) = E_P(Y \\mid A=1) - E_P(Y \\mid A=0)\n",
    "$$\n",
    "Having evaluated the above code, we can approximate the value of the parameter at this particular $P$ as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.320453538187597"
      ],
      "text/latex": [
       "0.320453538187597"
      ],
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       "0.320453538187597"
      ],
      "text/plain": [
       "[1] 0.3204535"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cond_mean_Y1 - cond_mean_Y0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the naive causal effect does not equal the true causal effect even at this effectively infinite sample size. That is, the parameter $\\Psi_{naive}$ does not identify $\\Psi_{cf}$. This should come as no surprise, as we can see that there is confounding in this example. That is, $W_1$ and $W_2$ affect both the probability of having $A = 1$, as well as the value of the outcome $Y$. Let's now explicity confirm that the parameter\n",
    "$$\n",
    "\\Psi(P) = E\\{E(Y \\mid A=1, W) - E(Y \\mid A=0, W)\\}\n",
    "$$\n",
    "does indeed identify the causal effect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>ate_w</th><th scope=col>p_w</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1.0222572</td><td>0.24952  </td></tr>\n",
       "\t<tr><td>1.0205086</td><td>0.24973  </td></tr>\n",
       "\t<tr><td>0.9859355</td><td>0.25136  </td></tr>\n",
       "\t<tr><td>1.0020733</td><td>0.24939  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " ate\\_w & p\\_w\\\\\n",
       "\\hline\n",
       "\t 1.0222572 & 0.24952  \\\\\n",
       "\t 1.0205086 & 0.24973  \\\\\n",
       "\t 0.9859355 & 0.25136  \\\\\n",
       "\t 1.0020733 & 0.24939  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "ate_w | p_w | \n",
       "|---|---|---|---|\n",
       "| 1.0222572 | 0.24952   | \n",
       "| 1.0205086 | 0.24973   | \n",
       "| 0.9859355 | 0.25136   | \n",
       "| 1.0020733 | 0.24939   | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  ate_w     p_w    \n",
       "1 1.0222572 0.24952\n",
       "2 1.0205086 0.24973\n",
       "3 0.9859355 0.25136\n",
       "4 1.0020733 0.24939"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# write a loop to get mean of Y | A = 1, W = w minus\n",
    "# mean of Y | A = 0, W = w in each strata of W\n",
    "out <- NULL\n",
    "for (w1 in c(0, 1)) {\n",
    "    for (w2 in c(0, 1)) {\n",
    "        ate_this_W <- with(big_obs,\n",
    "            mean(Y[A == 1 & W_1 == w1 & W_2 == w2]) -\n",
    "                mean(Y[A == 0 & W_1 == w1 & W_2 == w2])\n",
    "            )\n",
    "        p_this_W <- with(big_obs,\n",
    "            sum(W_1 == w1 & W_2 == w2) / nrow(big_obs)\n",
    "            )\n",
    "        out <- rbind(out, c(ate_this_W, p_this_W))\n",
    "    }\n",
    "}\n",
    "out <- as_tibble(out)\n",
    "colnames(out) <- c(\"ate_w\",\"p_w\")\n",
    "\n",
    "# let's look at the matrix we just made\n",
    "out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrix `out` contains a column that shows the average treatment effect in each strata, as well as the proportion of observations falling in that strata. A few observations stand out:\n",
    "\n",
    "- The average causal effect in each strata is equal to the overall average causal effect. Why? \n",
    "\n",
    "- The proportion falling in each strata is approximately equal. Why?\n",
    "\n",
    "We can use these values to get back the overall causal effect:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.00765702853328"
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       "1.00765702853328"
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       "1.00765702853328"
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      "text/plain": [
       "[1] 1.007657"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sum(out$ate_w * out$p_w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## III. Optimal Treatment Rules\n",
    "\n",
    "Let's consider a slight modification of the first example\n",
    "\\begin{align*}\n",
    "U_{W,1} &\\sim \\mbox{Bernoulli}(1/2) \\\\\n",
    "U_{W,2} &\\sim \\mbox{Bernoulli}(1/2) \\\\\n",
    "U_A &\\sim \\mbox{Normal}(0,1) \\\\\n",
    "U_Y &\\sim \\mbox{Normal}(0,1) \\ ,\n",
    "\\end{align*}\n",
    "and structural equations \\begin{align*}\n",
    "f_{W,1}(U_{W,1}) &= U_{W,1} \\\\\n",
    "f_{W,2}(U_{W,2}) &= U_{W,2} \\\\\n",
    "f_A(W_1, W_2, U_A) &= I(\\mbox{expit}(W_1 - W_2 + U_A) > 0.5)\\\\\n",
    "f_Y(W_1, W_2, A, U_Y) &= 2W_1 - 3 W_1W_2 A + W_2 A - A + U_Y\\ . \n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# this uses the same functions defined in section I above\n",
    "# for f_W1, f_W2, f_A, but we need to re-define f_Y\n",
    "f_Y <- function(W_1, W_2, A, U_Y) {\n",
    "    W_1 + W_2 - W_1 * W_2 * A + 3 * W_2 * A - A + U_Y\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we will study intervening in a graph according to a specified treatment rule. There are four possible covariate strata: $(W_1, W_2) \\in \\{ (0,0), (0,1), (1,0), (1,1) \\}$.\n",
    "\n",
    "Let's define a new variable\n",
    "$$\n",
    "W_S = f_S(W_1, W_2) = I(W_1 = 0, W_2 = 0) + 2 I(W_1 = 0, W_2 = 1) + 3 I(W_1 = 1, W_2 = 0) + 4 I(W_1 = 1, W_2 = 1) \\ . \n",
    "$$\n",
    "\n",
    "We are interested in the optimal treatment rule, which corresponds to picking a subset of the strata to treat. Let's write a function that can simulate from an intervened SCM according to a specified rule."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>W_1</th><th scope=col>W_2</th><th scope=col>W_S</th><th scope=col>A</th><th scope=col>Yd</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0         </td><td>0         </td><td>1         </td><td>1         </td><td>-1.7828683</td></tr>\n",
       "\t<tr><td>0         </td><td>0         </td><td>1         </td><td>1         </td><td>-0.8844762</td></tr>\n",
       "\t<tr><td>1         </td><td>0         </td><td>3         </td><td>0         </td><td> 0.3308235</td></tr>\n",
       "\t<tr><td>0         </td><td>1         </td><td>2         </td><td>0         </td><td> 1.6771381</td></tr>\n",
       "\t<tr><td>0         </td><td>0         </td><td>1         </td><td>1         </td><td>-1.9609462</td></tr>\n",
       "\t<tr><td>1         </td><td>0         </td><td>3         </td><td>0         </td><td> 2.0303488</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllll}\n",
       " W\\_1 & W\\_2 & W\\_S & A & Yd\\\\\n",
       "\\hline\n",
       "\t 0          & 0          & 1          & 1          & -1.7828683\\\\\n",
       "\t 0          & 0          & 1          & 1          & -0.8844762\\\\\n",
       "\t 1          & 0          & 3          & 0          &  0.3308235\\\\\n",
       "\t 0          & 1          & 2          & 0          &  1.6771381\\\\\n",
       "\t 0          & 0          & 1          & 1          & -1.9609462\\\\\n",
       "\t 1          & 0          & 3          & 0          &  2.0303488\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "W_1 | W_2 | W_S | A | Yd | \n",
       "|---|---|---|---|---|---|\n",
       "| 0          | 0          | 1          | 1          | -1.7828683 | \n",
       "| 0          | 0          | 1          | 1          | -0.8844762 | \n",
       "| 1          | 0          | 3          | 0          |  0.3308235 | \n",
       "| 0          | 1          | 2          | 0          |  1.6771381 | \n",
       "| 0          | 0          | 1          | 1          | -1.9609462 | \n",
       "| 1          | 0          | 3          | 0          |  2.0303488 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  W_1 W_2 W_S A Yd        \n",
       "1 0   0   1   1 -1.7828683\n",
       "2 0   0   1   1 -0.8844762\n",
       "3 1   0   3   0  0.3308235\n",
       "4 0   1   2   0  1.6771381\n",
       "5 0   0   1   1 -1.9609462\n",
       "6 1   0   3   0  2.0303488"
      ]
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     "data": {
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       "[1] TRUE"
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       "[1] TRUE"
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   ],
   "source": [
    "sim_rule_int_scm <- function(n, strata_tx) {\n",
    "    ################################################################\n",
    "    # define a function that takes as input\n",
    "    # n = the number of observations to simulate\n",
    "    # strataTrt = the values of W_s to treat \n",
    "    # return a tibble according to the rule-specific intervened SCM \n",
    "    ################################################################\n",
    "\n",
    "    # first we draw the errors\n",
    "    ## draw U_{W,1}\n",
    "    U_W1 <- rbinom(n, 1, 0.5)\n",
    "    ## draw U_{W,2}\n",
    "    U_W2 <- rbinom(n, 1, 0.5)\n",
    "    ## draw U_A\n",
    "    U_A <- rnorm(n, 0, 1)\n",
    "    ## draw U_Y\n",
    "    U_Y <- rnorm(n, 0, 1)\n",
    "\n",
    "    # now we can evaluate the observations sequentially\n",
    "    ## evaluate W_1\n",
    "    W_1 <- f_W1(U_W1)\n",
    "    ## evaluate W_2\n",
    "    W_2 <- f_W2(U_W2)\n",
    "    ## evaluate W_S\n",
    "    W_S <- as.numeric(I(W_1 == 0 & W_2 == 0) + 2 * I(W_1 == 0 & W_2 == 1) + \n",
    "        3 * I(W_1 == 1 & W_2 == 0) + 4 * I(W_1 == 1 & W_2 == 1))\n",
    "    # set A according to strataTrt\n",
    "    A <- as.numeric(W_S %in% strata_tx)\n",
    "\n",
    "    # evaluate Y with the set values of A\n",
    "    Y <- f_Y(W_1 = W_1, W_2 = W_2, A = A, U_Y = U_Y)\n",
    "\n",
    "    # return a tibble object\n",
    "    out <- as_tibble(list(W_1 = W_1, W_2 = W_2, W_S = W_S, A = A, Yd = Y) )\n",
    "    return(out)\n",
    "}\n",
    "\n",
    "# simulate data treating strata 1\n",
    "rule_only_1 <- sim_rule_int_scm(n = 100, strata_tx = 1)\n",
    "# let's take a look\n",
    "head(rule_only_1)\n",
    "\n",
    "# confirm that anyone with W_S == 1 has A = 1\n",
    "all(rule_only_1$A[rule_only_1$W_S == 1] == 1)\n",
    "# and that anyone without W_S == 1 has A = 0 \n",
    "all(rule_only_1$A[rule_only_1$W_S != 1] == 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now perform a grid search to determine the optimal treatment rule under this distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.744640697024526</li>\n",
       "\t<li>1.49553173170272</li>\n",
       "\t<li>0.74630593609533</li>\n",
       "\t<li>1.25039032058969</li>\n",
       "\t<li>1.25441661420076</li>\n",
       "\t<li>0.501774407421421</li>\n",
       "\t<li>0.995384427545952</li>\n",
       "\t<li>1.24687517908915</li>\n",
       "\t<li>1.74724049790319</li>\n",
       "\t<li>1.00260034722492</li>\n",
       "\t<li>1.00481997469567</li>\n",
       "\t<li>1.49896032989568</li>\n",
       "\t<li>0.75392361887479</li>\n",
       "\t<li>1.49986413206916</li>\n",
       "\t<li>1.24039538950792</li>\n",
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      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.744640697024526\n",
       "\\item 1.49553173170272\n",
       "\\item 0.74630593609533\n",
       "\\item 1.25039032058969\n",
       "\\item 1.25441661420076\n",
       "\\item 0.501774407421421\n",
       "\\item 0.995384427545952\n",
       "\\item 1.24687517908915\n",
       "\\item 1.74724049790319\n",
       "\\item 1.00260034722492\n",
       "\\item 1.00481997469567\n",
       "\\item 1.49896032989568\n",
       "\\item 0.75392361887479\n",
       "\\item 1.49986413206916\n",
       "\\item 1.24039538950792\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.744640697024526\n",
       "2. 1.49553173170272\n",
       "3. 0.74630593609533\n",
       "4. 1.25039032058969\n",
       "5. 1.25441661420076\n",
       "6. 0.501774407421421\n",
       "7. 0.995384427545952\n",
       "8. 1.24687517908915\n",
       "9. 1.74724049790319\n",
       "10. 1.00260034722492\n",
       "11. 1.00481997469567\n",
       "12. 1.49896032989568\n",
       "13. 0.75392361887479\n",
       "14. 1.49986413206916\n",
       "15. 1.24039538950792\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0.7446407 1.4955317 0.7463059 1.2503903 1.2544166 0.5017744 0.9953844\n",
       " [8] 1.2468752 1.7472405 1.0026003 1.0048200 1.4989603 0.7539236 1.4998641\n",
       "[15] 1.2403954"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<ol>\n",
       "\t<li><ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>3</li>\n",
       "</ol>\n",
       "</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate}\n",
       "\\item \\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 3\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\end{enumerate}\n"
      ],
      "text/markdown": [
       "1. 1. 1\n",
       "2. 3\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n"
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       "[[1]]\n",
       "[1] 1 3\n"
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# first create a list that contains all possible \n",
    "# subsets of strata. We do this using the combn function\n",
    "strata_list <- vector(mode = \"list\", length = 0)\n",
    "for(i in seq_len(max(rule_only_1$W_S))){\n",
    "    strata_list <- c(strata_list, \n",
    "        combn(1:4, i, simplify = FALSE))\n",
    "}\n",
    "\n",
    "# you can look at the list to see what this command\n",
    "# did (omitted here for space)\n",
    "#strata_list\n",
    "\n",
    "# write a function that computes E(Y_d)\n",
    "get_EYd <- function(strata, n = 1e5) {\n",
    "    # simulate data according to rule\n",
    "    dat <- sim_rule_int_scm(n = n, strata_tx = strata)\n",
    "    # calculate mean of Yd\n",
    "    return(mean(dat$Yd))\n",
    "}\n",
    "\n",
    "# apply the function over strataList\n",
    "all_EYd <- unlist(lapply(strata_list, get_EYd))\n",
    "# look at all the values for E(Y_d)\n",
    "all_EYd\n",
    "# optimal rule is the smallest value of E(Y_d)\n",
    "opt_index <- which(all_EYd == min(all_EYd))\n",
    "# what rule is it \n",
    "strata_list[opt_index]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's confirm the identification result given in the notes. We can define a function to simulate observed data from the SCM, that is, without intervening:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sim_rule_obs_scm <- function(n) {\n",
    "    # first we draw the errors\n",
    "    ## draw U_{W,1}\n",
    "    U_W1 <- rbinom(n, 1, 0.5)\n",
    "    ## draw U_{W,2}\n",
    "    U_W2 <- rbinom(n, 1, 0.5)\n",
    "    ## draw U_A\n",
    "    U_A <- rnorm(n, 0, 1)\n",
    "    ## draw U_Y\n",
    "    U_Y <- rnorm(n, 0, 1)\n",
    "\n",
    "    # now we can evaluate the observations sequentially\n",
    "    ## evaluate W_1\n",
    "    W_1 <- f_W1(U_W1)\n",
    "    ## evaluate W_2\n",
    "    W_2 <- f_W2(U_W2)\n",
    "    ## evaluate W_S\n",
    "    W_S <- as.numeric(I(W_1 == 0 & W_2 == 0) + 2 * I(W_1 == 0 & W_2 == 1) + \n",
    "        3 * I(W_1 == 1 & W_2 == 0) + 4 * I(W_1 == 1 & W_2 == 1))\n",
    "    # evaluate A according to f_A \n",
    "    A <- f_A(W_1 = W_1, W_2 = W_2, U_A = U_A)\n",
    "\n",
    "    # evaluate Y with the set values of A\n",
    "    Y <- f_Y(W_1 = W_1, W_2 = W_2, A = A, U_Y = U_Y)\n",
    "\n",
    "    ## return a data.frame object\n",
    "    out <- data.frame(W_1 = W_1, W_2 = W_2, W_S = W_S, A = A, Y = Y) \n",
    "    return(out)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now can simulate data from the observed data distribution implied by $P_{U,X}$. Recall from class that the optimal rule could be defined as $I(B_0(W) > 0)$, where $$\n",
    "B_0(W) = E(Y \\mid A=1, W) - E(Y \\mid A=0, W)\n",
    "$$ \n",
    "is the \"BLIP\" function. Let's now write a function that simulates a large observed data set and computes the BLIP function in each strata."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "get_blip <- function(strata, n = 1e6) {\n",
    "    # simulate data based on rule\n",
    "    dat <- sim_rule_obs_scm(n = n)\n",
    "    \n",
    "    EY_A1_WS <- dat %>%\n",
    "      dplyr::filter(W_S %in% strata & A == 1) %>%\n",
    "      summarise(mean(Y)) %>%\n",
    "      as.numeric()\n",
    "    \n",
    "    EY_A0_WS <- dat %>%\n",
    "      dplyr::filter(W_S %in% strata & A == 0) %>%\n",
    "      summarise(mean(Y)) %>%\n",
    "      as.numeric()\n",
    "\n",
    "    blip <- EY_A1_WS - EY_A0_WS \n",
    "    return(blip)\n",
    "}\n",
    "\n",
    "# now compute the blip for each strata\n",
    "all_blips <- unlist(lapply(split(1:4, 1:4), get_blip))\n",
    "all_blips\n",
    "\n",
    "# which strata have blip < 0\n",
    "as.numeric(which(all_blips < 0))"
   ]
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