Overdispersion_for_binomial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Overdispersion for binomial\n",
    "Let's look at an example of how overdispersion can happen in a binomial setting and how to detect it.  Consider three binomial random variables, and three vectors of trials, one with plow = .1, one with phigh = .9, and one with pmiddle = .5phigh + .5 plow = .5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "trials <- 500\n",
    "n <- 100\n",
    "plow <- 0.1 \n",
    "xlow <- rbinom(trials,n,plow)\n",
    "phigh <- 0.9\n",
    "xhigh <- rbinom(trials,n,phigh)\n",
    "xmixed <- c(x,y)\n",
    "trials <- 1000\n",
    "pmiddle <- 0.5 \n",
    "xmiddle <- rbinom(trials,n,pmiddle)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The means and hence mles are are what we expect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.10038</li>\n",
       "\t<li>0.89862</li>\n",
       "\t<li>0.49996</li>\n",
       "\t<li>0.49885</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.10038\n",
       "\\item 0.89862\n",
       "\\item 0.49996\n",
       "\\item 0.49885\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.10038\n",
       "2. 0.89862\n",
       "3. 0.49996\n",
       "4. 0.49885\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.10038 0.89862 0.49996 0.49885"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c( mean(xlow)/n, mean(xhigh)/n, mean(xmiddle)/n, mean(xmixed)/n )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The unmixed distributions are just binomial, with variances given as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>9</li>\n",
       "\t<li>9</li>\n",
       "\t<li>25</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 9\n",
       "\\item 9\n",
       "\\item 25\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 9\n",
       "2. 9\n",
       "3. 25\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  9  9 25"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n*c( plow*(1-plow), phigh*(1-phigh), pmiddle*(1-pmiddle) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Which matches with the experiement:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>9.3031623246493</li>\n",
       "\t<li>9.02500601202405</li>\n",
       "\t<li>24.5825665665666</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 9.3031623246493\n",
       "\\item 9.02500601202405\n",
       "\\item 24.5825665665666\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 9.3031623246493\n",
       "2. 9.02500601202405\n",
       "3. 24.5825665665666\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  9.303162  9.025006 24.582567"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c( var(xlow),var(xhigh),var(xmiddle) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But have a look at the mle =mean/n and variance of the mixed vector, where half the trials have p=.1 and half p=.9:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.49885</li>\n",
       "\t<li>1613.24301801802</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.49885\n",
       "\\item 1613.24301801802\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.49885\n",
       "2. 1613.24301801802\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]    0.49885 1613.24302"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c( mean(xmixed)/n, var(xmixed))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This much bigger variance is _overdispersion_.  The histograms look as you'd expect:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "</svg>\n"
      ],
      "text/plain": [
       "Plot with title “Histogram of xmixed”"
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(xmixed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "<path style=\"fill:none;stroke-width:0.75;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;\" d=\"M 314.398438 416.800781 L 362.398438 416.800781 L 362.398438 295.855469 L 314.398438 295.855469 Z \"/>\n",
       "<path style=\"fill:none;stroke-width:0.75;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;\" d=\"M 362.398438 416.800781 L 410.398438 416.800781 L 410.398438 399.949219 L 362.398438 399.949219 Z \"/>\n",
       "<path style=\"fill:none;stroke-width:0.75;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;\" d=\"M 410.398438 416.800781 L 458.398438 416.800781 L 458.398438 415.808594 L 410.398438 415.808594 Z \"/>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Plot with title “Histogram of xmiddle”"
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(xmiddle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Plot with title “Histogram of xlow”"
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(xlow)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Plot with title “Histogram of xhigh”"
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    "hist(xhigh)"
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