pooling_samples.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Pooling strategies to reduce mean test counts.\n",
    "\n",
    "Pooling can be used to screen a large group of people all at one time for illness.  If the group comes back negative, then they can all be ruled out for having the disease after just one test.  If it comes back positive, refined testing of the group is needed to determine which are ill.  Here, we consider two alternatives for this refined search process:  (1) a bisection approach and (2) a simple approach. In the bisection method, we split the group into two and test each subgroup.  If either half tests negative, we can rule out that half.  Otherwise, we split it again in two, etc.  In this way we can identify exactly which people have the disease.  In the simple method, we simply skip to testing each individual when the initial group comes back positive. \n",
    "\n",
    "Our aim is to see which performs best in the sense of reducing the mean number of tests required.  This depends sensitively on the fraction of people in the testing population who have the disease.  If it is a large fraction, pooling does not help.  If p = 0.1, (perhaps typical for people who show up with symptoms), then we can cut the test count by about half and both methods work ok.  If p << 1, then we can get a dramatic reduction in test count needed and the bisection method works best.\n",
    "\n",
    "Jonathan Landy, March 29, 2020.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Code to get best pooling result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n",
      "Bisection: fraction of full testing: 0.43\n",
      "Simple: fraction of full testing: 0.84\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'mean tests / pooling size')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "%pylab inline\n",
    "\n",
    "K = 5\n",
    "P_POSITIVE = 0.05\n",
    "\n",
    "def theory_bisection(p=P_POSITIVE, K=K):\n",
    "    count = 1 + 2 * np.sum([2 ** k * (1 - (1 - p) ** (2 ** (K - k))) for k in range(K)] )\n",
    "    return count / 2 ** K\n",
    "\n",
    "def theory_simple(p=P_POSITIVE, K=K):\n",
    "    n = 2 ** K\n",
    "    p0 = (1 - p) ** n\n",
    "    count = 1 * p0 + (1 + n) * (1 - p0)\n",
    "    return count / n\n",
    "\n",
    "print 'Bisection: fraction of full testing: %2.2f' % (theory_bisection())\n",
    "print 'Simple: fraction of full testing: %2.2f' % (theory_simple())\n",
    "\n",
    "p = 0.001\n",
    "data = [theory_bisection(p, k) for k in range(15)]\n",
    "min_index = np.argmin(data)\n",
    "plot(data, 'o--', label='bisection (min = %2.2f)'%data[min_index], alpha=0.5)\n",
    "plot(min_index, data[min_index], 'ro',alpha=0.5)\n",
    "\n",
    "data = [theory_simple(p, k) for k in range(15)]\n",
    "min_index = np.argmin(data)\n",
    "plot(data, 'o--', label='simple (min = %2.2f)'%data[min_index],alpha=0.5)\n",
    "plot(min_index, data[min_index], 'go',alpha=0.5)\n",
    "plt.legend()\n",
    "plt.title('Test count reduction vs log_2 pooling size, p = %0.3f' %p)\n",
    "plt.xlabel('log_2 pooling size')\n",
    "plt.ylabel('mean tests / pooling size')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sample to get histogram, confirm theoretical forms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "total size = 512\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "19"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def test_count(sequence):\n",
    "    \"\"\"\n",
    "    Recursively check the number of tests required to find all\n",
    "    positive cases in a given sequence.\n",
    "    \n",
    "    inputs\n",
    "    ------\n",
    "    sequence : list\n",
    "        List of 0,1 values with 0 = no disease, 1 = disease.\n",
    "        The ith element is the hidden result for the ith person.\n",
    "    \"\"\"\n",
    "    #print sequence\n",
    "    test_size = len(sequence)\n",
    "    if np.log2(test_size) != np.log2(test_size) // 1.0:\n",
    "        msg = 'Sequence must have a power of two length.'\n",
    "        raise ValueError(msg) \n",
    "    \n",
    "    # if just one person left, must test\n",
    "    if test_size == 1:\n",
    "        return 1\n",
    "    \n",
    "    # more than one person case:\n",
    "    if 1 not in sequence:\n",
    "        # no positives on this test, so no 'children' tests\n",
    "        return 1\n",
    "    else:\n",
    "        # at least one positive, must test two children\n",
    "        left_count = test_count(sequence[:test_size//2])\n",
    "        right_count = test_count(sequence[test_size//2:])\n",
    "        return 1 + left_count + right_count\n",
    "\n",
    "# create an example sequence:\n",
    "K = 9\n",
    "seq = [0 for i in range(2 ** K)]\n",
    "seq[12] = 1\n",
    "print 'total size = %i' % len(seq)\n",
    "test_count(seq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "# now we will generate many fake samples with a given p value\n",
    "\n",
    "def generate_random_seq(p=P_POSITIVE, K=K):\n",
    "    a = [0, 1]\n",
    "    seq = np.random.choice(a=a, size=2**K, p=[1 - p, p])\n",
    "    return seq\n",
    "\n",
    "checks = 10000\n",
    "test_counts_needed = []\n",
    "for _ in range(checks):\n",
    "    seq = generate_random_seq(p=0.001, K=9)\n",
    "    test_counts_needed.append(test_count(seq))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Histogram of needed test counts, p=0.001, K=9')"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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MrNP8SWAzs0I5AMzMCuUAMDMrlAPAzKxQDgAzs0I5AMzMCuUAMDMrlAPAzKxQDgAzs0I5AMzMCuUAMDMrlAPAzKxQDgAzs0I5AMzMCuUAMDMrlAPAzKxQDgAzs0I5AMzMCuUAMDMrlAPAzKxQDgAzs0I5AMzMCuUAMDMrlAPAzKxQDgAzs0I5AMzMCtVyAEjaVtI9km7M83tKukPSKklfk7R9bt8hz6/K/X2VbZyT2x+SdGynd8bMzFrXziuAjwErKvMXARdHxN7AU8Ds3D4beCq3X5yXQ9J04FRgf+A44AuSth1e+WZmNlQtBYCkqcDbgS/neQFHA4vyIguAk/P0SXme3H9MXv4kYGFEvBgRvwRWAYd0YifMzKx9rb4CuAT4OPBynn8t8HREbMrza4ApeXoKsBog9z+Tl/9de4N1fkfSHEnLJC3r7+9vY1fMzKwdgwaApBOB9RFx9wjUQ0TMi4gZETFj4sSJI3GVZmZFGtPCMkcA75R0ArAjsCtwKTBO0pj8LH8qsDYvvxbYA1gjaQywG/Bkpb2muo6ZmY2wQV8BRMQ5ETE1IvpIb+LeEhGnA7cC78qLzQJuyNOL8zy5/5aIiNx+aj5LaE9gGnBnx/bEzMza0sorgGb+Blgo6QLgHuCK3H4FcLWkVcAGUmgQEcslXQc8AGwCzoyIl4Zx/WZmNgxtBUBE3AbclqcfpsFZPBHxG+DdTdb/DPCZdos0M7PO8yeBzcwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyvUoAEgaUdJd0r6qaTlkj6d2/eUdIekVZK+Jmn73L5Dnl+V+/sq2zontz8k6dgttVNmZja4MS0s8yJwdERslLQd8ANJ3wL+Erg4IhZK+iIwG7g8/30qIvaWdCpwEfBnkqYDpwL7A68Dvitpn4h4aQvslw1B39xvDnndRy58ewcrMbORMOgrgEg25tnt8iWAo4FFuX0BcHKePinPk/uPkaTcvjAiXoyIXwKrgEM6shdmZta2lt4DkLStpHuB9cAS4BfA0xGxKS+yBpiSp6cAqwFy/zPAa6vtDdapXtccScskLevv729/j8zMrCUtBUBEvBQRBwJTSc/a99tSBUXEvIiYEREzJk6cuKWuxsyseG2dBRQRTwO3AocD4yTV3kOYCqzN02uBPQBy/27Ak9X2BuuYmdkIa+UsoImSxuXpnYC3AStIQfCuvNgs4IY8vTjPk/tviYjI7afms4T2BKYBd3ZqR8zMrD2tnAU0GVggaVtSYFwXETdKegBYKOkC4B7girz8FcDVklYBG0hn/hARyyVdBzwAbALO9BlAZmbdM2gARMR9wEEN2h+mwVk8EfEb4N1NtvUZ4DPtl2lmZp3mTwKbmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRXKAWBmVigHgJlZoRwAZmaFcgCYmRVq0ACQtIekWyU9IGm5pI/l9t0lLZG0Mv8dn9sl6TJJqyTdJ+ngyrZm5eVXSpq15XbLzMwG08orgE3A2RExHTgMOFPSdGAusDQipgFL8zzA8cC0fJkDXA4pMIBzgUOBQ4Bza6FhZmYjb9AAiIjHI+Inefo5YAUwBTgJWJAXWwCcnKdPAq6K5HZgnKTJwLHAkojYEBFPAUuA4zq6N2Zm1rK23gOQ1AccBNwBTIqIx3PXOmBSnp4CrK6stia3NWuvv445kpZJWtbf399OeWZm1oaWA0DSzsC/AmdFxLPVvogIIDpRUETMi4gZETFj4sSJndikmZk10FIASNqOdPC/JiK+npufyEM75L/rc/taYI/K6lNzW7N2MzPrglbOAhJwBbAiIj5X6VoM1M7kmQXcUGk/I58NdBjwTB4quhmYKWl8fvN3Zm4zM7MuGNPCMkcA7wV+June3PYJ4ELgOkmzgUeBU3LfTcAJwCrgBeD9ABGxQdL5wF15ufMiYkNH9sLMzNo2aABExA8ANek+psHyAZzZZFvzgfntFGhmZluGPwlsZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEcAGZmhXIAmJkVygFgZlYoB4CZWaEGDQBJ8yWtl3R/pW13SUskrcx/x+d2SbpM0ipJ90k6uLLOrLz8SkmztszumJlZq1p5BXAlcFxd21xgaURMA5bmeYDjgWn5Mge4HFJgAOcChwKHAOfWQsPMzLpj0ACIiO8BG+qaTwIW5OkFwMmV9qsiuR0YJ2kycCywJCI2RMRTwBJeHSpmZjaChvoewKSIeDxPrwMm5ekpwOrKcmtyW7P2V5E0R9IyScv6+/uHWJ6ZmQ1m2G8CR0QA0YFaatubFxEzImLGxIkTO7VZMzOrM9QAeCIP7ZD/rs/ta4E9KstNzW3N2s3MrEuGGgCLgdqZPLOAGyrtZ+SzgQ4DnslDRTcDMyWNz2/+zsxtZmbWJWMGW0DStcBRwARJa0hn81wIXCdpNvAocEpe/CbgBGAV8ALwfoCI2CDpfOCuvNx5EVH/xrKZmY2gQQMgIk5r0nVMg2UDOLPJduYD89uqzszMthh/EtjMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwK5QAwMyuUA8DMrFAOADOzQjkAzMwKNejvAZhZY31zvznkdR+58O0drMRsaPwKwMysUA4AM7NCOQDMzArlADAzK5QDwMysUD4LyIo2nDN5zEY7vwIwMyuUA8DMrFAeArJRzUM4ZkPnVwBmZoUa8QCQdJykhyStkjR3pK/fzMySER0CkrQt8HngbcAa4C5JiyPigZGsw3qLh3Ha083vIPL3H21dRvo9gEOAVRHxMICkhcBJgANglPNBvD3dur18P1nVSAfAFGB1ZX4NcGh1AUlzgDl5dqOkh1rc9gTgV8OusFbHRZ3a0oA6WvMIGW01j7Z6YSuteYT+p9ox2m7ndup9fSsL9dxZQBExD5jX7nqSlkXEjC1Q0hbjmre80VYvuOaRMtpq3hL1jvSbwGuBPSrzU3ObmZmNsJEOgLuAaZL2lLQ9cCqweIRrMDMzRngIKCI2SfoIcDOwLTA/IpZ3aPNtDxv1ANe85Y22esE1j5TRVnPH61VEdHqbZmY2CviTwGZmhXIAmJkVaqsIgNHw9RKS5ktaL+n+StvukpZIWpn/ju9mjVWS9pB0q6QHJC2X9LHc3ss17yjpTkk/zTV/OrfvKemO/Pj4Wj4BoWdI2lbSPZJuzPO9Xu8jkn4m6V5Jy3Jbzz4uACSNk7RI0oOSVkg6vJdrlrRvvn1rl2clndXpmkd9AFS+XuJ4YDpwmqTp3a2qoSuB4+ra5gJLI2IasDTP94pNwNkRMR04DDgz3669XPOLwNERcQBwIHCcpMOAi4CLI2Jv4ClgdhdrbORjwIrKfK/XC/DWiDiwcl56Lz8uAC4Fvh0R+wEHkG7vnq05Ih7Kt++BwJuBF4Dr6XTNETGqL8DhwM2V+XOAc7pdV5Na+4D7K/MPAZPz9GTgoW7XOEDtN5C+w2lU1Ay8BvgJ6ZPmvwLGNHq8dPtC+izMUuBo4EZAvVxvrukRYEJdW88+LoDdgF+ST3oZDTXX1TkT+OGWqHnUvwKg8ddLTOlSLe2aFBGP5+l1wKRuFtOMpD7gIOAOerzmPJxyL7AeWAL8Ang6IjblRXrt8XEJ8HHg5Tz/Wnq7XoAAviPp7vzVLdDbj4s9gX7gK3mo7cuSxtLbNVedClybpzta89YQAFuFSJHec+fkStoZ+FfgrIh4ttrXizVHxEuRXjZPJX354H5dLqkpSScC6yPi7m7X0qYjI+Jg0rDrmZL+qNrZg4+LMcDBwOURcRDwPHVDJz1YMwD5/Z93Av9S39eJmreGABjNXy/xhKTJAPnv+i7XsxlJ25EO/tdExNdzc0/XXBMRTwO3koZQxkmqfeixlx4fRwDvlPQIsJA0DHQpvVsvABGxNv9dTxqXPoTeflysAdZExB15fhEpEHq55prjgZ9ExBN5vqM1bw0BMJq/XmIxMCtPzyKNs/cESQKuAFZExOcqXb1c80RJ4/L0TqT3LFaQguBdebGeqTkizomIqRHRR3rc3hIRp9Oj9QJIGitpl9o0aXz6fnr4cRER64DVkvbNTceQvoK+Z2uuOI1Xhn+g0zV3+w2ODr1JcgLwc9J47992u54mNV4LPA78lvSMZDZpvHcpsBL4LrB7t+us1Hsk6eXlfcC9+XJCj9f8JuCeXPP9wN/n9r2AO4FVpJfSO3S71ga1HwXc2Ov15tp+mi/La/9vvfy4yPUdCCzLj41vAONHQc1jgSeB3SptHa3ZXwVhZlaorWEIyMzMhsABYGZWKAeAmVmhHABmZoVyAJiZFcoBYGZWKAeAmVmh/j872BtXmRODKQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(test_counts_needed, bins=20)\n",
    "plt.title('Histogram of needed test counts, p=%0.3f, K=%i'%(p, K))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}