{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "d83256ca-2a1e-4ed0-9199-97697020bd02",
   "metadata": {},
   "source": [
    "# ANOVA\n",
    "\n",
    "Ron Fisher recognized the value of the $t$-test as created by William Sealy Gossett and developed the mathematical extension needed to use the same ideas with more than 2 samples."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45515245-ba2c-4832-89d1-fc3cca65d71e",
   "metadata": {},
   "source": [
    "## Example: Coping Humor\n",
    "\n",
    "Let's question whether, in the personality data set, the average coping humor (CHS) varies based upon Primary Humor Style (PHS). We will first load and then subset the data properly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4a834966-01d3-423f-b08c-8c4bcb24a4db",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>CHS</th><th scope=col>PHS</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>28</td><td>SE</td></tr>\n",
       "\t<tr><td>29</td><td>SE</td></tr>\n",
       "\t<tr><td>30</td><td>AG</td></tr>\n",
       "\t<tr><td>27</td><td>SE</td></tr>\n",
       "\t<tr><td>24</td><td>AF</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " CHS & PHS\\\\\n",
       "\\hline\n",
       "\t 28 & SE\\\\\n",
       "\t 29 & SE\\\\\n",
       "\t 30 & AG\\\\\n",
       "\t 27 & SE\\\\\n",
       "\t 24 & AF\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| CHS | PHS |\n",
       "|---|---|\n",
       "| 28 | SE |\n",
       "| 29 | SE |\n",
       "| 30 | AG |\n",
       "| 27 | SE |\n",
       "| 24 | AF |\n",
       "\n"
      ],
      "text/plain": [
       "  CHS PHS\n",
       "1 28  SE \n",
       "2 29  SE \n",
       "3 30  AG \n",
       "4 27  SE \n",
       "5 24  AF "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pers <- read.csv('https://faculty.ung.edu/rsinn/data/personality.csv')\n",
    "humor <- pers[ , c('CHS','PHS')]\n",
    "head(humor,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e5f7e51-6d69-4211-ada9-201338192d81",
   "metadata": {},
   "source": [
    "### Hypotheses\n",
    "\n",
    "The null hypothesis states that the four group means will all be approximately equal. The four primary humor styles are *Affiliative*, *Aggressive*, *Self-Defeating* and *Self-Enhancing* and will be abbreviate as **af**, **ag**, **sd** and **se** respectively. The alternative is usually included in words as shown:\n",
    "\n",
    "$$\\begin{align} H_0 &: \\mu_{af} = \\mu_{ag} = \\mu_{sd} = \\mu_{se}\\\\H_a &: \\text{At least 1 group mean differs from the others}\\end{align}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed0ba52c-d1e9-4b54-8475-5a664d45b9df",
   "metadata": {},
   "source": [
    "### Verifying the Assumptions\n",
    "\n",
    "We need to perform the following steps:\n",
    "1. Create the linear model usinig the function **lm()**.\n",
    "2. Create a vector of residuals based upon the model.\n",
    "3. Display a density plot of the residuals (checking for evidence of normality).\n",
    "4. Display a QQ plot for the residuals (checking that all dots are reasonably close to the QQ line).\n",
    "\n",
    "These steps are all performed in the single code below directly below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "ad63713b-4624-4fd2-a32b-297c3301dc25",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAANlBMVEUAAABNTU1oaGh8fHyM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\nAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"QQ Plot: Humor\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mod = lm(CHS ~ PHS, data = humor)\n",
    "res = residuals(mod)\n",
    "layout(matrix(c(1,2), ncol = 2), lcm(8))\n",
    "plot(density(res), main = 'Density Plot: Humor')\n",
    "plot2 <- { qqnorm(mod$residual, main = 'QQ Plot: Humor')\n",
    "qqline(mod$residual, col = 'red') }"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e06580d6-8d15-4df2-a95a-125bdf4da966",
   "metadata": {},
   "source": [
    "**Analysis**\n",
    "\n",
    "The QQ plot is slightly concerning as the lower tail seems quite heavy, yet the density plot shows evidence these data were drawn from a normally distributed population. These data are appropriate for ANOVA methods though the heavy tail does indicate a small degree of concern about accuracy of $p$-values."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "981013f7-77a0-43ce-b68e-8eb473229304",
   "metadata": {},
   "source": [
    "### Conducting the ANOVA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "336036b7-5845-400d-982b-f313f62905ae",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "             Df Sum Sq Mean Sq F value   Pr(>F)    \n",
       "PHS           3  418.7  139.55   7.743 8.74e-05 ***\n",
       "Residuals   125 2252.8   18.02                     \n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "anova <- aov(mod)\n",
    "summary(anova)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7ba8180-345f-4459-b8bf-865f39cb9daa",
   "metadata": {},
   "source": [
    "### Reporting Out\n",
    "\n",
    "With $p = 8.74\\times 10^{-5} < 0.05 = \\alpha$, we reject the null. We have evidence, therefore, that at least one of the group means is significantly different than the others."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b8714d17-e9c0-4313-867c-ef6428a037ae",
   "metadata": {},
   "source": [
    "## Tukey HSD *Post Hoc* Testing\n",
    "\n",
    "Whenever we reject the null in an ANOVA, we must conduct a *post hoc* test to ferret out all the significant differences between group means."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "886735dc-0875-4acc-8fb8-39449cb19533",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "  Tukey multiple comparisons of means\n",
       "    95% family-wise confidence level\n",
       "\n",
       "Fit: aov(formula = mod)\n",
       "\n",
       "$PHS\n",
       "           diff        lwr       upr     p adj\n",
       "AG-AF -1.385338 -4.1385491 1.3678724 0.5581859\n",
       "SD-AF -2.100649 -4.9409691 0.7396704 0.2225615\n",
       "SE-AF  2.669048 -0.2357284 5.5738237 0.0837472\n",
       "SD-AG -0.715311 -3.3456923 1.9150703 0.8937175\n",
       "SE-AG  4.054386  1.3545316 6.7542403 0.0008566\n",
       "SE-SD  4.769697  1.9810664 7.5583275 0.0001080\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "TukeyHSD(anova)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3713018f-8ee1-4af4-88cf-98b24e9e1f88",
   "metadata": {},
   "source": [
    "The significant differences are indicated in rows 3, 5 and 6 where:\n",
    "1. The upper and lower bounds of the confidence interval estimates both have the same sign, or\n",
    "2. The indicated $p$-value is less than $\\alpha = 0.05$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9ad4671-54e4-4e94-a7c4-518f09c44230",
   "metadata": {},
   "source": [
    "Thus, we have evidence for a difference between the means of the following pairs of groups:\n",
    "- Self-enhancing vs. Affiliative\n",
    "- Self-enhancing vs. Aggressive\n",
    "- Self-enhancing vs. Self-defeating"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9562628c-bfe7-484f-b6e0-4f37dae9faab",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}