Matching on propensity score.ipynb

{
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  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import pyAgrum as gum\n",
    "import pyAgrum.lib.notebook as gnb\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## We are given this database coming from the historical records of a hospital"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
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       "<p>1000000 rows × 3 columns</p>\n",
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      "text/plain": [
       "        smoker  treatment  dead\n",
       "0            0          0     0\n",
       "1            1          1     0\n",
       "2            1          1     0\n",
       "3            0          1     0\n",
       "4            0          0     0\n",
       "...        ...        ...   ...\n",
       "999995       0          0     0\n",
       "999996       1          1     0\n",
       "999997       0          0     0\n",
       "999998       0          0     0\n",
       "999999       0          0     0\n",
       "\n",
       "[1000000 rows x 3 columns]"
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     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv(\"sample\")\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## And we are asked: \n",
    "## Question: How much will number of deads change if we decide treat everybody from now on with respect to not treating anyone? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## We can use matching on propensity score\n",
    "### We start computing the propensities of each sample. \n",
    "### Recall that from the paper\n",
    "\\begin{align}\n",
    "e(x)=p(z=1|x)\n",
    "\\end{align}\n",
    "### in our case:\n",
    "\\begin{align}\n",
    "e(smoker)=p(treatment=1|smoker)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### We start by computing the propensity of each patient"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.551550247155304\n"
     ]
    }
   ],
   "source": [
    "n_smokers = len(df[df[\"smoker\"]==1])\n",
    "n_treated_smokers = len(df[(df[\"smoker\"]==1) & (df[\"treatment\"]==1)])\n",
    "e_smoker = n_treated_smokers/n_smokers\n",
    "print(e_smoker)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.20002946321174753\n"
     ]
    }
   ],
   "source": [
    "n_non_smokers = len(df[df[\"smoker\"]==0])\n",
    "n_treated_non_smokers = len(df[(df[\"smoker\"]==0) & (df[\"treatment\"]==1)])\n",
    "e_non_smoker = n_treated_non_smokers/n_non_smokers\n",
    "print(e_non_smoker)"
   ]
  },
  {
   "cell_type": "code",
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       "        smoker  treatment  dead  propensity\n",
       "0            0          0     0    0.200029\n",
       "1            1          1     0    0.551550\n",
       "2            1          1     0    0.551550\n",
       "3            0          1     0    0.200029\n",
       "4            0          0     0    0.200029\n",
       "...        ...        ...   ...         ...\n",
       "999995       0          0     0    0.200029\n",
       "999996       1          1     0    0.551550\n",
       "999997       0          0     0    0.200029\n",
       "999998       0          0     0    0.200029\n",
       "999999       0          0     0    0.200029\n",
       "\n",
       "[1000000 rows x 4 columns]"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df[\"propensity\"] = df[\"smoker\"]*e_smoker + (1-df[\"smoker\"])*e_non_smoker\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### We match each treated patient with a control patient that has the same propensity score. However, we reduce the number of treated patients for efficiency reasons."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
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       "    smoker  treatment  dead  propensity\n",
       "0        1          1     0    0.551550\n",
       "1        1          1     1    0.551550\n",
       "2        1          1     1    0.551550\n",
       "3        0          1     0    0.200029\n",
       "4        0          1     0    0.200029\n",
       "..     ...        ...   ...         ...\n",
       "95       0          1     0    0.200029\n",
       "96       1          1     0    0.551550\n",
       "97       1          1     0    0.551550\n",
       "98       1          1     0    0.551550\n",
       "99       1          1     0    0.551550\n",
       "\n",
       "[100 rows x 4 columns]"
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "treated = df[df[\"treatment\"]==1].sample(100)\n",
    "treated = treated.reset_index(drop=True)\n",
    "treated"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### We match each treated patient with a randomly sampled untreated patient with the same propensity score"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
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       "<p>100 rows × 4 columns</p>\n",
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      ],
      "text/plain": [
       "    smoker  treatment  dead  propensity\n",
       "0      1.0        0.0   1.0    0.551550\n",
       "1      1.0        0.0   0.0    0.551550\n",
       "2      1.0        0.0   1.0    0.551550\n",
       "3      0.0        0.0   0.0    0.200029\n",
       "4      0.0        0.0   0.0    0.200029\n",
       "..     ...        ...   ...         ...\n",
       "95     0.0        0.0   0.0    0.200029\n",
       "96     1.0        0.0   1.0    0.551550\n",
       "97     1.0        0.0   1.0    0.551550\n",
       "98     1.0        0.0   0.0    0.551550\n",
       "99     1.0        0.0   0.0    0.551550\n",
       "\n",
       "[100 rows x 4 columns]"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "untreated = df[df[\"treatment\"]==0]\n",
    "matched_control = []\n",
    "def add_matched_control(patient):\n",
    "    control_patient = untreated[untreated[\"propensity\"]==patient[\"propensity\"]].sample().iloc[0]\n",
    "    matched_control.append(control_patient)\n",
    "    \n",
    "treated.apply(add_matched_control, axis=1)\n",
    "matched_control_df = pd.DataFrame(matched_control).reset_index(drop=True)\n",
    "matched_control_df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
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   "outputs": [
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       "    smoker  treatment  dead  propensity  smoker_control  treatment_control  \\\n",
       "0        1          1     0    0.551550             1.0                0.0   \n",
       "1        1          1     1    0.551550             1.0                0.0   \n",
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       "4        0          1     0    0.200029             0.0                0.0   \n",
       "..     ...        ...   ...         ...             ...                ...   \n",
       "95       0          1     0    0.200029             0.0                0.0   \n",
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       "98       1          1     0    0.551550             1.0                0.0   \n",
       "99       1          1     0    0.551550             1.0                0.0   \n",
       "\n",
       "    dead_control  propensity_control  \n",
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       "1            0.0            0.551550  \n",
       "2            1.0            0.551550  \n",
       "3            0.0            0.200029  \n",
       "4            0.0            0.200029  \n",
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       "97           1.0            0.551550  \n",
       "98           0.0            0.551550  \n",
       "99           0.0            0.551550  \n",
       "\n",
       "[100 rows x 8 columns]"
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     "execution_count": 155,
     "metadata": {},
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    }
   ],
   "source": [
    "paired_sample = treated.join(matched_control_df, rsuffix=\"_control\")\n",
    "paired_sample\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.17"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "ATE = (paired_sample[\"dead\"]-paired_sample[\"dead_control\"]).mean()\n",
    "ATE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Response: The percentage of deads is estimated to decrease!!!\n",
    "## Consequently, we should treat our population."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### However, we know that the real ATE is 0.083. Why is this calculation so inacurate? ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Let's take an alternative path to use propensity score matching\n",
    "### We start by taking a look at the distribution of propensity scores"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
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   "source": [
    "df.hist(\"propensity\");"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### We can split patients into two groups, those with high propensity (> 0.5) and those with low propensity (<=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.699177, 0.300823])"
      ]
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
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    "high_propensity = df[df[\"propensity\"]>0.5]\n",
    "low_propensity = df[df[\"propensity\"]<=0.5]\n",
    "counts = np.array([len(low_propensity), len(high_propensity)])\n",
    "percentages= counts / np.sum(counts)\n",
    "percentages"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Build a paired sample "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_samples=1000\n",
    "samples = []\n",
    "for i in range(n_samples):\n",
    "    is_high_propensity = random.random()>percentages[0]\n",
    "    if is_high_propensity:\n",
    "        treated_patient = high_propensity[high_propensity[\"treatment\"]==1].sample().iloc[0]\n",
    "        untreated_patient = high_propensity[high_propensity[\"treatment\"]==0].sample().iloc[0]\n",
    "    else:\n",
    "        treated_patient = low_propensity[low_propensity[\"treatment\"]==1].sample().iloc[0]\n",
    "        untreated_patient = low_propensity[low_propensity[\"treatment\"]==0].sample().iloc[0]\n",
    "    samples.append((treated_patient, untreated_patient))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [],
   "source": [
    "individual_treatment_effect = np.zeros(n_samples)\n",
    "i = 0\n",
    "for t, c in samples:\n",
    "    individual_treatment_effect[i] = t[\"dead\"]-c[\"dead\"]\n",
    "    i+=1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.074"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "ATE=individual_treatment_effect.mean()\n",
    "ATE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### We have managed to reduce the bias by sampling first from the distribution of propensity scores. \n",
    "### Can we do better?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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