Week 4 - Data Analysis (R).ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "16ff7c79",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "Attaching package: 'dplyr'\n",
      "\n",
      "\n",
      "The following objects are masked from 'package:stats':\n",
      "\n",
      "    filter, lag\n",
      "\n",
      "\n",
      "The following objects are masked from 'package:base':\n",
      "\n",
      "    intersect, setdiff, setequal, union\n",
      "\n",
      "\n",
      "Loading required package: carData\n",
      "\n",
      "\n",
      "Attaching package: 'car'\n",
      "\n",
      "\n",
      "The following object is masked from 'package:dplyr':\n",
      "\n",
      "    recode\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Load libraries\n",
    "library(dplyr)\n",
    "library(rcompanion)\n",
    "library(mvnormtest)\n",
    "library(car)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "5e5cb944",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 10 × 17</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>instant</th><th scope=col>dteday</th><th scope=col>season</th><th scope=col>yr</th><th scope=col>mnth</th><th scope=col>hr</th><th scope=col>holiday</th><th scope=col>weekday</th><th scope=col>workingday</th><th scope=col>weathersit</th><th scope=col>temp</th><th scope=col>atemp</th><th scope=col>hum</th><th scope=col>windspeed</th><th scope=col>casual</th><th scope=col>registered</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 1</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.81</td><td>0.0000</td><td>3</td><td>13</td><td>16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td> 2</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>1</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>8</td><td>32</td><td>40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td> 3</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>2</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>5</td><td>27</td><td>32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 4</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>3</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>3</td><td>10</td><td>13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td> 5</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>4</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>0</td><td> 1</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td> 6</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>5</td><td>0</td><td>6</td><td>0</td><td>2</td><td>0.24</td><td>0.2576</td><td>0.75</td><td>0.0896</td><td>0</td><td> 1</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td> 7</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>6</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>2</td><td> 0</td><td> 2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td> 8</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>7</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.20</td><td>0.2576</td><td>0.86</td><td>0.0000</td><td>1</td><td> 2</td><td> 3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td> 9</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>8</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>1</td><td> 7</td><td> 8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>10</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>9</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.32</td><td>0.3485</td><td>0.76</td><td>0.0000</td><td>8</td><td> 6</td><td>14</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 10 × 17\n",
       "\\begin{tabular}{r|lllllllllllllllll}\n",
       "  & instant & dteday & season & yr & mnth & hr & holiday & weekday & workingday & weathersit & temp & atemp & hum & windspeed & casual & registered & cnt\\\\\n",
       "  & <int> & <chr> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <dbl> & <dbl> & <dbl> & <dbl> & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 &  1 & 2011-01-01 & 1 & 0 & 1 & 0 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.81 & 0.0000 & 3 & 13 & 16\\\\\n",
       "\t2 &  2 & 2011-01-01 & 1 & 0 & 1 & 1 & 0 & 6 & 0 & 1 & 0.22 & 0.2727 & 0.80 & 0.0000 & 8 & 32 & 40\\\\\n",
       "\t3 &  3 & 2011-01-01 & 1 & 0 & 1 & 2 & 0 & 6 & 0 & 1 & 0.22 & 0.2727 & 0.80 & 0.0000 & 5 & 27 & 32\\\\\n",
       "\t4 &  4 & 2011-01-01 & 1 & 0 & 1 & 3 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.75 & 0.0000 & 3 & 10 & 13\\\\\n",
       "\t5 &  5 & 2011-01-01 & 1 & 0 & 1 & 4 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.75 & 0.0000 & 0 &  1 &  1\\\\\n",
       "\t6 &  6 & 2011-01-01 & 1 & 0 & 1 & 5 & 0 & 6 & 0 & 2 & 0.24 & 0.2576 & 0.75 & 0.0896 & 0 &  1 &  1\\\\\n",
       "\t7 &  7 & 2011-01-01 & 1 & 0 & 1 & 6 & 0 & 6 & 0 & 1 & 0.22 & 0.2727 & 0.80 & 0.0000 & 2 &  0 &  2\\\\\n",
       "\t8 &  8 & 2011-01-01 & 1 & 0 & 1 & 7 & 0 & 6 & 0 & 1 & 0.20 & 0.2576 & 0.86 & 0.0000 & 1 &  2 &  3\\\\\n",
       "\t9 &  9 & 2011-01-01 & 1 & 0 & 1 & 8 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.75 & 0.0000 & 1 &  7 &  8\\\\\n",
       "\t10 & 10 & 2011-01-01 & 1 & 0 & 1 & 9 & 0 & 6 & 0 & 1 & 0.32 & 0.3485 & 0.76 & 0.0000 & 8 &  6 & 14\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 10 × 17\n",
       "\n",
       "| <!--/--> | instant &lt;int&gt; | dteday &lt;chr&gt; | season &lt;int&gt; | yr &lt;int&gt; | mnth &lt;int&gt; | hr &lt;int&gt; | holiday &lt;int&gt; | weekday &lt;int&gt; | workingday &lt;int&gt; | weathersit &lt;int&gt; | temp &lt;dbl&gt; | atemp &lt;dbl&gt; | hum &lt;dbl&gt; | windspeed &lt;dbl&gt; | casual &lt;int&gt; | registered &lt;int&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| 1 |  1 | 2011-01-01 | 1 | 0 | 1 | 0 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.81 | 0.0000 | 3 | 13 | 16 |\n",
       "| 2 |  2 | 2011-01-01 | 1 | 0 | 1 | 1 | 0 | 6 | 0 | 1 | 0.22 | 0.2727 | 0.80 | 0.0000 | 8 | 32 | 40 |\n",
       "| 3 |  3 | 2011-01-01 | 1 | 0 | 1 | 2 | 0 | 6 | 0 | 1 | 0.22 | 0.2727 | 0.80 | 0.0000 | 5 | 27 | 32 |\n",
       "| 4 |  4 | 2011-01-01 | 1 | 0 | 1 | 3 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.75 | 0.0000 | 3 | 10 | 13 |\n",
       "| 5 |  5 | 2011-01-01 | 1 | 0 | 1 | 4 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.75 | 0.0000 | 0 |  1 |  1 |\n",
       "| 6 |  6 | 2011-01-01 | 1 | 0 | 1 | 5 | 0 | 6 | 0 | 2 | 0.24 | 0.2576 | 0.75 | 0.0896 | 0 |  1 |  1 |\n",
       "| 7 |  7 | 2011-01-01 | 1 | 0 | 1 | 6 | 0 | 6 | 0 | 1 | 0.22 | 0.2727 | 0.80 | 0.0000 | 2 |  0 |  2 |\n",
       "| 8 |  8 | 2011-01-01 | 1 | 0 | 1 | 7 | 0 | 6 | 0 | 1 | 0.20 | 0.2576 | 0.86 | 0.0000 | 1 |  2 |  3 |\n",
       "| 9 |  9 | 2011-01-01 | 1 | 0 | 1 | 8 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.75 | 0.0000 | 1 |  7 |  8 |\n",
       "| 10 | 10 | 2011-01-01 | 1 | 0 | 1 | 9 | 0 | 6 | 0 | 1 | 0.32 | 0.3485 | 0.76 | 0.0000 | 8 |  6 | 14 |\n",
       "\n"
      ],
      "text/plain": [
       "   instant dteday     season yr mnth hr holiday weekday workingday weathersit\n",
       "1   1      2011-01-01 1      0  1    0  0       6       0          1         \n",
       "2   2      2011-01-01 1      0  1    1  0       6       0          1         \n",
       "3   3      2011-01-01 1      0  1    2  0       6       0          1         \n",
       "4   4      2011-01-01 1      0  1    3  0       6       0          1         \n",
       "5   5      2011-01-01 1      0  1    4  0       6       0          1         \n",
       "6   6      2011-01-01 1      0  1    5  0       6       0          2         \n",
       "7   7      2011-01-01 1      0  1    6  0       6       0          1         \n",
       "8   8      2011-01-01 1      0  1    7  0       6       0          1         \n",
       "9   9      2011-01-01 1      0  1    8  0       6       0          1         \n",
       "10 10      2011-01-01 1      0  1    9  0       6       0          1         \n",
       "   temp atemp  hum  windspeed casual registered cnt\n",
       "1  0.24 0.2879 0.81 0.0000    3      13         16 \n",
       "2  0.22 0.2727 0.80 0.0000    8      32         40 \n",
       "3  0.22 0.2727 0.80 0.0000    5      27         32 \n",
       "4  0.24 0.2879 0.75 0.0000    3      10         13 \n",
       "5  0.24 0.2879 0.75 0.0000    0       1          1 \n",
       "6  0.24 0.2576 0.75 0.0896    0       1          1 \n",
       "7  0.22 0.2727 0.80 0.0000    2       0          2 \n",
       "8  0.20 0.2576 0.86 0.0000    1       2          3 \n",
       "9  0.24 0.2879 0.75 0.0000    1       7          8 \n",
       "10 0.32 0.3485 0.76 0.0000    8       6         14 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Read in bike rental data\n",
    "bikes <- read.csv(\"hour.csv\")\n",
    "head(bikes, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6733847e",
   "metadata": {},
   "source": [
    "# Run a Multiple Linear Regression on the Continuous Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d43425d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 10 × 6</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>dteday</th><th scope=col>temp</th><th scope=col>atemp</th><th scope=col>hum</th><th scope=col>windspeed</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.81</td><td>0.0000</td><td>16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>2011-01-01</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>2011-01-01</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>2011-01-01</td><td>0.24</td><td>0.2576</td><td>0.75</td><td>0.0896</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>2011-01-01</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td> 2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>2011-01-01</td><td>0.20</td><td>0.2576</td><td>0.86</td><td>0.0000</td><td> 3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td> 8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>2011-01-01</td><td>0.32</td><td>0.3485</td><td>0.76</td><td>0.0000</td><td>14</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 10 × 6\n",
       "\\begin{tabular}{r|llllll}\n",
       "  & dteday & temp & atemp & hum & windspeed & cnt\\\\\n",
       "  & <chr> & <dbl> & <dbl> & <dbl> & <dbl> & <int>\\\\\n",
       "\\hline\n",
       "\t1 & 2011-01-01 & 0.24 & 0.2879 & 0.81 & 0.0000 & 16\\\\\n",
       "\t2 & 2011-01-01 & 0.22 & 0.2727 & 0.80 & 0.0000 & 40\\\\\n",
       "\t3 & 2011-01-01 & 0.22 & 0.2727 & 0.80 & 0.0000 & 32\\\\\n",
       "\t4 & 2011-01-01 & 0.24 & 0.2879 & 0.75 & 0.0000 & 13\\\\\n",
       "\t5 & 2011-01-01 & 0.24 & 0.2879 & 0.75 & 0.0000 &  1\\\\\n",
       "\t6 & 2011-01-01 & 0.24 & 0.2576 & 0.75 & 0.0896 &  1\\\\\n",
       "\t7 & 2011-01-01 & 0.22 & 0.2727 & 0.80 & 0.0000 &  2\\\\\n",
       "\t8 & 2011-01-01 & 0.20 & 0.2576 & 0.86 & 0.0000 &  3\\\\\n",
       "\t9 & 2011-01-01 & 0.24 & 0.2879 & 0.75 & 0.0000 &  8\\\\\n",
       "\t10 & 2011-01-01 & 0.32 & 0.3485 & 0.76 & 0.0000 & 14\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 10 × 6\n",
       "\n",
       "| <!--/--> | dteday &lt;chr&gt; | temp &lt;dbl&gt; | atemp &lt;dbl&gt; | hum &lt;dbl&gt; | windspeed &lt;dbl&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|---|---|---|\n",
       "| 1 | 2011-01-01 | 0.24 | 0.2879 | 0.81 | 0.0000 | 16 |\n",
       "| 2 | 2011-01-01 | 0.22 | 0.2727 | 0.80 | 0.0000 | 40 |\n",
       "| 3 | 2011-01-01 | 0.22 | 0.2727 | 0.80 | 0.0000 | 32 |\n",
       "| 4 | 2011-01-01 | 0.24 | 0.2879 | 0.75 | 0.0000 | 13 |\n",
       "| 5 | 2011-01-01 | 0.24 | 0.2879 | 0.75 | 0.0000 |  1 |\n",
       "| 6 | 2011-01-01 | 0.24 | 0.2576 | 0.75 | 0.0896 |  1 |\n",
       "| 7 | 2011-01-01 | 0.22 | 0.2727 | 0.80 | 0.0000 |  2 |\n",
       "| 8 | 2011-01-01 | 0.20 | 0.2576 | 0.86 | 0.0000 |  3 |\n",
       "| 9 | 2011-01-01 | 0.24 | 0.2879 | 0.75 | 0.0000 |  8 |\n",
       "| 10 | 2011-01-01 | 0.32 | 0.3485 | 0.76 | 0.0000 | 14 |\n",
       "\n"
      ],
      "text/plain": [
       "   dteday     temp atemp  hum  windspeed cnt\n",
       "1  2011-01-01 0.24 0.2879 0.81 0.0000    16 \n",
       "2  2011-01-01 0.22 0.2727 0.80 0.0000    40 \n",
       "3  2011-01-01 0.22 0.2727 0.80 0.0000    32 \n",
       "4  2011-01-01 0.24 0.2879 0.75 0.0000    13 \n",
       "5  2011-01-01 0.24 0.2879 0.75 0.0000     1 \n",
       "6  2011-01-01 0.24 0.2576 0.75 0.0896     1 \n",
       "7  2011-01-01 0.22 0.2727 0.80 0.0000     2 \n",
       "8  2011-01-01 0.20 0.2576 0.86 0.0000     3 \n",
       "9  2011-01-01 0.24 0.2879 0.75 0.0000     8 \n",
       "10 2011-01-01 0.32 0.3485 0.76 0.0000    14 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Remove `instant`, `season`, `yr`, `mnth`, `hr`, `holiday`, `weekday`, `workingday`, `weathersit`, `casual`, `registered` so we are left with only the continuous predictor variables with `cnt` as DV\n",
    "bikes1 <- subset(bikes, select = -c(instant, season, yr, mnth, hr, holiday, weekday, workingday, weathersit, casual, registered))\n",
    "head(bikes1, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1183ce83",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 10 × 6</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>dteday</th><th scope=col>temp</th><th scope=col>atemp</th><th scope=col>hum</th><th scope=col>windspeed</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;date&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.81</td><td>0.0000</td><td>16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>2011-01-01</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>2011-01-01</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>2011-01-01</td><td>0.24</td><td>0.2576</td><td>0.75</td><td>0.0896</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>2011-01-01</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td> 2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>2011-01-01</td><td>0.20</td><td>0.2576</td><td>0.86</td><td>0.0000</td><td> 3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td>2011-01-01</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td> 8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>2011-01-01</td><td>0.32</td><td>0.3485</td><td>0.76</td><td>0.0000</td><td>14</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 10 × 6\n",
       "\\begin{tabular}{r|llllll}\n",
       "  & dteday & temp & atemp & hum & windspeed & cnt\\\\\n",
       "  & <date> & <dbl> & <dbl> & <dbl> & <dbl> & <int>\\\\\n",
       "\\hline\n",
       "\t1 & 2011-01-01 & 0.24 & 0.2879 & 0.81 & 0.0000 & 16\\\\\n",
       "\t2 & 2011-01-01 & 0.22 & 0.2727 & 0.80 & 0.0000 & 40\\\\\n",
       "\t3 & 2011-01-01 & 0.22 & 0.2727 & 0.80 & 0.0000 & 32\\\\\n",
       "\t4 & 2011-01-01 & 0.24 & 0.2879 & 0.75 & 0.0000 & 13\\\\\n",
       "\t5 & 2011-01-01 & 0.24 & 0.2879 & 0.75 & 0.0000 &  1\\\\\n",
       "\t6 & 2011-01-01 & 0.24 & 0.2576 & 0.75 & 0.0896 &  1\\\\\n",
       "\t7 & 2011-01-01 & 0.22 & 0.2727 & 0.80 & 0.0000 &  2\\\\\n",
       "\t8 & 2011-01-01 & 0.20 & 0.2576 & 0.86 & 0.0000 &  3\\\\\n",
       "\t9 & 2011-01-01 & 0.24 & 0.2879 & 0.75 & 0.0000 &  8\\\\\n",
       "\t10 & 2011-01-01 & 0.32 & 0.3485 & 0.76 & 0.0000 & 14\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 10 × 6\n",
       "\n",
       "| <!--/--> | dteday &lt;date&gt; | temp &lt;dbl&gt; | atemp &lt;dbl&gt; | hum &lt;dbl&gt; | windspeed &lt;dbl&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|---|---|---|\n",
       "| 1 | 2011-01-01 | 0.24 | 0.2879 | 0.81 | 0.0000 | 16 |\n",
       "| 2 | 2011-01-01 | 0.22 | 0.2727 | 0.80 | 0.0000 | 40 |\n",
       "| 3 | 2011-01-01 | 0.22 | 0.2727 | 0.80 | 0.0000 | 32 |\n",
       "| 4 | 2011-01-01 | 0.24 | 0.2879 | 0.75 | 0.0000 | 13 |\n",
       "| 5 | 2011-01-01 | 0.24 | 0.2879 | 0.75 | 0.0000 |  1 |\n",
       "| 6 | 2011-01-01 | 0.24 | 0.2576 | 0.75 | 0.0896 |  1 |\n",
       "| 7 | 2011-01-01 | 0.22 | 0.2727 | 0.80 | 0.0000 |  2 |\n",
       "| 8 | 2011-01-01 | 0.20 | 0.2576 | 0.86 | 0.0000 |  3 |\n",
       "| 9 | 2011-01-01 | 0.24 | 0.2879 | 0.75 | 0.0000 |  8 |\n",
       "| 10 | 2011-01-01 | 0.32 | 0.3485 | 0.76 | 0.0000 | 14 |\n",
       "\n"
      ],
      "text/plain": [
       "   dteday     temp atemp  hum  windspeed cnt\n",
       "1  2011-01-01 0.24 0.2879 0.81 0.0000    16 \n",
       "2  2011-01-01 0.22 0.2727 0.80 0.0000    40 \n",
       "3  2011-01-01 0.22 0.2727 0.80 0.0000    32 \n",
       "4  2011-01-01 0.24 0.2879 0.75 0.0000    13 \n",
       "5  2011-01-01 0.24 0.2879 0.75 0.0000     1 \n",
       "6  2011-01-01 0.24 0.2576 0.75 0.0896     1 \n",
       "7  2011-01-01 0.22 0.2727 0.80 0.0000     2 \n",
       "8  2011-01-01 0.20 0.2576 0.86 0.0000     3 \n",
       "9  2011-01-01 0.24 0.2879 0.75 0.0000     8 \n",
       "10 2011-01-01 0.32 0.3485 0.76 0.0000    14 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bikes2 <- bikes1\n",
    "\n",
    "# Coerce `dteday` to date\n",
    "bikes2$dteday <- as.Date(bikes2$dteday)\n",
    "\n",
    "# Coerce `\n",
    "head(bikes2, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a32ab805",
   "metadata": {},
   "source": [
    "## Backward Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "9491054a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = cnt ~ ., data = bikes2)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-350.47 -101.19  -25.01   70.26  644.73 \n",
       "\n",
       "Coefficients:\n",
       "              Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) -2.943e+03  8.461e+01 -34.783  < 2e-16 ***\n",
       "dteday       2.030e-01  5.518e-03  36.797  < 2e-16 ***\n",
       "temp         5.896e+01  3.926e+01   1.502    0.133    \n",
       "atemp        3.108e+02  4.403e+01   7.058 1.75e-12 ***\n",
       "hum         -2.749e+02  6.228e+00 -44.138  < 2e-16 ***\n",
       "windspeed    6.786e+01  1.009e+01   6.726 1.80e-11 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 151 on 17373 degrees of freedom\n",
       "Multiple R-squared:  0.3074,\tAdjusted R-squared:  0.3072 \n",
       "F-statistic:  1542 on 5 and 17373 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Run a linear regression model with all possible variables included\n",
    "FitAll = lm(cnt ~ ., data = bikes2)\n",
    "summary(FitAll)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0cf68592",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Start:  AIC=174392.1\n",
      "cnt ~ dteday + temp + atemp + hum + windspeed\n",
      "\n",
      "            Df Sum of Sq       RSS    AIC\n",
      "<none>                   396018340 174392\n",
      "- temp       1     51420 396069760 174392\n",
      "- windspeed  1   1031194 397049534 174435\n",
      "- atemp      1   1135559 397153899 174440\n",
      "- dteday     1  30864489 426882829 175694\n",
      "- hum        1  44407699 440426039 176237\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = cnt ~ dteday + temp + atemp + hum + windspeed, data = bikes2)\n",
       "\n",
       "Coefficients:\n",
       "(Intercept)       dteday         temp        atemp          hum    windspeed  \n",
       "  -2942.877        0.203       58.963      310.781     -274.878       67.860  \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Run a backwards regression\n",
    "step(FitAll, direction = 'backward')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "0644d975",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = cnt ~ dteday + temp + atemp + hum + windspeed, data = bikes2)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-350.47 -101.19  -25.01   70.26  644.73 \n",
       "\n",
       "Coefficients:\n",
       "              Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept) -2.943e+03  8.461e+01 -34.783  < 2e-16 ***\n",
       "dteday       2.030e-01  5.518e-03  36.797  < 2e-16 ***\n",
       "temp         5.896e+01  3.926e+01   1.502    0.133    \n",
       "atemp        3.108e+02  4.403e+01   7.058 1.75e-12 ***\n",
       "hum         -2.749e+02  6.228e+00 -44.138  < 2e-16 ***\n",
       "windspeed    6.786e+01  1.009e+01   6.726 1.80e-11 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 151 on 17373 degrees of freedom\n",
       "Multiple R-squared:  0.3074,\tAdjusted R-squared:  0.3072 \n",
       "F-statistic:  1542 on 5 and 17373 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fitsome <- lm(cnt ~ dteday + temp + atemp + hum + windspeed, data = bikes2)\n",
    "summary(fitsome)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73bee100",
   "metadata": {},
   "source": [
    "## Forward Regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b5fb7b63",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = cnt ~ 1, data = bikes2)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-188.46 -149.46  -47.46   91.54  787.54 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  189.463      1.376   137.7   <2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 181.4 on 17378 degrees of freedom\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fitstart <- lm(cnt ~ 1, data = bikes2)\n",
    "summary(fitstart)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "b5e62658",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Start:  AIC=180764.7\n",
      "cnt ~ 1\n",
      "\n",
      "            Df Sum of Sq       RSS    AIC\n",
      "+ temp       1  93677759 478083832 177657\n",
      "+ atemp      1  91907421 479854170 177721\n",
      "+ hum        1  59618351 512143240 178853\n",
      "+ dteday     1  44109491 527652100 179371\n",
      "+ windspeed  1   4970060 566791531 180615\n",
      "<none>                   571761591 180765\n",
      "\n",
      "Step:  AIC=177657\n",
      "cnt ~ temp\n",
      "\n",
      "            Df Sum of Sq       RSS    AIC\n",
      "+ hum        1  49874585 428209247 175744\n",
      "+ dteday     1  28884878 449198954 176576\n",
      "+ windspeed  1   6021342 472062489 177439\n",
      "<none>                   478083832 177657\n",
      "+ atemp      1     30698 478053133 177658\n",
      "\n",
      "Step:  AIC=175744.3\n",
      "cnt ~ temp + hum\n",
      "\n",
      "            Df Sum of Sq       RSS    AIC\n",
      "+ dteday     1  30430343 397778904 174465\n",
      "+ atemp      1    910902 427298346 175709\n",
      "+ windspeed  1    164674 428044573 175740\n",
      "<none>                   428209247 175744\n",
      "\n",
      "Step:  AIC=174465.2\n",
      "cnt ~ temp + hum + dteday\n",
      "\n",
      "            Df Sum of Sq       RSS    AIC\n",
      "+ atemp      1    729370 397049534 174435\n",
      "+ windspeed  1    625004 397153899 174440\n",
      "<none>                   397778904 174465\n",
      "\n",
      "Step:  AIC=174435.3\n",
      "cnt ~ temp + hum + dteday + atemp\n",
      "\n",
      "            Df Sum of Sq       RSS    AIC\n",
      "+ windspeed  1   1031194 396018340 174392\n",
      "<none>                   397049534 174435\n",
      "\n",
      "Step:  AIC=174392.1\n",
      "cnt ~ temp + hum + dteday + atemp + windspeed\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = cnt ~ temp + hum + dteday + atemp + windspeed, data = bikes2)\n",
       "\n",
       "Coefficients:\n",
       "(Intercept)         temp          hum       dteday        atemp    windspeed  \n",
       "  -2942.877       58.963     -274.878        0.203      310.781       67.860  \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "step(fitstart, direction = 'forward', scope = (formula(FitAll)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0689407",
   "metadata": {},
   "source": [
    "Both forward and backward regression have shown that all continuous variables in the dataset are good predictors of the `cnt` dependent variable."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ddcee10",
   "metadata": {},
   "source": [
    "# Run a MANOVA on the Categorical Variables with all 3 possible DVs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "4c64f1bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 10 × 17</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>instant</th><th scope=col>dteday</th><th scope=col>season</th><th scope=col>yr</th><th scope=col>mnth</th><th scope=col>hr</th><th scope=col>holiday</th><th scope=col>weekday</th><th scope=col>workingday</th><th scope=col>weathersit</th><th scope=col>temp</th><th scope=col>atemp</th><th scope=col>hum</th><th scope=col>windspeed</th><th scope=col>casual</th><th scope=col>registered</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 1</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.81</td><td>0.0000</td><td>3</td><td>13</td><td>16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td> 2</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>1</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>8</td><td>32</td><td>40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td> 3</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>2</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>5</td><td>27</td><td>32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 4</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>3</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>3</td><td>10</td><td>13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td> 5</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>4</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>0</td><td> 1</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td> 6</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>5</td><td>0</td><td>6</td><td>0</td><td>2</td><td>0.24</td><td>0.2576</td><td>0.75</td><td>0.0896</td><td>0</td><td> 1</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td> 7</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>6</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.22</td><td>0.2727</td><td>0.80</td><td>0.0000</td><td>2</td><td> 0</td><td> 2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td> 8</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>7</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.20</td><td>0.2576</td><td>0.86</td><td>0.0000</td><td>1</td><td> 2</td><td> 3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td> 9</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>8</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.24</td><td>0.2879</td><td>0.75</td><td>0.0000</td><td>1</td><td> 7</td><td> 8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>10</td><td>2011-01-01</td><td>1</td><td>0</td><td>1</td><td>9</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0.32</td><td>0.3485</td><td>0.76</td><td>0.0000</td><td>8</td><td> 6</td><td>14</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 10 × 17\n",
       "\\begin{tabular}{r|lllllllllllllllll}\n",
       "  & instant & dteday & season & yr & mnth & hr & holiday & weekday & workingday & weathersit & temp & atemp & hum & windspeed & casual & registered & cnt\\\\\n",
       "  & <int> & <chr> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <dbl> & <dbl> & <dbl> & <dbl> & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 &  1 & 2011-01-01 & 1 & 0 & 1 & 0 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.81 & 0.0000 & 3 & 13 & 16\\\\\n",
       "\t2 &  2 & 2011-01-01 & 1 & 0 & 1 & 1 & 0 & 6 & 0 & 1 & 0.22 & 0.2727 & 0.80 & 0.0000 & 8 & 32 & 40\\\\\n",
       "\t3 &  3 & 2011-01-01 & 1 & 0 & 1 & 2 & 0 & 6 & 0 & 1 & 0.22 & 0.2727 & 0.80 & 0.0000 & 5 & 27 & 32\\\\\n",
       "\t4 &  4 & 2011-01-01 & 1 & 0 & 1 & 3 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.75 & 0.0000 & 3 & 10 & 13\\\\\n",
       "\t5 &  5 & 2011-01-01 & 1 & 0 & 1 & 4 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.75 & 0.0000 & 0 &  1 &  1\\\\\n",
       "\t6 &  6 & 2011-01-01 & 1 & 0 & 1 & 5 & 0 & 6 & 0 & 2 & 0.24 & 0.2576 & 0.75 & 0.0896 & 0 &  1 &  1\\\\\n",
       "\t7 &  7 & 2011-01-01 & 1 & 0 & 1 & 6 & 0 & 6 & 0 & 1 & 0.22 & 0.2727 & 0.80 & 0.0000 & 2 &  0 &  2\\\\\n",
       "\t8 &  8 & 2011-01-01 & 1 & 0 & 1 & 7 & 0 & 6 & 0 & 1 & 0.20 & 0.2576 & 0.86 & 0.0000 & 1 &  2 &  3\\\\\n",
       "\t9 &  9 & 2011-01-01 & 1 & 0 & 1 & 8 & 0 & 6 & 0 & 1 & 0.24 & 0.2879 & 0.75 & 0.0000 & 1 &  7 &  8\\\\\n",
       "\t10 & 10 & 2011-01-01 & 1 & 0 & 1 & 9 & 0 & 6 & 0 & 1 & 0.32 & 0.3485 & 0.76 & 0.0000 & 8 &  6 & 14\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 10 × 17\n",
       "\n",
       "| <!--/--> | instant &lt;int&gt; | dteday &lt;chr&gt; | season &lt;int&gt; | yr &lt;int&gt; | mnth &lt;int&gt; | hr &lt;int&gt; | holiday &lt;int&gt; | weekday &lt;int&gt; | workingday &lt;int&gt; | weathersit &lt;int&gt; | temp &lt;dbl&gt; | atemp &lt;dbl&gt; | hum &lt;dbl&gt; | windspeed &lt;dbl&gt; | casual &lt;int&gt; | registered &lt;int&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| 1 |  1 | 2011-01-01 | 1 | 0 | 1 | 0 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.81 | 0.0000 | 3 | 13 | 16 |\n",
       "| 2 |  2 | 2011-01-01 | 1 | 0 | 1 | 1 | 0 | 6 | 0 | 1 | 0.22 | 0.2727 | 0.80 | 0.0000 | 8 | 32 | 40 |\n",
       "| 3 |  3 | 2011-01-01 | 1 | 0 | 1 | 2 | 0 | 6 | 0 | 1 | 0.22 | 0.2727 | 0.80 | 0.0000 | 5 | 27 | 32 |\n",
       "| 4 |  4 | 2011-01-01 | 1 | 0 | 1 | 3 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.75 | 0.0000 | 3 | 10 | 13 |\n",
       "| 5 |  5 | 2011-01-01 | 1 | 0 | 1 | 4 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.75 | 0.0000 | 0 |  1 |  1 |\n",
       "| 6 |  6 | 2011-01-01 | 1 | 0 | 1 | 5 | 0 | 6 | 0 | 2 | 0.24 | 0.2576 | 0.75 | 0.0896 | 0 |  1 |  1 |\n",
       "| 7 |  7 | 2011-01-01 | 1 | 0 | 1 | 6 | 0 | 6 | 0 | 1 | 0.22 | 0.2727 | 0.80 | 0.0000 | 2 |  0 |  2 |\n",
       "| 8 |  8 | 2011-01-01 | 1 | 0 | 1 | 7 | 0 | 6 | 0 | 1 | 0.20 | 0.2576 | 0.86 | 0.0000 | 1 |  2 |  3 |\n",
       "| 9 |  9 | 2011-01-01 | 1 | 0 | 1 | 8 | 0 | 6 | 0 | 1 | 0.24 | 0.2879 | 0.75 | 0.0000 | 1 |  7 |  8 |\n",
       "| 10 | 10 | 2011-01-01 | 1 | 0 | 1 | 9 | 0 | 6 | 0 | 1 | 0.32 | 0.3485 | 0.76 | 0.0000 | 8 |  6 | 14 |\n",
       "\n"
      ],
      "text/plain": [
       "   instant dteday     season yr mnth hr holiday weekday workingday weathersit\n",
       "1   1      2011-01-01 1      0  1    0  0       6       0          1         \n",
       "2   2      2011-01-01 1      0  1    1  0       6       0          1         \n",
       "3   3      2011-01-01 1      0  1    2  0       6       0          1         \n",
       "4   4      2011-01-01 1      0  1    3  0       6       0          1         \n",
       "5   5      2011-01-01 1      0  1    4  0       6       0          1         \n",
       "6   6      2011-01-01 1      0  1    5  0       6       0          2         \n",
       "7   7      2011-01-01 1      0  1    6  0       6       0          1         \n",
       "8   8      2011-01-01 1      0  1    7  0       6       0          1         \n",
       "9   9      2011-01-01 1      0  1    8  0       6       0          1         \n",
       "10 10      2011-01-01 1      0  1    9  0       6       0          1         \n",
       "   temp atemp  hum  windspeed casual registered cnt\n",
       "1  0.24 0.2879 0.81 0.0000    3      13         16 \n",
       "2  0.22 0.2727 0.80 0.0000    8      32         40 \n",
       "3  0.22 0.2727 0.80 0.0000    5      27         32 \n",
       "4  0.24 0.2879 0.75 0.0000    3      10         13 \n",
       "5  0.24 0.2879 0.75 0.0000    0       1          1 \n",
       "6  0.24 0.2576 0.75 0.0896    0       1          1 \n",
       "7  0.22 0.2727 0.80 0.0000    2       0          2 \n",
       "8  0.20 0.2576 0.86 0.0000    1       2          3 \n",
       "9  0.24 0.2879 0.75 0.0000    1       7          8 \n",
       "10 0.32 0.3485 0.76 0.0000    8       6         14 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Make a new copy of the original dataset\n",
    "bikes3 <- bikes\n",
    "head(bikes3, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "6cb25985",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 10 × 11</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>season</th><th scope=col>yr</th><th scope=col>mnth</th><th scope=col>hr</th><th scope=col>holiday</th><th scope=col>weekday</th><th scope=col>workingday</th><th scope=col>weathersit</th><th scope=col>casual</th><th scope=col>registered</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td><td>6</td><td>0</td><td>1</td><td>3</td><td>13</td><td>16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>1</td><td>0</td><td>1</td><td>1</td><td>0</td><td>6</td><td>0</td><td>1</td><td>8</td><td>32</td><td>40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>1</td><td>0</td><td>1</td><td>2</td><td>0</td><td>6</td><td>0</td><td>1</td><td>5</td><td>27</td><td>32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>1</td><td>0</td><td>1</td><td>3</td><td>0</td><td>6</td><td>0</td><td>1</td><td>3</td><td>10</td><td>13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>1</td><td>0</td><td>1</td><td>4</td><td>0</td><td>6</td><td>0</td><td>1</td><td>0</td><td> 1</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>1</td><td>0</td><td>1</td><td>5</td><td>0</td><td>6</td><td>0</td><td>2</td><td>0</td><td> 1</td><td> 1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>1</td><td>0</td><td>1</td><td>6</td><td>0</td><td>6</td><td>0</td><td>1</td><td>2</td><td> 0</td><td> 2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>1</td><td>0</td><td>1</td><td>7</td><td>0</td><td>6</td><td>0</td><td>1</td><td>1</td><td> 2</td><td> 3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td>1</td><td>0</td><td>1</td><td>8</td><td>0</td><td>6</td><td>0</td><td>1</td><td>1</td><td> 7</td><td> 8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>1</td><td>0</td><td>1</td><td>9</td><td>0</td><td>6</td><td>0</td><td>1</td><td>8</td><td> 6</td><td>14</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 10 × 11\n",
       "\\begin{tabular}{r|lllllllllll}\n",
       "  & season & yr & mnth & hr & holiday & weekday & workingday & weathersit & casual & registered & cnt\\\\\n",
       "  & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 & 1 & 0 & 1 & 0 & 0 & 6 & 0 & 1 & 3 & 13 & 16\\\\\n",
       "\t2 & 1 & 0 & 1 & 1 & 0 & 6 & 0 & 1 & 8 & 32 & 40\\\\\n",
       "\t3 & 1 & 0 & 1 & 2 & 0 & 6 & 0 & 1 & 5 & 27 & 32\\\\\n",
       "\t4 & 1 & 0 & 1 & 3 & 0 & 6 & 0 & 1 & 3 & 10 & 13\\\\\n",
       "\t5 & 1 & 0 & 1 & 4 & 0 & 6 & 0 & 1 & 0 &  1 &  1\\\\\n",
       "\t6 & 1 & 0 & 1 & 5 & 0 & 6 & 0 & 2 & 0 &  1 &  1\\\\\n",
       "\t7 & 1 & 0 & 1 & 6 & 0 & 6 & 0 & 1 & 2 &  0 &  2\\\\\n",
       "\t8 & 1 & 0 & 1 & 7 & 0 & 6 & 0 & 1 & 1 &  2 &  3\\\\\n",
       "\t9 & 1 & 0 & 1 & 8 & 0 & 6 & 0 & 1 & 1 &  7 &  8\\\\\n",
       "\t10 & 1 & 0 & 1 & 9 & 0 & 6 & 0 & 1 & 8 &  6 & 14\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 10 × 11\n",
       "\n",
       "| <!--/--> | season &lt;int&gt; | yr &lt;int&gt; | mnth &lt;int&gt; | hr &lt;int&gt; | holiday &lt;int&gt; | weekday &lt;int&gt; | workingday &lt;int&gt; | weathersit &lt;int&gt; | casual &lt;int&gt; | registered &lt;int&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| 1 | 1 | 0 | 1 | 0 | 0 | 6 | 0 | 1 | 3 | 13 | 16 |\n",
       "| 2 | 1 | 0 | 1 | 1 | 0 | 6 | 0 | 1 | 8 | 32 | 40 |\n",
       "| 3 | 1 | 0 | 1 | 2 | 0 | 6 | 0 | 1 | 5 | 27 | 32 |\n",
       "| 4 | 1 | 0 | 1 | 3 | 0 | 6 | 0 | 1 | 3 | 10 | 13 |\n",
       "| 5 | 1 | 0 | 1 | 4 | 0 | 6 | 0 | 1 | 0 |  1 |  1 |\n",
       "| 6 | 1 | 0 | 1 | 5 | 0 | 6 | 0 | 2 | 0 |  1 |  1 |\n",
       "| 7 | 1 | 0 | 1 | 6 | 0 | 6 | 0 | 1 | 2 |  0 |  2 |\n",
       "| 8 | 1 | 0 | 1 | 7 | 0 | 6 | 0 | 1 | 1 |  2 |  3 |\n",
       "| 9 | 1 | 0 | 1 | 8 | 0 | 6 | 0 | 1 | 1 |  7 |  8 |\n",
       "| 10 | 1 | 0 | 1 | 9 | 0 | 6 | 0 | 1 | 8 |  6 | 14 |\n",
       "\n"
      ],
      "text/plain": [
       "   season yr mnth hr holiday weekday workingday weathersit casual registered\n",
       "1  1      0  1    0  0       6       0          1          3      13        \n",
       "2  1      0  1    1  0       6       0          1          8      32        \n",
       "3  1      0  1    2  0       6       0          1          5      27        \n",
       "4  1      0  1    3  0       6       0          1          3      10        \n",
       "5  1      0  1    4  0       6       0          1          0       1        \n",
       "6  1      0  1    5  0       6       0          2          0       1        \n",
       "7  1      0  1    6  0       6       0          1          2       0        \n",
       "8  1      0  1    7  0       6       0          1          1       2        \n",
       "9  1      0  1    8  0       6       0          1          1       7        \n",
       "10 1      0  1    9  0       6       0          1          8       6        \n",
       "   cnt\n",
       "1  16 \n",
       "2  40 \n",
       "3  32 \n",
       "4  13 \n",
       "5   1 \n",
       "6   1 \n",
       "7   2 \n",
       "8   3 \n",
       "9   8 \n",
       "10 14 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Subset to only the categorical IVs and the 3 possible DVs\n",
    "bikes4 <- subset(bikes3, select = -c(instant, dteday, temp, atemp, hum, windspeed))\n",
    "head(bikes4, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "eb3dc40c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 17379 × 11</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>season</th><th scope=col>yr</th><th scope=col>mnth</th><th scope=col>hr</th><th scope=col>holiday</th><th scope=col>weekday</th><th scope=col>workingday</th><th scope=col>weathersit</th><th scope=col>casual</th><th scope=col>registered</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;fct&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>1</td><td>0</td><td>1</td><td>0 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 3</td><td>13</td><td> 16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>1</td><td>0</td><td>1</td><td>1 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 8</td><td>32</td><td> 40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>1</td><td>0</td><td>1</td><td>2 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 5</td><td>27</td><td> 32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td>1</td><td>0</td><td>1</td><td>3 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 3</td><td>10</td><td> 13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>1</td><td>0</td><td>1</td><td>4 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 0</td><td> 1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>1</td><td>0</td><td>1</td><td>5 </td><td>0</td><td>6</td><td>0</td><td>2</td><td> 0</td><td> 1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>1</td><td>0</td><td>1</td><td>6 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 2</td><td> 0</td><td>  2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>1</td><td>0</td><td>1</td><td>7 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 1</td><td> 2</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td>1</td><td>0</td><td>1</td><td>8 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 1</td><td> 7</td><td>  8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td>1</td><td>0</td><td>1</td><td>9 </td><td>0</td><td>6</td><td>0</td><td>1</td><td> 8</td><td> 6</td><td> 14</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>1</td><td>0</td><td>1</td><td>10</td><td>0</td><td>6</td><td>0</td><td>1</td><td>12</td><td>24</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>12</th><td>1</td><td>0</td><td>1</td><td>11</td><td>0</td><td>6</td><td>0</td><td>1</td><td>26</td><td>30</td><td> 56</td></tr>\n",
       "\t<tr><th scope=row>13</th><td>1</td><td>0</td><td>1</td><td>12</td><td>0</td><td>6</td><td>0</td><td>1</td><td>29</td><td>55</td><td> 84</td></tr>\n",
       "\t<tr><th scope=row>14</th><td>1</td><td>0</td><td>1</td><td>13</td><td>0</td><td>6</td><td>0</td><td>2</td><td>47</td><td>47</td><td> 94</td></tr>\n",
       "\t<tr><th scope=row>15</th><td>1</td><td>0</td><td>1</td><td>14</td><td>0</td><td>6</td><td>0</td><td>2</td><td>35</td><td>71</td><td>106</td></tr>\n",
       "\t<tr><th scope=row>16</th><td>1</td><td>0</td><td>1</td><td>15</td><td>0</td><td>6</td><td>0</td><td>2</td><td>40</td><td>70</td><td>110</td></tr>\n",
       "\t<tr><th scope=row>17</th><td>1</td><td>0</td><td>1</td><td>16</td><td>0</td><td>6</td><td>0</td><td>2</td><td>41</td><td>52</td><td> 93</td></tr>\n",
       "\t<tr><th scope=row>18</th><td>1</td><td>0</td><td>1</td><td>17</td><td>0</td><td>6</td><td>0</td><td>2</td><td>15</td><td>52</td><td> 67</td></tr>\n",
       "\t<tr><th scope=row>19</th><td>1</td><td>0</td><td>1</td><td>18</td><td>0</td><td>6</td><td>0</td><td>3</td><td> 9</td><td>26</td><td> 35</td></tr>\n",
       "\t<tr><th scope=row>20</th><td>1</td><td>0</td><td>1</td><td>19</td><td>0</td><td>6</td><td>0</td><td>3</td><td> 6</td><td>31</td><td> 37</td></tr>\n",
       "\t<tr><th scope=row>21</th><td>1</td><td>0</td><td>1</td><td>20</td><td>0</td><td>6</td><td>0</td><td>2</td><td>11</td><td>25</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>22</th><td>1</td><td>0</td><td>1</td><td>21</td><td>0</td><td>6</td><td>0</td><td>2</td><td> 3</td><td>31</td><td> 34</td></tr>\n",
       "\t<tr><th scope=row>23</th><td>1</td><td>0</td><td>1</td><td>22</td><td>0</td><td>6</td><td>0</td><td>2</td><td>11</td><td>17</td><td> 28</td></tr>\n",
       "\t<tr><th scope=row>24</th><td>1</td><td>0</td><td>1</td><td>23</td><td>0</td><td>6</td><td>0</td><td>2</td><td>15</td><td>24</td><td> 39</td></tr>\n",
       "\t<tr><th scope=row>25</th><td>1</td><td>0</td><td>1</td><td>0 </td><td>0</td><td>0</td><td>0</td><td>2</td><td> 4</td><td>13</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>26</th><td>1</td><td>0</td><td>1</td><td>1 </td><td>0</td><td>0</td><td>0</td><td>2</td><td> 1</td><td>16</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>27</th><td>1</td><td>0</td><td>1</td><td>2 </td><td>0</td><td>0</td><td>0</td><td>2</td><td> 1</td><td> 8</td><td>  9</td></tr>\n",
       "\t<tr><th scope=row>28</th><td>1</td><td>0</td><td>1</td><td>3 </td><td>0</td><td>0</td><td>0</td><td>2</td><td> 2</td><td> 4</td><td>  6</td></tr>\n",
       "\t<tr><th scope=row>29</th><td>1</td><td>0</td><td>1</td><td>4 </td><td>0</td><td>0</td><td>0</td><td>2</td><td> 2</td><td> 1</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>30</th><td>1</td><td>0</td><td>1</td><td>6 </td><td>0</td><td>0</td><td>0</td><td>3</td><td> 0</td><td> 2</td><td>  2</td></tr>\n",
       "\t<tr><th scope=row>⋮</th><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr>\n",
       "\t<tr><th scope=row>17350</th><td>1</td><td>1</td><td>12</td><td>18</td><td>0</td><td>0</td><td>0</td><td>2</td><td>12</td><td>113</td><td>125</td></tr>\n",
       "\t<tr><th scope=row>17351</th><td>1</td><td>1</td><td>12</td><td>19</td><td>0</td><td>0</td><td>0</td><td>1</td><td>16</td><td> 86</td><td>102</td></tr>\n",
       "\t<tr><th scope=row>17352</th><td>1</td><td>1</td><td>12</td><td>20</td><td>0</td><td>0</td><td>0</td><td>1</td><td> 9</td><td> 63</td><td> 72</td></tr>\n",
       "\t<tr><th scope=row>17353</th><td>1</td><td>1</td><td>12</td><td>21</td><td>0</td><td>0</td><td>0</td><td>1</td><td> 5</td><td> 42</td><td> 47</td></tr>\n",
       "\t<tr><th scope=row>17354</th><td>1</td><td>1</td><td>12</td><td>22</td><td>0</td><td>0</td><td>0</td><td>1</td><td> 6</td><td> 30</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>17355</th><td>1</td><td>1</td><td>12</td><td>23</td><td>0</td><td>0</td><td>0</td><td>1</td><td>10</td><td> 39</td><td> 49</td></tr>\n",
       "\t<tr><th scope=row>17356</th><td>1</td><td>1</td><td>12</td><td>0 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 4</td><td> 30</td><td> 34</td></tr>\n",
       "\t<tr><th scope=row>17357</th><td>1</td><td>1</td><td>12</td><td>1 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 6</td><td> 13</td><td> 19</td></tr>\n",
       "\t<tr><th scope=row>17358</th><td>1</td><td>1</td><td>12</td><td>2 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 3</td><td>  8</td><td> 11</td></tr>\n",
       "\t<tr><th scope=row>17359</th><td>1</td><td>1</td><td>12</td><td>3 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 0</td><td>  1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>17360</th><td>1</td><td>1</td><td>12</td><td>4 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 0</td><td>  3</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>17361</th><td>1</td><td>1</td><td>12</td><td>5 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 0</td><td>  9</td><td>  9</td></tr>\n",
       "\t<tr><th scope=row>17362</th><td>1</td><td>1</td><td>12</td><td>6 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 0</td><td> 40</td><td> 40</td></tr>\n",
       "\t<tr><th scope=row>17363</th><td>1</td><td>1</td><td>12</td><td>7 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 2</td><td> 83</td><td> 85</td></tr>\n",
       "\t<tr><th scope=row>17364</th><td>1</td><td>1</td><td>12</td><td>8 </td><td>0</td><td>1</td><td>1</td><td>1</td><td> 9</td><td>187</td><td>196</td></tr>\n",
       "\t<tr><th scope=row>17365</th><td>1</td><td>1</td><td>12</td><td>9 </td><td>0</td><td>1</td><td>1</td><td>2</td><td>13</td><td>144</td><td>157</td></tr>\n",
       "\t<tr><th scope=row>17366</th><td>1</td><td>1</td><td>12</td><td>10</td><td>0</td><td>1</td><td>1</td><td>2</td><td>33</td><td> 87</td><td>120</td></tr>\n",
       "\t<tr><th scope=row>17367</th><td>1</td><td>1</td><td>12</td><td>11</td><td>0</td><td>1</td><td>1</td><td>2</td><td>43</td><td>114</td><td>157</td></tr>\n",
       "\t<tr><th scope=row>17368</th><td>1</td><td>1</td><td>12</td><td>12</td><td>0</td><td>1</td><td>1</td><td>2</td><td>52</td><td>172</td><td>224</td></tr>\n",
       "\t<tr><th scope=row>17369</th><td>1</td><td>1</td><td>12</td><td>13</td><td>0</td><td>1</td><td>1</td><td>2</td><td>38</td><td>165</td><td>203</td></tr>\n",
       "\t<tr><th scope=row>17370</th><td>1</td><td>1</td><td>12</td><td>14</td><td>0</td><td>1</td><td>1</td><td>2</td><td>62</td><td>185</td><td>247</td></tr>\n",
       "\t<tr><th scope=row>17371</th><td>1</td><td>1</td><td>12</td><td>15</td><td>0</td><td>1</td><td>1</td><td>2</td><td>69</td><td>246</td><td>315</td></tr>\n",
       "\t<tr><th scope=row>17372</th><td>1</td><td>1</td><td>12</td><td>16</td><td>0</td><td>1</td><td>1</td><td>2</td><td>30</td><td>184</td><td>214</td></tr>\n",
       "\t<tr><th scope=row>17373</th><td>1</td><td>1</td><td>12</td><td>17</td><td>0</td><td>1</td><td>1</td><td>2</td><td>14</td><td>150</td><td>164</td></tr>\n",
       "\t<tr><th scope=row>17374</th><td>1</td><td>1</td><td>12</td><td>18</td><td>0</td><td>1</td><td>1</td><td>2</td><td>10</td><td>112</td><td>122</td></tr>\n",
       "\t<tr><th scope=row>17375</th><td>1</td><td>1</td><td>12</td><td>19</td><td>0</td><td>1</td><td>1</td><td>2</td><td>11</td><td>108</td><td>119</td></tr>\n",
       "\t<tr><th scope=row>17376</th><td>1</td><td>1</td><td>12</td><td>20</td><td>0</td><td>1</td><td>1</td><td>2</td><td> 8</td><td> 81</td><td> 89</td></tr>\n",
       "\t<tr><th scope=row>17377</th><td>1</td><td>1</td><td>12</td><td>21</td><td>0</td><td>1</td><td>1</td><td>1</td><td> 7</td><td> 83</td><td> 90</td></tr>\n",
       "\t<tr><th scope=row>17378</th><td>1</td><td>1</td><td>12</td><td>22</td><td>0</td><td>1</td><td>1</td><td>1</td><td>13</td><td> 48</td><td> 61</td></tr>\n",
       "\t<tr><th scope=row>17379</th><td>1</td><td>1</td><td>12</td><td>23</td><td>0</td><td>1</td><td>1</td><td>1</td><td>12</td><td> 37</td><td> 49</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 17379 × 11\n",
       "\\begin{tabular}{r|lllllllllll}\n",
       "  & season & yr & mnth & hr & holiday & weekday & workingday & weathersit & casual & registered & cnt\\\\\n",
       "  & <fct> & <fct> & <fct> & <fct> & <fct> & <fct> & <fct> & <fct> & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 & 1 & 0 & 1 & 0  & 0 & 6 & 0 & 1 &  3 & 13 &  16\\\\\n",
       "\t2 & 1 & 0 & 1 & 1  & 0 & 6 & 0 & 1 &  8 & 32 &  40\\\\\n",
       "\t3 & 1 & 0 & 1 & 2  & 0 & 6 & 0 & 1 &  5 & 27 &  32\\\\\n",
       "\t4 & 1 & 0 & 1 & 3  & 0 & 6 & 0 & 1 &  3 & 10 &  13\\\\\n",
       "\t5 & 1 & 0 & 1 & 4  & 0 & 6 & 0 & 1 &  0 &  1 &   1\\\\\n",
       "\t6 & 1 & 0 & 1 & 5  & 0 & 6 & 0 & 2 &  0 &  1 &   1\\\\\n",
       "\t7 & 1 & 0 & 1 & 6  & 0 & 6 & 0 & 1 &  2 &  0 &   2\\\\\n",
       "\t8 & 1 & 0 & 1 & 7  & 0 & 6 & 0 & 1 &  1 &  2 &   3\\\\\n",
       "\t9 & 1 & 0 & 1 & 8  & 0 & 6 & 0 & 1 &  1 &  7 &   8\\\\\n",
       "\t10 & 1 & 0 & 1 & 9  & 0 & 6 & 0 & 1 &  8 &  6 &  14\\\\\n",
       "\t11 & 1 & 0 & 1 & 10 & 0 & 6 & 0 & 1 & 12 & 24 &  36\\\\\n",
       "\t12 & 1 & 0 & 1 & 11 & 0 & 6 & 0 & 1 & 26 & 30 &  56\\\\\n",
       "\t13 & 1 & 0 & 1 & 12 & 0 & 6 & 0 & 1 & 29 & 55 &  84\\\\\n",
       "\t14 & 1 & 0 & 1 & 13 & 0 & 6 & 0 & 2 & 47 & 47 &  94\\\\\n",
       "\t15 & 1 & 0 & 1 & 14 & 0 & 6 & 0 & 2 & 35 & 71 & 106\\\\\n",
       "\t16 & 1 & 0 & 1 & 15 & 0 & 6 & 0 & 2 & 40 & 70 & 110\\\\\n",
       "\t17 & 1 & 0 & 1 & 16 & 0 & 6 & 0 & 2 & 41 & 52 &  93\\\\\n",
       "\t18 & 1 & 0 & 1 & 17 & 0 & 6 & 0 & 2 & 15 & 52 &  67\\\\\n",
       "\t19 & 1 & 0 & 1 & 18 & 0 & 6 & 0 & 3 &  9 & 26 &  35\\\\\n",
       "\t20 & 1 & 0 & 1 & 19 & 0 & 6 & 0 & 3 &  6 & 31 &  37\\\\\n",
       "\t21 & 1 & 0 & 1 & 20 & 0 & 6 & 0 & 2 & 11 & 25 &  36\\\\\n",
       "\t22 & 1 & 0 & 1 & 21 & 0 & 6 & 0 & 2 &  3 & 31 &  34\\\\\n",
       "\t23 & 1 & 0 & 1 & 22 & 0 & 6 & 0 & 2 & 11 & 17 &  28\\\\\n",
       "\t24 & 1 & 0 & 1 & 23 & 0 & 6 & 0 & 2 & 15 & 24 &  39\\\\\n",
       "\t25 & 1 & 0 & 1 & 0  & 0 & 0 & 0 & 2 &  4 & 13 &  17\\\\\n",
       "\t26 & 1 & 0 & 1 & 1  & 0 & 0 & 0 & 2 &  1 & 16 &  17\\\\\n",
       "\t27 & 1 & 0 & 1 & 2  & 0 & 0 & 0 & 2 &  1 &  8 &   9\\\\\n",
       "\t28 & 1 & 0 & 1 & 3  & 0 & 0 & 0 & 2 &  2 &  4 &   6\\\\\n",
       "\t29 & 1 & 0 & 1 & 4  & 0 & 0 & 0 & 2 &  2 &  1 &   3\\\\\n",
       "\t30 & 1 & 0 & 1 & 6  & 0 & 0 & 0 & 3 &  0 &  2 &   2\\\\\n",
       "\t⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮\\\\\n",
       "\t17350 & 1 & 1 & 12 & 18 & 0 & 0 & 0 & 2 & 12 & 113 & 125\\\\\n",
       "\t17351 & 1 & 1 & 12 & 19 & 0 & 0 & 0 & 1 & 16 &  86 & 102\\\\\n",
       "\t17352 & 1 & 1 & 12 & 20 & 0 & 0 & 0 & 1 &  9 &  63 &  72\\\\\n",
       "\t17353 & 1 & 1 & 12 & 21 & 0 & 0 & 0 & 1 &  5 &  42 &  47\\\\\n",
       "\t17354 & 1 & 1 & 12 & 22 & 0 & 0 & 0 & 1 &  6 &  30 &  36\\\\\n",
       "\t17355 & 1 & 1 & 12 & 23 & 0 & 0 & 0 & 1 & 10 &  39 &  49\\\\\n",
       "\t17356 & 1 & 1 & 12 & 0  & 0 & 1 & 1 & 1 &  4 &  30 &  34\\\\\n",
       "\t17357 & 1 & 1 & 12 & 1  & 0 & 1 & 1 & 1 &  6 &  13 &  19\\\\\n",
       "\t17358 & 1 & 1 & 12 & 2  & 0 & 1 & 1 & 1 &  3 &   8 &  11\\\\\n",
       "\t17359 & 1 & 1 & 12 & 3  & 0 & 1 & 1 & 1 &  0 &   1 &   1\\\\\n",
       "\t17360 & 1 & 1 & 12 & 4  & 0 & 1 & 1 & 1 &  0 &   3 &   3\\\\\n",
       "\t17361 & 1 & 1 & 12 & 5  & 0 & 1 & 1 & 1 &  0 &   9 &   9\\\\\n",
       "\t17362 & 1 & 1 & 12 & 6  & 0 & 1 & 1 & 1 &  0 &  40 &  40\\\\\n",
       "\t17363 & 1 & 1 & 12 & 7  & 0 & 1 & 1 & 1 &  2 &  83 &  85\\\\\n",
       "\t17364 & 1 & 1 & 12 & 8  & 0 & 1 & 1 & 1 &  9 & 187 & 196\\\\\n",
       "\t17365 & 1 & 1 & 12 & 9  & 0 & 1 & 1 & 2 & 13 & 144 & 157\\\\\n",
       "\t17366 & 1 & 1 & 12 & 10 & 0 & 1 & 1 & 2 & 33 &  87 & 120\\\\\n",
       "\t17367 & 1 & 1 & 12 & 11 & 0 & 1 & 1 & 2 & 43 & 114 & 157\\\\\n",
       "\t17368 & 1 & 1 & 12 & 12 & 0 & 1 & 1 & 2 & 52 & 172 & 224\\\\\n",
       "\t17369 & 1 & 1 & 12 & 13 & 0 & 1 & 1 & 2 & 38 & 165 & 203\\\\\n",
       "\t17370 & 1 & 1 & 12 & 14 & 0 & 1 & 1 & 2 & 62 & 185 & 247\\\\\n",
       "\t17371 & 1 & 1 & 12 & 15 & 0 & 1 & 1 & 2 & 69 & 246 & 315\\\\\n",
       "\t17372 & 1 & 1 & 12 & 16 & 0 & 1 & 1 & 2 & 30 & 184 & 214\\\\\n",
       "\t17373 & 1 & 1 & 12 & 17 & 0 & 1 & 1 & 2 & 14 & 150 & 164\\\\\n",
       "\t17374 & 1 & 1 & 12 & 18 & 0 & 1 & 1 & 2 & 10 & 112 & 122\\\\\n",
       "\t17375 & 1 & 1 & 12 & 19 & 0 & 1 & 1 & 2 & 11 & 108 & 119\\\\\n",
       "\t17376 & 1 & 1 & 12 & 20 & 0 & 1 & 1 & 2 &  8 &  81 &  89\\\\\n",
       "\t17377 & 1 & 1 & 12 & 21 & 0 & 1 & 1 & 1 &  7 &  83 &  90\\\\\n",
       "\t17378 & 1 & 1 & 12 & 22 & 0 & 1 & 1 & 1 & 13 &  48 &  61\\\\\n",
       "\t17379 & 1 & 1 & 12 & 23 & 0 & 1 & 1 & 1 & 12 &  37 &  49\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 17379 × 11\n",
       "\n",
       "| <!--/--> | season &lt;fct&gt; | yr &lt;fct&gt; | mnth &lt;fct&gt; | hr &lt;fct&gt; | holiday &lt;fct&gt; | weekday &lt;fct&gt; | workingday &lt;fct&gt; | weathersit &lt;fct&gt; | casual &lt;int&gt; | registered &lt;int&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| 1 | 1 | 0 | 1 | 0  | 0 | 6 | 0 | 1 |  3 | 13 |  16 |\n",
       "| 2 | 1 | 0 | 1 | 1  | 0 | 6 | 0 | 1 |  8 | 32 |  40 |\n",
       "| 3 | 1 | 0 | 1 | 2  | 0 | 6 | 0 | 1 |  5 | 27 |  32 |\n",
       "| 4 | 1 | 0 | 1 | 3  | 0 | 6 | 0 | 1 |  3 | 10 |  13 |\n",
       "| 5 | 1 | 0 | 1 | 4  | 0 | 6 | 0 | 1 |  0 |  1 |   1 |\n",
       "| 6 | 1 | 0 | 1 | 5  | 0 | 6 | 0 | 2 |  0 |  1 |   1 |\n",
       "| 7 | 1 | 0 | 1 | 6  | 0 | 6 | 0 | 1 |  2 |  0 |   2 |\n",
       "| 8 | 1 | 0 | 1 | 7  | 0 | 6 | 0 | 1 |  1 |  2 |   3 |\n",
       "| 9 | 1 | 0 | 1 | 8  | 0 | 6 | 0 | 1 |  1 |  7 |   8 |\n",
       "| 10 | 1 | 0 | 1 | 9  | 0 | 6 | 0 | 1 |  8 |  6 |  14 |\n",
       "| 11 | 1 | 0 | 1 | 10 | 0 | 6 | 0 | 1 | 12 | 24 |  36 |\n",
       "| 12 | 1 | 0 | 1 | 11 | 0 | 6 | 0 | 1 | 26 | 30 |  56 |\n",
       "| 13 | 1 | 0 | 1 | 12 | 0 | 6 | 0 | 1 | 29 | 55 |  84 |\n",
       "| 14 | 1 | 0 | 1 | 13 | 0 | 6 | 0 | 2 | 47 | 47 |  94 |\n",
       "| 15 | 1 | 0 | 1 | 14 | 0 | 6 | 0 | 2 | 35 | 71 | 106 |\n",
       "| 16 | 1 | 0 | 1 | 15 | 0 | 6 | 0 | 2 | 40 | 70 | 110 |\n",
       "| 17 | 1 | 0 | 1 | 16 | 0 | 6 | 0 | 2 | 41 | 52 |  93 |\n",
       "| 18 | 1 | 0 | 1 | 17 | 0 | 6 | 0 | 2 | 15 | 52 |  67 |\n",
       "| 19 | 1 | 0 | 1 | 18 | 0 | 6 | 0 | 3 |  9 | 26 |  35 |\n",
       "| 20 | 1 | 0 | 1 | 19 | 0 | 6 | 0 | 3 |  6 | 31 |  37 |\n",
       "| 21 | 1 | 0 | 1 | 20 | 0 | 6 | 0 | 2 | 11 | 25 |  36 |\n",
       "| 22 | 1 | 0 | 1 | 21 | 0 | 6 | 0 | 2 |  3 | 31 |  34 |\n",
       "| 23 | 1 | 0 | 1 | 22 | 0 | 6 | 0 | 2 | 11 | 17 |  28 |\n",
       "| 24 | 1 | 0 | 1 | 23 | 0 | 6 | 0 | 2 | 15 | 24 |  39 |\n",
       "| 25 | 1 | 0 | 1 | 0  | 0 | 0 | 0 | 2 |  4 | 13 |  17 |\n",
       "| 26 | 1 | 0 | 1 | 1  | 0 | 0 | 0 | 2 |  1 | 16 |  17 |\n",
       "| 27 | 1 | 0 | 1 | 2  | 0 | 0 | 0 | 2 |  1 |  8 |   9 |\n",
       "| 28 | 1 | 0 | 1 | 3  | 0 | 0 | 0 | 2 |  2 |  4 |   6 |\n",
       "| 29 | 1 | 0 | 1 | 4  | 0 | 0 | 0 | 2 |  2 |  1 |   3 |\n",
       "| 30 | 1 | 0 | 1 | 6  | 0 | 0 | 0 | 3 |  0 |  2 |   2 |\n",
       "| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |\n",
       "| 17350 | 1 | 1 | 12 | 18 | 0 | 0 | 0 | 2 | 12 | 113 | 125 |\n",
       "| 17351 | 1 | 1 | 12 | 19 | 0 | 0 | 0 | 1 | 16 |  86 | 102 |\n",
       "| 17352 | 1 | 1 | 12 | 20 | 0 | 0 | 0 | 1 |  9 |  63 |  72 |\n",
       "| 17353 | 1 | 1 | 12 | 21 | 0 | 0 | 0 | 1 |  5 |  42 |  47 |\n",
       "| 17354 | 1 | 1 | 12 | 22 | 0 | 0 | 0 | 1 |  6 |  30 |  36 |\n",
       "| 17355 | 1 | 1 | 12 | 23 | 0 | 0 | 0 | 1 | 10 |  39 |  49 |\n",
       "| 17356 | 1 | 1 | 12 | 0  | 0 | 1 | 1 | 1 |  4 |  30 |  34 |\n",
       "| 17357 | 1 | 1 | 12 | 1  | 0 | 1 | 1 | 1 |  6 |  13 |  19 |\n",
       "| 17358 | 1 | 1 | 12 | 2  | 0 | 1 | 1 | 1 |  3 |   8 |  11 |\n",
       "| 17359 | 1 | 1 | 12 | 3  | 0 | 1 | 1 | 1 |  0 |   1 |   1 |\n",
       "| 17360 | 1 | 1 | 12 | 4  | 0 | 1 | 1 | 1 |  0 |   3 |   3 |\n",
       "| 17361 | 1 | 1 | 12 | 5  | 0 | 1 | 1 | 1 |  0 |   9 |   9 |\n",
       "| 17362 | 1 | 1 | 12 | 6  | 0 | 1 | 1 | 1 |  0 |  40 |  40 |\n",
       "| 17363 | 1 | 1 | 12 | 7  | 0 | 1 | 1 | 1 |  2 |  83 |  85 |\n",
       "| 17364 | 1 | 1 | 12 | 8  | 0 | 1 | 1 | 1 |  9 | 187 | 196 |\n",
       "| 17365 | 1 | 1 | 12 | 9  | 0 | 1 | 1 | 2 | 13 | 144 | 157 |\n",
       "| 17366 | 1 | 1 | 12 | 10 | 0 | 1 | 1 | 2 | 33 |  87 | 120 |\n",
       "| 17367 | 1 | 1 | 12 | 11 | 0 | 1 | 1 | 2 | 43 | 114 | 157 |\n",
       "| 17368 | 1 | 1 | 12 | 12 | 0 | 1 | 1 | 2 | 52 | 172 | 224 |\n",
       "| 17369 | 1 | 1 | 12 | 13 | 0 | 1 | 1 | 2 | 38 | 165 | 203 |\n",
       "| 17370 | 1 | 1 | 12 | 14 | 0 | 1 | 1 | 2 | 62 | 185 | 247 |\n",
       "| 17371 | 1 | 1 | 12 | 15 | 0 | 1 | 1 | 2 | 69 | 246 | 315 |\n",
       "| 17372 | 1 | 1 | 12 | 16 | 0 | 1 | 1 | 2 | 30 | 184 | 214 |\n",
       "| 17373 | 1 | 1 | 12 | 17 | 0 | 1 | 1 | 2 | 14 | 150 | 164 |\n",
       "| 17374 | 1 | 1 | 12 | 18 | 0 | 1 | 1 | 2 | 10 | 112 | 122 |\n",
       "| 17375 | 1 | 1 | 12 | 19 | 0 | 1 | 1 | 2 | 11 | 108 | 119 |\n",
       "| 17376 | 1 | 1 | 12 | 20 | 0 | 1 | 1 | 2 |  8 |  81 |  89 |\n",
       "| 17377 | 1 | 1 | 12 | 21 | 0 | 1 | 1 | 1 |  7 |  83 |  90 |\n",
       "| 17378 | 1 | 1 | 12 | 22 | 0 | 1 | 1 | 1 | 13 |  48 |  61 |\n",
       "| 17379 | 1 | 1 | 12 | 23 | 0 | 1 | 1 | 1 | 12 |  37 |  49 |\n",
       "\n"
      ],
      "text/plain": [
       "      season yr mnth hr holiday weekday workingday weathersit casual registered\n",
       "1     1      0  1    0  0       6       0          1           3     13        \n",
       "2     1      0  1    1  0       6       0          1           8     32        \n",
       "3     1      0  1    2  0       6       0          1           5     27        \n",
       "4     1      0  1    3  0       6       0          1           3     10        \n",
       "5     1      0  1    4  0       6       0          1           0      1        \n",
       "6     1      0  1    5  0       6       0          2           0      1        \n",
       "7     1      0  1    6  0       6       0          1           2      0        \n",
       "8     1      0  1    7  0       6       0          1           1      2        \n",
       "9     1      0  1    8  0       6       0          1           1      7        \n",
       "10    1      0  1    9  0       6       0          1           8      6        \n",
       "11    1      0  1    10 0       6       0          1          12     24        \n",
       "12    1      0  1    11 0       6       0          1          26     30        \n",
       "13    1      0  1    12 0       6       0          1          29     55        \n",
       "14    1      0  1    13 0       6       0          2          47     47        \n",
       "15    1      0  1    14 0       6       0          2          35     71        \n",
       "16    1      0  1    15 0       6       0          2          40     70        \n",
       "17    1      0  1    16 0       6       0          2          41     52        \n",
       "18    1      0  1    17 0       6       0          2          15     52        \n",
       "19    1      0  1    18 0       6       0          3           9     26        \n",
       "20    1      0  1    19 0       6       0          3           6     31        \n",
       "21    1      0  1    20 0       6       0          2          11     25        \n",
       "22    1      0  1    21 0       6       0          2           3     31        \n",
       "23    1      0  1    22 0       6       0          2          11     17        \n",
       "24    1      0  1    23 0       6       0          2          15     24        \n",
       "25    1      0  1    0  0       0       0          2           4     13        \n",
       "26    1      0  1    1  0       0       0          2           1     16        \n",
       "27    1      0  1    2  0       0       0          2           1      8        \n",
       "28    1      0  1    3  0       0       0          2           2      4        \n",
       "29    1      0  1    4  0       0       0          2           2      1        \n",
       "30    1      0  1    6  0       0       0          3           0      2        \n",
       "⋮     ⋮      ⋮  ⋮    ⋮  ⋮       ⋮       ⋮          ⋮          ⋮      ⋮         \n",
       "17350 1      1  12   18 0       0       0          2          12     113       \n",
       "17351 1      1  12   19 0       0       0          1          16      86       \n",
       "17352 1      1  12   20 0       0       0          1           9      63       \n",
       "17353 1      1  12   21 0       0       0          1           5      42       \n",
       "17354 1      1  12   22 0       0       0          1           6      30       \n",
       "17355 1      1  12   23 0       0       0          1          10      39       \n",
       "17356 1      1  12   0  0       1       1          1           4      30       \n",
       "17357 1      1  12   1  0       1       1          1           6      13       \n",
       "17358 1      1  12   2  0       1       1          1           3       8       \n",
       "17359 1      1  12   3  0       1       1          1           0       1       \n",
       "17360 1      1  12   4  0       1       1          1           0       3       \n",
       "17361 1      1  12   5  0       1       1          1           0       9       \n",
       "17362 1      1  12   6  0       1       1          1           0      40       \n",
       "17363 1      1  12   7  0       1       1          1           2      83       \n",
       "17364 1      1  12   8  0       1       1          1           9     187       \n",
       "17365 1      1  12   9  0       1       1          2          13     144       \n",
       "17366 1      1  12   10 0       1       1          2          33      87       \n",
       "17367 1      1  12   11 0       1       1          2          43     114       \n",
       "17368 1      1  12   12 0       1       1          2          52     172       \n",
       "17369 1      1  12   13 0       1       1          2          38     165       \n",
       "17370 1      1  12   14 0       1       1          2          62     185       \n",
       "17371 1      1  12   15 0       1       1          2          69     246       \n",
       "17372 1      1  12   16 0       1       1          2          30     184       \n",
       "17373 1      1  12   17 0       1       1          2          14     150       \n",
       "17374 1      1  12   18 0       1       1          2          10     112       \n",
       "17375 1      1  12   19 0       1       1          2          11     108       \n",
       "17376 1      1  12   20 0       1       1          2           8      81       \n",
       "17377 1      1  12   21 0       1       1          1           7      83       \n",
       "17378 1      1  12   22 0       1       1          1          13      48       \n",
       "17379 1      1  12   23 0       1       1          1          12      37       \n",
       "      cnt\n",
       "1      16\n",
       "2      40\n",
       "3      32\n",
       "4      13\n",
       "5       1\n",
       "6       1\n",
       "7       2\n",
       "8       3\n",
       "9       8\n",
       "10     14\n",
       "11     36\n",
       "12     56\n",
       "13     84\n",
       "14     94\n",
       "15    106\n",
       "16    110\n",
       "17     93\n",
       "18     67\n",
       "19     35\n",
       "20     37\n",
       "21     36\n",
       "22     34\n",
       "23     28\n",
       "24     39\n",
       "25     17\n",
       "26     17\n",
       "27      9\n",
       "28      6\n",
       "29      3\n",
       "30      2\n",
       "⋮     ⋮  \n",
       "17350 125\n",
       "17351 102\n",
       "17352  72\n",
       "17353  47\n",
       "17354  36\n",
       "17355  49\n",
       "17356  34\n",
       "17357  19\n",
       "17358  11\n",
       "17359   1\n",
       "17360   3\n",
       "17361   9\n",
       "17362  40\n",
       "17363  85\n",
       "17364 196\n",
       "17365 157\n",
       "17366 120\n",
       "17367 157\n",
       "17368 224\n",
       "17369 203\n",
       "17370 247\n",
       "17371 315\n",
       "17372 214\n",
       "17373 164\n",
       "17374 122\n",
       "17375 119\n",
       "17376  89\n",
       "17377  90\n",
       "17378  61\n",
       "17379  49"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Coerce the categorical variables to factors\n",
    "bikes4$season <- as.factor(bikes4$season)\n",
    "bikes4$yr <- as.factor(bikes4$yr)\n",
    "bikes4$mnth <- as.factor(bikes4$mnth)\n",
    "bikes4$hr <- as.factor(bikes4$hr)\n",
    "bikes4$holiday <- as.factor(bikes4$holiday)\n",
    "bikes4$weekday <- as.factor(bikes4$weekday)\n",
    "bikes4$workingday <- as.factor(bikes4$workingday)\n",
    "bikes4$weathersit <- as.factor(bikes4$weathersit)\n",
    "\n",
    "bikes4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c67deb35",
   "metadata": {},
   "source": [
    "## Test for Assumptions for MANOVA\n",
    "\n",
    "### Sample Size\n",
    "\"There must be more cases than dependent variables in every cell. In addition, there must be at least 20 cases per independent variable.\"\n",
    "\n",
    "Given that there are 17,379 observations, we know that we far exceed the 20 cases per independent variable. With 8 IVs, we would need a minimum of 160 cases, so we far exceed that minimum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8e89bfca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "   1    2    3    4 \n",
       "4242 4409 4496 4232 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "   0    1 \n",
       "8645 8734 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "   1    2    3    4    5    6    7    8    9   10   11   12 \n",
       "1429 1341 1473 1437 1488 1440 1488 1475 1437 1451 1437 1483 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19 \n",
       "726 724 715 697 697 717 725 727 727 727 727 727 728 729 729 729 730 730 728 728 \n",
       " 20  21  22  23 \n",
       "728 728 728 728 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "    0     1 \n",
       "16879   500 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "   0    1    2    3    4    5    6 \n",
       "2502 2479 2453 2475 2471 2487 2512 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "    0     1 \n",
       " 5514 11865 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "    1     2     3     4 \n",
       "11413  4544  1419     3 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(bikes4$season)\n",
    "table(bikes4$yr)\n",
    "table(bikes4$mnth)\n",
    "table(bikes4$hr)\n",
    "table(bikes4$holiday)\n",
    "table(bikes4$weekday)\n",
    "table(bikes4$workingday)\n",
    "table(bikes4$weathersit)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5b70af9",
   "metadata": {},
   "source": [
    "Nearly every cell of the table counts has more than the minimum of 3 required for the 3 DVs we will be testing. The only exception is the 3 (equal, rather than greater) in the `4` of `weathersit`. We will continue with our analysis anyway."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a4ed2e9",
   "metadata": {},
   "source": [
    "### Multivariate Normality\n",
    "We begin by subsetting to only our DVs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "7a5d9854",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 17379 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>casual</th><th scope=col>registered</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 3</td><td>13</td><td> 16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td> 8</td><td>32</td><td> 40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td> 5</td><td>27</td><td> 32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 3</td><td>10</td><td> 13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td> 0</td><td> 1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td> 0</td><td> 1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td> 2</td><td> 0</td><td>  2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td> 1</td><td> 2</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td> 1</td><td> 7</td><td>  8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td> 8</td><td> 6</td><td> 14</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>12</td><td>24</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>12</th><td>26</td><td>30</td><td> 56</td></tr>\n",
       "\t<tr><th scope=row>13</th><td>29</td><td>55</td><td> 84</td></tr>\n",
       "\t<tr><th scope=row>14</th><td>47</td><td>47</td><td> 94</td></tr>\n",
       "\t<tr><th scope=row>15</th><td>35</td><td>71</td><td>106</td></tr>\n",
       "\t<tr><th scope=row>16</th><td>40</td><td>70</td><td>110</td></tr>\n",
       "\t<tr><th scope=row>17</th><td>41</td><td>52</td><td> 93</td></tr>\n",
       "\t<tr><th scope=row>18</th><td>15</td><td>52</td><td> 67</td></tr>\n",
       "\t<tr><th scope=row>19</th><td> 9</td><td>26</td><td> 35</td></tr>\n",
       "\t<tr><th scope=row>20</th><td> 6</td><td>31</td><td> 37</td></tr>\n",
       "\t<tr><th scope=row>21</th><td>11</td><td>25</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>22</th><td> 3</td><td>31</td><td> 34</td></tr>\n",
       "\t<tr><th scope=row>23</th><td>11</td><td>17</td><td> 28</td></tr>\n",
       "\t<tr><th scope=row>24</th><td>15</td><td>24</td><td> 39</td></tr>\n",
       "\t<tr><th scope=row>25</th><td> 4</td><td>13</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>26</th><td> 1</td><td>16</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>27</th><td> 1</td><td> 8</td><td>  9</td></tr>\n",
       "\t<tr><th scope=row>28</th><td> 2</td><td> 4</td><td>  6</td></tr>\n",
       "\t<tr><th scope=row>29</th><td> 2</td><td> 1</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>30</th><td> 0</td><td> 2</td><td>  2</td></tr>\n",
       "\t<tr><th scope=row>⋮</th><td>⋮</td><td>⋮</td><td>⋮</td></tr>\n",
       "\t<tr><th scope=row>17350</th><td>12</td><td>113</td><td>125</td></tr>\n",
       "\t<tr><th scope=row>17351</th><td>16</td><td> 86</td><td>102</td></tr>\n",
       "\t<tr><th scope=row>17352</th><td> 9</td><td> 63</td><td> 72</td></tr>\n",
       "\t<tr><th scope=row>17353</th><td> 5</td><td> 42</td><td> 47</td></tr>\n",
       "\t<tr><th scope=row>17354</th><td> 6</td><td> 30</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>17355</th><td>10</td><td> 39</td><td> 49</td></tr>\n",
       "\t<tr><th scope=row>17356</th><td> 4</td><td> 30</td><td> 34</td></tr>\n",
       "\t<tr><th scope=row>17357</th><td> 6</td><td> 13</td><td> 19</td></tr>\n",
       "\t<tr><th scope=row>17358</th><td> 3</td><td>  8</td><td> 11</td></tr>\n",
       "\t<tr><th scope=row>17359</th><td> 0</td><td>  1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>17360</th><td> 0</td><td>  3</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>17361</th><td> 0</td><td>  9</td><td>  9</td></tr>\n",
       "\t<tr><th scope=row>17362</th><td> 0</td><td> 40</td><td> 40</td></tr>\n",
       "\t<tr><th scope=row>17363</th><td> 2</td><td> 83</td><td> 85</td></tr>\n",
       "\t<tr><th scope=row>17364</th><td> 9</td><td>187</td><td>196</td></tr>\n",
       "\t<tr><th scope=row>17365</th><td>13</td><td>144</td><td>157</td></tr>\n",
       "\t<tr><th scope=row>17366</th><td>33</td><td> 87</td><td>120</td></tr>\n",
       "\t<tr><th scope=row>17367</th><td>43</td><td>114</td><td>157</td></tr>\n",
       "\t<tr><th scope=row>17368</th><td>52</td><td>172</td><td>224</td></tr>\n",
       "\t<tr><th scope=row>17369</th><td>38</td><td>165</td><td>203</td></tr>\n",
       "\t<tr><th scope=row>17370</th><td>62</td><td>185</td><td>247</td></tr>\n",
       "\t<tr><th scope=row>17371</th><td>69</td><td>246</td><td>315</td></tr>\n",
       "\t<tr><th scope=row>17372</th><td>30</td><td>184</td><td>214</td></tr>\n",
       "\t<tr><th scope=row>17373</th><td>14</td><td>150</td><td>164</td></tr>\n",
       "\t<tr><th scope=row>17374</th><td>10</td><td>112</td><td>122</td></tr>\n",
       "\t<tr><th scope=row>17375</th><td>11</td><td>108</td><td>119</td></tr>\n",
       "\t<tr><th scope=row>17376</th><td> 8</td><td> 81</td><td> 89</td></tr>\n",
       "\t<tr><th scope=row>17377</th><td> 7</td><td> 83</td><td> 90</td></tr>\n",
       "\t<tr><th scope=row>17378</th><td>13</td><td> 48</td><td> 61</td></tr>\n",
       "\t<tr><th scope=row>17379</th><td>12</td><td> 37</td><td> 49</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 17379 × 3\n",
       "\\begin{tabular}{r|lll}\n",
       "  & casual & registered & cnt\\\\\n",
       "  & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 &  3 & 13 &  16\\\\\n",
       "\t2 &  8 & 32 &  40\\\\\n",
       "\t3 &  5 & 27 &  32\\\\\n",
       "\t4 &  3 & 10 &  13\\\\\n",
       "\t5 &  0 &  1 &   1\\\\\n",
       "\t6 &  0 &  1 &   1\\\\\n",
       "\t7 &  2 &  0 &   2\\\\\n",
       "\t8 &  1 &  2 &   3\\\\\n",
       "\t9 &  1 &  7 &   8\\\\\n",
       "\t10 &  8 &  6 &  14\\\\\n",
       "\t11 & 12 & 24 &  36\\\\\n",
       "\t12 & 26 & 30 &  56\\\\\n",
       "\t13 & 29 & 55 &  84\\\\\n",
       "\t14 & 47 & 47 &  94\\\\\n",
       "\t15 & 35 & 71 & 106\\\\\n",
       "\t16 & 40 & 70 & 110\\\\\n",
       "\t17 & 41 & 52 &  93\\\\\n",
       "\t18 & 15 & 52 &  67\\\\\n",
       "\t19 &  9 & 26 &  35\\\\\n",
       "\t20 &  6 & 31 &  37\\\\\n",
       "\t21 & 11 & 25 &  36\\\\\n",
       "\t22 &  3 & 31 &  34\\\\\n",
       "\t23 & 11 & 17 &  28\\\\\n",
       "\t24 & 15 & 24 &  39\\\\\n",
       "\t25 &  4 & 13 &  17\\\\\n",
       "\t26 &  1 & 16 &  17\\\\\n",
       "\t27 &  1 &  8 &   9\\\\\n",
       "\t28 &  2 &  4 &   6\\\\\n",
       "\t29 &  2 &  1 &   3\\\\\n",
       "\t30 &  0 &  2 &   2\\\\\n",
       "\t⋮ & ⋮ & ⋮ & ⋮\\\\\n",
       "\t17350 & 12 & 113 & 125\\\\\n",
       "\t17351 & 16 &  86 & 102\\\\\n",
       "\t17352 &  9 &  63 &  72\\\\\n",
       "\t17353 &  5 &  42 &  47\\\\\n",
       "\t17354 &  6 &  30 &  36\\\\\n",
       "\t17355 & 10 &  39 &  49\\\\\n",
       "\t17356 &  4 &  30 &  34\\\\\n",
       "\t17357 &  6 &  13 &  19\\\\\n",
       "\t17358 &  3 &   8 &  11\\\\\n",
       "\t17359 &  0 &   1 &   1\\\\\n",
       "\t17360 &  0 &   3 &   3\\\\\n",
       "\t17361 &  0 &   9 &   9\\\\\n",
       "\t17362 &  0 &  40 &  40\\\\\n",
       "\t17363 &  2 &  83 &  85\\\\\n",
       "\t17364 &  9 & 187 & 196\\\\\n",
       "\t17365 & 13 & 144 & 157\\\\\n",
       "\t17366 & 33 &  87 & 120\\\\\n",
       "\t17367 & 43 & 114 & 157\\\\\n",
       "\t17368 & 52 & 172 & 224\\\\\n",
       "\t17369 & 38 & 165 & 203\\\\\n",
       "\t17370 & 62 & 185 & 247\\\\\n",
       "\t17371 & 69 & 246 & 315\\\\\n",
       "\t17372 & 30 & 184 & 214\\\\\n",
       "\t17373 & 14 & 150 & 164\\\\\n",
       "\t17374 & 10 & 112 & 122\\\\\n",
       "\t17375 & 11 & 108 & 119\\\\\n",
       "\t17376 &  8 &  81 &  89\\\\\n",
       "\t17377 &  7 &  83 &  90\\\\\n",
       "\t17378 & 13 &  48 &  61\\\\\n",
       "\t17379 & 12 &  37 &  49\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 17379 × 3\n",
       "\n",
       "| <!--/--> | casual &lt;int&gt; | registered &lt;int&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|\n",
       "| 1 |  3 | 13 |  16 |\n",
       "| 2 |  8 | 32 |  40 |\n",
       "| 3 |  5 | 27 |  32 |\n",
       "| 4 |  3 | 10 |  13 |\n",
       "| 5 |  0 |  1 |   1 |\n",
       "| 6 |  0 |  1 |   1 |\n",
       "| 7 |  2 |  0 |   2 |\n",
       "| 8 |  1 |  2 |   3 |\n",
       "| 9 |  1 |  7 |   8 |\n",
       "| 10 |  8 |  6 |  14 |\n",
       "| 11 | 12 | 24 |  36 |\n",
       "| 12 | 26 | 30 |  56 |\n",
       "| 13 | 29 | 55 |  84 |\n",
       "| 14 | 47 | 47 |  94 |\n",
       "| 15 | 35 | 71 | 106 |\n",
       "| 16 | 40 | 70 | 110 |\n",
       "| 17 | 41 | 52 |  93 |\n",
       "| 18 | 15 | 52 |  67 |\n",
       "| 19 |  9 | 26 |  35 |\n",
       "| 20 |  6 | 31 |  37 |\n",
       "| 21 | 11 | 25 |  36 |\n",
       "| 22 |  3 | 31 |  34 |\n",
       "| 23 | 11 | 17 |  28 |\n",
       "| 24 | 15 | 24 |  39 |\n",
       "| 25 |  4 | 13 |  17 |\n",
       "| 26 |  1 | 16 |  17 |\n",
       "| 27 |  1 |  8 |   9 |\n",
       "| 28 |  2 |  4 |   6 |\n",
       "| 29 |  2 |  1 |   3 |\n",
       "| 30 |  0 |  2 |   2 |\n",
       "| ⋮ | ⋮ | ⋮ | ⋮ |\n",
       "| 17350 | 12 | 113 | 125 |\n",
       "| 17351 | 16 |  86 | 102 |\n",
       "| 17352 |  9 |  63 |  72 |\n",
       "| 17353 |  5 |  42 |  47 |\n",
       "| 17354 |  6 |  30 |  36 |\n",
       "| 17355 | 10 |  39 |  49 |\n",
       "| 17356 |  4 |  30 |  34 |\n",
       "| 17357 |  6 |  13 |  19 |\n",
       "| 17358 |  3 |   8 |  11 |\n",
       "| 17359 |  0 |   1 |   1 |\n",
       "| 17360 |  0 |   3 |   3 |\n",
       "| 17361 |  0 |   9 |   9 |\n",
       "| 17362 |  0 |  40 |  40 |\n",
       "| 17363 |  2 |  83 |  85 |\n",
       "| 17364 |  9 | 187 | 196 |\n",
       "| 17365 | 13 | 144 | 157 |\n",
       "| 17366 | 33 |  87 | 120 |\n",
       "| 17367 | 43 | 114 | 157 |\n",
       "| 17368 | 52 | 172 | 224 |\n",
       "| 17369 | 38 | 165 | 203 |\n",
       "| 17370 | 62 | 185 | 247 |\n",
       "| 17371 | 69 | 246 | 315 |\n",
       "| 17372 | 30 | 184 | 214 |\n",
       "| 17373 | 14 | 150 | 164 |\n",
       "| 17374 | 10 | 112 | 122 |\n",
       "| 17375 | 11 | 108 | 119 |\n",
       "| 17376 |  8 |  81 |  89 |\n",
       "| 17377 |  7 |  83 |  90 |\n",
       "| 17378 | 13 |  48 |  61 |\n",
       "| 17379 | 12 |  37 |  49 |\n",
       "\n"
      ],
      "text/plain": [
       "      casual registered cnt\n",
       "1      3     13          16\n",
       "2      8     32          40\n",
       "3      5     27          32\n",
       "4      3     10          13\n",
       "5      0      1           1\n",
       "6      0      1           1\n",
       "7      2      0           2\n",
       "8      1      2           3\n",
       "9      1      7           8\n",
       "10     8      6          14\n",
       "11    12     24          36\n",
       "12    26     30          56\n",
       "13    29     55          84\n",
       "14    47     47          94\n",
       "15    35     71         106\n",
       "16    40     70         110\n",
       "17    41     52          93\n",
       "18    15     52          67\n",
       "19     9     26          35\n",
       "20     6     31          37\n",
       "21    11     25          36\n",
       "22     3     31          34\n",
       "23    11     17          28\n",
       "24    15     24          39\n",
       "25     4     13          17\n",
       "26     1     16          17\n",
       "27     1      8           9\n",
       "28     2      4           6\n",
       "29     2      1           3\n",
       "30     0      2           2\n",
       "⋮     ⋮      ⋮          ⋮  \n",
       "17350 12     113        125\n",
       "17351 16      86        102\n",
       "17352  9      63         72\n",
       "17353  5      42         47\n",
       "17354  6      30         36\n",
       "17355 10      39         49\n",
       "17356  4      30         34\n",
       "17357  6      13         19\n",
       "17358  3       8         11\n",
       "17359  0       1          1\n",
       "17360  0       3          3\n",
       "17361  0       9          9\n",
       "17362  0      40         40\n",
       "17363  2      83         85\n",
       "17364  9     187        196\n",
       "17365 13     144        157\n",
       "17366 33      87        120\n",
       "17367 43     114        157\n",
       "17368 52     172        224\n",
       "17369 38     165        203\n",
       "17370 62     185        247\n",
       "17371 69     246        315\n",
       "17372 30     184        214\n",
       "17373 14     150        164\n",
       "17374 10     112        122\n",
       "17375 11     108        119\n",
       "17376  8      81         89\n",
       "17377  7      83         90\n",
       "17378 13      48         61\n",
       "17379 12      37         49"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bikes4dv <- subset(bikes4, select = c(casual, registered, cnt))\n",
    "bikes4dv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "cbc75a15",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 5000 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>casual</th><th scope=col>registered</th><th scope=col>cnt</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 3</td><td>13</td><td> 16</td></tr>\n",
       "\t<tr><th scope=row>2</th><td> 8</td><td>32</td><td> 40</td></tr>\n",
       "\t<tr><th scope=row>3</th><td> 5</td><td>27</td><td> 32</td></tr>\n",
       "\t<tr><th scope=row>4</th><td> 3</td><td>10</td><td> 13</td></tr>\n",
       "\t<tr><th scope=row>5</th><td> 0</td><td> 1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>6</th><td> 0</td><td> 1</td><td>  1</td></tr>\n",
       "\t<tr><th scope=row>7</th><td> 2</td><td> 0</td><td>  2</td></tr>\n",
       "\t<tr><th scope=row>8</th><td> 1</td><td> 2</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>9</th><td> 1</td><td> 7</td><td>  8</td></tr>\n",
       "\t<tr><th scope=row>10</th><td> 8</td><td> 6</td><td> 14</td></tr>\n",
       "\t<tr><th scope=row>11</th><td>12</td><td>24</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>12</th><td>26</td><td>30</td><td> 56</td></tr>\n",
       "\t<tr><th scope=row>13</th><td>29</td><td>55</td><td> 84</td></tr>\n",
       "\t<tr><th scope=row>14</th><td>47</td><td>47</td><td> 94</td></tr>\n",
       "\t<tr><th scope=row>15</th><td>35</td><td>71</td><td>106</td></tr>\n",
       "\t<tr><th scope=row>16</th><td>40</td><td>70</td><td>110</td></tr>\n",
       "\t<tr><th scope=row>17</th><td>41</td><td>52</td><td> 93</td></tr>\n",
       "\t<tr><th scope=row>18</th><td>15</td><td>52</td><td> 67</td></tr>\n",
       "\t<tr><th scope=row>19</th><td> 9</td><td>26</td><td> 35</td></tr>\n",
       "\t<tr><th scope=row>20</th><td> 6</td><td>31</td><td> 37</td></tr>\n",
       "\t<tr><th scope=row>21</th><td>11</td><td>25</td><td> 36</td></tr>\n",
       "\t<tr><th scope=row>22</th><td> 3</td><td>31</td><td> 34</td></tr>\n",
       "\t<tr><th scope=row>23</th><td>11</td><td>17</td><td> 28</td></tr>\n",
       "\t<tr><th scope=row>24</th><td>15</td><td>24</td><td> 39</td></tr>\n",
       "\t<tr><th scope=row>25</th><td> 4</td><td>13</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>26</th><td> 1</td><td>16</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>27</th><td> 1</td><td> 8</td><td>  9</td></tr>\n",
       "\t<tr><th scope=row>28</th><td> 2</td><td> 4</td><td>  6</td></tr>\n",
       "\t<tr><th scope=row>29</th><td> 2</td><td> 1</td><td>  3</td></tr>\n",
       "\t<tr><th scope=row>30</th><td> 0</td><td> 2</td><td>  2</td></tr>\n",
       "\t<tr><th scope=row>⋮</th><td>⋮</td><td>⋮</td><td>⋮</td></tr>\n",
       "\t<tr><th scope=row>4971</th><td> 33</td><td> 80</td><td>113</td></tr>\n",
       "\t<tr><th scope=row>4972</th><td>  8</td><td> 71</td><td> 79</td></tr>\n",
       "\t<tr><th scope=row>4973</th><td> 19</td><td> 48</td><td> 67</td></tr>\n",
       "\t<tr><th scope=row>4974</th><td> 24</td><td> 26</td><td> 50</td></tr>\n",
       "\t<tr><th scope=row>4975</th><td>  6</td><td>  7</td><td> 13</td></tr>\n",
       "\t<tr><th scope=row>4976</th><td>  4</td><td>  4</td><td>  8</td></tr>\n",
       "\t<tr><th scope=row>4977</th><td>  5</td><td>  8</td><td> 13</td></tr>\n",
       "\t<tr><th scope=row>4978</th><td> 19</td><td> 26</td><td> 45</td></tr>\n",
       "\t<tr><th scope=row>4979</th><td> 54</td><td> 88</td><td>142</td></tr>\n",
       "\t<tr><th scope=row>4980</th><td> 55</td><td>128</td><td>183</td></tr>\n",
       "\t<tr><th scope=row>4981</th><td> 91</td><td>163</td><td>254</td></tr>\n",
       "\t<tr><th scope=row>4982</th><td>112</td><td>163</td><td>275</td></tr>\n",
       "\t<tr><th scope=row>4983</th><td>146</td><td>192</td><td>338</td></tr>\n",
       "\t<tr><th scope=row>4984</th><td>142</td><td>152</td><td>294</td></tr>\n",
       "\t<tr><th scope=row>4985</th><td>130</td><td>196</td><td>326</td></tr>\n",
       "\t<tr><th scope=row>4986</th><td>109</td><td>184</td><td>293</td></tr>\n",
       "\t<tr><th scope=row>4987</th><td> 98</td><td>208</td><td>306</td></tr>\n",
       "\t<tr><th scope=row>4988</th><td>114</td><td>206</td><td>320</td></tr>\n",
       "\t<tr><th scope=row>4989</th><td> 91</td><td>213</td><td>304</td></tr>\n",
       "\t<tr><th scope=row>4990</th><td> 87</td><td>187</td><td>274</td></tr>\n",
       "\t<tr><th scope=row>4991</th><td> 70</td><td>205</td><td>275</td></tr>\n",
       "\t<tr><th scope=row>4992</th><td> 68</td><td> 96</td><td>164</td></tr>\n",
       "\t<tr><th scope=row>4993</th><td> 25</td><td> 71</td><td> 96</td></tr>\n",
       "\t<tr><th scope=row>4994</th><td> 14</td><td> 56</td><td> 70</td></tr>\n",
       "\t<tr><th scope=row>4995</th><td>  7</td><td> 22</td><td> 29</td></tr>\n",
       "\t<tr><th scope=row>4996</th><td>  5</td><td> 12</td><td> 17</td></tr>\n",
       "\t<tr><th scope=row>4997</th><td>  4</td><td>  7</td><td> 11</td></tr>\n",
       "\t<tr><th scope=row>4998</th><td>  0</td><td>  4</td><td>  4</td></tr>\n",
       "\t<tr><th scope=row>4999</th><td>  2</td><td>  2</td><td>  4</td></tr>\n",
       "\t<tr><th scope=row>5000</th><td>  2</td><td> 24</td><td> 26</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 5000 × 3\n",
       "\\begin{tabular}{r|lll}\n",
       "  & casual & registered & cnt\\\\\n",
       "  & <int> & <int> & <int>\\\\\n",
       "\\hline\n",
       "\t1 &  3 & 13 &  16\\\\\n",
       "\t2 &  8 & 32 &  40\\\\\n",
       "\t3 &  5 & 27 &  32\\\\\n",
       "\t4 &  3 & 10 &  13\\\\\n",
       "\t5 &  0 &  1 &   1\\\\\n",
       "\t6 &  0 &  1 &   1\\\\\n",
       "\t7 &  2 &  0 &   2\\\\\n",
       "\t8 &  1 &  2 &   3\\\\\n",
       "\t9 &  1 &  7 &   8\\\\\n",
       "\t10 &  8 &  6 &  14\\\\\n",
       "\t11 & 12 & 24 &  36\\\\\n",
       "\t12 & 26 & 30 &  56\\\\\n",
       "\t13 & 29 & 55 &  84\\\\\n",
       "\t14 & 47 & 47 &  94\\\\\n",
       "\t15 & 35 & 71 & 106\\\\\n",
       "\t16 & 40 & 70 & 110\\\\\n",
       "\t17 & 41 & 52 &  93\\\\\n",
       "\t18 & 15 & 52 &  67\\\\\n",
       "\t19 &  9 & 26 &  35\\\\\n",
       "\t20 &  6 & 31 &  37\\\\\n",
       "\t21 & 11 & 25 &  36\\\\\n",
       "\t22 &  3 & 31 &  34\\\\\n",
       "\t23 & 11 & 17 &  28\\\\\n",
       "\t24 & 15 & 24 &  39\\\\\n",
       "\t25 &  4 & 13 &  17\\\\\n",
       "\t26 &  1 & 16 &  17\\\\\n",
       "\t27 &  1 &  8 &   9\\\\\n",
       "\t28 &  2 &  4 &   6\\\\\n",
       "\t29 &  2 &  1 &   3\\\\\n",
       "\t30 &  0 &  2 &   2\\\\\n",
       "\t⋮ & ⋮ & ⋮ & ⋮\\\\\n",
       "\t4971 &  33 &  80 & 113\\\\\n",
       "\t4972 &   8 &  71 &  79\\\\\n",
       "\t4973 &  19 &  48 &  67\\\\\n",
       "\t4974 &  24 &  26 &  50\\\\\n",
       "\t4975 &   6 &   7 &  13\\\\\n",
       "\t4976 &   4 &   4 &   8\\\\\n",
       "\t4977 &   5 &   8 &  13\\\\\n",
       "\t4978 &  19 &  26 &  45\\\\\n",
       "\t4979 &  54 &  88 & 142\\\\\n",
       "\t4980 &  55 & 128 & 183\\\\\n",
       "\t4981 &  91 & 163 & 254\\\\\n",
       "\t4982 & 112 & 163 & 275\\\\\n",
       "\t4983 & 146 & 192 & 338\\\\\n",
       "\t4984 & 142 & 152 & 294\\\\\n",
       "\t4985 & 130 & 196 & 326\\\\\n",
       "\t4986 & 109 & 184 & 293\\\\\n",
       "\t4987 &  98 & 208 & 306\\\\\n",
       "\t4988 & 114 & 206 & 320\\\\\n",
       "\t4989 &  91 & 213 & 304\\\\\n",
       "\t4990 &  87 & 187 & 274\\\\\n",
       "\t4991 &  70 & 205 & 275\\\\\n",
       "\t4992 &  68 &  96 & 164\\\\\n",
       "\t4993 &  25 &  71 &  96\\\\\n",
       "\t4994 &  14 &  56 &  70\\\\\n",
       "\t4995 &   7 &  22 &  29\\\\\n",
       "\t4996 &   5 &  12 &  17\\\\\n",
       "\t4997 &   4 &   7 &  11\\\\\n",
       "\t4998 &   0 &   4 &   4\\\\\n",
       "\t4999 &   2 &   2 &   4\\\\\n",
       "\t5000 &   2 &  24 &  26\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 5000 × 3\n",
       "\n",
       "| <!--/--> | casual &lt;int&gt; | registered &lt;int&gt; | cnt &lt;int&gt; |\n",
       "|---|---|---|---|\n",
       "| 1 |  3 | 13 |  16 |\n",
       "| 2 |  8 | 32 |  40 |\n",
       "| 3 |  5 | 27 |  32 |\n",
       "| 4 |  3 | 10 |  13 |\n",
       "| 5 |  0 |  1 |   1 |\n",
       "| 6 |  0 |  1 |   1 |\n",
       "| 7 |  2 |  0 |   2 |\n",
       "| 8 |  1 |  2 |   3 |\n",
       "| 9 |  1 |  7 |   8 |\n",
       "| 10 |  8 |  6 |  14 |\n",
       "| 11 | 12 | 24 |  36 |\n",
       "| 12 | 26 | 30 |  56 |\n",
       "| 13 | 29 | 55 |  84 |\n",
       "| 14 | 47 | 47 |  94 |\n",
       "| 15 | 35 | 71 | 106 |\n",
       "| 16 | 40 | 70 | 110 |\n",
       "| 17 | 41 | 52 |  93 |\n",
       "| 18 | 15 | 52 |  67 |\n",
       "| 19 |  9 | 26 |  35 |\n",
       "| 20 |  6 | 31 |  37 |\n",
       "| 21 | 11 | 25 |  36 |\n",
       "| 22 |  3 | 31 |  34 |\n",
       "| 23 | 11 | 17 |  28 |\n",
       "| 24 | 15 | 24 |  39 |\n",
       "| 25 |  4 | 13 |  17 |\n",
       "| 26 |  1 | 16 |  17 |\n",
       "| 27 |  1 |  8 |   9 |\n",
       "| 28 |  2 |  4 |   6 |\n",
       "| 29 |  2 |  1 |   3 |\n",
       "| 30 |  0 |  2 |   2 |\n",
       "| ⋮ | ⋮ | ⋮ | ⋮ |\n",
       "| 4971 |  33 |  80 | 113 |\n",
       "| 4972 |   8 |  71 |  79 |\n",
       "| 4973 |  19 |  48 |  67 |\n",
       "| 4974 |  24 |  26 |  50 |\n",
       "| 4975 |   6 |   7 |  13 |\n",
       "| 4976 |   4 |   4 |   8 |\n",
       "| 4977 |   5 |   8 |  13 |\n",
       "| 4978 |  19 |  26 |  45 |\n",
       "| 4979 |  54 |  88 | 142 |\n",
       "| 4980 |  55 | 128 | 183 |\n",
       "| 4981 |  91 | 163 | 254 |\n",
       "| 4982 | 112 | 163 | 275 |\n",
       "| 4983 | 146 | 192 | 338 |\n",
       "| 4984 | 142 | 152 | 294 |\n",
       "| 4985 | 130 | 196 | 326 |\n",
       "| 4986 | 109 | 184 | 293 |\n",
       "| 4987 |  98 | 208 | 306 |\n",
       "| 4988 | 114 | 206 | 320 |\n",
       "| 4989 |  91 | 213 | 304 |\n",
       "| 4990 |  87 | 187 | 274 |\n",
       "| 4991 |  70 | 205 | 275 |\n",
       "| 4992 |  68 |  96 | 164 |\n",
       "| 4993 |  25 |  71 |  96 |\n",
       "| 4994 |  14 |  56 |  70 |\n",
       "| 4995 |   7 |  22 |  29 |\n",
       "| 4996 |   5 |  12 |  17 |\n",
       "| 4997 |   4 |   7 |  11 |\n",
       "| 4998 |   0 |   4 |   4 |\n",
       "| 4999 |   2 |   2 |   4 |\n",
       "| 5000 |   2 |  24 |  26 |\n",
       "\n"
      ],
      "text/plain": [
       "     casual registered cnt\n",
       "1     3     13          16\n",
       "2     8     32          40\n",
       "3     5     27          32\n",
       "4     3     10          13\n",
       "5     0      1           1\n",
       "6     0      1           1\n",
       "7     2      0           2\n",
       "8     1      2           3\n",
       "9     1      7           8\n",
       "10    8      6          14\n",
       "11   12     24          36\n",
       "12   26     30          56\n",
       "13   29     55          84\n",
       "14   47     47          94\n",
       "15   35     71         106\n",
       "16   40     70         110\n",
       "17   41     52          93\n",
       "18   15     52          67\n",
       "19    9     26          35\n",
       "20    6     31          37\n",
       "21   11     25          36\n",
       "22    3     31          34\n",
       "23   11     17          28\n",
       "24   15     24          39\n",
       "25    4     13          17\n",
       "26    1     16          17\n",
       "27    1      8           9\n",
       "28    2      4           6\n",
       "29    2      1           3\n",
       "30    0      2           2\n",
       "⋮    ⋮      ⋮          ⋮  \n",
       "4971  33     80        113\n",
       "4972   8     71         79\n",
       "4973  19     48         67\n",
       "4974  24     26         50\n",
       "4975   6      7         13\n",
       "4976   4      4          8\n",
       "4977   5      8         13\n",
       "4978  19     26         45\n",
       "4979  54     88        142\n",
       "4980  55    128        183\n",
       "4981  91    163        254\n",
       "4982 112    163        275\n",
       "4983 146    192        338\n",
       "4984 142    152        294\n",
       "4985 130    196        326\n",
       "4986 109    184        293\n",
       "4987  98    208        306\n",
       "4988 114    206        320\n",
       "4989  91    213        304\n",
       "4990  87    187        274\n",
       "4991  70    205        275\n",
       "4992  68     96        164\n",
       "4993  25     71         96\n",
       "4994  14     56         70\n",
       "4995   7     22         29\n",
       "4996   5     12         17\n",
       "4997   4      7         11\n",
       "4998   0      4          4\n",
       "4999   2      2          4\n",
       "5000   2     24         26"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Limit number of rows because the Wilks-Shapiro test cannot take too much data\n",
    "bikes4dv1 <- bikes4dv[1:5000,]\n",
    "bikes4dv1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "bf1c7d6d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tShapiro-Wilk normality test\n",
       "\n",
       "data:  Z\n",
       "W = 0.7431, p-value < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Run Wilks-Shapiro test for Multivariate Normality\n",
    "mshapiro.test(t(bikes4dv1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d4031eb",
   "metadata": {},
   "source": [
    "We do NOT meet the assumption of multivariate normality that is required for MANOVA. However, we were never taught what to actually do in that case, only to continue on for the sake of learning. So, that is what we will do."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "500364e8",
   "metadata": {},
   "source": [
    "### Homogeneity of Variance\n",
    "\n",
    "We have previously discovered that our DVs are not normally distributed. Therefore, we will use the Fligner-Killeen test for homogeniety of variance, as it is known to be the most forgiving option when dealing with non-normal data.\n",
    "\n",
    "Although we have 8 possible categorical IVs, I have chosen only 3 to check below as if 3 are not homogenous in variance, then the full 8 certainly will not be either."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "e49a2d69",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tFligner-Killeen test of homogeneity of variances\n",
       "\n",
       "data:  casual by interaction(season, weekday, holiday)\n",
       "Fligner-Killeen:med chi-squared = 5173.8, df = 36, p-value < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fligner.test(casual ~ interaction(season, weekday, holiday), data = bikes4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17bbd9a9",
   "metadata": {},
   "source": [
    "Unfortunately, the `casual` DV does not meet the assumption of homogeneity of variance. Therefore, we expect that `registered` and `cnt` will not meet it either, as we already know that the 3 DVs are closely related. However, we can check them quickly as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "11fc8c24",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tFligner-Killeen test of homogeneity of variances\n",
       "\n",
       "data:  registered by interaction(season, weekday, holiday)\n",
       "Fligner-Killeen:med chi-squared = 1335.6, df = 36, p-value < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "\tFligner-Killeen test of homogeneity of variances\n",
       "\n",
       "data:  cnt by interaction(season, weekday, holiday)\n",
       "Fligner-Killeen:med chi-squared = 1363.5, df = 36, p-value < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fligner.test(registered ~ interaction(season, weekday, holiday), data = bikes4)\n",
    "fligner.test(cnt ~ interaction(season, weekday, holiday), data = bikes4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b602139",
   "metadata": {},
   "source": [
    "Indeed, we see that these data do NOT meet the assumption of homogeniety of variance in order to proceed with the MANOVA. However, for exploration purposes, we will continue with the knowledge that our final analysis may not be completely reliable.\n",
    "\n",
    "### Absence of Multicollinearity\n",
    "We run a correlation on all combinations of the DVs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "4f77eca7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tPearson's product-moment correlation\n",
       "\n",
       "data:  bikes4dv$casual and bikes4dv$registered\n",
       "t = 77.459, df = 17377, p-value < 2.2e-16\n",
       "alternative hypothesis: true correlation is not equal to 0\n",
       "95 percent confidence interval:\n",
       " 0.4954822 0.5175867\n",
       "sample estimates:\n",
       "      cor \n",
       "0.5066177 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "\tPearson's product-moment correlation\n",
       "\n",
       "data:  bikes4dv$casual and bikes4dv$cnt\n",
       "t = 127.27, df = 17377, p-value < 2.2e-16\n",
       "alternative hypothesis: true correlation is not equal to 0\n",
       "95 percent confidence interval:\n",
       " 0.6867886 0.7021806\n",
       "sample estimates:\n",
       "      cor \n",
       "0.6945641 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "\tPearson's product-moment correlation\n",
       "\n",
       "data:  bikes4dv$registered and bikes4dv$cnt\n",
       "t = 546.82, df = 17377, p-value < 2.2e-16\n",
       "alternative hypothesis: true correlation is not equal to 0\n",
       "95 percent confidence interval:\n",
       " 0.9713222 0.9729557\n",
       "sample estimates:\n",
       "      cor \n",
       "0.9721507 \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor.test(bikes4dv$casual, bikes4dv$registered, method = 'pearson', use = 'complete.obs')\n",
    "cor.test(bikes4dv$casual, bikes4dv$cnt, method = 'pearson', use = 'complete.obs')\n",
    "cor.test(bikes4dv$registered, bikes4dv$cnt, method = 'pearson', use = 'complete.obs')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae563f65",
   "metadata": {},
   "source": [
    "We can see that only `casual` and `registered` meet the assumption of the absence of multicollinearity. Indeed, `casual` is very close to the 0.7 boundary of multicollinearity with `cnt`, while `registered` is extremely closely correlated with `cnt`. Because of these relationships, we will use only the `casual` and `registered` variables to run the MANOVA. (In other words, `registered` would account for nearly all of the variance in `cnt` as well, so there is no need to include both variables.)\n",
    "\n",
    "## Run the MANOVA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "fd882887",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "             Df  Pillai approx F num Df den Df    Pr(>F)    \n",
       "ind_vars      8 0.48387   692.95     16  34740 < 2.2e-16 ***\n",
       "Residuals 17370                                             \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ind_vars <- cbind(bikes4$season, bikes4$yr, bikes4$mnth, bikes4$hr, bikes4$holiday, bikes4$workingday, bikes4$weekday, bikes4$weathersit)\n",
    "\n",
    "MANOVA <- manova(cbind(casual, registered) ~ ind_vars, data = bikes4)\n",
    "summary(MANOVA)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "176e6051",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       " Response casual :\n",
       "               Df   Sum Sq Mean Sq F value    Pr(>F)    \n",
       "ind_vars        8 10243057 1280382  694.95 < 2.2e-16 ***\n",
       "Residuals   17370 32002618    1842                      \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       " Response registered :\n",
       "               Df    Sum Sq  Mean Sq F value    Pr(>F)    \n",
       "ind_vars        8 106950304 13368788  797.55 < 2.2e-16 ***\n",
       "Residuals   17370 291162785    16762                      \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary.aov(MANOVA, test = 'wilks')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5c569e2",
   "metadata": {},
   "source": [
    "We see from the post-hocs that the independent variables are significant predictors for both `casual` and `registered` users.\n",
    "\n",
    "# Conclusion\n",
    "It appears that all continuous and all categorical variables are significant predictors of the `casual` and `registered` bike rental counts. We have excluded the `cnt` variable as too closely related to `registered` and therefore redundant. We will take these conclusions back to our Python notebook in order to continue the Machine Learning portion of the analysis."
   ]
  }
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