MultivariateData.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multivariate Data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Let's use the bdims dataset \n",
    "https://www.openintro.org/stat/data/bdims.php"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "download.file(\"http://www.openintro.org/stat/data/bdims.RData\", destfile = \"bdims.RData\")\n",
    "load(\"bdims.RData\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>bia.di</th><th scope=col>bii.di</th><th scope=col>bit.di</th><th scope=col>che.de</th><th scope=col>che.di</th><th scope=col>elb.di</th><th scope=col>wri.di</th><th scope=col>kne.di</th><th scope=col>ank.di</th><th scope=col>sho.gi</th><th scope=col>...</th><th scope=col>bic.gi</th><th scope=col>for.gi</th><th scope=col>kne.gi</th><th scope=col>cal.gi</th><th scope=col>ank.gi</th><th scope=col>wri.gi</th><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th><th scope=col>sex</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>42.9 </td><td>26.0 </td><td>31.5 </td><td>17.7 </td><td>28.0 </td><td>13.1 </td><td>10.4 </td><td>18.8 </td><td>14.1 </td><td>106.2</td><td>...  </td><td>32.5 </td><td>26.0 </td><td>34.5 </td><td>36.5 </td><td>23.5 </td><td>16.5 </td><td>21   </td><td>65.6 </td><td>174.0</td><td>1    </td></tr>\n",
       "\t<tr><td>43.7 </td><td>28.5 </td><td>33.5 </td><td>16.9 </td><td>30.8 </td><td>14.0 </td><td>11.8 </td><td>20.6 </td><td>15.1 </td><td>110.5</td><td>...  </td><td>34.4 </td><td>28.0 </td><td>36.5 </td><td>37.5 </td><td>24.5 </td><td>17.0 </td><td>23   </td><td>71.8 </td><td>175.3</td><td>1    </td></tr>\n",
       "\t<tr><td>40.1 </td><td>28.2 </td><td>33.3 </td><td>20.9 </td><td>31.7 </td><td>13.9 </td><td>10.9 </td><td>19.7 </td><td>14.1 </td><td>115.1</td><td>...  </td><td>33.4 </td><td>28.8 </td><td>37.0 </td><td>37.3 </td><td>21.9 </td><td>16.9 </td><td>28   </td><td>80.7 </td><td>193.5</td><td>1    </td></tr>\n",
       "\t<tr><td>44.3 </td><td>29.9 </td><td>34.0 </td><td>18.4 </td><td>28.2 </td><td>13.9 </td><td>11.2 </td><td>20.9 </td><td>15.0 </td><td>104.5</td><td>...  </td><td>31.0 </td><td>26.2 </td><td>37.0 </td><td>34.8 </td><td>23.0 </td><td>16.6 </td><td>23   </td><td>72.6 </td><td>186.5</td><td>1    </td></tr>\n",
       "\t<tr><td>42.5 </td><td>29.9 </td><td>34.0 </td><td>21.5 </td><td>29.4 </td><td>15.2 </td><td>11.6 </td><td>20.7 </td><td>14.9 </td><td>107.5</td><td>...  </td><td>32.0 </td><td>28.4 </td><td>37.7 </td><td>38.6 </td><td>24.4 </td><td>18.0 </td><td>22   </td><td>78.8 </td><td>187.2</td><td>1    </td></tr>\n",
       "\t<tr><td>43.3 </td><td>27.0 </td><td>31.5 </td><td>19.6 </td><td>31.3 </td><td>14.0 </td><td>11.5 </td><td>18.8 </td><td>13.9 </td><td>119.8</td><td>...  </td><td>33.0 </td><td>28.0 </td><td>36.6 </td><td>36.1 </td><td>23.5 </td><td>16.9 </td><td>21   </td><td>74.8 </td><td>181.5</td><td>1    </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllllllllllllllllllllll}\n",
       " bia.di & bii.di & bit.di & che.de & che.di & elb.di & wri.di & kne.di & ank.di & sho.gi & ... & bic.gi & for.gi & kne.gi & cal.gi & ank.gi & wri.gi & age & wgt & hgt & sex\\\\\n",
       "\\hline\n",
       "\t 42.9  & 26.0  & 31.5  & 17.7  & 28.0  & 13.1  & 10.4  & 18.8  & 14.1  & 106.2 & ...   & 32.5  & 26.0  & 34.5  & 36.5  & 23.5  & 16.5  & 21    & 65.6  & 174.0 & 1    \\\\\n",
       "\t 43.7  & 28.5  & 33.5  & 16.9  & 30.8  & 14.0  & 11.8  & 20.6  & 15.1  & 110.5 & ...   & 34.4  & 28.0  & 36.5  & 37.5  & 24.5  & 17.0  & 23    & 71.8  & 175.3 & 1    \\\\\n",
       "\t 40.1  & 28.2  & 33.3  & 20.9  & 31.7  & 13.9  & 10.9  & 19.7  & 14.1  & 115.1 & ...   & 33.4  & 28.8  & 37.0  & 37.3  & 21.9  & 16.9  & 28    & 80.7  & 193.5 & 1    \\\\\n",
       "\t 44.3  & 29.9  & 34.0  & 18.4  & 28.2  & 13.9  & 11.2  & 20.9  & 15.0  & 104.5 & ...   & 31.0  & 26.2  & 37.0  & 34.8  & 23.0  & 16.6  & 23    & 72.6  & 186.5 & 1    \\\\\n",
       "\t 42.5  & 29.9  & 34.0  & 21.5  & 29.4  & 15.2  & 11.6  & 20.7  & 14.9  & 107.5 & ...   & 32.0  & 28.4  & 37.7  & 38.6  & 24.4  & 18.0  & 22    & 78.8  & 187.2 & 1    \\\\\n",
       "\t 43.3  & 27.0  & 31.5  & 19.6  & 31.3  & 14.0  & 11.5  & 18.8  & 13.9  & 119.8 & ...   & 33.0  & 28.0  & 36.6  & 36.1  & 23.5  & 16.9  & 21    & 74.8  & 181.5 & 1    \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "bia.di | bii.di | bit.di | che.de | che.di | elb.di | wri.di | kne.di | ank.di | sho.gi | ... | bic.gi | for.gi | kne.gi | cal.gi | ank.gi | wri.gi | age | wgt | hgt | sex | \n",
       "|---|---|---|---|---|---|\n",
       "| 42.9  | 26.0  | 31.5  | 17.7  | 28.0  | 13.1  | 10.4  | 18.8  | 14.1  | 106.2 | ...   | 32.5  | 26.0  | 34.5  | 36.5  | 23.5  | 16.5  | 21    | 65.6  | 174.0 | 1     | \n",
       "| 43.7  | 28.5  | 33.5  | 16.9  | 30.8  | 14.0  | 11.8  | 20.6  | 15.1  | 110.5 | ...   | 34.4  | 28.0  | 36.5  | 37.5  | 24.5  | 17.0  | 23    | 71.8  | 175.3 | 1     | \n",
       "| 40.1  | 28.2  | 33.3  | 20.9  | 31.7  | 13.9  | 10.9  | 19.7  | 14.1  | 115.1 | ...   | 33.4  | 28.8  | 37.0  | 37.3  | 21.9  | 16.9  | 28    | 80.7  | 193.5 | 1     | \n",
       "| 44.3  | 29.9  | 34.0  | 18.4  | 28.2  | 13.9  | 11.2  | 20.9  | 15.0  | 104.5 | ...   | 31.0  | 26.2  | 37.0  | 34.8  | 23.0  | 16.6  | 23    | 72.6  | 186.5 | 1     | \n",
       "| 42.5  | 29.9  | 34.0  | 21.5  | 29.4  | 15.2  | 11.6  | 20.7  | 14.9  | 107.5 | ...   | 32.0  | 28.4  | 37.7  | 38.6  | 24.4  | 18.0  | 22    | 78.8  | 187.2 | 1     | \n",
       "| 43.3  | 27.0  | 31.5  | 19.6  | 31.3  | 14.0  | 11.5  | 18.8  | 13.9  | 119.8 | ...   | 33.0  | 28.0  | 36.6  | 36.1  | 23.5  | 16.9  | 21    | 74.8  | 181.5 | 1     | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  bia.di bii.di bit.di che.de che.di elb.di wri.di kne.di ank.di sho.gi ...\n",
       "1 42.9   26.0   31.5   17.7   28.0   13.1   10.4   18.8   14.1   106.2  ...\n",
       "2 43.7   28.5   33.5   16.9   30.8   14.0   11.8   20.6   15.1   110.5  ...\n",
       "3 40.1   28.2   33.3   20.9   31.7   13.9   10.9   19.7   14.1   115.1  ...\n",
       "4 44.3   29.9   34.0   18.4   28.2   13.9   11.2   20.9   15.0   104.5  ...\n",
       "5 42.5   29.9   34.0   21.5   29.4   15.2   11.6   20.7   14.9   107.5  ...\n",
       "6 43.3   27.0   31.5   19.6   31.3   14.0   11.5   18.8   13.9   119.8  ...\n",
       "  bic.gi for.gi kne.gi cal.gi ank.gi wri.gi age wgt  hgt   sex\n",
       "1 32.5   26.0   34.5   36.5   23.5   16.5   21  65.6 174.0 1  \n",
       "2 34.4   28.0   36.5   37.5   24.5   17.0   23  71.8 175.3 1  \n",
       "3 33.4   28.8   37.0   37.3   21.9   16.9   28  80.7 193.5 1  \n",
       "4 31.0   26.2   37.0   34.8   23.0   16.6   23  72.6 186.5 1  \n",
       "5 32.0   28.4   37.7   38.6   24.4   18.0   22  78.8 187.2 1  \n",
       "6 33.0   28.0   36.6   36.1   23.5   16.9   21  74.8 181.5 1  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(bdims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>1</th><th scope=col>2</th><th scope=col>3</th><th scope=col>4</th><th scope=col>5</th><th scope=col>6</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>bia.di</th><td>42.9 </td><td>43.7 </td><td>40.1 </td><td>44.3 </td><td>42.5 </td><td>43.3 </td></tr>\n",
       "\t<tr><th scope=row>bii.di</th><td>26.0 </td><td>28.5 </td><td>28.2 </td><td>29.9 </td><td>29.9 </td><td>27.0 </td></tr>\n",
       "\t<tr><th scope=row>bit.di</th><td>31.5 </td><td>33.5 </td><td>33.3 </td><td>34.0 </td><td>34.0 </td><td>31.5 </td></tr>\n",
       "\t<tr><th scope=row>che.de</th><td>17.7 </td><td>16.9 </td><td>20.9 </td><td>18.4 </td><td>21.5 </td><td>19.6 </td></tr>\n",
       "\t<tr><th scope=row>che.di</th><td>28.0 </td><td>30.8 </td><td>31.7 </td><td>28.2 </td><td>29.4 </td><td>31.3 </td></tr>\n",
       "\t<tr><th scope=row>elb.di</th><td>13.1 </td><td>14.0 </td><td>13.9 </td><td>13.9 </td><td>15.2 </td><td>14.0 </td></tr>\n",
       "\t<tr><th scope=row>wri.di</th><td>10.4 </td><td>11.8 </td><td>10.9 </td><td>11.2 </td><td>11.6 </td><td>11.5 </td></tr>\n",
       "\t<tr><th scope=row>kne.di</th><td>18.8 </td><td>20.6 </td><td>19.7 </td><td>20.9 </td><td>20.7 </td><td>18.8 </td></tr>\n",
       "\t<tr><th scope=row>ank.di</th><td>14.1 </td><td>15.1 </td><td>14.1 </td><td>15.0 </td><td>14.9 </td><td>13.9 </td></tr>\n",
       "\t<tr><th scope=row>sho.gi</th><td>106.2</td><td>110.5</td><td>115.1</td><td>104.5</td><td>107.5</td><td>119.8</td></tr>\n",
       "\t<tr><th scope=row>che.gi</th><td>89.5 </td><td>97.0 </td><td>97.5 </td><td>97.0 </td><td>97.5 </td><td>99.9 </td></tr>\n",
       "\t<tr><th scope=row>wai.gi</th><td>71.5 </td><td>79.0 </td><td>83.2 </td><td>77.8 </td><td>80.0 </td><td>82.5 </td></tr>\n",
       "\t<tr><th scope=row>nav.gi</th><td>74.5 </td><td>86.5 </td><td>82.9 </td><td>78.8 </td><td>82.5 </td><td>80.1 </td></tr>\n",
       "\t<tr><th scope=row>hip.gi</th><td>93.5 </td><td>94.8 </td><td>95.0 </td><td>94.0 </td><td>98.5 </td><td>95.3 </td></tr>\n",
       "\t<tr><th scope=row>thi.gi</th><td>51.5 </td><td>51.5 </td><td>57.3 </td><td>53.0 </td><td>55.4 </td><td>57.5 </td></tr>\n",
       "\t<tr><th scope=row>bic.gi</th><td>32.5 </td><td>34.4 </td><td>33.4 </td><td>31.0 </td><td>32.0 </td><td>33.0 </td></tr>\n",
       "\t<tr><th scope=row>for.gi</th><td>26.0 </td><td>28.0 </td><td>28.8 </td><td>26.2 </td><td>28.4 </td><td>28.0 </td></tr>\n",
       "\t<tr><th scope=row>kne.gi</th><td>34.5 </td><td>36.5 </td><td>37.0 </td><td>37.0 </td><td>37.7 </td><td>36.6 </td></tr>\n",
       "\t<tr><th scope=row>cal.gi</th><td>36.5 </td><td>37.5 </td><td>37.3 </td><td>34.8 </td><td>38.6 </td><td>36.1 </td></tr>\n",
       "\t<tr><th scope=row>ank.gi</th><td>23.5 </td><td>24.5 </td><td>21.9 </td><td>23.0 </td><td>24.4 </td><td>23.5 </td></tr>\n",
       "\t<tr><th scope=row>wri.gi</th><td>16.5 </td><td>17.0 </td><td>16.9 </td><td>16.6 </td><td>18.0 </td><td>16.9 </td></tr>\n",
       "\t<tr><th scope=row>age</th><td>21   </td><td>23   </td><td>28   </td><td>23   </td><td>22   </td><td>21   </td></tr>\n",
       "\t<tr><th scope=row>wgt</th><td>65.6 </td><td>71.8 </td><td>80.7 </td><td>72.6 </td><td>78.8 </td><td>74.8 </td></tr>\n",
       "\t<tr><th scope=row>hgt</th><td>174.0</td><td>175.3</td><td>193.5</td><td>186.5</td><td>187.2</td><td>181.5</td></tr>\n",
       "\t<tr><th scope=row>sex</th><td>1    </td><td>1    </td><td>1    </td><td>1    </td><td>1    </td><td>1    </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllll}\n",
       "  & 1 & 2 & 3 & 4 & 5 & 6\\\\\n",
       "\\hline\n",
       "\tbia.di & 42.9  & 43.7  & 40.1  & 44.3  & 42.5  & 43.3 \\\\\n",
       "\tbii.di & 26.0  & 28.5  & 28.2  & 29.9  & 29.9  & 27.0 \\\\\n",
       "\tbit.di & 31.5  & 33.5  & 33.3  & 34.0  & 34.0  & 31.5 \\\\\n",
       "\tche.de & 17.7  & 16.9  & 20.9  & 18.4  & 21.5  & 19.6 \\\\\n",
       "\tche.di & 28.0  & 30.8  & 31.7  & 28.2  & 29.4  & 31.3 \\\\\n",
       "\telb.di & 13.1  & 14.0  & 13.9  & 13.9  & 15.2  & 14.0 \\\\\n",
       "\twri.di & 10.4  & 11.8  & 10.9  & 11.2  & 11.6  & 11.5 \\\\\n",
       "\tkne.di & 18.8  & 20.6  & 19.7  & 20.9  & 20.7  & 18.8 \\\\\n",
       "\tank.di & 14.1  & 15.1  & 14.1  & 15.0  & 14.9  & 13.9 \\\\\n",
       "\tsho.gi & 106.2 & 110.5 & 115.1 & 104.5 & 107.5 & 119.8\\\\\n",
       "\tche.gi & 89.5  & 97.0  & 97.5  & 97.0  & 97.5  & 99.9 \\\\\n",
       "\twai.gi & 71.5  & 79.0  & 83.2  & 77.8  & 80.0  & 82.5 \\\\\n",
       "\tnav.gi & 74.5  & 86.5  & 82.9  & 78.8  & 82.5  & 80.1 \\\\\n",
       "\thip.gi & 93.5  & 94.8  & 95.0  & 94.0  & 98.5  & 95.3 \\\\\n",
       "\tthi.gi & 51.5  & 51.5  & 57.3  & 53.0  & 55.4  & 57.5 \\\\\n",
       "\tbic.gi & 32.5  & 34.4  & 33.4  & 31.0  & 32.0  & 33.0 \\\\\n",
       "\tfor.gi & 26.0  & 28.0  & 28.8  & 26.2  & 28.4  & 28.0 \\\\\n",
       "\tkne.gi & 34.5  & 36.5  & 37.0  & 37.0  & 37.7  & 36.6 \\\\\n",
       "\tcal.gi & 36.5  & 37.5  & 37.3  & 34.8  & 38.6  & 36.1 \\\\\n",
       "\tank.gi & 23.5  & 24.5  & 21.9  & 23.0  & 24.4  & 23.5 \\\\\n",
       "\twri.gi & 16.5  & 17.0  & 16.9  & 16.6  & 18.0  & 16.9 \\\\\n",
       "\tage & 21    & 23    & 28    & 23    & 22    & 21   \\\\\n",
       "\twgt & 65.6  & 71.8  & 80.7  & 72.6  & 78.8  & 74.8 \\\\\n",
       "\thgt & 174.0 & 175.3 & 193.5 & 186.5 & 187.2 & 181.5\\\\\n",
       "\tsex & 1     & 1     & 1     & 1     & 1     & 1    \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | 1 | 2 | 3 | 4 | 5 | 6 | \n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| bia.di | 42.9  | 43.7  | 40.1  | 44.3  | 42.5  | 43.3  | \n",
       "| bii.di | 26.0  | 28.5  | 28.2  | 29.9  | 29.9  | 27.0  | \n",
       "| bit.di | 31.5  | 33.5  | 33.3  | 34.0  | 34.0  | 31.5  | \n",
       "| che.de | 17.7  | 16.9  | 20.9  | 18.4  | 21.5  | 19.6  | \n",
       "| che.di | 28.0  | 30.8  | 31.7  | 28.2  | 29.4  | 31.3  | \n",
       "| elb.di | 13.1  | 14.0  | 13.9  | 13.9  | 15.2  | 14.0  | \n",
       "| wri.di | 10.4  | 11.8  | 10.9  | 11.2  | 11.6  | 11.5  | \n",
       "| kne.di | 18.8  | 20.6  | 19.7  | 20.9  | 20.7  | 18.8  | \n",
       "| ank.di | 14.1  | 15.1  | 14.1  | 15.0  | 14.9  | 13.9  | \n",
       "| sho.gi | 106.2 | 110.5 | 115.1 | 104.5 | 107.5 | 119.8 | \n",
       "| che.gi | 89.5  | 97.0  | 97.5  | 97.0  | 97.5  | 99.9  | \n",
       "| wai.gi | 71.5  | 79.0  | 83.2  | 77.8  | 80.0  | 82.5  | \n",
       "| nav.gi | 74.5  | 86.5  | 82.9  | 78.8  | 82.5  | 80.1  | \n",
       "| hip.gi | 93.5  | 94.8  | 95.0  | 94.0  | 98.5  | 95.3  | \n",
       "| thi.gi | 51.5  | 51.5  | 57.3  | 53.0  | 55.4  | 57.5  | \n",
       "| bic.gi | 32.5  | 34.4  | 33.4  | 31.0  | 32.0  | 33.0  | \n",
       "| for.gi | 26.0  | 28.0  | 28.8  | 26.2  | 28.4  | 28.0  | \n",
       "| kne.gi | 34.5  | 36.5  | 37.0  | 37.0  | 37.7  | 36.6  | \n",
       "| cal.gi | 36.5  | 37.5  | 37.3  | 34.8  | 38.6  | 36.1  | \n",
       "| ank.gi | 23.5  | 24.5  | 21.9  | 23.0  | 24.4  | 23.5  | \n",
       "| wri.gi | 16.5  | 17.0  | 16.9  | 16.6  | 18.0  | 16.9  | \n",
       "| age | 21    | 23    | 28    | 23    | 22    | 21    | \n",
       "| wgt | 65.6  | 71.8  | 80.7  | 72.6  | 78.8  | 74.8  | \n",
       "| hgt | 174.0 | 175.3 | 193.5 | 186.5 | 187.2 | 181.5 | \n",
       "| sex | 1     | 1     | 1     | 1     | 1     | 1     | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "       1     2     3     4     5     6    \n",
       "bia.di 42.9  43.7  40.1  44.3  42.5  43.3 \n",
       "bii.di 26.0  28.5  28.2  29.9  29.9  27.0 \n",
       "bit.di 31.5  33.5  33.3  34.0  34.0  31.5 \n",
       "che.de 17.7  16.9  20.9  18.4  21.5  19.6 \n",
       "che.di 28.0  30.8  31.7  28.2  29.4  31.3 \n",
       "elb.di 13.1  14.0  13.9  13.9  15.2  14.0 \n",
       "wri.di 10.4  11.8  10.9  11.2  11.6  11.5 \n",
       "kne.di 18.8  20.6  19.7  20.9  20.7  18.8 \n",
       "ank.di 14.1  15.1  14.1  15.0  14.9  13.9 \n",
       "sho.gi 106.2 110.5 115.1 104.5 107.5 119.8\n",
       "che.gi 89.5  97.0  97.5  97.0  97.5  99.9 \n",
       "wai.gi 71.5  79.0  83.2  77.8  80.0  82.5 \n",
       "nav.gi 74.5  86.5  82.9  78.8  82.5  80.1 \n",
       "hip.gi 93.5  94.8  95.0  94.0  98.5  95.3 \n",
       "thi.gi 51.5  51.5  57.3  53.0  55.4  57.5 \n",
       "bic.gi 32.5  34.4  33.4  31.0  32.0  33.0 \n",
       "for.gi 26.0  28.0  28.8  26.2  28.4  28.0 \n",
       "kne.gi 34.5  36.5  37.0  37.0  37.7  36.6 \n",
       "cal.gi 36.5  37.5  37.3  34.8  38.6  36.1 \n",
       "ank.gi 23.5  24.5  21.9  23.0  24.4  23.5 \n",
       "wri.gi 16.5  17.0  16.9  16.6  18.0  16.9 \n",
       "age    21    23    28    23    22    21   \n",
       "wgt    65.6  71.8  80.7  72.6  78.8  74.8 \n",
       "hgt    174.0 175.3 193.5 186.5 187.2 181.5\n",
       "sex    1     1     1     1     1     1    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t(head(bdims))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# reduce to only 3 columns\n",
    "lessData <- bdims[c(\"age\", \"wgt\", \"hgt\")]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>21   </td><td>65.6 </td><td>174.0</td></tr>\n",
       "\t<tr><td>23   </td><td>71.8 </td><td>175.3</td></tr>\n",
       "\t<tr><td>28   </td><td>80.7 </td><td>193.5</td></tr>\n",
       "\t<tr><td>23   </td><td>72.6 </td><td>186.5</td></tr>\n",
       "\t<tr><td>22   </td><td>78.8 </td><td>187.2</td></tr>\n",
       "\t<tr><td>21   </td><td>74.8 </td><td>181.5</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       " age & wgt & hgt\\\\\n",
       "\\hline\n",
       "\t 21    & 65.6  & 174.0\\\\\n",
       "\t 23    & 71.8  & 175.3\\\\\n",
       "\t 28    & 80.7  & 193.5\\\\\n",
       "\t 23    & 72.6  & 186.5\\\\\n",
       "\t 22    & 78.8  & 187.2\\\\\n",
       "\t 21    & 74.8  & 181.5\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "age | wgt | hgt | \n",
       "|---|---|---|---|---|---|\n",
       "| 21    | 65.6  | 174.0 | \n",
       "| 23    | 71.8  | 175.3 | \n",
       "| 28    | 80.7  | 193.5 | \n",
       "| 23    | 72.6  | 186.5 | \n",
       "| 22    | 78.8  | 187.2 | \n",
       "| 21    | 74.8  | 181.5 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  age wgt  hgt  \n",
       "1 21  65.6 174.0\n",
       "2 23  71.8 175.3\n",
       "3 28  80.7 193.5\n",
       "4 23  72.6 186.5\n",
       "5 22  78.8 187.2\n",
       "6 21  74.8 181.5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(lessData)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      age             wgt              hgt       \n",
       " Min.   :18.00   Min.   : 42.00   Min.   :147.2  \n",
       " 1st Qu.:23.00   1st Qu.: 58.40   1st Qu.:163.8  \n",
       " Median :27.00   Median : 68.20   Median :170.3  \n",
       " Mean   :30.18   Mean   : 69.15   Mean   :171.1  \n",
       " 3rd Qu.:36.00   3rd Qu.: 78.85   3rd Qu.:177.8  \n",
       " Max.   :67.00   Max.   :116.40   Max.   :198.1  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(lessData)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visual exploration\n",
    "\n",
    "Let's build out some histograms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "library(gridExtra) # provides grid.arrange function\n",
    "library(ggplot2) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MkwKU4QIiGSYVKcIERCJMOkOEGIhEiG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\nRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "grid.arrange(\n",
    "    \n",
    "    # use only 10 bins to group the data\n",
    "    qplot(lessData$age, bins = 10, xlab = \"Age\"),\n",
    "    \n",
    "    qplot(lessData$wgt, bins = 10, xlab = \"Weight\"),\n",
    "    \n",
    "    qplot(lessData$hgt, bins = 10, xlab = \"Height\"),\n",
    "    \n",
    "    ncol = 1\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Moving to multiple variables...multivariate\n",
    "\n",
    "The next question for multivariate statistics is to attempt to understand if there are any relationships between variables. \n",
    "\n",
    "* Covariance is the corrleated dispersion of the 2 variables. \n",
    "* Correlation measures the relative growth rates of the 2 variables. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"Weight and Height\"\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "90.0545152840471"
      ],
      "text/latex": [
       "90.0545152840471"
      ],
      "text/markdown": [
       "90.0545152840471"
      ],
      "text/plain": [
       "[1] 90.05452"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Weight and Height\")\n",
    "# cov() with 2 arguments treat them as 2 lists variables values\n",
    "cov(lessData$wgt, lessData$hgt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.717301078724164"
      ],
      "text/latex": [
       "0.717301078724164"
      ],
      "text/markdown": [
       "0.717301078724164"
      ],
      "text/plain": [
       "[1] 0.7173011"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(lessData$wgt, lessData$hgt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "26.5781162538688"
      ],
      "text/latex": [
       "26.5781162538688"
      ],
      "text/markdown": [
       "26.5781162538688"
      ],
      "text/plain": [
       "[1] 26.57812"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cov(lessData$wgt, lessData$age)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.207265236366444"
      ],
      "text/latex": [
       "0.207265236366444"
      ],
      "text/markdown": [
       "0.207265236366444"
      ],
      "text/plain": [
       "[1] 0.2072652"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(lessData$wgt, lessData$age)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "6.13591224828683"
      ],
      "text/latex": [
       "6.13591224828683"
      ],
      "text/markdown": [
       "6.13591224828683"
      ],
      "text/plain": [
       "[1] 6.135912"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cov(lessData$hgt, lessData$age)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.0678834933061811"
      ],
      "text/latex": [
       "0.0678834933061811"
      ],
      "text/markdown": [
       "0.0678834933061811"
      ],
      "text/plain": [
       "[1] 0.06788349"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(lessData$hgt, lessData$age)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>age</th><td>92.322739</td><td> 26.57812</td><td> 6.135912</td></tr>\n",
       "\t<tr><th scope=row>wgt</th><td>26.578116</td><td>178.10938</td><td>90.054515</td></tr>\n",
       "\t<tr><th scope=row>hgt</th><td> 6.135912</td><td> 90.05452</td><td>88.495510</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & age & wgt & hgt\\\\\n",
       "\\hline\n",
       "\tage & 92.322739 &  26.57812 &  6.135912\\\\\n",
       "\twgt & 26.578116 & 178.10938 & 90.054515\\\\\n",
       "\thgt &  6.135912 &  90.05452 & 88.495510\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | age | wgt | hgt | \n",
       "|---|---|---|\n",
       "| age | 92.322739 |  26.57812 |  6.135912 | \n",
       "| wgt | 26.578116 | 178.10938 | 90.054515 | \n",
       "| hgt |  6.135912 |  90.05452 | 88.495510 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "    age       wgt       hgt      \n",
       "age 92.322739  26.57812  6.135912\n",
       "wgt 26.578116 178.10938 90.054515\n",
       "hgt  6.135912  90.05452 88.495510"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cov(lessData)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>age</th><td>1.00000000</td><td>0.2072652 </td><td>0.06788349</td></tr>\n",
       "\t<tr><th scope=row>wgt</th><td>0.20726524</td><td>1.0000000 </td><td>0.71730108</td></tr>\n",
       "\t<tr><th scope=row>hgt</th><td>0.06788349</td><td>0.7173011 </td><td>1.00000000</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       "  & age & wgt & hgt\\\\\n",
       "\\hline\n",
       "\tage & 1.00000000 & 0.2072652  & 0.06788349\\\\\n",
       "\twgt & 0.20726524 & 1.0000000  & 0.71730108\\\\\n",
       "\thgt & 0.06788349 & 0.7173011  & 1.00000000\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | age | wgt | hgt | \n",
       "|---|---|---|\n",
       "| age | 1.00000000 | 0.2072652  | 0.06788349 | \n",
       "| wgt | 0.20726524 | 1.0000000  | 0.71730108 | \n",
       "| hgt | 0.06788349 | 0.7173011  | 1.00000000 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "    age        wgt       hgt       \n",
       "age 1.00000000 0.2072652 0.06788349\n",
       "wgt 0.20726524 1.0000000 0.71730108\n",
       "hgt 0.06788349 0.7173011 1.00000000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(lessData)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpreting correlations\n",
    "\n",
    "From the above, we see the following correlations:\n",
    "\n",
    "1) Weight and Height = 0.717\n",
    "2) Weight and Age = 0.207\n",
    "3) Height and Age = 0.067\n",
    "\n",
    "These numbers tell us the following:\n",
    "\n",
    "1) There is a strong positive relationship between weight and height. The more a person weights, typically the taller they are (?)\n",
    "2) There is a weak positive relationship between weight and age\n",
    "3) There is little to no determination of height based on age. \n",
    "\n",
    "Regarding the last one...how can this be? We all know that as someone grows up, they get taller. The reason our data does not have a strong height to age correlation is because the data covers adults, not children. So, presumably they have nearly completed growing. \n",
    "\n",
    "* Postive correlation - We expect a trend from lower left toward upper right if we plot the 2 variables against each other. \n",
    "* Negative correlation - We expect a trend from upper left toward lower right if we plot the 2 variables against each other. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gxIpTiifquBrELYhbAQOQRAUkICJFYA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\nA3f6TQlBZWtYAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "wh <- ggplot(lessData, aes(x = wgt, y = hgt))\n",
    "wa <- ggplot(lessData, aes(x = wgt, y = age))\n",
    "ha <- ggplot(lessData, aes(x = hgt, y = age))\n",
    "\n",
    "grid.arrange(\n",
    "    wh + geom_point(color = \"red\"),\n",
    "    wa + geom_point(color = \"blue\"),\n",
    "    ha + geom_point(), # default color is black\n",
    "    ncol = 3\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Use case: multiple regression\n",
    "\n",
    "Based on the exploration above and the discovered correlation of 0.72 between height and weight, we expect that we can use height to predict weight or weight to predict weight. This prediction is the concept of linear regression. \n",
    "\n",
    "But what if we want to use 2 variables to predict a third?\n",
    "Or even better, using 100s of variables to make a numerical prediction?!?!\n",
    "\n",
    "### Multiple Regression\n",
    "\n",
    "Multiple regression is using 2 or more predictors (independent) variables to predict the value of a dependent variable. Since we have wgt as kg and hgt as cm, we can compute thi value and add it as a column to the dataframe. \n",
    "\n",
    "Conversion of cm to m = lessData$hght/100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "lessData$QueteletIndex <- with(lessData, lessData$wgt/(lessData$hgt/100) ^2) \n",
    "# aka BMI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th><th scope=col>QueteletIndex</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>21      </td><td>65.6    </td><td>174.0   </td><td>21.66733</td></tr>\n",
       "\t<tr><td>23      </td><td>71.8    </td><td>175.3   </td><td>23.36472</td></tr>\n",
       "\t<tr><td>28      </td><td>80.7    </td><td>193.5   </td><td>21.55319</td></tr>\n",
       "\t<tr><td>23      </td><td>72.6    </td><td>186.5   </td><td>20.87272</td></tr>\n",
       "\t<tr><td>22      </td><td>78.8    </td><td>187.2   </td><td>22.48612</td></tr>\n",
       "\t<tr><td>21      </td><td>74.8    </td><td>181.5   </td><td>22.70640</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       " age & wgt & hgt & QueteletIndex\\\\\n",
       "\\hline\n",
       "\t 21       & 65.6     & 174.0    & 21.66733\\\\\n",
       "\t 23       & 71.8     & 175.3    & 23.36472\\\\\n",
       "\t 28       & 80.7     & 193.5    & 21.55319\\\\\n",
       "\t 23       & 72.6     & 186.5    & 20.87272\\\\\n",
       "\t 22       & 78.8     & 187.2    & 22.48612\\\\\n",
       "\t 21       & 74.8     & 181.5    & 22.70640\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "age | wgt | hgt | QueteletIndex | \n",
       "|---|---|---|---|---|---|\n",
       "| 21       | 65.6     | 174.0    | 21.66733 | \n",
       "| 23       | 71.8     | 175.3    | 23.36472 | \n",
       "| 28       | 80.7     | 193.5    | 21.55319 | \n",
       "| 23       | 72.6     | 186.5    | 20.87272 | \n",
       "| 22       | 78.8     | 187.2    | 22.48612 | \n",
       "| 21       | 74.8     | 181.5    | 22.70640 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  age wgt  hgt   QueteletIndex\n",
       "1 21  65.6 174.0 21.66733     \n",
       "2 23  71.8 175.3 23.36472     \n",
       "3 28  80.7 193.5 21.55319     \n",
       "4 23  72.6 186.5 20.87272     \n",
       "5 22  78.8 187.2 22.48612     \n",
       "6 21  74.8 181.5 22.70640     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(lessData)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      age             wgt              hgt        QueteletIndex  \n",
       " Min.   :18.00   Min.   : 42.00   Min.   :147.2   Min.   :16.88  \n",
       " 1st Qu.:23.00   1st Qu.: 58.40   1st Qu.:163.8   1st Qu.:20.96  \n",
       " Median :27.00   Median : 68.20   Median :170.3   Median :23.16  \n",
       " Mean   :30.18   Mean   : 69.15   Mean   :171.1   Mean   :23.46  \n",
       " 3rd Qu.:36.00   3rd Qu.: 78.85   3rd Qu.:177.8   3rd Qu.:25.47  \n",
       " Max.   :67.00   Max.   :116.40   Max.   :198.1   Max.   :38.19  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(lessData)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th><th scope=col>QueteletIndex</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>age</th><td>92.322739</td><td> 26.57812</td><td> 6.135912</td><td> 7.541478</td></tr>\n",
       "\t<tr><th scope=row>wgt</th><td>26.578116</td><td>178.10938</td><td>90.054515</td><td>36.023441</td></tr>\n",
       "\t<tr><th scope=row>hgt</th><td> 6.135912</td><td> 90.05452</td><td>88.495510</td><td> 6.299680</td></tr>\n",
       "\t<tr><th scope=row>QueteletIndex</th><td> 7.541478</td><td> 36.02344</td><td> 6.299680</td><td>10.714224</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       "  & age & wgt & hgt & QueteletIndex\\\\\n",
       "\\hline\n",
       "\tage & 92.322739 &  26.57812 &  6.135912 &  7.541478\\\\\n",
       "\twgt & 26.578116 & 178.10938 & 90.054515 & 36.023441\\\\\n",
       "\thgt &  6.135912 &  90.05452 & 88.495510 &  6.299680\\\\\n",
       "\tQueteletIndex &  7.541478 &  36.02344 &  6.299680 & 10.714224\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | age | wgt | hgt | QueteletIndex | \n",
       "|---|---|---|---|\n",
       "| age | 92.322739 |  26.57812 |  6.135912 |  7.541478 | \n",
       "| wgt | 26.578116 | 178.10938 | 90.054515 | 36.023441 | \n",
       "| hgt |  6.135912 |  90.05452 | 88.495510 |  6.299680 | \n",
       "| QueteletIndex |  7.541478 |  36.02344 |  6.299680 | 10.714224 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "              age       wgt       hgt       QueteletIndex\n",
       "age           92.322739  26.57812  6.135912  7.541478    \n",
       "wgt           26.578116 178.10938 90.054515 36.023441    \n",
       "hgt            6.135912  90.05452 88.495510  6.299680    \n",
       "QueteletIndex  7.541478  36.02344  6.299680 10.714224    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cov(lessData)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>age</th><th scope=col>wgt</th><th scope=col>hgt</th><th scope=col>QueteletIndex</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>age</th><td>1.00000000</td><td>0.2072652 </td><td>0.06788349</td><td>0.2397849 </td></tr>\n",
       "\t<tr><th scope=row>wgt</th><td>0.20726524</td><td>1.0000000 </td><td>0.71730108</td><td>0.8246344 </td></tr>\n",
       "\t<tr><th scope=row>hgt</th><td>0.06788349</td><td>0.7173011 </td><td>1.00000000</td><td>0.2045867 </td></tr>\n",
       "\t<tr><th scope=row>QueteletIndex</th><td>0.23978488</td><td>0.8246344 </td><td>0.20458675</td><td>1.0000000 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llll}\n",
       "  & age & wgt & hgt & QueteletIndex\\\\\n",
       "\\hline\n",
       "\tage & 1.00000000 & 0.2072652  & 0.06788349 & 0.2397849 \\\\\n",
       "\twgt & 0.20726524 & 1.0000000  & 0.71730108 & 0.8246344 \\\\\n",
       "\thgt & 0.06788349 & 0.7173011  & 1.00000000 & 0.2045867 \\\\\n",
       "\tQueteletIndex & 0.23978488 & 0.8246344  & 0.20458675 & 1.0000000 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | age | wgt | hgt | QueteletIndex | \n",
       "|---|---|---|---|\n",
       "| age | 1.00000000 | 0.2072652  | 0.06788349 | 0.2397849  | \n",
       "| wgt | 0.20726524 | 1.0000000  | 0.71730108 | 0.8246344  | \n",
       "| hgt | 0.06788349 | 0.7173011  | 1.00000000 | 0.2045867  | \n",
       "| QueteletIndex | 0.23978488 | 0.8246344  | 0.20458675 | 1.0000000  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "              age        wgt       hgt        QueteletIndex\n",
       "age           1.00000000 0.2072652 0.06788349 0.2397849    \n",
       "wgt           0.20726524 1.0000000 0.71730108 0.8246344    \n",
       "hgt           0.06788349 0.7173011 1.00000000 0.2045867    \n",
       "QueteletIndex 0.23978488 0.8246344 0.20458675 1.0000000    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(lessData)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Strong correlations?\n",
    "* We see that correlation between weight and height is 0.72\n",
    "* We see that correlation between weight and quetelet index is 0.82\n",
    "\n",
    "Can we predict weight based on someone's height and quetelet index (BMI)? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ldrU6dTdyw+KqvpuU4aMRZ45KFGeBkc2VSP6R5xxPndLy3DJhM8iVSKsDvkcYS7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\nsLOSxD2u8sl5jr2jZO8zA6m++rUOwY8HwSOMQecB88SKlNVIsY1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BlNNzoe9cc5RgqAYAFr/wMXPMs28JJHQQ+zglwZDZjyZYuhAsTFBU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\n/wAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(lessData)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using a linear model solver, lm()\n",
    "\n",
    "LM function computes a linear regression\n",
    "\n",
    "* LHS ~ RHS is the model to compute\n",
    "    * LHS is the dependent variable\n",
    "    * RHS is the independent variable (predictors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# fit variable will hold a statistical model\n",
    "fit <- lm(wgt ~ hgt + QueteletIndex, data = lessData)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = wgt ~ hgt + QueteletIndex, data = lessData)\n",
       "\n",
       "Coefficients:\n",
       "  (Intercept)            hgt  QueteletIndex  \n",
       "    -137.5505         0.8123         2.8846  \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fit # model coefficients"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = wgt ~ hgt + QueteletIndex, data = lessData)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-4.4684 -0.5174 -0.2086  0.4846  5.0896 \n",
       "\n",
       "Coefficients:\n",
       "                Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   -137.55047    0.84580  -162.6   <2e-16 ***\n",
       "hgt              0.81227    0.00495   164.1   <2e-16 ***\n",
       "QueteletIndex    2.88461    0.01422   202.8   <2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 1.025 on 504 degrees of freedom\n",
       "Multiple R-squared:  0.9941,\tAdjusted R-squared:  0.9941 \n",
       "F-statistic: 4.262e+04 on 2 and 504 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### A key line is the R-squared test\n",
    "\n",
    "Multiple R-squared:  0.9941,\tAdjusted R-squared:  0.9941 \n",
    "\n",
    "The R-squared test measures the proportion of the variance in the dependent variable that is predictable from the independent variable. \n",
    "\n",
    "Reviewing the solved model.\n",
    "```\n",
    "Coefficients:\n",
    "  (Intercept)            hgt  QueteletIndex  \n",
    "    -137.5505         0.8123         2.8846  \n",
    "```\n",
    "That implies that   \n",
    "$wgt = 0.8123*hgt + 2.8846*QueteletIndex - 137.5505$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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CJphU+m10nCo+W0cERSCheezmDD4+W0cETSCZ+cNiEJj5jTwhFJJTyiRymXRTEc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\nOPAFzQ8QyRj21lwa4MAXNN9fJPPwkebSAAe+oPneIv21DsNqLg1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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(ggplot2)\n",
    "\n",
    "########################\n",
    "#  adapted from: \n",
    "# https://susanejohnston.wordpress.com/2012/08/09/a-quick-and-easy-function-to-plot-lm-results-in-r/\n",
    "########################\n",
    "\n",
    "    # Note that fit is an object with various things, such as a model.\n",
    "ggplot(lessData, \n",
    "       aes_string(\n",
    "                x = (0.8123*lessData$hgt + 2.8846*lessData$QueteletIndex - 137.5505), \n",
    "                y = names(fit$model)[1]\n",
    "       ) # end of aes_string\n",
    "  ) + \n",
    "  geom_point() +\n",
    "  stat_smooth(method = \"lm\", col = \"red\") +\n",
    "  labs(title = paste(\"Adj R2 = \",signif(summary(fit)$adj.r.squared, 5),\n",
    "                     \"Intercept =\",signif(fit$coef[[1]],5 ),\n",
    "                     \" Slope =\",signif(fit$coef[[2]], 5),\n",
    "                     \" P =\",signif(summary(fit)$coef[2,4], 5)\n",
    "                ) # end of title string concatenations\n",
    "       , x = \"hgt + QueteletIndex\"\n",
    "      ) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.4.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}