{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basics of Simple Linear Regression\n",
    "\n",
    "本课程前置需要装的包："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": [
     "hide-output"
    ],
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "载入需要的程辑包：s20x\n",
      "\n"
     ]
    }
   ],
   "source": [
    "require(s20x)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 分析数据过程\n",
    "\n",
    "### 读取数据\n",
    "\n",
    "读取数据表格，`header=TRUE` 表示第一行是表头，`sep=\",\"` 表示分隔符是逗号。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 6 × 15</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>Grade</th><th scope=col>Pass</th><th scope=col>Exam</th><th scope=col>Degree</th><th scope=col>Gender</th><th scope=col>Attend</th><th scope=col>Assign</th><th scope=col>Test</th><th scope=col>B</th><th scope=col>C</th><th scope=col>MC</th><th scope=col>Colour</th><th scope=col>Stage1</th><th scope=col>Years.Since</th><th scope=col>Repeat</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</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;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;chr&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td>C</td><td>Yes</td><td>42</td><td>BSc  </td><td>Male  </td><td>Yes</td><td>17.2</td><td> 9.1</td><td> 5</td><td>13</td><td>12</td><td>Blue  </td><td>C</td><td>2.5</td><td>Yes</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>B</td><td>Yes</td><td>58</td><td>BCom </td><td>Female</td><td>Yes</td><td>17.2</td><td>13.6</td><td>12</td><td>12</td><td>17</td><td>Yellow</td><td>A</td><td>2.0</td><td>No </td></tr>\n",
       "\t<tr><th scope=row>3</th><td>A</td><td>Yes</td><td>81</td><td>Other</td><td>Female</td><td>Yes</td><td>17.2</td><td>14.5</td><td>14</td><td>17</td><td>25</td><td>Blue  </td><td>A</td><td>3.0</td><td>No </td></tr>\n",
       "\t<tr><th scope=row>4</th><td>A</td><td>Yes</td><td>86</td><td>Other</td><td>Female</td><td>Yes</td><td>19.6</td><td>19.1</td><td>15</td><td>17</td><td>27</td><td>Yellow</td><td>A</td><td>0.0</td><td>No </td></tr>\n",
       "\t<tr><th scope=row>5</th><td>D</td><td>No </td><td>35</td><td>Other</td><td>Male  </td><td>No </td><td> 8.0</td><td> 8.2</td><td> 4</td><td> 1</td><td>15</td><td>Blue  </td><td>C</td><td>3.0</td><td>No </td></tr>\n",
       "\t<tr><th scope=row>6</th><td>A</td><td>Yes</td><td>72</td><td>BCom </td><td>Female</td><td>Yes</td><td>18.4</td><td>12.7</td><td>15</td><td>17</td><td>20</td><td>Blue  </td><td>A</td><td>1.5</td><td>No </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 15\n",
       "\\begin{tabular}{r|lllllllllllllll}\n",
       "  & Grade & Pass & Exam & Degree & Gender & Attend & Assign & Test & B & C & MC & Colour & Stage1 & Years.Since & Repeat\\\\\n",
       "  & <chr> & <chr> & <int> & <chr> & <chr> & <chr> & <dbl> & <dbl> & <int> & <int> & <int> & <chr> & <chr> & <dbl> & <chr>\\\\\n",
       "\\hline\n",
       "\t1 & C & Yes & 42 & BSc   & Male   & Yes & 17.2 &  9.1 &  5 & 13 & 12 & Blue   & C & 2.5 & Yes\\\\\n",
       "\t2 & B & Yes & 58 & BCom  & Female & Yes & 17.2 & 13.6 & 12 & 12 & 17 & Yellow & A & 2.0 & No \\\\\n",
       "\t3 & A & Yes & 81 & Other & Female & Yes & 17.2 & 14.5 & 14 & 17 & 25 & Blue   & A & 3.0 & No \\\\\n",
       "\t4 & A & Yes & 86 & Other & Female & Yes & 19.6 & 19.1 & 15 & 17 & 27 & Yellow & A & 0.0 & No \\\\\n",
       "\t5 & D & No  & 35 & Other & Male   & No  &  8.0 &  8.2 &  4 &  1 & 15 & Blue   & C & 3.0 & No \\\\\n",
       "\t6 & A & Yes & 72 & BCom  & Female & Yes & 18.4 & 12.7 & 15 & 17 & 20 & Blue   & A & 1.5 & No \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 15\n",
       "\n",
       "| <!--/--> | Grade &lt;chr&gt; | Pass &lt;chr&gt; | Exam &lt;int&gt; | Degree &lt;chr&gt; | Gender &lt;chr&gt; | Attend &lt;chr&gt; | Assign &lt;dbl&gt; | Test &lt;dbl&gt; | B &lt;int&gt; | C &lt;int&gt; | MC &lt;int&gt; | Colour &lt;chr&gt; | Stage1 &lt;chr&gt; | Years.Since &lt;dbl&gt; | Repeat &lt;chr&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| 1 | C | Yes | 42 | BSc   | Male   | Yes | 17.2 |  9.1 |  5 | 13 | 12 | Blue   | C | 2.5 | Yes |\n",
       "| 2 | B | Yes | 58 | BCom  | Female | Yes | 17.2 | 13.6 | 12 | 12 | 17 | Yellow | A | 2.0 | No  |\n",
       "| 3 | A | Yes | 81 | Other | Female | Yes | 17.2 | 14.5 | 14 | 17 | 25 | Blue   | A | 3.0 | No  |\n",
       "| 4 | A | Yes | 86 | Other | Female | Yes | 19.6 | 19.1 | 15 | 17 | 27 | Yellow | A | 0.0 | No  |\n",
       "| 5 | D | No  | 35 | Other | Male   | No  |  8.0 |  8.2 |  4 |  1 | 15 | Blue   | C | 3.0 | No  |\n",
       "| 6 | A | Yes | 72 | BCom  | Female | Yes | 18.4 | 12.7 | 15 | 17 | 20 | Blue   | A | 1.5 | No  |\n",
       "\n"
      ],
      "text/plain": [
       "  Grade Pass Exam Degree Gender Attend Assign Test B  C  MC Colour Stage1\n",
       "1 C     Yes  42   BSc    Male   Yes    17.2    9.1  5 13 12 Blue   C     \n",
       "2 B     Yes  58   BCom   Female Yes    17.2   13.6 12 12 17 Yellow A     \n",
       "3 A     Yes  81   Other  Female Yes    17.2   14.5 14 17 25 Blue   A     \n",
       "4 A     Yes  86   Other  Female Yes    19.6   19.1 15 17 27 Yellow A     \n",
       "5 D     No   35   Other  Male   No      8.0    8.2  4  1 15 Blue   C     \n",
       "6 A     Yes  72   BCom   Female Yes    18.4   12.7 15 17 20 Blue   A     \n",
       "  Years.Since Repeat\n",
       "1 2.5         Yes   \n",
       "2 2.0         No    \n",
       "3 3.0         No    \n",
       "4 0.0         No    \n",
       "5 3.0         No    \n",
       "6 1.5         No    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>146</li><li>15</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 146\n",
       "\\item 15\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 146\n",
       "2. 15\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 146  15"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<style>\n",
       ".list-inline {list-style: none; margin:0; padding: 0}\n",
       ".list-inline>li {display: inline-block}\n",
       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
       "</style>\n",
       "<ol class=list-inline><li>42</li><li>58</li><li>81</li><li>86</li><li>35</li><li>72</li><li>42</li><li>25</li><li>36</li><li>48</li><li>29</li><li>54</li><li>49</li><li>52</li><li>28</li><li>34</li><li>51</li><li>81</li><li>80</li><li>41</li></ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 42\n",
       "\\item 58\n",
       "\\item 81\n",
       "\\item 86\n",
       "\\item 35\n",
       "\\item 72\n",
       "\\item 42\n",
       "\\item 25\n",
       "\\item 36\n",
       "\\item 48\n",
       "\\item 29\n",
       "\\item 54\n",
       "\\item 49\n",
       "\\item 52\n",
       "\\item 28\n",
       "\\item 34\n",
       "\\item 51\n",
       "\\item 81\n",
       "\\item 80\n",
       "\\item 41\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 42\n",
       "2. 58\n",
       "3. 81\n",
       "4. 86\n",
       "5. 35\n",
       "6. 72\n",
       "7. 42\n",
       "8. 25\n",
       "9. 36\n",
       "10. 48\n",
       "11. 29\n",
       "12. 54\n",
       "13. 49\n",
       "14. 52\n",
       "15. 28\n",
       "16. 34\n",
       "17. 51\n",
       "18. 81\n",
       "19. 80\n",
       "20. 41\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 42 58 81 86 35 72 42 25 36 48 29 54 49 52 28 34 51 81 80 41"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "course.df <- read.table(\"../data/STATS20x.txt\", header = TRUE, sep = \"\\t\")\n",
    "head(course.df) # 看前面大约10行的内容\n",
    "dim(course.df) # 看有多少行、多少列\n",
    "course.df$Exam[1:20] # 看前20行的Exam列"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 绘图观测数据\n",
    "\n",
    "对数据进行绘图分析，着重分析 `Exam` 和 `Test` 两个变量之间的关系。\n",
    "\n",
    "首先应当粗略查看两者的关系，如线性、二次、曲线、正弦等"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Plot of Exam vs. Test   (lowess+/-sd)\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(s20x)\n",
    "trendscatter(Exam ~ Test, data = course.df)\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 进行初步拟合\n",
    "\n",
    "可以看到整体大致呈线性关系，故我们采用线性回归模型。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(Exam ~ Test, data = course.df)\n",
    "# 绘制回归直线\n",
    "examtest.fit <- lm(Exam ~ Test, data = course.df)\n",
    "# lty = 2 表示虚线，col = \"red\" 表示红色\n",
    "abline(examtest.fit, lty = 2, col = \"red\")\n",
    "\n",
    "points(\n",
    "    0,\n",
    "    predict(examtest.fit, newdata = data.frame(Test = 0)),\n",
    "    col = \"blue\",\n",
    "    pch = 19\n",
    ")\n",
    "points(10, predict(examtest.fit,\n",
    "    newdata = data.frame(Test = 10)\n",
    "), col = \"blue\", pch = 19)\n",
    "points(20, predict(examtest.fit,\n",
    "    newdata = data.frame(Test = 20)\n",
    "), col = \"blue\", pch = 19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Exam ~ Test, data = course.df)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-39.980  -6.471   0.826   8.575  33.242 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   9.0845     3.2204   2.821  0.00547 ** \n",
       "Test          3.7859     0.2647  14.301  < 2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 12.05 on 144 degrees of freedom\n",
       "Multiple R-squared:  0.5868,\tAdjusted R-squared:  0.5839 \n",
       "F-statistic: 204.5 on 1 and 144 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(examtest.fit)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "notebookRunGroups": {
     "groupValue": ""
    }
   },
   "source": [
    "其中：\n",
    "\n",
    "- Call：表示回归方程，指明了自变量和因变量\n",
    "- Risiduals：残差，指明了残差的分布，如最大、最小、中值等\n",
    "- Coefficients：系数，此处即 $a_i$ 和 $b_i$ 的值\n",
    "- Residual standard error：残差标准差，即残差的标准差\n",
    "- Multiple R-squared：多元 $R^2$ 值\n",
    "- Adjusted R-squared：调整后的 $R^2$ 值\n",
    "- F-statistic：F 统计量，即 F 统计量。F 统计量的分子是回归平方和，分母是残差平方和。F 统计量的值越大，说明回归平方和越大，即回归模型的拟合效果越好。F 统计量的值越小，说明回归平方和越小，即回归模型的拟合效果越差。p-value 则相反。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 分析数据是否可以接受\n",
    "\n",
    "### 残差观测\n",
    "\n",
    "针对指定行分析预测值和残差："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A data.frame: 1 × 2</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>course.df.Test.1.</th><th scope=col>course.df.Exam.1.</th></tr>\n",
       "\t<tr><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>9.1</td><td>42</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 1 × 2\n",
       "\\begin{tabular}{ll}\n",
       " course.df.Test.1. & course.df.Exam.1.\\\\\n",
       " <dbl> & <int>\\\\\n",
       "\\hline\n",
       "\t 9.1 & 42\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 1 × 2\n",
       "\n",
       "| course.df.Test.1. &lt;dbl&gt; | course.df.Exam.1. &lt;int&gt; |\n",
       "|---|---|\n",
       "| 9.1 | 42 |\n",
       "\n"
      ],
      "text/plain": [
       "  course.df.Test.1. course.df.Exam.1.\n",
       "1 9.1               42               "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<strong>1:</strong> 43.5363712056028"
      ],
      "text/latex": [
       "\\textbf{1:} 43.5363712056028"
      ],
      "text/markdown": [
       "**1:** 43.5363712056028"
      ],
      "text/plain": [
       "       1 \n",
       "43.53637 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<strong>1:</strong> -1.53637120560281"
      ],
      "text/latex": [
       "\\textbf{1:} -1.53637120560281"
      ],
      "text/markdown": [
       "**1:** -1.53637120560281"
      ],
      "text/plain": [
       "        1 \n",
       "-1.536371 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data.frame(course.df$Test[1], course.df$Exam[1]) # 原第一行\n",
    "# 按照 tidyverse 的风格，也可以使用 dplyr 包的 select 函数来选择列\n",
    "# dplyr::select(course.df[1, ], Exam, Test)\n",
    "fitted(examtest.fit)[1] # 拟合值\n",
    "resid(examtest.fit)[1] # 残差"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "检验上，一个成功的拟合模型的残差应当有：\n",
    "\n",
    "1. 残差均值接近于 0\n",
    "2. 残差满足正态分布\n",
    "3. 没有或排除了异常点\n",
    "\n",
    "#### 残差均值接近于 0\n",
    "\n",
    "分析残差，看是否符合均值等于0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 其中 which = 1 表示残差直方图（histogram of residuals），\n",
    "# which = 2 表示残差QQ图（qqplot，即 normal quantile-quantile-plot），\n",
    "# which = 3 表示残差标准化图\n",
    "plot(examtest.fit, which = 1)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 残差满足正态分布\n",
    "\n",
    "残差在分布上在符合正态同分布：iid – independence（并且这是根据学生在考试中应该相互独立的表现）。残差应该有大致恒定的散布。这其实是 Equality Of Variance (EOV，方差相等) 原则。\n",
    "\n",
    "检查残差是否满足正态分布："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "normcheck(examtest.fit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Exam ~ Test, data = course2.df)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-90.251  -6.846   2.638   9.456  33.996 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  15.2374     3.7172   4.099 6.88e-05 ***\n",
       "Test          3.2006     0.3023  10.588  < 2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 14.34 on 145 degrees of freedom\n",
       "Multiple R-squared:  0.436,\tAdjusted R-squared:  0.4322 \n",
       "F-statistic: 112.1 on 1 and 145 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ZuymIjUCpEVJORm+uI6TUCCkno3fXEVJqhJSfMTNvzpESIyQjRs8hxl088n2EKIPxCAjJhPHT6rEvhO0NicF4FIRkwthnYqP3MvTfIE/6oiAkC8bOBiZ29O4GGUPEQUgWDP103zcFzb5BQoqDkCwY+Ome/t4MhJQaIZmYYbl2Ge8XMuM9TjhHSqb2kIzMsDzLaDsSbpA18icuTfUhJT5ej/5HpMvD0fQV9j/SmngMLk3lIRk5Y+hfRtvRaVbHz75thJT2gO/5Qmpn3oRkHSGlPeB7nmW4y7UjQrKt8pDsnCPdf31wvQS7+ArhNzekTeN8+yMDpZzayRcftozrL0/+Oko8ZTNwb+RoZkgb5whJtZCAL0VYxjFptsWYGZJz37Kl9BwiLiNP7V6EvUH+fFbvD/NmhyRbSd8hojIybHgRsplBwer9Yd/sp3YH2VJ6DhGVzR+coE1BEjbvjxzMHTasVnvVUvoOEZPJH5xTRwu9harJ+yMLc0Pa5j5sSHq8UXdTm9HY1x2pn1tzjhRoZkhf+U/trr9YOdaEjiIsnqldoJkhNblP7VIOl8f93378w1Gchw8TFwPyU/nULqWR5x9TOyr03srN7Kd2eU/tUhodkvgGkcLcYcPX6le1lL5DlEL+c09Ihsx+apf5sCEl+SkNIzY7CCkd+USMEZsdtb+MIi3f/2+CNjMwYrOCkIxYai8DNFQh/a7nrmTwECWbcPEIJs1+YR/nSAJ0lD3ZC/u2siUdqwuJjPI3e4vQz3Hl9vuVk15Osh+S8iH4/C5BNl4hi0CCLUJff49GO7eSLeloPyTt3LnNqPcGmXFnQRDS9rRxta5zpJSXVrnqmoWZIa3/ntrt3cfxt6qQUm72YR9QHmaGtD0FtDoNGz5lSzqa/6khJDybvWn19Hefzm1E63lzCHsICc/Y2RBCc97SmXlzjpQ7QgrhvQY9clr9cPHIc4PeqV3ayThz+H4zQ/q8vq5vX9n4O+znvuPxImxYmWkn48zhfeaOv5uf9q/fVU3t5j8TO2XU3e0d9vwt7bM+nmP6zAzpt3Hr/d/DkWtq2tkwezbwvCcobKKQdg7B1MNr9jnSl3Mb575Ey3l7CHMEIT2++IiQsjd/2PD3rE7+nlzG/2PNDun5RXyElD3RI5L2MpL5/1jyaTXnSLmbf460+jtHWtd1juTfY5pu/sbUzo65U7vLs7qfpqqpnTeJlFeEuI5kxcyQbp9Fcahpr52P91nfaVaX8Z8N/djZoOWbQ/D55AWbEZJ7fI6nWM3zIfLjCel6ETbfPxz6zQ7pUhAhnfWG1Gb0HyGVipDEes6RLh393U3T/2xhb+fAZCApQhLrGc21Gf339kthNxj+JcRASHLvHwpOD0ftl4RXXeXXhRGMkNJoOzqT7QOavVMJOoSUxt/TOkIqGSHF9fJyckIq06yQHiy8KpOeXgbb+XU8zpGyQEgRx8RPr9+T70xlamfHjJAiSvcDEPEH7s1b48t3pnIdyYrqQ4p2PD5ioiqVhxTxpPycUdAHWiI/hBTzgJfdDCgfIUU8IB3Vo/KQ/OdI82YDbzJiAFCs6kNST5Bv3/XaESPpgtUekudRIvD66fVVsK/P6rhIWjBC6hN2+nQderNtpzKE1Cfo5/528YiQKkNIfQJ+7jvXYAmpMoTUa/IpTdsRG0nrREi9Jg/ZzmMG3xhQvrsXVhCSR9BPffsOJ303R0ilIiQxz2YGntoVjJCkfJuCGDaUjJCUvJvrCKlkhDTfbebt36NKSCUjpLkvMr1fPLp05Nlz5Hlfu5QvnoVe7SHN3rTauQh7fyfVRG+xwHs22FF9SP3HGzNke309ufcGex+RAjfI+o4VcoMIVnlInvOWMac0fR1NvcGw0yf5DSIcIfUdcFxImhskpOwRUt8BU/5wE1L2Kg9Jf5oRdoOcI+UuaUi/X+t2t9l68ys/ROC0VzH4erh6FLZpNWwbHlM7OxKGdPjovMHxSnuIGT83cy/FvOxm6L+OpA5p/uKhkjCkjWt+du3v9tvGbaSHSPxMpjNkmPCWWzwTK1jCkBq3u/1+5xrlIdKeWz9cgx3/znXMBkqWMKSHpxr+5x2mQwrsiJCKxiPSRA/XYCe9kSohlSztOdJ23/7O1DnStJPyOZ8x4T1HCvmcZtiRcvy96kztPg7SQwRP7aZ+45yPamFaXbC015E27XWkZv1l5zpS6PGChL3CAhmofGeDkZMTI8tAODshRftAWu9Bn/66ECPLQLhFQhoMxV5Ij+dG6o89IqTsVR7SyHOkp1md/vPD2NmQu6QXZEc/e0sY0phx2WNHMT6HL3Rqx4Y6KxKG9NsYDGnEz+Lrw1GMz7MM27J6DMoPeimf2h3WbtVekTX01G5Y9Kd14XhCaEfac6Qf536O2YXU/Tt7Hdm6t6qVeNiwX7n1Ia+QHhnqiJAsST61+3LNNuOQLCEkQ9KPv3cfw+fV8h8NI9Mt8TLYWGTHEteRPlOHZGS6JV+GkT8Xjpa2CEU8RNB0a84+b+Ey/DfII5IVNYQUci4x53VHwmWkvUGEI6S3XjoSTOsIqWSE9Mbrw5Hi8hEhlayGkKaenLx5OJJcPopwjiS+QQSrI6Rp0603D0eSy7BM7QpWRUjzLuAIdwXJL2cZuT6GWkKawdLuOphFSEPoCCMQEiBASHf6vQyoBiFdTdvLYGRswMe6WEFIF5M6MjLI5q1b7SCk1puMfEMGI5dWPd/FtdrECOnkbUf9IRnZ7OP5LnYPpUZIJ+8yGnxAIiTcEdI7A5uCCAnPCOmNwc0MnCPhSTEhBU97X79xeFOQ/K3+mdrlrpCQpB80NrwpKMJnZnAdKW+lhBT2bf/+yZ9ToU5lhBR2bt0ZesvO8lGpikM6dSQfl6FS1YZ0fjgiJGiUEdL0k5br0zrvN3o+O3ni4VC6UkKaOrW7nh11v/FpWsdwGeMVEpLiOtLz5aOwByvUqZiQ5nrZFcSJECYgpLPX3XWEhAkIqfVmVxAhYYLKQup5Gezb3XWM5jBeJSGdZwP/+l5P/nZ3HaM5jFdFSJckJn9UC6M5jFVHSKdfInziEXBVQ0jnW6MjRFRPSMwNEFHdIfG+3hCpIaTeQfbQa8qNvJ2q/WOhkpDeDrKHPj/MyNup2j8WjjWE1M4Y3vzvefBz+OQXZFNe4eVqcmKlh9Q79B75llvCtaSceTBfSa3wkMI7IiRMUXRInmuwI95yS7sWQipbySHN3MvAORLGKyakN+OEmVsZmNphvEJCivNzw3UkjFVKSGHfBoiUERLn1lhYbSGxuw5RlBiSZ8gwfPkICFFGSN1zJO/FozcdMQCAQCkh3aZ2EztiJA2JQkK6/s/edw128bcKYrRYsGJCagV2xLYdzFVaSP1fe/+WWzMPOAUhlayskIIPREiYp/KQOEeCRvUh6Sdp/R9PxtSuXLWHJL+2462F60jFKiOkyUOGeHj+VqcSQvK+gC/xpiAmCpUqIKSBi0dLPCARUnWyD2no4SjxJlVCqlTuIVl6WtfyniOFDRsYUWQg/5D6v7bIayY8U7uw8TdD8yzkHpLPQq896r+O1Pl1ws0FfRcSKzkkY8JOnzjpygMhJUNIJSOkZAipZJmGlOXnWHKOVLAsQ/qX6PPJU27DE38XEssxpKGMXqZ1Zn6CTfSMGPILafDR6PXyEc+pEFuWIfm+9c2uIM7yEV1+Ifm9fcutoFslJExQWEjCtwoiJExQVkg9u+s4R0JspYX0/uasTO1QrHxCmvcxlsydEVUuIc38PFggrkxCoiPYlkVIZATrcgiJjmBeHiEN/fuDr4X1jA2MTBSMLAOBcghpyOB7BcnfR0HOyDIQrICQht9zy3Np1chVVyPLQLDsQxrx1nWezT5G9gEZWQbC5R7SmLeAJCREZzmkMbO6MW+5RUiIzm5IwqF32nOkkPlb0nMkBoQRmA1JefEo5dTO/AZZBoRRGA1JfQ023XWk0MeWZA8TDAijsBlSvnsZzJ/tmF9gpqyGFP8YcZj/OTW/wEzZDGn4EAu9Qf4g8z+n5heYqUxDWuQTW0YxfwpifoF5yjKk9B/E96L/s1usD8XMLzBPOYa0fEfeH0bzl2nMLzBHGYa0eEY8PcKL/EIy0xEl4S7HkOIffwAh4Vl+IRlASHhGSCE4R8ITQgrBCBlPCCkMI2Q8ICRAIJ+QDEzrgD65hLT8bgbAI5OQ6Ai2JQ3p92vtTtab30mHICNYlzCkw4e7W004BB3BvIQhbVzzs2t/t982bjP+EGQE8xKG1Ljd7fc718Q4BLCQhCE9XML0X88kJGSGRyRAIO050nbf/m7iOVLnn0fYmMNmHwikHH+vOlO7j8P0Q8TYKsr2U0ikvY60aa8jNeuvUdeRnqd1MV68wAsiIGF4Z8Pz1aMYL6fjJXrQsBOS63p3FZaQYFfKkA6fzq22lxsZGH+/2c1ASLAr5Rah5rzR7nwj/pDebgriHAlmJR1/f//V9N202+z8Ib3fXMfUDmYlvSDb/mXffOyHQ+o5NNeRYNMCW4QOqxVbhFCYhCF9uOtF2I8VIaEsCUP6dp+X3+3dipBQlJTj782tnu3AiQkhITNJL8ju1tff7T+DQmIwAKPs7GwYPgSjapiVVUipDg9MlVFIbOeBXYQECBASIJBRSJwjwa6sQmJqB6tyConrSDArr5AAowgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBCoJiY9DR1xVhNRWREqIqI6QYtwo0FFDSO7pr4AcIQEChAQI1BAS50iIro6QmNohsipC4joSYqskJCAuQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJGOvfv399XyIkYJx/Z++/SEjAOIQEzPfvn68kQgJGISRAgJAABc6RAAFCAiS4jgTERUiAACEBAoQECBASIEBIgAAhAQKEBAgQEiBASIAAIQEChAQIEBIgQEiAACEBAoQECBASIEBIgIDRkIDMBPyU68OJwOYqTa7K5KJsrkq6KJN/whc2V2lyVSYXZXNVhGSEyVWZXJTNVRGSESZXZXJRNldFSEaYXJXJRdlcFSEZYXJVJhdlc1WEZITJVZlclM1VEZIRJldlclE2V0VIRphclclF2VwVIRlhclUmF2VzVYRkhMlVmVyUzVURkhEmV2VyUTZXVWFIgHGEBAgQEiBASIAAIQEChAQIEBIgQEiAACEBAoQECBASIEBIgAAhAQKEBAgQEiBASICA/ZCC39Y8ou/rcjaNazaHRddyc12UpTvs++N2/9i5q+6LEt5VNu5vj52ln4uL3XU5q3ZpH8uu5uK6KEt32KZdSXP6obVzV90XpbyrTNzfPju3XnoJz3bN5b7/dc3u9He/Cy/o5LYoQ3fYzn0eTo+Un5buqs6ilHeV+ZC+3dfSS3jy7VaXn9mN2/79+mNhhfdFGbrD1ucVnRZm567qLEp5V2UQ0vfSS3jiNsfLz+za7Y9GHgLuizJ4hzlTd9XZOSTdXWU+pLXbfv6dGy69jI7d8foz+/iXRd0XZe4OO7iVqbuq1S5KeVdZ+YP1Wp/PB1dLr+OBvZCOnZCM3WHfp2d1tu6qy6KUd5WVP1gv537+/v+xsfV8xXJI1u6wfXN6OmfrrrotSndXGfmDDTmYGJzeWA7pzMwddmja/+Hbuqsui7r8jeSusvEHG2bkv8DFZTWNqZ+Ox2UYWdRxdf4ptXVXrR7SkSzKxh9smJH/AhcPU7u9kVGUxZD2H6t9+xtLd9VtURd1hNS403VxG/8Fbi53/Vd7cWTrbIzIbg+Tdu6w7e1E3tBddV+U8q4yH9LmdN8fztfzzDC4s+G2KEN32P4+ELNzV3UWpbyrzId0aNoRpYH/k3Vcnwx8WJo0XxZl6A77dPfNbGbuqs6ilHeV+ZD+/o/RuA8rs9yLa0iHdkvzsmu56S7KxsRE/IQAAAHiSURBVB3mOiGZuaueFyW6q+yHBGSAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQ8uA6ev4VAx8aWzFCysNwSB/8p1wS935Geh+MBr+I2Lj3M0JIdnHvZ+TeyveHa86fxr1dObfaXp77LbYycN9n5FbKus1m9fe77/Np0zchLYz7PiPXUrZudTgeVu7vgahxu+Pxx33w1G5h3PsZubaydoe/Xw9uffpH26cvYhHc+xm5ttIZhG+cW+923S9iEdz7GXkT0vGr+ftrsyekhXHvZ+QeUvefbjcfnCMtjns/I/dzpO3rFwhpUdz7Gbm28uOa3WnyvT5tDPq5Te32y66uboSUkduDzqo9QzqdGf2cT5Z+T0m5ZtHV1Y2QMvKws8F9to9A7c6Gv46Ovx+EtCBCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRA4H+ln/hSyxBwzAAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 创造一个包含异常点的数据集并验证异常点对回归直线的影响\n",
    "n <- nrow(course.df)\n",
    "# 复制一数据集的最后一行\n",
    "course2.df <- course.df[c(1:n, n), ]\n",
    "# 修改新数据集的最后一行的 Test 和 Exam 列的值，故意创造一个差异极大的观测值\n",
    "course2.df[n + 1, c(\"Test\", \"Exam\")] <- c(25, 5)\n",
    "# 画出散点图\n",
    "plot(Exam ~ Test, data = course2.df)\n",
    "## 并标记我们创建的新的观测点\n",
    "points(25, 5, pch = 19, col = \"red\")\n",
    "\n",
    "# 如果有的观测值是异常值，那么回归直线就会受到影响\n",
    "examtest2.fit <- lm(Exam ~ Test, data = course2.df)\n",
    "summary(examtest2.fit)\n",
    "\n",
    "# 或者直接画图验证该点造成的影响\n",
    "abline(examtest.fit, lty = 2, lwd = 2, col = \"blue\")\n",
    "abline(examtest2.fit, lty = 2, lwd = 2, col = \"red\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对其进行观测值差异分析："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Cook's Distance plot\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Cook's Distance plot\""
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 画出异常值的影响\n",
    "cooks20x(examtest2.fit)\n",
    "# 对比原来的值影响\n",
    "cooks20x(examtest.fit)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### R 方观测\n",
    "\n",
    "R Squared 即 R 平方，是回归平方和与总平方和的比值，即 $R^2 = \\frac{SSR}{SST}$，其中 SSR 为回归平方和，SST 为总平方和。R 平方的值越大，说明回归平方和越大，即回归模型的拟合效果越好。R 平方的值越小，说明回归平方和越小，即回归模型的拟合效果越差。\n",
    "\n",
    "SSR 即回归平方和，是因变量的预测值与因变量的均值之差的平方和，即 $SSR = \\sum_{i=1}^n (y_i - \\bar{y})^2$，其中 $y_i$ 为第 $i$ 个观测值，$\\bar{y}$ 为因变量的均值。下面将简要介绍 SSR 的计算方法。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Exam ~ 1, data = course.df)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-41.877 -12.877  -1.377  15.623  40.123 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   52.877      1.546   34.21   <2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 18.68 on 145 degrees of freedom\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Exam ~ Test, data = course.df)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-39.980  -6.471   0.826   8.575  33.242 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   9.0845     3.2204   2.821  0.00547 ** \n",
       "Test          3.7859     0.2647  14.301  < 2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 12.05 on 144 degrees of freedom\n",
       "Multiple R-squared:  0.5868,\tAdjusted R-squared:  0.5839 \n",
       "F-statistic: 204.5 on 1 and 144 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 消除一次项\n",
    "examnull.fit <- lm(Exam ~ 1, data = course.df)\n",
    "summary(examnull.fit)\n",
    "# 对比之前的 Summary\n",
    "summary(examtest.fit)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "此时我们可以得到 SS（Null）的值 18.68，以及 SS（Test）的值 12.05。\n",
    "\n",
    "R 方的值即 1 - SS（Null）/SS（Test）的值，即 0.5868。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "置信区间：$[a_i - 2SE(a_i), a_i + 2SE(a_i)]$，即 $[a_i - 2\\sqrt{Var(a_i)}, a_i + 2\\sqrt{Var(a_i)}]$，其中 $Var(a_i)$ 为 $a_i$ 的方差。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 每一个拟合值的 T 检验\n",
    "\n",
    "> 知道看什么，什么意思，怎么看"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = Exam ~ Test, data = course.df)\n",
       "\n",
       "Residuals:\n",
       "    Min      1Q  Median      3Q     Max \n",
       "-39.980  -6.471   0.826   8.575  33.242 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   9.0845     3.2204   2.821  0.00547 ** \n",
       "Test          3.7859     0.2647  14.301  < 2e-16 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 12.05 on 144 degrees of freedom\n",
       "Multiple R-squared:  0.5868,\tAdjusted R-squared:  0.5839 \n",
       "F-statistic: 204.5 on 1 and 144 DF,  p-value: < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(examtest.fit)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以看出 Test 行的 Pr（P-value）的值小于 2.2x10^-16，远小于 0.05，故拒绝原假设，即拟合值的系（旁边的3颗*也表示可信度极高，即该斜率的线性拟合极好）\n",
    "\n",
    "- 零假设 $H_0$：Test 和 Exam 之间的线性关系系数为 0（没有线性关系），即 即 $a_i$ 的系数为 0\n",
    "- 备择假设 $H_1$：Test 和 Exam 之间的线性关系系数不为 0（有线性关系），即 即 $a_i$ 的系数不为 0\n",
    "\n",
    "我们对于斜率的置信程度，是由标准误差决定的，即 $SE(a_i)$，即 $SE(a_i) = \\sqrt{\\frac{SSE}{n-2}}$，其中 SSE 为残差平方和，即 $SSE = \\sum_{i=1}^n (y_i - \\hat{y_i})^2$，其中 $\\hat{y_i}$ 为第 $i$ 个观测值的预测值，即 $\\hat{y_i} = a_i + b_i x_i$，$x_i$ 为第 $i$ 个观测值的自变量值。此处的 $se(a)$ 为 0.2647。于是我们有：\n",
    "\n",
    "$$\n",
    "\\frac{3.7859 - 0}{0.2647} = 14.34\n",
    "$$\n",
    "\n",
    "此结果表示偏离此结果的标准差，这个数字越大，代表我们对于斜率的置信程度越高。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 利用分析结果做预测\n",
    "\n",
    "### 拟合值的置信区间"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
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   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 2 × 2 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>2.719020</td><td>15.449907</td></tr>\n",
       "\t<tr><th scope=row>Test</th><td>3.262659</td><td> 4.309189</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 2 × 2 of type dbl\n",
       "\\begin{tabular}{r|ll}\n",
       "  & 2.5 \\% & 97.5 \\%\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 2.719020 & 15.449907\\\\\n",
       "\tTest & 3.262659 &  4.309189\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 2 × 2 of type dbl\n",
       "\n",
       "| <!--/--> | 2.5 % | 97.5 % |\n",
       "|---|---|---|\n",
       "| (Intercept) | 2.719020 | 15.449907 |\n",
       "| Test | 3.262659 |  4.309189 |\n",
       "\n"
      ],
      "text/plain": [
       "            2.5 %    97.5 %   \n",
       "(Intercept) 2.719020 15.449907\n",
       "Test        3.262659  4.309189"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 2 × 2 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>0.5 %</th><th scope=col>99.5 %</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>(Intercept)</th><td>0.6778171</td><td>17.491110</td></tr>\n",
       "\t<tr><th scope=row>Test</th><td>3.0948635</td><td> 4.476984</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 2 × 2 of type dbl\n",
       "\\begin{tabular}{r|ll}\n",
       "  & 0.5 \\% & 99.5 \\%\\\\\n",
       "\\hline\n",
       "\t(Intercept) & 0.6778171 & 17.491110\\\\\n",
       "\tTest & 3.0948635 &  4.476984\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 2 × 2 of type dbl\n",
       "\n",
       "| <!--/--> | 0.5 % | 99.5 % |\n",
       "|---|---|---|\n",
       "| (Intercept) | 0.6778171 | 17.491110 |\n",
       "| Test | 3.0948635 |  4.476984 |\n",
       "\n"
      ],
      "text/plain": [
       "            0.5 %     99.5 %   \n",
       "(Intercept) 0.6778171 17.491110\n",
       "Test        3.0948635  4.476984"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "confint(examtest.fit)\n",
    "# Intercept 即截距，Test 即斜率\n",
    "# 也可以自己修改置信水平\n",
    "confint(examtest.fit, level = 0.99)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 预测\n",
    "\n",
    "1. 准确预测值\n",
    "2. 预测的均值范围\n",
    "3. 预测每一个个体的取值范围\n",
    "\n",
    "区间估计和点估计的区别：\n",
    "\n",
    "- 区间估计：给出一个区间，表示参数的可能取值范围\n",
    "- 点估计：给出一个点，表示参数的可能取值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 3 × 3 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>fit</th><th scope=col>lwr</th><th scope=col>upr</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 9.084463</td><td> 2.71902</td><td>15.44991</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>46.943703</td><td>44.80912</td><td>49.07828</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>84.802942</td><td>79.97021</td><td>89.63568</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 3 × 3 of type dbl\n",
       "\\begin{tabular}{r|lll}\n",
       "  & fit & lwr & upr\\\\\n",
       "\\hline\n",
       "\t1 &  9.084463 &  2.71902 & 15.44991\\\\\n",
       "\t2 & 46.943703 & 44.80912 & 49.07828\\\\\n",
       "\t3 & 84.802942 & 79.97021 & 89.63568\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 3 × 3 of type dbl\n",
       "\n",
       "| <!--/--> | fit | lwr | upr |\n",
       "|---|---|---|---|\n",
       "| 1 |  9.084463 |  2.71902 | 15.44991 |\n",
       "| 2 | 46.943703 | 44.80912 | 49.07828 |\n",
       "| 3 | 84.802942 | 79.97021 | 89.63568 |\n",
       "\n"
      ],
      "text/plain": [
       "  fit       lwr      upr     \n",
       "1  9.084463  2.71902 15.44991\n",
       "2 46.943703 44.80912 49.07828\n",
       "3 84.802942 79.97021 89.63568"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A matrix: 3 × 3 of type dbl</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>fit</th><th scope=col>lwr</th><th scope=col>upr</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>1</th><td> 9.084463</td><td>-15.56475</td><td> 33.73368</td></tr>\n",
       "\t<tr><th scope=row>2</th><td>46.943703</td><td> 23.03510</td><td> 70.85231</td></tr>\n",
       "\t<tr><th scope=row>3</th><td>84.802942</td><td> 60.50438</td><td>109.10151</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 3 × 3 of type dbl\n",
       "\\begin{tabular}{r|lll}\n",
       "  & fit & lwr & upr\\\\\n",
       "\\hline\n",
       "\t1 &  9.084463 & -15.56475 &  33.73368\\\\\n",
       "\t2 & 46.943703 &  23.03510 &  70.85231\\\\\n",
       "\t3 & 84.802942 &  60.50438 & 109.10151\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 3 × 3 of type dbl\n",
       "\n",
       "| <!--/--> | fit | lwr | upr |\n",
       "|---|---|---|---|\n",
       "| 1 |  9.084463 | -15.56475 |  33.73368 |\n",
       "| 2 | 46.943703 |  23.03510 |  70.85231 |\n",
       "| 3 | 84.802942 |  60.50438 | 109.10151 |\n",
       "\n"
      ],
      "text/plain": [
       "  fit       lwr       upr      \n",
       "1  9.084463 -15.56475  33.73368\n",
       "2 46.943703  23.03510  70.85231\n",
       "3 84.802942  60.50438 109.10151"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# 区间估计\n",
    "preds.df <- data.frame(Test = seq(0, 20, by = 10))\n",
    "predict(examtest.fit, newdata = preds.df, interval = \"confidence\")\n",
    "# 点估计\n",
    "predict(examtest.fit, newdata = preds.df, interval = \"prediction\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "其中：\n",
    "\n",
    "- 区间估计表格的 [2,2:3] 表示所有半期考试10分，期末考试的分数的均值的范围\n",
    "- 区间估计表格的 [2,2:3] 表示所有半期考试10分个体的分数的范围，落在这个范围即为正常值"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 总结\n",
    "\n",
    "遇到此类问题，通用思路（适用于分析x和y两个未知数的某种关系）：\n",
    "\n",
    "- 绘制数据散点图并简要查看自变量与因变量之间是哪种关系（如果有关系），最好是能够通过工具分析（也可能会有一份研究意图的声明可以被指导）。提出适当的研究方式。在上边的例子中，我们就决定采用了线性模型：\n",
    "\n",
    "    $$\n",
    "    y = β_0 + β_1x_i + ε_i, ε_i ∼ N(0, σ^2) (where β_1 > 0)\n",
    "    $$\n",
    "\n",
    "- 使用 `lm` 函数进行模型拟合。\n",
    "- 检查我们提出的假设进行合适方式的验证。\n",
    "    - Independence OK? (how were the data collected?)\n",
    "    - EOV Okay? Using `plot(examtest.fit, which = 1)`.\n",
    "    - Normality Okay? Using `normcheck`.\n",
    "    \n",
    "    If these are okay then go to next step.\n",
    "- 尝试适时删除任何不重要的解释变量（后面会讲）。如果能删除，请检查新的研究方式。\n",
    "- 确保个别要点不会产生过分的不适当的影响，并尝试删除/纠正它们。Using `cooks20x`.\n",
    "- 做出结论/预测，讨论极限，并回答相关的研究问题。\n",
    "\n",
    "注意：在上述步骤中，在对当前步骤满意之前，切记不要匆忙进行下一步。"
   ]
  }
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