number_of_fills.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Number of fills for drugs with reject code 76\n",
    "\n",
    "Pharmacy claims might be rejected with code 76 because a patient fills some drug to its limited number. In this notebook, we try to determine the limitation on number of fills for such drugs. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Assumptions.** Suppose drug X for a payer can be rejected with code 76. \n",
    "1. The number of fills  for X of all patients that filed claims for X are identically independently drew from some Possion distribution\n",
    "$$P(n;\\mu) = \\frac{e^{-\\mu} \\mu^{n-1}}{(n-1)!},$$\n",
    "where $\\mu$ is the unknown average number of fills.\n",
    "2. The limitation $L$ times of fills is the same for all patitents.\n",
    "3. Total pharmacy claims approval rate for drug X with a fixed payer is known, say $N$, and the total approved claims is also known to be $M$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability of a pharmacy claim getting approved is\n",
    "$$P(1 \\le n \\le L; \\mu).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can estimate the probability of a pharmacy claim getting approved to be $\\frac{M}{N}$, so $\\mu$ and $L$ is related by\n",
    "$$P(1 \\le n \\le L; \\mu) \\approx \\frac{M}{N},$$\n",
    "which allows us to determine $L$ from $\\mu$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note: The estimation between $L$ and $\\mu$ relies critically on the distribution $P(n; \\mu)$. So the estimation can be poor if the actual distribution of $P(n; \\mu)$ is not a Possion distribution.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import poisson\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "\n",
    "sns.set_style('whitegrid')\n",
    "\n",
    "import utils"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability of approval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>bin</th>\n",
       "      <th>drug</th>\n",
       "      <th>total</th>\n",
       "      <th>approved</th>\n",
       "      <th>rate</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>417380</td>\n",
       "      <td>B</td>\n",
       "      <td>44371</td>\n",
       "      <td>39952</td>\n",
       "      <td>0.900408</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>417614</td>\n",
       "      <td>C</td>\n",
       "      <td>72140</td>\n",
       "      <td>65055</td>\n",
       "      <td>0.901788</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>417740</td>\n",
       "      <td>A</td>\n",
       "      <td>108768</td>\n",
       "      <td>97951</td>\n",
       "      <td>0.900550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>999001</td>\n",
       "      <td>A</td>\n",
       "      <td>325532</td>\n",
       "      <td>293040</td>\n",
       "      <td>0.900188</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>999001</td>\n",
       "      <td>B</td>\n",
       "      <td>164658</td>\n",
       "      <td>148286</td>\n",
       "      <td>0.900570</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>999001</td>\n",
       "      <td>C</td>\n",
       "      <td>150550</td>\n",
       "      <td>135341</td>\n",
       "      <td>0.898977</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      bin drug   total  approved      rate\n",
       "0  417380    B   44371     39952  0.900408\n",
       "1  417614    C   72140     65055  0.901788\n",
       "2  417740    A  108768     97951  0.900550\n",
       "3  999001    A  325532    293040  0.900188\n",
       "4  999001    B  164658    148286  0.900570\n",
       "5  999001    C  150550    135341  0.898977"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = utils.load_tables()['full']\n",
    "df76 = df[(df.reject_code != 70) & (df.reject_code != 75)]\n",
    "df76 = df76.groupby(['bin', 'drug']).agg(\n",
    "    total=('pharmacy_claim_approved', 'count'),\n",
    "    approved=('pharmacy_claim_approved', 'sum'),\n",
    ").reset_index()\n",
    "df76['rate'] = df76.approved / df76.total\n",
    "df76"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For all drugs that rejected with code 76 for all payers in this dataset, the probability of claims being approved is about 0.9. In a mathematical form,\n",
    "$$P(1 \\le n \\le L; \\mu) \\approx 0.9,$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $L$ as a function of $\\mu$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def solve_L(mu, p=0.9):\n",
    "    '''For a given $\\mu$, find the best $L$, such that\n",
    "    $P(1 \\le n \\le L; \\mu)$ is close to p.\n",
    "    '''\n",
    "    \n",
    "    L = 1\n",
    "    loss = abs(poisson.cdf(L, mu, 1) - p)\n",
    "    while loss > abs(poisson.cdf(L+1, mu, 1) - p):\n",
    "        loss = abs(poisson.cdf(L+1, mu, 1) - p)\n",
    "        L += 1\n",
    "    return L, loss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Average number of fills</th>\n",
       "      <th>Limitation of fills</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>7</td>\n",
       "      <td>11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>8</td>\n",
       "      <td>12</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>9</td>\n",
       "      <td>13</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>10</td>\n",
       "      <td>15</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Average number of fills  Limitation of fills\n",
       "0                        1                    3\n",
       "1                        2                    4\n",
       "2                        3                    6\n",
       "3                        4                    7\n",
       "4                        5                    9\n",
       "5                        6                   10\n",
       "6                        7                   11\n",
       "7                        8                   12\n",
       "8                        9                   13\n",
       "9                       10                   15"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compute L for mu = 1, 2, ... 10\n",
    "mu = np.arange(1, 11)\n",
    "L = np.array([solve_L(a, 0.9)[0] for a in mu])\n",
    "\n",
    "df_temp = pd.DataFrame(\n",
    "    data={\n",
    "        'Average number of fills': mu,\n",
    "        'Limitation of fills': L\n",
    "    }\n",
    ")\n",
    "df_temp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the average number of fills is 4, the the estimated limitation of fills is 7 times. That is to say, a patient can fill this drug for 7 times without getting rejected with code 76."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(mu, L)\n",
    "plt.xlabel('Average number of fills', fontsize=16)\n",
    "plt.ylabel('Limitation of fills', fontsize=16)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Limitations of fills are roughly linear to average numbers of fills."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The best $\\mu$ and $L$\n",
    "\n",
    "In our dataset, the average rate of approval for drugs associated with code 76 is 0.9. We can find the best pair ($\\mu$, $L$) that minimizes\n",
    "$$C(\\mu, L) = \\left|0.9 - P(1\\le n \\le L; \\mu)\\right|.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the `solve_L` function, it not only returns the best $L$ for a given $\\mu$ (and $p$), but also returns $C(\\mu, L)$. To find the best pair of $(\\mu, L)$, we can search a range of $\\mu$s, and find the pair with smallest $C(\\mu, L)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(7, 11, 0.0014792058890872317),\n",
       " (4, 7, 0.01067397840257367),\n",
       " (8, 12, 0.011924001018518537),\n",
       " (6, 10, 0.01607598300512414),\n",
       " (3, 6, 0.01608205796869655),\n",
       " (10, 15, 0.01654152706533718),\n",
       " (1, 3, 0.019698602928605813),\n",
       " (9, 13, 0.02422657082903512),\n",
       " (5, 9, 0.031906365278151605),\n",
       " (2, 4, 0.04287653950145298)]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = 0.9\n",
    "search_range = np.arange(1, 11)\n",
    "candidates = [(m, *solve_L(m, p)) for m in search_range]\n",
    "candidates.sort(key=lambda x: x[-1])\n",
    "candidates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The best pair of (mu, L) is (7, 11).\n"
     ]
    }
   ],
   "source": [
    "print(\n",
    "    f'The best pair of (mu, L) is {tuple(candidates[0][:2])}.'\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Potential Application\n",
    "\n",
    "This estimate can help payers to make strategy on the limition of fills, and thus affect the number of prior authorizations they need to process. It can applied to balance the cost between handling prior authorizations and the cost of approving a drug."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def approved_proba(mu, L):\n",
    "    return poisson.cdf(L, mu, 1)\n",
    "\n",
    "mu = 7\n",
    "L = np.arange(1, 21)\n",
    "p = [approved_proba(mu, l) for l in L]\n",
    "\n",
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(L, p, label=f'$\\mu$={mu}')\n",
    "plt.title(f'$\\mu$={mu}', fontsize=16)\n",
    "plt.xlabel('limitation of fills', fontsize=16)\n",
    "plt.xticks(L, L)\n",
    "plt.yticks(np.arange(0, 1.1, 0.1), np.arange(0, 1.1, 0.1).round(1))\n",
    "plt.ylabel('approved probability', fontsize=16)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot shows how the approved probability changes as limitation of fills changes, if the average number of fills is 7. We see that if the limitation of fills is 11, the approved probability is roughly 90%. It means that for 100 pharmacy claims on this drug for a payer, 10 prior authorizations will be filed to the payer. If we increase the limitation of fills to 12, then only 5 prior authorizatinos will be filed because the approved probability increases to roughly 95%. It effectively reduces half of the prior authorizations the payer needs to process.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}