{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": [],
      "toc_visible": true
    },
    "kernelspec": {
      "name": "ir",
      "display_name": "R"
    },
    "language_info": {
      "name": "R"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# 第6回 推定と検定"
      ],
      "metadata": {
        "id": "obnzB1V21pax"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "[![open in colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/slt666666/biostatistics_text_wed/blob/main/source/_static/colab_notebook/chapter6.ipynb)\n",
        "\n",
        "※Web上ではテーブルや記号など一部LaTeXが反映されず見にくくなってしまっていますが、Google Colabだとちゃんと見えます。"
      ],
      "metadata": {
        "id": "k4J0fH6lGRX3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## はじめに\n",
        "\n",
        "ある母集団の特徴を知りたいときに、集団全体のデータからではなく、一部の標本データから母集団の特徴(母数など)を推測することが推測統計学の1つの目的でした。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter5/sampling2.png?raw=true\" alt=\"title\" height=\"250px\">\n",
        "\n",
        "今回はこの母集団の特徴を知る「推定」や「検定」を扱います。また、推定・検定を行うために重要となってくる標本分布についてもより詳しく見ていきます。"
      ],
      "metadata": {
        "id": "2bLFQcxy824O"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 正規分布からの標本\n",
        "\n",
        "生物学では多くの母集団の分布を正規分布とみなす場合が多いです。\n",
        "\n",
        "また、正規分布だと仮定した場合、理論や応用が非常にスムーズであることがよく知られており、多くの推定手法や検定手法が正規分布に基づいて構築されています。\n",
        "\n",
        "そこで、まず正規分布の母集団から抽出した標本の**標本分布**についてより詳しく見ていきます。"
      ],
      "metadata": {
        "id": "kGgTx5SRbfFN"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 正規分布の性質\n",
        "\n",
        "前回学習した通り、確率変数$X$が正規分布に従うとき、その密度関数は\n",
        "\n",
        "$f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}}e^{\\lbrace-\\frac{(x-\\mu)^2}{2\\sigma^2}\\rbrace}$\n",
        "\n",
        "で、その平均値(期待値)、分散は$E(X) = \\mu$,    $V(X) = \\sigma^2$となります。\n",
        "\n",
        "つまり、正規分布の形は平均$\\mu$と分散$\\sigma^2$の2つのパラメータ(**母数**)で決まります。\n",
        "\n",
        "正規分布は実用上非常に重要な分布で、統計分析を行う際には正規分布を仮定することが非常に多いです。\n",
        "\n",
        "その理由としては、\n",
        "\n",
        "* 草丈の分布や実験誤差の分布など、生物学において分析対象となる現象は正規分布で表せるものが多い。\n",
        "* 標本平均などの多くの統計量は、確率変数の和の形になっているものが多い。このような統計量は中心極限定理により正規分布として近似することができる。\n",
        "\n",
        "といった点が挙げられます。ほかにも、変数の変換や数学的な扱いが容易である点等、様々な扱いやすい特徴を持っています。\n"
      ],
      "metadata": {
        "id": "rt-AN8Y04-Kj"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "では続いて、正規分布している母集団から標本を抽出し、そこから得られる統計量の分布について見ていきます。\n",
        "\n",
        "標本から得られる統計量の情報は、母集団の推定や検定に必要な情報となります。\n",
        "\n",
        "### 分散が既知の時の標本平均の標本分布\n",
        "\n",
        "まず、標本平均はどのような分布に従うかを見てみます。\n",
        "\n",
        "標本平均$\\bar{X}$は、確率変数$X_1, X_2, ... X_n$の和を$n$で割った値になります。\n",
        "\n",
        "$X_1, X_2, ... X_n$は各々$N(\\mu, \\sigma^2)$に従うため、$\\bar{X}$の分布についても正規分布であり、\n",
        "\n",
        "平均は$E(\\bar{X}) = \\mu \\times n / n = \\mu$、\n",
        "\n",
        "分散は$V(\\bar{X}) = \\sigma^2 \\times n/n^2 = \\sigma^2/n$\n",
        "\n",
        "となります。\n",
        "\n",
        "つまり、$\\bar{X}$の分布は$N(\\mu, \\sigma^2/n)$に従います。\n",
        "\n",
        "これは$n$が増加するにしたがって$\\bar{X}$の分散が小さくなり、$\\bar{X}$が$\\mu$のより正確な推定値となることを示しています。\n",
        "\n",
        "$\\bar{X}$は単独の$X_i$よりも分散が小さいという意味で優れた測定値だと考えることができます。\n",
        "\n",
        "この結果は、**平均値の区間推定**を行う際に用いられます。\n",
        "\n",
        "<br>\n",
        "\n",
        "実際に$n$が増えるほど分散が小さくなる例として、\n",
        "\n",
        "$n$人が受ける試験の点数が平均$\\mu=50$、分散$\\sigma^2=10^2$の正規分布に従っているとすると、\n",
        "\n",
        "この試験を何度も繰り返したときに、受験者集が増えることで平均点の分布がどうなっていくのか見てみます。\n",
        "\n",
        "正規分布からランダム抽出できる`rnorm`関数で確認してみます。\n"
      ],
      "metadata": {
        "id": "pIZ204X9D1Z1"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 試験の平均点の分布をみる\n",
        "\n",
        "n = 10 # n人試験を受ける\n",
        "\n",
        "mean_scores <- c()\n",
        "for (i in 1:1000) {                  # 試験を1000回繰り返した場合のデータを作成\n",
        "  scores <- rnorm(n, mean=50, sd=10) # 平均50分散10^2の正規分布から標本nサンプリング\n",
        "  mean_scores <- c(mean_scores, c(mean(scores)))\n",
        "}\n",
        "mean_scores <- data.frame(x=mean_scores)\n",
        "\n",
        "# 平均点のグラフ化\n",
        "library(ggplot2)\n",
        "g <- ggplot(mean_scores, aes(x = x))\n",
        "g <- g + geom_histogram()\n",
        "g <- g + xlim(40, 60)\n",
        "g <- g + theme(text = element_text(size = 18))\n",
        "g"
      ],
      "metadata": {
        "id": "MNV6g7enq08m",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 549
        },
        "outputId": "c1b755af-38b5-4434-f30d-e6b8b795787e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "\u001b[1m\u001b[22m`stat_bin()` using `bins = 30`. Pick better value `binwidth`.\n",
            "Warning message:\n",
            "“\u001b[1m\u001b[22mRemoved 1 row containing non-finite outside the scale range (`stat_bin()`).”\n",
            "Warning message:\n",
            "“\u001b[1m\u001b[22mRemoved 2 rows containing missing values or values outside the scale range\n",
            "(`geom_bar()`).”\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "plot without title"
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAACx1BMVEUAAAABAQECAgIDAwME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\nSBCQiAQBiUjQ/wfkqP4ZfESbSgAAAABJRU5ErkJggg=="
          },
          "metadata": {
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "という形で、$n$の増加に伴って分散が小さくなることが分かります。\n",
        "\n",
        "<br>\n",
        "\n",
        "$\\bar{X}$の分布は$N(\\mu, \\sigma^2/n)$に従うことが分かったので、\n",
        "\n",
        "例えば分散が$\\sigma^2=5^2$である母集団において、平均値$\\mu=170$だと仮定したとします。\n",
        "\n",
        "そこで、10標本調査した結果、標本平均が$\\bar{X} = 175$だったとすると、$\\mu=170$という仮定がどのくらい正しいか考えてみます。\n",
        "\n",
        "標本平均は$N(170, 5^2/10)$に従うので、この正規分布における$175$がどの位置かグラフで見てみると\n"
      ],
      "metadata": {
        "id": "cIrV5kbhSM3H"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 175以下の領域を可視化\n",
        "library(ggplot2)\n",
        "\n",
        "x <- seq(160,180,0.1)\n",
        "y <- dnorm(x, 170, sqrt(25/10))\n",
        "data <- data.frame(x=x, y=y)\n",
        "\n",
        "x_color <- seq(160,175,0.1)\n",
        "y_color <- dnorm(x_color, 170, sqrt(25/10))\n",
        "data_color <- data.frame(x=x_color, y=y_color)\n",
        "\n",
        "g <- ggplot()\n",
        "g <- g + geom_path(data=data, aes(x=x, y=y))\n",
        "g <- g + geom_ribbon(data=data_color, aes(x=x, y=y, ymin=0, ymax=y))\n",
        "g"
      ],
      "metadata": {
        "id": "ehRfjwHQTUiu",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 437
        },
        "outputId": "ae018e59-e571-4080-9139-6e95b971bf47"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "plot without title"
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAADAFBMVEUAAAABAQECAgIDAwME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\nEpFAQCIS6P8DfR8nCNnvQKgAAAAASUVORK5CYII="
          },
          "metadata": {
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "また、`pnorm`関数で標本平均$\\bar{X} \\geqq 175$の確率を求めてみると"
      ],
      "metadata": {
        "id": "wP95yuOYUSMm"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 標本平均が175以上になる確率を求める\n",
        "1 - pnorm(175, mean=170, sd=sqrt(25/10))"
      ],
      "metadata": {
        "id": "O2PaOMiMUhWS",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "0b79a2a1-bd37-4a16-978a-2c8803cfff26"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "0.000782701129001251"
            ],
            "text/markdown": "0.000782701129001251",
            "text/latex": "0.000782701129001251",
            "text/plain": [
              "[1] 0.0007827011"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "ということで、もし$\\mu = 170$であったら、$\\bar{X} = 175$となるのは非常に低い確率だと分かります。\n",
        "\n",
        "よって、もともとの仮定であった$\\mu = 170$がおかしかったのでは？ということが分かったりします。"
      ],
      "metadata": {
        "id": "fqocLz0qUnaR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 標本分散の標本分布\n",
        "\n",
        "次に標本分散がどの様な分布に従うかを見てみます。\n",
        "\n",
        "標本分散は前回扱った通り、\n",
        "\n",
        "$s^2=\\dfrac{1}{n-1}\\lbrace(X_1-\\bar{X})^2+(X_2-\\bar{X})^2+...+(X_n-\\bar{X})^2\\rbrace$\n",
        "\n",
        "によって計算され、$E(s^2)=\\sigma^2$と、期待値が母集団の母分散と一致する不偏性を持ちます。\n",
        "\n",
        "### $\\chi^2$分布\n",
        "\n",
        "ここで、標本分散を扱う際に必ず関係してくる確率分布である$\\chi^2$**分布**を紹介します。\n",
        "\n",
        "標準正規分布$N(0,1)$に従う確率変数$Z_1, Z_2,...Z_k$があったとき、\n",
        "\n",
        "$\\chi^2 = Z_1^2+Z_2^2+...+Z_k^2$が従う確率分布を**自由度$k$の$\\chi^2$分布**と呼びます。\n",
        "\n",
        "自由度$k$の$\\chi^2$分布は$\\chi^2(k)$と表します。\n",
        "\n",
        "$\\chi^2$分布は自由度をパラメータとする分布で、正規分布に並び、分散の区間推定や適合度検定など、多くの統計的な問題に利用される分布になります。\n",
        "\n",
        "$\\chi^2$分布についてもＲで確率を計算を行うことができ、`dchisq`や`pchisq`関数などで計算できます。\n",
        "\n",
        "`dchisq`関数で$\\chi^2$分布の確率密度の計算が出来ます。\n",
        "\n",
        "```\n",
        "dchisq(x, df=自由度)\n",
        "```\n",
        "\n",
        "$\\chi^2$分布は自由度によって形が決まります。\n",
        "\n",
        "下のコードで自由度に応じてどんな形状になるか確認することができます。"
      ],
      "metadata": {
        "id": "F28bLpoOIvcz"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# カイ2乗分布の描写\n",
        "\n",
        "df <- 3 #自由度\n",
        "\n",
        "library(ggplot2)\n",
        "x <- seq(0,20,0.01)             # x = 0 ~ 10の範囲を描写\n",
        "m <- dchisq(x, df=df)           # 自由度dfのカイ二乗分布\n",
        "data <- data.frame(x=x,y=m)\n",
        "g <- ggplot(data,aes(x=x, y=y))\n",
        "g <- g + geom_line()\n",
        "g"
      ],
      "metadata": {
        "id": "1r9yXdUee-rY",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 437
        },
        "outputId": "09acff52-e16e-415a-c93c-183cf850fd75"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "plot without title"
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAADAFBMVEUAAAABAQECAgIDAwME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"
          },
          "metadata": {
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "ここで、標本分散$s^2$について、\n",
        "\n",
        "$s^2=\\dfrac{1}{n-1}\\lbrace(X_1-\\bar{X})^2+(X_2-\\bar{X})^2+...+(X_n-\\bar{X})^2\\rbrace$\n",
        "\n",
        "とすると、\n",
        "\n",
        "$(n-1)s^2/\\sigma^2$は自由度$n-1$の$\\chi^2$分布$\\chi^2(n-1)$に従う、\n",
        "\n",
        "と$\\chi^2$分布を用いて、標本分散$s^2$の標本分布を表す事が出来ます。\n",
        "\n",
        "<br>\n",
        "\n",
        "そのため、この$\\chi^2$分布を用いることで、標本分散の変動具合などを考えることができます。\n",
        "\n",
        "例えば、ある植物の草丈が母平均$\\mu=50$ (cm)、母分散$\\sigma^2=5^2$だとします。\n",
        "\n",
        "この植物を10標本採集した際に、標本分散$s^2$が50を超える確率はどのくらいか考えてみます。\n",
        "\n",
        "$s^2 > 50$となる確率$P(s^2 > 50)$は、先ほどの$\\chi^2=(n-1)s^2/\\sigma^2$に変形すると、\n",
        "\n",
        "$P(s^2 > 50)$\n",
        "\n",
        "$= P(\\chi^2>(10-1)\\times50/25)$\n",
        "\n",
        "$=P(\\chi^2>18) (自由度=9)$\n",
        "\n",
        "となります。\n",
        "\n",
        "自由度$n$の$\\chi^2$分布で$\\chi^2$値が$x$より大きくなる確率は`pchisq`関数を用いて計算します。\n",
        "\n",
        "`pchisq`関数では$\\chi^2$分布に従う確率変数がある値以下になる確率を計算することができるので、\n",
        "\n",
        "```\n",
        "pchisq(x, df=自由度)\n",
        "```\n",
        "\n",
        "$P(\\chi^2>18) (自由度=9)$は"
      ],
      "metadata": {
        "id": "WwoEGMSfeyKC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度9のカイ二乗分布で、\"18より大きくなる\"確率を求める\n",
        "1 - pchisq(18, df=9)"
      ],
      "metadata": {
        "id": "wUAkRFVUdjX2",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "ea073710-5966-4e5b-dcc5-5fa0846c74f6"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "0.0351735394669848"
            ],
            "text/markdown": "0.0351735394669848",
            "text/latex": "0.0351735394669848",
            "text/plain": [
              "[1] 0.03517354"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "ということで、母分散$s^2=25$の2倍を超える様な分散が標本分散に現れる確率は、3.5%くらいだと考えられることが分かります。"
      ],
      "metadata": {
        "id": "p7dP0Wlbd2Fa"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "#### 自由度の簡単な説明\n",
        "\n",
        "$(n-1)s^2/\\sigma^2$は自由度$n-1$の$\\chi^2$分布$\\chi^2(n-1)$に従うということでしたが、\n",
        "\n",
        "この理由を簡単に説明すると、\n",
        "\n",
        "標本分散の式を変形すると、\n",
        "\n",
        "$(n-1)s^2 = (X_1-\\bar{X})^2+(X_2-\\bar{X})^2+...+(X_n-\\bar{X})^2$\n",
        "\n",
        "となり、この式は、\n",
        "\n",
        "$\\dfrac{(n-1)s^2}{\\sigma^2} = (\\dfrac{X_1-\\bar{X}}{\\sigma})^2+(\\dfrac{X_2-\\bar{X}}{\\sigma})^2+...+(\\dfrac{X_n-\\bar{X}}{\\sigma})^2$\n",
        "\n",
        "と形を整理することができます。\n",
        "\n",
        "$\\chi^2$分布の定義は、標準正規分布$N(0,1)$に従う確率変数$Z_1, Z_2,...Z_k$があったとき、\n",
        "\n",
        "$\\chi^2 = Z_1^2+Z_2^2+...+Z_k^2$が従う確率分布を**自由度$k$の$\\chi^2$分布**と呼ぶ、ということでした。\n",
        "\n",
        "$Z_i = \\dfrac{X_i-\\mu}{\\sigma}$と$\\dfrac{X_i-\\bar{X}}{\\sigma}$を比べると、$\\mu$と$\\bar{X}$以外同じ形です。\n",
        "\n",
        "ただし、$\\bar{X}$の場合、$X_1, X_2, ... X_{n-1}$まで値が決まれば、標本平均$\\bar{X}$から最後の$X_n$の値が決まってしまうことになります。\n",
        "\n",
        "言い換えると、**自由に動ける変数の個数**が$n-1$個ということで、自由度が$n-1$となります。\n",
        "\n"
      ],
      "metadata": {
        "id": "4b5TD7jEbxN0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 分散が未知の時の標本平均の標本分布\n",
        "\n",
        "分散が既知の場合の標本平均$\\bar{X}$の分布については先ほど考えました。\n",
        "\n",
        "しかし、母分散$\\sigma^2$が既知の場合というのはあまり現実的ではありません。\n",
        "\n",
        "さきほど見た通り、標本平均$\\bar{X}$の標本分布は正規分布$N(\\mu, \\sigma^2/n)$に従います。\n",
        "\n",
        "標本平均$\\bar{X}$を標準化した$Z=\\dfrac{\\bar{X}-\\mu}{\\sqrt{\\sigma^2/n}}$は標準正規分布$N(0,1)$に従いますが、\n",
        "\n",
        "$\\sigma^2$が分からないので計算できません。\n",
        "\n",
        "そこで、母分散$\\sigma^2$の代わりに標本分散$s^2$で代用した値を考えます。\n",
        "\n",
        "$\\sigma^2$を$s^2$で代用すると、$t=\\dfrac{\\bar{X}-\\mu}{\\sqrt{s^2/n}}$となります。\n",
        "\n",
        "この値はスチューデントの**t統計量**と定義されます。\n",
        "\n",
        "当然、式が変わってしまったのでこの値は$N(0,1)$には従いません。\n",
        "\n",
        "そこで、$t$がどのような確率分布に従うか考えてみます。\n",
        "\n",
        "$t$を式変形して、\n",
        "\n",
        "$t=\\dfrac{\\bar{X}-\\mu}{\\sqrt{\\sigma^2/n}}/\\sqrt{\\dfrac{s^2}{\\sigma^2}}$\n",
        "\n",
        "とおき、\n",
        "\n",
        "さらに分母を変形すると、\n",
        "\n",
        "$t=\\dfrac{\\bar{X}-\\mu}{\\sqrt{\\sigma^2/n}}/\\sqrt{\\dfrac{(n-1)s^2}{\\sigma^2}/(n-1)}$\n",
        "\n",
        "となります。式中の、\n",
        "\n",
        "$\\dfrac{\\bar{X}-\\mu}{\\sqrt{\\sigma^2/n}}$は標準正規分布$N(0,1)$に従い、\n",
        "\n",
        "$\\dfrac{(n-1)s^2}{\\sigma^2}$は自由度$n-1$の$\\chi^2$分布$\\chi^2(n-1)$に従うので、\n",
        "\n",
        "$t$は$N(0,1)$と$\\chi^2(n-1)$が組み合わさって決まることが分かります。\n",
        "\n",
        "この様な分布を**t分布**と呼びます。定義としては、\n",
        "\n",
        "1. 確率変数$Z$が標準正規分布に従う。$Z \\sim N(0,1)$\n",
        "2. 確率変数$Y$が自由度$k$の$\\chi^2$分布に従う。$Y \\sim \\chi^2(k)$\n",
        "3. $Y$と$Z$は独立\n",
        "\n",
        "この時、$t=\\dfrac{Z}{\\sqrt{Y/k}}$と定義される$t$が従う確率分布を**自由度$k$のt分布**と呼びます。この分布は$t(k)$で表します。\n",
        "\n",
        "今回最初に扱った$t=\\dfrac{\\bar{X}-\\mu}{\\sqrt{s^2/n}}$は、自由度$n-1$の$t$分布$t(n-1)$に従うことになります。\n",
        "\n",
        "$t$分布は分散が未知の場合に、標準正規分布の代用品として扱うことになりますが、自由度$k$が大きい場合は、標準正規分布とほとんど変わらなくなるという特徴を持っています。\n",
        "\n",
        "$t$統計量は、平均値の比較を行うt検定や平均値の区間推定に用いられる重要な統計量なので、度々登場します。\n",
        "\n",
        "当然ですが、$t$分布に関しても、Rで計算をすることが可能です。\n",
        "\n",
        "`dt`や`pt`関数などで計算できます。\n",
        "\n",
        "`dt`関数で$t$分布の確率密度の計算が出来ます。\n",
        "\n",
        "```\n",
        "dt(x, df=自由度)\n",
        "```\n",
        "\n",
        "$t$分布も自由度によって形が変わるので、\n",
        "\n",
        "下のコードで自由度に応じて$t$分布がどんな形状になるかと、標準正規分布との類似性の確認することができます。\n"
      ],
      "metadata": {
        "id": "twBL_Sb6gGJs"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度dfのt分布+標準正規分布のグラフを描写\n",
        "\n",
        "df <- 3 #自由度\n",
        "\n",
        "library(ggplot2)\n",
        "x <- seq(-5,5,0.01)            # x = 0 ~ 10の範囲を描写\n",
        "m1 <- dt(x, df=df)             # 自由度dfのt分布\n",
        "m2 <- dnorm(x, mean=0, sd=1)   # 標準正規分布\n",
        "data <- data.frame(x=x,y1=m1, y2=m2)\n",
        "g <- ggplot(data)\n",
        "g <- g + geom_line(aes(x=x, y=y1), colour=\"red\")   # t分布を赤線で\n",
        "g <- g + geom_line(aes(x=x, y=y2), colour=\"black\") # 標準正規分布を黒線で\n",
        "g"
      ],
      "metadata": {
        "id": "ZLbDtiSWXYl9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 437
        },
        "outputId": "bbcbabb5-654e-49b9-eb67-fbcc11db92c4"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "plot without title"
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAIAAAByhViMAAAACXBIWXMAABJ0AAASdAHeZh94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\nBwZO+CtCmAWcAAAAAElFTkSuQmCC"
          },
          "metadata": {
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "分散が既知であれば、先ほどの様に、$\\mu=170$は妥当かどうか？の様な評価が出来ましたが、\n",
        "\n",
        "分散が未知の場合は、この$t$統計量を用いて母平均の推定などを行います。\n",
        "\n",
        "$t$統計量を用いた母平均の推定や検定はこの後扱っていきます。"
      ],
      "metadata": {
        "id": "3QpdRowbX8lC"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 推定\n",
        "\n",
        "それではこれまでに得た知識を踏まえて、実際に母集団の特徴を調べる方法を学んでいきます。\n",
        "\n",
        "**推定**とは、標本から得られたデータを基に、標本が抽出された母集団の分布の母数である、母平均や母分散などの値を求めることを指します。\n",
        "\n",
        "例えば、多くの生物の特徴(草丈や種子数など)が正規分布$N(\\mu, \\sigma^2)$に従うことが分かっていても、その平均$\\mu$や分散$\\sigma^2$が分からなければ分布の形は分かりません。\n",
        "\n",
        "また、珍しい現象がポアソン分布$Po(\\lambda)$に従うことが多いと分かっていても、ポアソン分布は$\\lambda$によって分布の形は大きく変わってしまうので、$\\lambda$をなんとかして知る必要があります。\n",
        "\n",
        "そして多くの場合、$\\mu,\\sigma^2,\\lambda$といった母数は未知なので、これらを標本$X_1, X_2, ..., X_n$から定める必要があります。\n",
        "\n",
        "これを母数の**推定**と言います。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/estimate.png?raw=true\" alt=\"title\" height=\"250px\">\n",
        "\n",
        "母数を推定するために標本から得られた統計量(標本平均など)を一般に**推定量**と呼び、\n",
        "\n",
        "推定しようとするパラメータ$\\theta$の推定量は`^`をつけて$\\hat{\\theta}$と表すことが多いです。"
      ],
      "metadata": {
        "id": "6cdNfhMsnyff"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 点推定と区間推定\n",
        "\n",
        "推定は点推定と区間推定の2つに大きく分類されます。\n",
        "\n",
        "台風の進路はその分かりやすい例になります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/typhoon.png?raw=true\" alt=\"title\" height=\"250px\">\n",
        "\n",
        "<small>https://newsdig.tbs.co.jp/articles/-/1357569より</small>\n",
        "\n",
        "母平均$\\mu$を標本平均$\\bar{X}$で推定する場合の様に、母数$\\theta$をある一つの値$\\hat{\\theta}$で指定する方法を**点推定**と呼びます。\n",
        "\n",
        "点推定では、不偏推定量などの考え方が登場します。\n",
        "\n",
        "また、標本$X_1, X_2, ..., X_n$は確率変数なので、標本から推定された$\\hat{\\theta}$は当然実際の$\\theta$からは多少なりともズレる(誤差がある)ことになります。\n",
        "\n",
        "そこで、どのくらい誤差があるかを推定量の標本分布をもとに考える必要があります。\n",
        "\n",
        "この時用いられる**区間推定**は、一つの値を求めるのではなく、母数がある確率以上で含まれることが保証される区間を推定するもので、ある程度誤差を含んだ値の範囲を推定する手法になります。\n",
        "\n",
        "例えば$P(A \\leqq \\mu \\leqq B) \\geqq 95\\%$の様に、$\\mu$が95%の確率で入ることが保証される区間がA~Bの範囲である、という形で区間を推定します。"
      ],
      "metadata": {
        "id": "M01CE232FTfI"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 点推定\n",
        "\n",
        "点推定はこれまでにも何度か触れてきました。\n",
        "\n",
        "例えば、標本平均$\\bar{X}=(X_1+X_2+...+X_n)/n$は母平均$\\mu$の推定量と考えられます。\n",
        "\n",
        "この推定量を実際の観測値を基に計算した値を**推定値**と呼びます。\n",
        "\n",
        "観測データが異なれば推定値も異なるので、推定値は推定量がとりうる値の一つが実現したものと考えられます。\n",
        "\n",
        "実用上意味のある推定量は真の母数$\\theta$付近にあることを示す基準を満たしていることが理想とされ、\n",
        "\n",
        "この理想を満たす推定量かどうかという点に関するいくつかの指標があります。\n",
        "\n",
        "**不偏性**と**一致性**はその代表的な指標になります。\n",
        "\n",
        "**不偏性**\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/huhen.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "$\\hat{\\theta}$の分布は$\\theta$の周辺に集まっている必要があり、その基準の一つが不偏性になります。\n",
        "\n",
        "不偏性の定義としては、推定量の期待値が真の母数の値と一致する性質を指します。\n",
        "\n",
        "$E(\\hat{\\theta})=\\theta$\n",
        "\n",
        "この性質を満たす推定量を**不偏推定量**と呼びます。\n",
        "\n",
        "以前扱った通り、\n",
        "\n",
        "標本平均$\\bar{X}=(X_1+X_2+...+X_n)/n$や、\n",
        "\n",
        "標本分散$s^2=\\dfrac{1}{n-1}\\lbrace(X_1-\\bar{X})^2+(X_2-\\bar{X})^2+...+(X_n-\\bar{X})^2\\rbrace$は、\n",
        "\n",
        "$E(\\bar{X}) = \\mu$,    $E(s^2) = \\sigma^2$となるので、いずれも不偏推定量と言えます。\n",
        "\n",
        "**一致性**\n",
        "\n",
        "一致性は、標本の大きさ$n$が大きくなるにつれ、推定量$\\hat{\\theta}$が真の母数の値$\\theta$に近付いていく性質を指します。\n",
        "\n",
        "この様な性質を持つ推定量を**一致推定量**と呼び、標本平均$\\bar{X}$は母平均$\\mu$の一致推定量、標本分散$s^2$は母分散$\\sigma^2$の一致推定量となります。\n",
        "\n",
        "標本平均は先ほど見た通り、一致性を示す推定量になります。\n",
        "\n"
      ],
      "metadata": {
        "id": "cyj38eBaOZhK"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 試験の平均点の分布をみる\n",
        "\n",
        "n = 100 # n人試験を受ける\n",
        "\n",
        "mean_scores <- c()\n",
        "for (i in 1:1000) {                  # 試験を1000回繰り返した場合のデータを作成\n",
        "  scores <- rnorm(n, mean=50, sd=10) # 平均50分散10^2の正規分布から標本nサンプリング\n",
        "  mean_scores <- c(mean_scores, c(mean(scores)))\n",
        "}\n",
        "mean_scores <- data.frame(x=mean_scores)\n",
        "\n",
        "# 平均点のグラフ化\n",
        "library(ggplot2)\n",
        "g <- ggplot(mean_scores, aes(x = x))\n",
        "g <- g + geom_histogram()\n",
        "g <- g + xlim(40, 60)\n",
        "g <- g + theme(text = element_text(size = 18))\n",
        "g"
      ],
      "metadata": {
        "id": "bqj770hcAH5N",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 512
        },
        "outputId": "53688d40-a283-4ab3-b0a5-b9a2ced98194"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "\u001b[1m\u001b[22m`stat_bin()` using `bins = 30`. Pick better value `binwidth`.\n",
            "Warning message:\n",
            "“\u001b[1m\u001b[22mRemoved 2 rows containing missing values or values outside the scale range\n",
            "(`geom_bar()`).”\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "plot without title"
            ],
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAACwVBMVEUAAAABAQECAgIDAwME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\nQmCC"
          },
          "metadata": {
            "image/png": {
              "width": 420,
              "height": 420
            }
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "不偏性と一致性は、推定量が最低限満たす必要のある性質になります。\n",
        "\n",
        "今回はこの性質を満たす統計量として標本平均、標本分散を例として取り上げましたが、\n",
        "\n",
        "性質を満たす推定量を求める方法として、モーメント法や最尤法などがあり、これらはまた後半の講義で扱う予定です。"
      ],
      "metadata": {
        "id": "fjdv-HMfAMbc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 区間推定\n",
        "\n",
        "点推定では$\\theta$をある一つの値として推定していましたが、\n",
        "\n",
        "区間推定では$\\theta$に対し、真の母数の値$\\theta$がある区間$[L, U]$に入ることが保証される確率が$1-\\alpha$以上になる、という考え方をします。\n",
        "\n",
        "※真の母数の値$\\theta$がある区間$[L, U]$に$1-\\alpha$の確率で入る…とは違うのですが、この違いの説明は後程。\n",
        "\n",
        "具体的には、$P(L \\leqq \\theta \\leqq U) \\geqq 1-\\alpha$　となる確率変数$L, U$を求めます。\n",
        "\n",
        "$L$は下限信頼限界、$U$は上限信頼限界、区間$[L, U]$を$100(1-\\alpha)\\%$**信頼区間**と呼びます。\n",
        "\n",
        "$1-\\alpha$としては、$95\\%$信頼区間や$99\\%$信頼区間がよく用いられます。\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "k5jsHOALW5Di"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 母平均の区間推定\n",
        "\n",
        "まずは母平均の区間推定を行っていきます。"
      ],
      "metadata": {
        "id": "R8B-xf3XHClb"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##### 分散が既知の場合\n",
        "\n",
        "母平均$\\mu$、母分散$\\sigma^2$の正規分布$N(\\mu, \\sigma^2)$に従う母集団から抽出した標本の標本平均$\\bar{X}$は平均$\\mu$、分散$\\sigma^2/n$の正規分布に従いました。\n",
        "\n",
        "$\\bar{X} \\sim N(\\mu, \\sigma^2/n)$\n",
        "\n",
        "これを標準化すると、標準正規分布に従うようになります。\n",
        "\n",
        "$Z=\\dfrac{\\sqrt{n}(\\bar{X}-\\mu)}{\\sigma} \\sim N(0,1)$\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/trust_norm.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "図のように、この$Z$は確率$1-\\alpha$で区間$[-Z_{\\alpha/2}, Z_{\\alpha/2}]$に含まれる形になります。\n",
        "\n",
        "つまり、\n",
        "\n",
        "$P(-Z_{\\alpha/2} \\leqq \\dfrac{\\sqrt{n}(\\bar{X}-\\mu)}{\\sigma} \\leqq Z_{\\alpha/2})=1-\\alpha$\n",
        "\n",
        "となる。この式を母平均$\\mu$について解くと、\n",
        "\n",
        "$P(\\bar{X}-Z_{\\alpha/2}\\dfrac{\\sigma}{\\sqrt{n}} \\leqq \\mu \\leqq \\bar{X}+Z_{\\alpha/2}\\dfrac{\\sigma}{\\sqrt{n}})$\n",
        "\n",
        "となる。よって、$\\mu$の$100(1-\\alpha)\\%$信頼区間は、\n",
        "\n",
        "$[\\bar{X}-Z_{\\alpha/2}\\dfrac{\\sigma}{\\sqrt{n}}, \\bar{X}+Z_{\\alpha/2}\\dfrac{\\sigma}{\\sqrt{n}}]$\n",
        "\n",
        "と得られます。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/trust_mu.png?raw=true\" alt=\"title\" height=\"200px\">\n",
        "\n",
        "Rの`qnorm`関数を使用することで、今回の$Z_{\\alpha/2}$の様な標準正規分布において確率が$1-\\alpha/2$となる位置にある$Z$値を計算できました。\n",
        "\n",
        "```\n",
        "qnorm(確率, mean=平均, sd=標準偏差)\n",
        "```\n",
        "\n",
        "$\\alpha=0.05$の時の$Z_{\\alpha/2}$や$-Z_{\\alpha/2}$を求めてみると"
      ],
      "metadata": {
        "id": "lAW9Smwvg4A9"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# α=0.05におけるZ_α/2を求める\n",
        "qnorm(0.975, mean=0, sd=1)"
      ],
      "metadata": {
        "id": "8u7Lv2HhlFBH",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "3c1b5c23-702f-4f21-df75-691cf239fdfa"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "1.95996398454005"
            ],
            "text/markdown": "1.95996398454005",
            "text/latex": "1.95996398454005",
            "text/plain": [
              "[1] 1.959964"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "と、Rを使うことで$Z_{\\alpha/2}$や$-Z_{\\alpha/2}$を求める事が出来ます。"
      ],
      "metadata": {
        "id": "eU74eH0hFY3R"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "実際に信頼区間を計算をしてみます。\n",
        "\n",
        "例えば、母分散$\\sigma^2=9$の集団から20個体採集して得られた集団の値が、\n",
        "\n",
        "99, 111, 92, 109, 92, 101, 108, 96, 84, 109, 99, 99, 111, 86, 96, 116, 115, 92, 85, 100\n",
        "\n",
        "だったとします。この結果から、母平均の95%信頼区間を計算してみましょう。\n",
        "\n",
        "信頼区間は下の式で計算できるので、先ほど得られた$Z_{\\alpha/2}$や$-Z_{\\alpha/2}$を使用して計算してみましょう。\n",
        "\n",
        "$[\\bar{X}-Z_{\\alpha/2}\\dfrac{\\sigma}{\\sqrt{n}}, \\bar{X}+Z_{\\alpha/2}\\dfrac{\\sigma}{\\sqrt{n}}]$"
      ],
      "metadata": {
        "id": "qImZKxSNnpE0"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 得られたデータの母平均の95%信頼区間を求める\n",
        "data <- c(99, 111, 92, 109, 92, 101, 108, 96, 84, 109, 99, 99, 111, 86, 96, 116, 115, 92, 85, 100)\n",
        "mean(data) - qnorm(0.975, mean=0, sd=1)*3/sqrt(20)\n",
        "mean(data) + qnorm(0.975, mean=0, sd=1)*3/sqrt(20)"
      ],
      "metadata": {
        "id": "AMGs1K1olE9v",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "outputId": "6d30d9ea-e250-495c-dd7a-6eed57a4a64b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "98.6852161891351"
            ],
            "text/markdown": "98.6852161891351",
            "text/latex": "98.6852161891351",
            "text/plain": [
              "[1] 98.68522"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "101.314783810865"
            ],
            "text/markdown": "101.314783810865",
            "text/latex": "101.314783810865",
            "text/plain": [
              "[1] 101.3148"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "よって今回のデータから母平均の$95\\%$信頼区間は$[98.68521..., 101.31478...]$と計算できました。"
      ],
      "metadata": {
        "id": "XoIToZJsGv-W"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 信頼区間について\n",
        "\n",
        "少しややこしいですが、この信頼区間の$95\\%$の意味は、母平均が$95\\%$の確率でこの範囲にある、ということではありません。\n",
        "\n",
        "厳密には、$n=20$の標本抽出を100回繰り返すと、100回標本平均が計算され、その標本平均を基に100個の信頼区間が得られます。\n",
        "\n",
        "この100個の信頼区間のうち、母平均$\\mu$が含まれる区間が$95\\%$ある、という意味を表しています。"
      ],
      "metadata": {
        "id": "y11crcY5oNAu"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##### 分散が未知の場合\n",
        "\n",
        "さて、ここまでの説明では、母分散$\\sigma^2$が既知のものとして計算を進めてきました。\n",
        "\n",
        "$P(-Z_{\\alpha/2} \\leqq \\dfrac{\\sqrt{n}(\\bar{X}-\\mu)}{\\sigma} \\leqq Z_{\\alpha/2})=1-\\alpha$\n",
        "\n",
        "しかし、現実には母分散$\\sigma^2$が既知な状況というのはほとんどないので、上の式には$\\sigma$が含まれているので計算できない値になります。\n",
        "\n",
        "この場合は、標本分散$s^2$を代わりに用います。\n",
        "\n",
        "先ほどの式の$\\sigma$を$s$に置き換えると、\n",
        "\n",
        "$\\dfrac{\\sqrt{n}(\\bar{X}-\\mu)}{s}$は標準正規分布$N(0,1)$には従わず、自由度$n-1$の$t$分布$t(n-1)$に従うので、\n",
        "\n",
        "$t_{\\alpha/2}$の様な、$t$分布において確率が$1-\\alpha/2$となる位置にある$t$値を用いると、\n",
        "\n",
        "$P(-t_{\\alpha/2}(n-1) \\leqq \\dfrac{\\sqrt{n}(\\bar{X}-\\mu)}{s} \\leqq t_{\\alpha/2}(n-1))=1-\\alpha$\n",
        "\n",
        "となる。\n",
        "\n",
        "この式を母平均$\\mu$について解くと、\n",
        "\n",
        "$P(\\bar{X}-t_{\\alpha/2}(n-1)\\times\\dfrac{s}{\\sqrt{n}} \\leqq \\mu \\leqq \\bar{X}+t_{\\alpha/2}(n-1)\\times\\dfrac{s}{\\sqrt{n}})$\n",
        "\n",
        "となる。よって、$\\mu$の$100(1-\\alpha)\\%$信頼区間は、\n",
        "\n",
        "$[\\bar{X}-t_{\\alpha/2}(n-1)\\times\\dfrac{s}{\\sqrt{n}}, \\bar{X}+t_{\\alpha/2}(n-1)\\times\\dfrac{s}{\\sqrt{n}}]$\n",
        "\n",
        "として得ることができます。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/trust_t.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "こちらも先ほどと同じように、$t$分布において確率が$1-\\alpha/2$となる位置にある$t$値はRで計算できます。\n",
        "\n",
        "Rの`qt`関数を使用することで計算できました。\n",
        "\n",
        "```\n",
        "qt(確率, 自由度)\n",
        "```\n",
        "\n",
        "$\\alpha=0.05$の時の自由度$19$の$t_{\\alpha/2}$を求めてみると\n"
      ],
      "metadata": {
        "id": "TbFzawPEG4RA"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# α=0.05の時のt_α/2を求める\n",
        "qt(0.975, 19)"
      ],
      "metadata": {
        "id": "51bxhZ5ckhtw",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "0c4ec447-7f9c-4c09-e363-f107e8788527"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "2.09302405440831"
            ],
            "text/markdown": "2.09302405440831",
            "text/latex": "2.09302405440831",
            "text/plain": [
              "[1] 2.093024"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "先ほどと同じデータから、分散が未知の場合の母平均の95%信頼区間を計算してみましょう。\n",
        "\n",
        "$[\\bar{X}-t_{\\alpha/2}(n-1)\\times\\dfrac{s}{\\sqrt{n}}, \\bar{X}+t_{\\alpha/2}(n-1)\\times\\dfrac{s}{\\sqrt{n}}]$"
      ],
      "metadata": {
        "id": "dD-8cxAgsejb"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "data <- c(99, 111, 92, 109, 92, 101, 108, 96, 84, 109, 99, 99, 111, 86, 96, 116, 115, 92, 85, 100)\n",
        "mean(data) - qt(0.975, 19)*sd(data)/sqrt(20)\n",
        "mean(data) + qt(0.975, 19)*sd(data)/sqrt(20)"
      ],
      "metadata": {
        "id": "hQ7XrFn0lDPM",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "outputId": "96b13b61-05b7-4587-ac13-f343657663f5"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "95.3768574342647"
            ],
            "text/markdown": "95.3768574342647",
            "text/latex": "95.3768574342647",
            "text/plain": [
              "[1] 95.37686"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "104.623142565735"
            ],
            "text/markdown": "104.623142565735",
            "text/latex": "104.623142565735",
            "text/plain": [
              "[1] 104.6231"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "よって今回のデータから母平均の$95\\%$信頼区間は$[95.37685..., 104.62314...]$と計算できました。"
      ],
      "metadata": {
        "id": "cj0Xrrg9JtlD"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 母分散の区間推定\n",
        "\n",
        "次に、母分散の区間推定をしてみましょう。\n",
        "\n",
        "正規分布$N(\\mu, \\sigma^2)$に従う母集団から得た標本サイズ$n$の標本から得られた標本分散$s^2$は、\n",
        "\n",
        "少し前に学んだ通り、$(n-1)s^2/\\sigma^2$が自由度$n-1$の$\\chi^2$分布$\\chi^2(n-1)$に従うということを利用します。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/trust_chi.png?raw=true\" alt=\"title\" height=\"200px\">\n",
        "\n",
        "先ほどと同じように、上図の自由度$n-1$の$\\chi^2$分布の上側$1-\\alpha/2$点と下側$\\alpha/2$点に挟まれた部分が$1-\\alpha$となるので、\n",
        "\n",
        "$(n-1)s^2/\\sigma^2$は確率$1-\\alpha$で区間$[\\chi^2_{1-\\alpha/2}(n-1), \\chi^2_{\\alpha/2}(n-1)]$に含まれる形になります。\n",
        "\n",
        "つまり、\n",
        "\n",
        "$P(\\chi^2_{1-\\alpha/2}(n-1) \\leqq (n-1)s^2/\\sigma^2 \\leqq \\chi^2_{\\alpha/2}(n-1)) = 1-\\alpha$\n",
        "\n",
        "となります。\n",
        "\n",
        "これを$\\sigma^2$について解くと、\n",
        "\n",
        "$P(\\dfrac{(n-1)s^2}{\\chi^2_{\\alpha/2}(n-1)} \\leqq \\sigma^2 \\leqq \\dfrac{(n-1)s^2}{\\chi^2_{1-\\alpha/2}(n-1)}) = 1-\\alpha$\n",
        "\n",
        "となるので、母分散$\\sigma^2$の$100(1-\\alpha)\\%$信頼区間は、\n",
        "\n",
        "$[\\dfrac{(n-1)s^2}{\\chi^2_{\\alpha/2}(n-1)}, \\dfrac{(n-1)s^2}{\\chi^2_{1-\\alpha/2}(n-1)}]$\n",
        "\n",
        "で計算できます。\n",
        "\n",
        "$\\chi^2$分布に関しても、上側$1-\\alpha/2$点と下側$\\alpha/2$点はRの`qchisq`関数で計算することができました。\n",
        "\n",
        "```\n",
        "qchisq(確率, 自由度)\n",
        "```\n",
        "\n",
        "$\\alpha=0.05$の時の、自由度$19$の$\\chi^2$分布における$\\chi^2_{1-\\alpha/2}$と$\\chi^2_{\\alpha/2}$を求めてみると\n"
      ],
      "metadata": {
        "id": "ukzmEIuBtlvS"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# α=0.05, df=19の時の1-α/2点とα/2点を求める\n",
        "qchisq(0.975, 19)\n",
        "qchisq(0.025, 19)"
      ],
      "metadata": {
        "id": "WMZdfOb1tg8F",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "outputId": "9079068e-7aac-4f00-a3a9-adf36c6e6862"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "32.8523268617297"
            ],
            "text/markdown": "32.8523268617297",
            "text/latex": "32.8523268617297",
            "text/plain": [
              "[1] 32.85233"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "8.90651648198797"
            ],
            "text/markdown": "8.90651648198797",
            "text/latex": "8.90651648198797",
            "text/plain": [
              "[1] 8.906516"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "得られた$\\chi^2_{1-\\alpha/2}$と$\\chi^2_{\\alpha/2}$を用いて、\n",
        "\n",
        "先ほどとのデータから、母分散の$95\\%$信頼区間を計算してみましょう。\n",
        "\n",
        "$[\\dfrac{(n-1)s^2}{\\chi^2_{\\alpha/2}(n-1)}, \\dfrac{(n-1)s^2}{\\chi^2_{1-\\alpha/2}(n-1)}]$"
      ],
      "metadata": {
        "id": "DcoIlq9VxqB9"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 母分散の95%信頼区間を求める。\n",
        "data <- c(99, 111, 92, 109, 92, 101, 108, 96, 84, 109, 99, 99, 111, 86, 96, 116, 115, 92, 85, 100)\n",
        "(19 * sd(data)^2) / qchisq(0.975, 19)\n",
        "(19 * sd(data)^2) / qchisq(0.025, 19)"
      ],
      "metadata": {
        "id": "NRsPbYnmxaSA",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "outputId": "40c2c4c1-82a6-4ab2-8b41-8f6fd6c14bc1"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "56.4343587534361"
            ],
            "text/markdown": "56.4343587534361",
            "text/latex": "56.4343587534361",
            "text/plain": [
              "[1] 56.43436"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "208.1621926765"
            ],
            "text/markdown": "208.1621926765",
            "text/latex": "208.1621926765",
            "text/plain": [
              "[1] 208.1622"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "よって、母分散の$95\\%$信頼区間は、$[56.43435..., 208.16219]$と計算できました。\n",
        "\n",
        "この様な形で、標本分布の情報から母平均や母分散の信頼区間を計算することができます。"
      ],
      "metadata": {
        "id": "gbsnGVLVyXCj"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 検定\n",
        "\n",
        "続いて、検定を扱います。\n",
        "\n",
        "検定とは、母集団に対してある仮説をたて、その仮説の下で期待される結果と、実際に観測した結果の違いが、偶然によるものなのかどうかを、確率の基準で評価することになります。\n",
        "\n",
        "まずは検定の考え方である「有意性検定」の考え方に触れ、その後、対立仮説という考えに基づく「仮説検定」を扱います。"
      ],
      "metadata": {
        "id": "SFnkRxdBydn0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 有意性検定\n",
        "\n",
        "検定の目的は、母集団についてたてられた仮説を、得られた標本に基づいて検証することにあります。\n",
        "\n",
        "よく用いられる例としてメンデル遺伝のエンドウ豆のケースを考えてみましょう。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/mendel.png?raw=true\" alt=\"title\" height=\"300px\">\n",
        "\n",
        "黄色&丸い豆のエンドウ豆と緑色&しわのあるエンドウ豆を交配し、F2集団を作成すると、\n",
        "\n",
        "各形質は理論上、$9:3:3:1$の割合で出現するというものです。\n",
        "\n",
        "当然、実際にF2集団を作成しても、厳密に$9:3:3:1$になることは稀で、理論比から多少ずれた結果が得られることが予想されます。\n",
        "\n",
        "ここで重要なのが、この「ずれ」が誤差の範囲内なのか、それ以上に意味のあるものなのか、という所です。\n",
        "\n",
        "もし誤差として考え得る以上に大幅なズレ方をしていた場合、統計学では仮説からのずれは**有意**であると言います。\n",
        "\n",
        "また、立てられた仮説($9:3:3:1$になるはず)を**統計的仮説**や単に**仮説**と呼びます。\n",
        "\n",
        "つまり、今回学ぶ**仮説検定**では、統計的仮説の有意性を検定をすることになります。\n",
        "\n",
        "<br>\n",
        "\n",
        "有意性は標本が有意なずれを見せる確率で表されます。\n",
        "\n",
        "ここで標本分布が重要になってきます。\n",
        "\n",
        "例えば、あるコインを投げた時に表裏が出る確率は同じだ、と仮説を立て検証をしてみます。\n",
        "\n",
        "実際にコインを20回投げてみると16回表が出ました。\n",
        "\n",
        "この時「このコインは表裏が同じ確率で出る」という仮説はどの程度支持できるでしょうか？\n",
        "\n",
        "**もし表裏が同じ確率で出るというのが正しければ**、コインが表になる回数$X$は二項分布$Bi(20, 1/2)$に従う確率変数です。\n",
        "\n",
        "そこでコインの表が16回以上出る確率$P(X \\geqq 16)$を、Rの二項分布の関数`pbinom`を使用して計算してみましょう。\n",
        "\n",
        "```\n",
        "# ex) 確率0.5の事象を20回試行したときに成功する回数がx回以下となる確率\n",
        "\n",
        "pbinom(x, size=20, prob=0.5)\n",
        "```"
      ],
      "metadata": {
        "id": "O7qbta-me9a8"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# コインの表が16回以上出る確率を求める\n",
        "1 - pbinom(15, size=20, prob=0.5)"
      ],
      "metadata": {
        "id": "0UXo7x3xydAX",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "32de38ff-4a88-4221-c601-d3e03e02f439"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "0.00590896606445312"
            ],
            "text/markdown": "0.00590896606445312",
            "text/latex": "0.00590896606445312",
            "text/plain": [
              "[1] 0.005908966"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "$P(X \\geqq 16) = 1 - 0.9941 = 0.0059$と、とんでもなく低い確率になります。\n",
        "\n",
        "つまり、$X$が$16$回という観測データは仮説(表裏$50\\%$)からすれば到底出現する可能性の低い、\n",
        "\n",
        "かなりずれた(**=有意にずれた**)値ということになります。\n",
        "\n",
        "つまり、そもそも仮説である表裏$50\\%$が誤っている可能性が高い、と判断されることになります。\n",
        "\n",
        "このことを**仮説は棄却された**、と言います。\n",
        "\n",
        "ここで、$0.0059(0.59\\%)$がとんでもなく低い確率と考えましたが、ではどのくらい低かったら**有意に低い**と判断して良いのでしょうか？\n",
        "\n",
        "この基準とする確率を**有意水準**といい、有意水準$\\alpha$と表すことが多いです。\n",
        "\n",
        "有意水準$\\alpha$は$0.05$や$0.01$が使用されることが多く、**仮説は有意水準$\\alpha=0.01$のもとで棄却された**などと表現します。\n",
        "\n",
        "例えば、先ほどの例で表の回数が$13$回だった時、同じように$P(X \\geqq 13)$を計算してみると"
      ],
      "metadata": {
        "id": "VMX969UPxC2O"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# コインの表が13回以上出る確率を求める\n",
        "1 - pbinom(12, size=20, prob=0.5)"
      ],
      "metadata": {
        "id": "MpJhM3lfwHCr",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "d6258aba-0390-4f97-b231-4ea782e4c181"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "0.131587982177734"
            ],
            "text/markdown": "0.131587982177734",
            "text/latex": "0.131587982177734",
            "text/plain": [
              "[1] 0.131588"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "$P(X \\geqq 13) = 1 - 0.86841 = 0.13159$となるので、\n",
        "\n",
        "有意水準$\\alpha=0.05$のもとでは有意ではない(仮説の下で十分起こりうる)と判断されることになります。\n",
        "\n",
        "この様に、仮説検定では、有意水準という基準を設け、観測値に基づいて得られた値がその基準を超えるときに仮説を棄却します。\n",
        "\n",
        "このとき、下図のように、基準を超える部分は帰無仮説を棄却する範囲なので**棄却域**と呼びます。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject1.png?raw=true\" alt=\"title\" height=\"180px\">"
      ],
      "metadata": {
        "id": "1QwpPzH5zFeS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 帰無仮説と対立仮説\n",
        "\n",
        "先ほどのコインの例だと、\"コインの表裏が同じ確率で出る\"という仮説は有意水準$\\alpha=0.05$のもと棄却されました。\n",
        "\n",
        "これは\"コインの表裏は同じ確率では出ない\"という別の仮説が**採択**された形になります。\n",
        "\n",
        "このとき、元の仮説(表裏同じ確率)を**帰無仮説**、これと対立する仮説を**対立仮説**と呼び、それぞれ$H_0$、$H_1$と記します。\n",
        "\n",
        "今回の場合だと\n",
        "\n",
        "* 帰無仮説$H_0$: コインの表が出る確率$p = 0.5$\n",
        "* 対立仮説$H_1$: コインの表が出る確率$p \\ne 0.5$\n",
        "\n",
        "となります。"
      ],
      "metadata": {
        "id": "Z7TE6SCxzmT7"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 過誤\n",
        "\n",
        "実験で得られる標本はあくまで母集団の一部なので、検定を行った結果が毎回100%正しいとは限りません。\n",
        "\n",
        "仮説検定は、採択する仮説に対し、ある程度誤りの可能性を含んだ上で行われます。\n",
        "\n",
        "この誤りを**過誤**といい、仮説検定では2種類の過誤が生じる可能性があります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/false_positive.png?raw=true\" alt=\"title\" height=\"250px\">\n",
        "\n",
        "本当は帰無仮説($H_0$)が正しいのに、対立仮説($H_1$)を採択してしまう過誤が**第一種の過誤**(Type I error, 偽陽性, False positive)で、\n",
        "\n",
        "その確率は$\\alpha$、つまり有意水準の割合で生じることになります。$\\alpha$の確率で生じる稀な標本を観測してしまったがために、本当は正しい帰無仮説を棄却してしまう場合になります。\n",
        "\n",
        "一方、本当は対立仮説($H_1$)が正しいのに、帰無仮説($H_0$)を採択してしまう過誤を**第二種の過誤**(Type II error, 偽陰性, False negative)と呼びます。\n",
        "\n",
        "第一種の過誤は、例えば本当は「薬品処理の効果は無い」のに「薬品処理の効果がある」と結論付けてしまう過誤となります。この誤りを小さくするためには有意水準$\\alpha$を小さく設定することになります。\n",
        "\n",
        "$\\alpha$については5%や1%がよく用いられますが、場合によって0.1%や10%なども用いられ、明確な基準があるわけでは無いです。第一種の過誤が生じた場合の損害等に応じて決定します。\n",
        "\n",
        "ただし、有意水準を厳しくするほど、第二種の過誤である、効果があるのに無いと判断される可能性が高くなります。\n",
        "\n",
        "そのため、例えば毒性試験など、出来るだけ効果を逃したくない場合は第二種の過誤を小さくすることが求められます。"
      ],
      "metadata": {
        "id": "IQwFYc3j7BLv"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 帰無仮説の採択について\n",
        "\n",
        "有意性検定は、仮説に基づく期待から、大幅なずれが出ていることを根拠に、仮説を否定(棄却)する手法となります。\n",
        "\n",
        "つまり、**仮説を棄却すること**が中心の手法となる。\n",
        "\n",
        "仮説が棄却されなかった場合に、**その仮説を証明しているわけでは無い**。\n",
        "\n",
        "得られた標本の結果が仮説に「矛盾していない」というだけのことになる。\n",
        "\n",
        "棄却されない = 仮説が正しい、では無い点は注意しておきましょう。\n",
        "\n",
        "コインの例で言うと、仮にコインの裏表の確率が同じという仮説が棄却されなかった、というのは裏表が同じ確率であることを示したわけでは無く、極端にズレているわけでは無い、と言うだけの話になります。"
      ],
      "metadata": {
        "id": "_auisNON49LE"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 両側検定と片側検定\n",
        "\n",
        "ある２つのグループの差を考えます。\n",
        "\n",
        "例えば島Aに生息するメヒシバの草丈と、島Bに生息するメヒシバの草丈を比較するとします。\n",
        "\n",
        "この時、島Aの草丈と島Bの草丈の差$x$は正の場合も負の場合もあります。(Aの方が大きい場合or小さい場合)\n",
        "\n",
        "島Aの草丈と島Bの草丈に差があるかないか、という検定の場合、この正負にかかわらず、0から遠いほど有意であると考える形になります。\n",
        "\n",
        "この場合は下図の様に分布の両側に棄却域が存在している状況になり、**両側検定**と呼ばれています。\n",
        "\n",
        "島Aの草丈と島Bの草丈の差$x$が有意かどうか検定する場合\n",
        "\n",
        "* 島Aの草丈が極端に大きい場合\n",
        "* 島Aの草丈が極端に小さい場合\n",
        "\n",
        "の２種類あるというわけです。\n",
        "\n",
        "一方、実験や調査の特性上、その正負まで含め検定したい場合があります。\n",
        "\n",
        "島AとBの間に差があるか、ではなく、島Aの方が島Bの草丈より大きいかどうか、という検定の場合です。\n",
        "\n",
        "このとき、棄却域は分布の右側にのみ存在し、**片側検定**と呼ばれています。\n",
        "\n",
        "同じ有意水準$\\alpha$でも、両側検定か片側検定かで厳しさが変わってきます。\n",
        "\n",
        "片側検定は正負も検定の対象となる意味で両側検定より厳しく、\n",
        "\n",
        "両側検定は有意と判断される値の大きさの点で片側検定より厳しいといえます。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/test_side.png?raw=true\" alt=\"title\" height=\"180px\">"
      ],
      "metadata": {
        "id": "DEjifY-A546c"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 正規母集団に対する仮説検定\n",
        "\n",
        "ここからは具体的に検定を実施する方法を見ていきます。\n",
        "\n",
        "まずは、もっともよく使用されている検定として、母集団の分布が正規分布である場合の仮説検定を見ていきます。"
      ],
      "metadata": {
        "id": "TGHCxCDqhoXT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 母平均に関する検定\n",
        "\n",
        "最初に、母平均$\\mu$、母分散$\\sigma^2$の正規分布$N(\\mu, \\sigma^2)$に従う母集団において、母平均$\\mu$に関する検定について見ていきます。"
      ],
      "metadata": {
        "id": "YEn0tbzTEeNh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 両側検定\n",
        "\n",
        "両側検定では、ある集団の観測値が得られたとき、その母集団の母平均が$\\mu_0$なのかどうかを検定します。\n",
        "\n",
        "帰無仮説、対立仮説は、\n",
        "\n",
        "* $H_0:\\mu=\\mu_0$\n",
        "* $H_1:\\mu\\ne\\mu_0$\n",
        "\n",
        "で与えられます。\n",
        "\n",
        "この仮説の下、得られたデータから計算される標本平均$\\bar{X}$が$\\mu_0$からどれだけ離れている(差・ずれがある)かによって検定が行われます。\n",
        "\n",
        "この差を判断する際に、分散$\\sigma^2$が既知かどうかで検定方法が異なってきます。\n",
        "\n",
        "<br>\n",
        "\n",
        "**分散$\\sigma^2$が既知の場合**\n",
        "\n",
        "検定において仮説とのずれ・差の判断に用いる統計量を**検定統計量**と呼びます。\n",
        "\n",
        "今回の検定統計量は、$\\bar{X}$を標準化した値になります。\n",
        "\n",
        "$\\bar{X}$を標準化すると、\n",
        "\n",
        "$Z=\\dfrac{\\bar{X}-\\mu}{\\sigma/\\sqrt{n}}$\n",
        "\n",
        "となり、この値は標準正規分布$N(0, 1)$に従う値になります。\n",
        "\n",
        "ここで、帰無仮説$\\mu=\\mu_0$が正しければ、\n",
        "\n",
        "$Z=\\dfrac{\\bar{X}-\\mu_0}{\\sigma/\\sqrt{n}}$\n",
        "\n",
        "となり、$Z$の値を計算する事が出来ます。\n",
        "\n",
        "得られる$Z$の値を標準正規分布から得られるパーセント点$Z_{\\alpha/2}$と比較して、\n",
        "\n",
        "$|Z|>Z_{\\alpha/2}$だと$H_0$を棄却することになり、$|Z|\\leqq Z_{\\alpha/2}$だと$H_0$は棄却されないことになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject2.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "<br>\n",
        "\n",
        "**分散$\\sigma^2$が未知の場合**\n",
        "\n",
        "しかし実際に検定をする際には、$\\sigma^2$が既知であることはほとんどなく、分散が未知の状況で検定を行う場合がほとんどです。\n",
        "\n",
        "そこで、分散が未知の場合には、母分散$\\sigma^2$を標本分散$s^2$で置き換えた$t$統計量\n",
        "\n",
        "$t = \\dfrac{\\bar{X}-\\mu}{s/\\sqrt{n}}$\n",
        "\n",
        "を検定統計量として用います。この$t$値は自由度$n-1$の$t$分布$t(n-1)$に従います。\n",
        "\n",
        "帰無仮説$\\mu=\\mu_0$が正しければ、\n",
        "\n",
        "$t = \\dfrac{\\bar{X}-\\mu_0}{s/\\sqrt{n}}$\n",
        "\n",
        "となり、$t$値を計算できるので、\n",
        "\n",
        "計算した$t$値が$t$分布から得られるパーセント点$t_{\\alpha/2}(n-1)$と比較して、\n",
        "\n",
        "$|t|>t_{\\alpha/2}(n-1)$であれば$H_0$を棄却することになり、$|t|\\leqq t_{\\alpha/2}$であれば$H_0$は棄却されないことになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject3.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "この$t$統計量を用いた検定をスチューデントの$t$**検定**と呼びます。\n",
        "\n"
      ],
      "metadata": {
        "id": "rxaNA41sw5Cm"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**[例]**\n",
        "\n",
        "A島において平均草丈が20cmとされている植物がいます。\n",
        "\n",
        "C島に生息するこの植物の草丈も概ね20cmなのかどうか調べたい。\n",
        "\n",
        "そこで、C島で10標本採集したところ、その草丈は23, 19, 26, 21, 25, 28, 22, 23, 27, 19であった。\n",
        "\n",
        "C島に生息する個体もA島と同じように20㎝なのかどうか検定を試みる。\n",
        "\n",
        "このとき、帰無仮説と対立仮説は\n",
        "\n",
        "* $H_0:\\mu=20$\n",
        "* $H_1:\\mu\\ne 20$\n",
        "\n",
        "と表せます。\n",
        "\n",
        "そこでこの帰無仮説のもと$t$値を計算してみると、\n",
        "\n",
        "$t = \\dfrac{\\bar{X}-\\mu_0}{s/\\sqrt{n}}$ で計算できるので\n",
        "\n"
      ],
      "metadata": {
        "id": "r9KVsLON44Au"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# t値を求める\n",
        "data <- c(23, 19, 26, 21, 25, 28, 22, 23, 27, 19)\n",
        "\n",
        "(mean(data) - 20) / (sd(data) / sqrt(10))"
      ],
      "metadata": {
        "id": "w5egALI76VV4",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "018a0a6e-a79e-4b13-906b-5315c00cc97d"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "3.29816819303121"
            ],
            "text/markdown": "3.29816819303121",
            "text/latex": "3.29816819303121",
            "text/plain": [
              "[1] 3.298168"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "この$t$値が仮説の下でそこそこあり得る値なのか、それともほとんどあり得ない極端な値なのかを有意水準を基準に調べます。\n",
        "\n",
        "例えば有意水準$5\\%$で検定を行う場合は、自由度$10-1=9$の$t$分布における$t_{\\alpha/2}$の値と得られた$t$値比較をします。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/t_test1.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "自由度$9$の$t分布$におけるパーセント点$t_{0.025}(9)$は、Rの$t$分布の関数`qt`で計算できます。\n",
        "\n",
        "```\n",
        "qt(累積確率, 自由度)\n",
        "```\n"
      ],
      "metadata": {
        "id": "CxvTvMbW80lg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度9のt分布におけるパーセント点を求める\n",
        "qt(0.975, 9)"
      ],
      "metadata": {
        "id": "uNmWMSZl9UmZ",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "99fb88e0-34a0-4e41-9740-0aa175872bc3"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "2.2621571627982"
            ],
            "text/markdown": "2.2621571627982",
            "text/latex": "2.2621571627982",
            "text/plain": [
              "[1] 2.262157"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "よって、$t_{0.025}(9) < t$となるので、有意水準$\\alpha=0.05$の下で、帰無仮説は棄却されることになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/p.png?raw=true\" alt=\"title\" height=\"180px\">\n",
        "\n",
        "また、帰無仮説が正しいと仮定したときに、観測されたデータ以上に極端な結果が得られる確率を**p値**と呼びます。\n",
        "\n",
        "$p$値が小さいほど、偶然では説明できないような結果だと判断され、帰無仮説を棄却することに繋がります。\n",
        "\n",
        "$p$値を`pt`関数で計算してみると\n",
        "\n",
        "```\n",
        "pt(t値, 自由度)\n",
        "```"
      ],
      "metadata": {
        "id": "Lo7jSo4P9_jm"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# P値の計算\n",
        "1 - pt(3.29816819303121, 9)"
      ],
      "metadata": {
        "id": "VV62eeobFELI",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "28ac2c58-e1cc-4eb1-ccce-b82f1ea09642"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "0.00462845775005205"
            ],
            "text/markdown": "0.00462845775005205",
            "text/latex": "0.00462845775005205",
            "text/plain": [
              "[1] 0.004628458"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "となり、$p$値が有意水準より小さいので帰無仮説は棄却される、と捉えることもできます。"
      ],
      "metadata": {
        "id": "1ccwV7A-H7ab"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 片側検定\n",
        "\n",
        "片側検定では、ある集団の観測値が得られたとき、その母集団の母平均が$\\mu_0$より大きいのかどうか、という検定で、\n",
        "\n",
        "帰無仮説、対立仮説は、\n",
        "\n",
        "* $H_0:\\mu=\\mu_0$\n",
        "* $H_1:\\mu > \\mu_0$\n",
        "\n",
        "で与えられます。\n",
        "\n",
        "帰無仮説が両側検定の時と同じなので、検定統計量$t$は同じです。\n",
        "\n",
        "ただし、対立仮説が異なるので、棄却域が異なることになります。\n",
        "\n",
        "分散が未知の場合を例にすると、$t$統計量を計算し、\n",
        "\n",
        "$t>t_{\\alpha}(n-1)$であれば$H_0$を棄却することになり、$t\\leqq t_{\\alpha}(n-1)$であれば$H_0$は棄却されないことになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject4.png?raw=true\" alt=\"title\" height=\"180px\">"
      ],
      "metadata": {
        "id": "a16G-23q-aEp"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "[例]\n",
        "\n",
        "さきほどの平均草丈が20cmとされている植物がいる。C島に生息するこの植物の草丈は20cm**より大きいのか**どうか調べたい。\n",
        "\n",
        "C島で10標本採集したところ、その草丈は23, 19, 26, 21, 25, 28, 22, 23, 27, 19であった。\n",
        "\n",
        "C島に生息する個体は20㎝より大きいのかどうか5%の有意水準で検定する。\n",
        "\n",
        "この検定における帰無仮説と対立仮説は\n",
        "\n",
        "* $H_0:μ=20$\n",
        "* $H_1:μ>20$\n",
        "\n",
        "と表せる。 先ほど計算した$t$値は$3.29816819303121$だったので、\n",
        "\n",
        "パーセント点$t_{0.05}(9)$を見てみると、"
      ],
      "metadata": {
        "id": "IFxsxDjWABWD"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度9のt分布におけるα=0.05のパーセント点を求める\n",
        "qt(0.95, 9)"
      ],
      "metadata": {
        "id": "OqTQHVP4A-Ks",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "aced04de-1a25-487b-e682-6c64b8b176a0"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "1.83311293265624"
            ],
            "text/markdown": "1.83311293265624",
            "text/latex": "1.83311293265624",
            "text/plain": [
              "[1] 1.833113"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "となり、有意水準5%で帰無仮説$H_0$は棄却された。\n",
        "\n",
        "よって、C島に生息する植物の平均値は20㎝より高かったことが認められる。"
      ],
      "metadata": {
        "id": "kDgVI2sIBOFs"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 母分散に対する仮説検定\n",
        "\n",
        "母集団の母分散に関しても、母分散$\\sigma^2$がある値かどうか検定することが出来ます。\n",
        "\n",
        "母分散の検定における帰無仮説・対立仮説は、\n",
        "\n",
        "* 帰無仮説$H_0:\\sigma^2=\\sigma_0^2$\n",
        "* 対立仮説$H_1:\\sigma^2\\ne\\sigma_0^2$\n",
        "\n",
        "となります。\n",
        "\n",
        "このとき、検定統計量は標本分散$s^2$による\n",
        "\n",
        "$\\chi^2=(n-1)s^2/\\sigma^2$\n",
        "\n",
        "を用いることになります。\n",
        "\n",
        "この検定統計量は自由度$n-1$の$\\chi^2$分布$\\chi^2(n-1)$に従います。\n",
        "\n",
        "帰無仮説のもとでは、$\\chi^2=(n-1)s^2/\\sigma_0^2$を計算する事が出来るので、\n",
        "\n",
        "自由度$n-1$の$\\chi^2$分布のパーセント点$\\chi_{1-\\alpha/2}^2(n-1)$および$\\chi_{\\alpha/2}^2(n-1)$を用いて、\n",
        "\n",
        "両側検定の場合は$\\chi_{1-\\alpha/2}^2(n-1) < \\chi^2 < \\chi_{\\alpha/2}^2(n-1)$のときに$H_0$を棄却せず、それ以外は棄却することになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject5.png?raw=true\" alt=\"title\" height=\"220px\">\n",
        "\n",
        "この検定を正規母集団の母分散についての$\\chi^2$**検定**と呼びます。\n",
        "\n",
        "今回のように分散そのものを検定する状況はあまりありませんが、$\\chi^2$検定はばらつきについて検定する際によく用いられ、また後程何度か登場します。\n",
        "\n"
      ],
      "metadata": {
        "id": "GpWNXOhIByyY"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 母平均の差の検定\n",
        "\n",
        "母平均がある特定の値かどうか、という検定は実際あまり実施されませんが、\n",
        "\n",
        "二つの正規母集団の母平均の差の検定は、非常によく用いられます。\n",
        "\n",
        "例えば、施肥効果を調べるために、植物を2つのグループ(A, B)に分けます。Aグループには施肥を行い、もう片方のBグループには何もせず、2つのグループの間に差があるかを確認する場合に用いられます。\n",
        "\n",
        "これを**2標本検定**といい、遺伝子をノックアウトした効果、実験である処理を行った効果...等々、生物学において適用例は多岐にわたります。\n",
        "\n",
        "より一般化していくと、\n",
        "\n",
        "2つの正規母集団がそれぞれ$N(\\mu_1, \\sigma_1^2), N(\\mu_2, \\sigma_2^2)$に従っていた時、\n",
        "\n",
        "それぞれの集団から大きさ$m,n$の標本$X_1, X_2, ..., X_m, Y_1, Y_2, ..., Y_n$を抽出したとします。\n",
        "\n",
        "これらの集団の母平均に差があるか検定する場合は、\n",
        "\n",
        "帰無仮説$H_0:\\mu_1 = \\mu_2$\n",
        "\n",
        "となり、対立仮説は\n",
        "\n",
        "両側検定なら$H_1:\\mu_1 \\ne \\mu_2$\n",
        "\n",
        "片側検定なら$H_1:\\mu_1 > \\mu_2$または$H_1:\\mu_1 < \\mu_2$\n",
        "\n",
        "となります。\n",
        "\n",
        "この検定を実施する際に、使用する検定方法は2つの集団の母分散が等しいかどうかによって変わってきます。"
      ],
      "metadata": {
        "id": "rpP2R9u9EaF2"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##### **分散が等しい場合**\n",
        "\n",
        "2つの母集団の分散が等しく、$\\sigma_1^2=\\sigma_2^2=\\sigma^2$である場合、\n",
        "\n",
        "標本平均$\\bar{X}$と$\\bar{Y}$の分布はそれぞれ$N(\\mu_1, \\sigma_1^2/m), N(\\mu_2, \\sigma_2^2/n)$に従います。\n",
        "\n",
        "このとき標本平均の差$\\bar{X} - \\bar{Y}$の分布は、平均は差、分散は和になるので、\n",
        "\n",
        "$N(\\mu_1-\\mu_2, (\\sigma_1^2/m)+(\\sigma_2^2/n))$となります。\n",
        "\n",
        "分散が等しい$\\sigma_1^2=\\sigma_2^2=\\sigma^2$とき、この平均値の差$\\bar{X}-\\bar{Y}$を標準化すると、\n",
        "\n",
        "$Z = \\dfrac{(\\bar{X}-\\bar{Y})-(\\mu_1-\\mu_2)}{\\sqrt{(\\dfrac{1}{m}+\\dfrac{1}{n})\\sigma^2}}$\n",
        "\n",
        "となり、この$Z$は標準正規分布$N(0,1)$に従います。\n",
        "\n",
        "しかし、$\\sigma^2$が未知なので、各々の標本分散を合併した値に置き換える必要があります。\n",
        "\n",
        "そこで、\n",
        "\n",
        "$s^2 = \\dfrac{\\sum_{i=1}^{m}(X_i-\\bar{X})^2+\\sum_{j=1}^{n}(Y_j-\\bar{Y})^2}{m+n-2}$\n",
        "\n",
        "$=\\dfrac{ (m-1)s_1^2+(n-1)s_2^2}{m+n-2}$\n",
        "\n",
        "この様に合併した標本分散$s^2$を定義すると、\n",
        "\n",
        "$(m+n-2)s^2/\\sigma^2$は自由度$m+n-2$の$\\chi^2$分布$\\chi^2(m+n-2)$に従う値になります。\n",
        "\n",
        "さきほどの$Z = \\dfrac{(\\bar{X}-\\bar{Y})-(\\mu_1-\\mu_2)}{\\sqrt{(\\dfrac{1}{m}+\\dfrac{1}{n})\\sigma^2}}$の$\\sigma^2$をこの$s^2$に置き換えると、\n",
        "\n",
        "$t = \\dfrac{(\\bar{X}-\\bar{Y})-(\\mu_1-\\mu_2)}{\\sqrt{(\\dfrac{1}{m}+\\dfrac{1}{n})s^2}}$\n",
        "\n",
        "となり、この$t$値は**2標本t統計量**と言い、自由度$m+n-2$の$t$分布$t(m+n-2)$に従います。\n",
        "\n",
        "帰無仮説($\\mu_1=\\mu_2$)のもとではこの2標本t統計量は\n",
        "\n",
        "$t = \\dfrac{(\\bar{X}-\\bar{Y})}{\\sqrt{(\\dfrac{1}{m}+\\dfrac{1}{n})s^2}}$\n",
        "\n",
        "となり、\n",
        "\n",
        "両側検定の場合$|t|>t_{\\alpha/2}(m+n-2)$のとき帰無仮説を棄却することになり、それ以外の場合は棄却しないことになります。\n",
        "\n",
        "片側検定の場合は$t>t_{\\alpha}(m+n-2)$のとき(または$t<-t_{\\alpha}(m+n-2)$のとき)帰無仮説を棄却することになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject6.png?raw=true\" alt=\"title\" height=\"220px\">"
      ],
      "metadata": {
        "id": "C0CHPDqzj25a"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**[例]**\n",
        "\n",
        "ある植物に対し、施肥を行ったグループと行わなかったグループの草丈を比較する。各グループ10標本の測定結果が以下の様になった際に、各グループの分散が等しい時、これらのグループ間で草丈の平均値に差はあると言って良いだろうか？\n",
        "\n",
        "施肥を行ったグループ: 13, 14, 12, 15, 13, 12, 15, 14, 16, 12\n",
        "\n",
        "施肥無しのグループ: 12, 11, 13, 12, 14, 12, 13, 12, 15, 13\n",
        "\n",
        "まず、このデータから$s^2$を計算し、その値を用いて$t$値を計算する。\n",
        "\n",
        "$s^2 = \\dfrac{ (m-1)s_1^2+(n-1)s_2^2}{m+n-2}$、\n",
        "$t = \\dfrac{(\\bar{X}-\\bar{Y})}{\\sqrt{(\\dfrac{1}{m}+\\dfrac{1}{n})s^2}}$\n",
        "\n"
      ],
      "metadata": {
        "id": "9fYP71-Eq2Py"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# GroupA, GroupBの平均値の差をt検定するため、t値を求める\n",
        "GroupA <- c(13, 14, 12, 15, 13, 12, 15, 14, 16, 12)\n",
        "GroupB <- c(12, 11, 13, 12, 14, 12, 13, 12, 15, 13)\n",
        "\n",
        "s2 = (9*sd(GroupA)^2 + 9*sd(GroupB)^2) / (10+10-2)\n",
        "(mean(GroupA) - mean(GroupB)) / sqrt((1/10+1/10)*s2)"
      ],
      "metadata": {
        "id": "t3frVyCbrzBQ",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "75c1328a-e4df-4109-ef09-7a23269b5ccd"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "1.54601550264744"
            ],
            "text/markdown": "1.54601550264744",
            "text/latex": "1.54601550264744",
            "text/plain": [
              "[1] 1.546016"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "この$t$値に対し、$t$分布のパーセント点$t_{\\alpha/2}(m+n-2)$を見ると"
      ],
      "metadata": {
        "id": "rZf4wM01uKYA"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度m+n-2のt分布のパーセント点を求める\n",
        "qt(0.975, 18)"
      ],
      "metadata": {
        "id": "jil4wVPTuKkj",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "b8450949-9df1-4942-c2c9-b4172e5636da"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "2.10092204024104"
            ],
            "text/markdown": "2.10092204024104",
            "text/latex": "2.10092204024104",
            "text/plain": [
              "[1] 2.100922"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "ということで、$|t|>t_{\\alpha/2}(m+n-2)$を満たさないので、棄却できない。\n",
        "\n",
        "よって、有意水準5%のもとでGroupAとGroupBの平均値が同じという帰無仮説を棄却できなかったので、\n",
        "\n",
        "施肥の効果は有意とは言えない、と評価できます。"
      ],
      "metadata": {
        "id": "_QR1OekTuclW"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "##### **分散が異なる場合**\n",
        "\n",
        "両集団の母分散が等しくない場合は複雑になります。\n",
        "\n",
        "この場合、2つの母集団の分散$\\sigma_1^2, \\sigma_2^2$を推定して$\\bar{X}-\\bar{Y}$の分布を求める必要があります。\n",
        "\n",
        "…が、どう頑張っても$\\sigma_1^2, \\sigma_2^2$無しの状態では、$\\bar{X}-\\bar{Y}$の正確な分布を求めることはできません。\n",
        "\n",
        "そこで、近似的に分布を求める**ウェルチの近似法**を使用します。\n",
        "\n",
        "これは、平均値の差$\\bar{X}-\\bar{Y}$を標準化した\n",
        "\n",
        "$Z = \\dfrac{(\\bar{X}-\\bar{Y})-(\\mu_1-\\mu_2)}{\\sqrt{(\\sigma_1^2/m)+(\\sigma_2^2/n)}}$\n",
        "\n",
        "に、各々の標本分散$s_1^2, s_2^2$を代入します。\n",
        "\n",
        "$t= \\dfrac{(\\bar{X}-\\bar{Y})-(\\mu_1-\\mu_2)}{\\sqrt{(s_1^2/m)+(s_2^2/n)}}$\n",
        "\n",
        "この$t$値が、近似的に自由度が\n",
        "\n",
        "$v=\\dfrac{(\\dfrac{s_1^2}{m}+\\dfrac{s_2^2}{n})^2}{\\dfrac{(s_1^2/m)^2}{m-1}+\\dfrac{(s_2^2/n)^2}{n-1}}$\n",
        "\n",
        "に最も近い整数$v^{*}$の$t$分布$t(v^{*})$に従うという近似です。\n",
        "\n",
        "ウェルチの近似法を用いると、両側検定の場合、$|t|>t_{\\alpha/2}(v^*)$の時に、帰無仮説を棄却することになります。。\n",
        "\n",
        "片側検定の場合、$t>t_{\\alpha}(v^*)$(または$t<-t_{\\alpha}(v^*)$)の時、帰無仮説を棄却することになります。\n",
        "\n",
        "この検定は**ウェルチのt検定**と呼ばれます。"
      ],
      "metadata": {
        "id": "q2ovJhabj4vV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 母分散の比検定\n",
        "\n",
        "2つの集団の母平均が等しいかどうかを検定する方法は分散が等しいかどうかで異なりました。\n",
        "\n",
        "そこで、そもそも2つの集団の分散が等しい(比が1)かどうかを検定する必要があります。\n",
        "\n",
        "この検定における帰無仮説と対立仮説は\n",
        "\n",
        "帰無仮説としては$H_0: \\sigma_1^2=\\sigma_2^2$、\n",
        "\n",
        "対立仮説としては$H_1: \\sigma_1^2 \\ne \\sigma_2^2$\n",
        "\n",
        "となります。\n",
        "\n",
        "この帰無仮説は、$\\sigma_1^2/\\sigma_2^2=1$と置き換えて、標本分散に基づく$s_1^2/s_2^2$から評価していきます。\n",
        "\n",
        "$s_1^2,s_2^2$はそれぞれ独立に$\\chi^2$分布に従うので、$\\chi^2$分布に従う確率変数の**比**がどんな確率分布に従うか知る必要がある。\n",
        "\n",
        "ここで、$F$**分布**が登場します。$F$分布とは、\n",
        "\n",
        "独立な確率変数$U, V$がそれぞれ\n",
        "\n",
        "* $U$は自由度$k_1$の$\\chi^2$分布$\\chi^2(k_1)$に従う\n",
        "* $V$は自由度$k_2$の$\\chi^2$分布$\\chi^2(k_2)$に従う\n",
        "\n",
        "この$U, V$をそれぞれの自由度で割った後の比(**フィッシャーの分散比**と呼ぶ)を、\n",
        "\n",
        "$F=\\dfrac{U/k_1}{V/k_2}$\n",
        "\n",
        "と定義すると、$F$の従う確率分布を**自由度($k_1, k_2$)の$F$分布**と呼び、$F(k_1, k_2)$と表します。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/F.png?raw=true\" alt=\"title\" height=\"220px\">\n",
        "\n",
        "独立である標本分散$s_1^2, s_2^2$それぞれの標本分布について考えてみると、\n",
        "\n",
        "* $(m-1)s_1^2/\\sigma_1^2$が自由度$m-1$の$\\chi^2$分布$\\chi^2(m-1)$に従う。\n",
        "* $(n-1)s_2^2/\\sigma_2^2$が自由度$n-1$の$\\chi^2$分布$\\chi^2(n-1)$に従う。\n",
        "\n",
        "ということになります。\n",
        "\n",
        "$F$分布の定義より、\n",
        "\n",
        "$F=\\dfrac{\\dfrac{(m-1)s_1^2}{\\sigma_1^2}/(m-1)}{\\dfrac{(n-1)s_2^2}{\\sigma_2^2}/(n-1)}=\\dfrac{\\sigma_2^2}{\\sigma_1^2}\\dfrac{s_1^2}{s_2^2}$\n",
        "\n",
        "が自由度$(m-1, n-1)$の$F$分布$F(m-1, n-1)$に従うことになります。\n",
        "\n",
        "これは帰無仮説($\\sigma_1^2=\\sigma_2^2$)において、**標本の分散比**\n",
        "\n",
        "$F=s_1^2/s_2^2$となる。\n",
        "\n",
        "つまり、分散比の検定では、\n",
        "\n",
        "$F_{1-\\alpha/2}(m-1, n-1) \\leqq F \\leqq F_{\\alpha/2}(m-1, n-1)$\n",
        "\n",
        "のときには帰無仮説は棄却されず、\n",
        "\n",
        "$F < F_{1-\\alpha/2}(m-1, n-1)$、または$F > F_{\\alpha/2}(m-1, n-1)$\n",
        "\n",
        "のときに帰無仮説を棄却することになります。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/reject7.png?raw=true\" alt=\"title\" height=\"220px\">\n",
        "\n",
        "この様な$F$統計量を検定統計量として用いる検定を$F$**検定**と呼びます。"
      ],
      "metadata": {
        "id": "mUu834PVvVdT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**[例]**\n",
        "\n",
        "先ほどのデータを用いて$F$検定を行ってみましょう。\n",
        "\n",
        "施肥を行ったグループ: 13, 14, 12, 15, 13, 12, 15, 14, 16, 12\n",
        "\n",
        "施肥無しのグループ: 12, 11, 13, 12, 14, 12, 13, 12, 15, 13\n",
        "\n",
        "各グループの分散が等しいかどうか検定するには、\n",
        "\n",
        "まず標本の分散比である$F$を計算します。\n",
        "\n",
        "$F=s_1^2/s_2^2$"
      ],
      "metadata": {
        "id": "wHGiTq2J6V17"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 標本の分散比を求める\n",
        "GroupA <- c(13, 14, 12, 15, 13, 12, 15, 14, 16, 12)\n",
        "GroupB <- c(12, 11, 13, 12, 14, 12, 13, 12, 15, 13)\n",
        "\n",
        "sd(GroupA)^2 / sd(GroupB)^2"
      ],
      "metadata": {
        "id": "T73buHVP6sK0",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "d3c2f73f-ca01-41af-d76e-56be2c300944"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "1.52066115702479"
            ],
            "text/markdown": "1.52066115702479",
            "text/latex": "1.52066115702479",
            "text/plain": [
              "[1] 1.520661"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "次に、$F_{1-\\alpha/2}(m-1, n-1), F_{\\alpha/2}(m-1, n-1)$の値を求めます。\n",
        "\n",
        "Rでは`qf`関数によって$F$分布のパーセント点を計算できます。\n",
        "\n",
        "```\n",
        "qf(0.975, 自由度1, 自由度2)\n",
        "```\n",
        "\n",
        "有意水準$\\alpha=5\\%$のもと、$F_{1-\\alpha/2}(m-1, n-1), F_{\\alpha/2}(m-1, n-1)$を求めてみると、\n"
      ],
      "metadata": {
        "id": "bxb_KNSq6_Bn"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度(9, 9)のF分布におけるパーセント点を求める\n",
        "qf(0.975, 9, 9)\n",
        "qf(0.025, 9, 9)"
      ],
      "metadata": {
        "id": "Bw6AAiQO7bu3",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "outputId": "8f02c398-a436-488e-fb0b-e5ad8875ee26"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "4.02599415828298"
            ],
            "text/markdown": "4.02599415828298",
            "text/latex": "4.02599415828298",
            "text/plain": [
              "[1] 4.025994"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "0.248385854694455"
            ],
            "text/markdown": "0.248385854694455",
            "text/latex": "0.248385854694455",
            "text/plain": [
              "[1] 0.2483859"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "ということで、$F_{1-\\alpha/2}(m-1, n-1) \\leqq F \\leqq F_{\\alpha/2}(m-1, n-1)$となり、\n",
        "\n",
        "帰無仮説は棄却されない。\n",
        "\n",
        "よってGroupAとGroupBの分散は有意水準$5\\%$のもとで異なるとは言えない。という判断が出来ます。\n",
        "\n",
        "この結果から分散が等しいと仮定した$t$検定を行う…といった形で進めます。"
      ],
      "metadata": {
        "id": "4rnSEB3M7mAt"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### **対応のある標本の平均値の比較**\n",
        "\n",
        "これまで扱ってきた2集団の母平均を比較する場合の検定は、2つの集団がそれぞれ独立している、つまり**対応の無い**検定と言えます。\n",
        "\n",
        "しかし、同じ標本に対して、処理を行った前後の比較をする場合、\n",
        "\n",
        "例えば同じ植物個体に施肥する前後で草丈を比較するなど、\n",
        "\n",
        "この様な場合は、施肥前の標本$X_1, X_2, ..., X_n$と施肥後の標本$Y_1, Y_2, ..., Y_n$が独立ではなく、\n",
        "\n",
        "何らかの関係性を持っていることになります。($X_k$の個体の施肥後が$Y_k$)\n",
        "\n",
        "この場合、施肥前と施肥後に差があるかを検定する際に、これまで扱っていた手法を適用することができません。\n",
        "\n",
        "<br>\n",
        "\n",
        "この検定では、施肥前後の**差の母平均**$\\delta$について考える必要があります。\n",
        "\n",
        "帰無仮説と対立仮説は、\n",
        "\n",
        "* $H_0: \\delta=0$\n",
        "* $H_1: \\delta\\ne 0$\n",
        "\n",
        "となります。\n",
        "\n",
        "標本のペア$X_k$と$Y_k$の差を$X_k - Y_k = d_k$とおいて、\n",
        "\n",
        "$d_k$の標本平均を$\\bar{d}$、標本分散を$s_d^2$とすると、検定統計量は\n",
        "\n",
        "$t=\\dfrac{\\bar{d}}{\\sqrt{s_d^2/n}}$\n",
        "\n",
        "で定義されます。これが自由度$n-1$の$t$分布$t(n-1)$に従うので、\n",
        "\n",
        "両側検定の場合$|t|>t_{\\alpha/2}(n-1)$であれば帰無仮説が棄却されます。\n",
        "\n",
        "**対応のない$t$検定**との一番の違いとして、自由度が異なることになります。\n"
      ],
      "metadata": {
        "id": "pPyNpPt6-rKM"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Rによるt検定\n",
        "\n",
        "ここまで、様々な状況下における$t$検定について扱い、いくつか式に沿って計算を行ってきましたが、\n",
        "\n",
        "これらの計算はすべてRの`t.test`関数によって実施することができます。\n",
        "\n",
        "例えば対応がない2集団で分散が等しい場合の$t$検定を両側検定で実施するには\n",
        "\n",
        "```\n",
        "t.test(x=集団1のデータ,y=集団2のデータ,alternative=\"t\",var.equal=T,paired=F)\n",
        "```\n",
        "の様な形で実行できます。\n",
        "\n",
        "先ほどの草丈のサンプルデータを用いて$t$検定をやってみると\n",
        "\n",
        "施肥を行ったグループ: 13, 14, 12, 15, 13, 12, 15, 14, 16, 12\n",
        "\n",
        "施肥無しのグループ: 12, 11, 13, 12, 14, 12, 13, 12, 15, 13"
      ],
      "metadata": {
        "id": "9TQmhKtz9ulC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# t.test関数を使用し、対応が無く分散が等しい場合のt検定を両側検定で実施する\n",
        "GroupA <- c(13, 14, 12, 15, 13, 12, 15, 14, 16, 12)\n",
        "GroupB <- c(12, 11, 13, 12, 14, 12, 13, 12, 15, 13)\n",
        "t.test(x=GroupA,y=GroupB,alternative=\"t\",var.equal=T,paired=F)"
      ],
      "metadata": {
        "id": "TAVLTDVEFd3v",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 223
        },
        "outputId": "683604cd-7383-411a-eca9-96a34b2b0eb0"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "\n",
              "\tTwo Sample t-test\n",
              "\n",
              "data:  GroupA and GroupB\n",
              "t = 1.546, df = 18, p-value = 0.1395\n",
              "alternative hypothesis: true difference in means is not equal to 0\n",
              "95 percent confidence interval:\n",
              " -0.3230342  2.1230342\n",
              "sample estimates:\n",
              "mean of x mean of y \n",
              "     13.6      12.7 \n"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "`t = XXX`という部分に統計量$t$値があり、`df = XXX`は自由度、\n",
        "\n",
        "`p-value = XXX`はこの$t$値よりも極端な値をとる確率($p$値)を示しています。\n",
        "\n",
        "この$p$値をもとに、有意水準と比較して有意かどうか判断します。\n",
        "\n",
        "`t.test`関数がある程度使えるようになれば、\n",
        "\n",
        "毎回$t$値を計算する式を書いたり、$t$分布のパーセント点を見たりする必要も無くなるので、必要に応じて習得してみてください。\n",
        "\n"
      ],
      "metadata": {
        "id": "2Sin4YaKGAgE"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 様々な$\\chi^2$検定\n",
        "\n",
        "$\\chi^2$分布は標本分散が従う分布であり、ばらつきについての検定によく用いられます。\n",
        "\n",
        "そのため、母分散に関する検定だけではなく、様々な検定に応用されています。\n",
        "\n",
        "その代表例として、今回は適合度検定と独立性検定を紹介します。"
      ],
      "metadata": {
        "id": "kZR9AR2-wE-p"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 適合度検定\n",
        "\n",
        "仮定された理論上の確率に対して、標本で得られた観察結果が適合しているか否かを検証する検定として、**適合度の$\\chi^2$検定**があります。\n",
        "\n",
        "例として、序盤に出てきたメンデルの法則に則るエンドウ豆のケースを考えてみましょう。\n",
        "\n",
        "<img src=\"https://github.com/slt666666/biostatistics_text_wed/blob/main/source/_static/images/chapter6/mendel.png?raw=true\" alt=\"title\" height=\"300px\">\n",
        "\n",
        "黄色&丸い豆のエンドウ豆と緑色&しわのあるエンドウ豆を交配し、F2集団を作成すると、\n",
        "\n",
        "各形質は理論上、$9:3:3:1$の割合で出現すると考えられます。\n",
        "\n",
        "そこで、実際にF2集団を作成して観察値が得られたときに、各形質の度数が理論的な割合と合致しているかを調べます。\n",
        "\n",
        "例えば556個体の標本を得て、下記の様なデータが得られたとします。\n",
        "\n",
        "\\begin{array}{lrr} \\hline\n",
        "   表現型 & 黄・丸 & 黄・しわ & 緑・丸 & 緑・しわ & 計 \\\\ \\hline\n",
        "   観測度数 & 315 & 101 & 108 & 32 & 556 \\\\\n",
        "   確率 & 9/16 & 3/16 & 3/16 & 1/16 & 1 \\\\\n",
        "   理論度数 & 312.75 & 104.25 & 104.25 & 34.75 & 556 \\\\ \\hline\n",
        "   度数の差 & 2.25 & -3.25 & 3.75 & -2.75 & 0 \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "こうしてみると、凡そ理論上の数値と近い観察結果が得られていることが分かります。\n",
        "\n",
        "この差が有意なのかどうかを調べる方法として、ピアソンの**適合度基準**を使用します。\n",
        "\n",
        "<br>\n",
        "\n",
        "\\begin{array}{lrr} \\hline\n",
        "   属性カテゴリー & A_1 & A_2 & ... & A_k & 計 \\\\ \\hline\n",
        "   観測度数 & f_1 & f_2 & ... & f_k & n \\\\\n",
        "   確率 & p_1 & p_2 & ... & p_k & 1 \\\\\n",
        "   理論度数 & np_1 & np_2 & ... & np_k & n \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "適合度基準とは、\n",
        "\n",
        "ある属性によって$n$個の個体が$k$種類のカテゴリー($A_1, A_2, ..., A_k$)に分類されるとき、\n",
        "\n",
        "標本から得られた各カテゴリー毎の度数を**観測度数**($f_1, f_2, ..., f_k$)とします。\n",
        "\n",
        "この観測度数が理論確率$p_1, p_2, ..., p_k$に適合するかを見るためには、\n",
        "\n",
        "理論確率通りに観察された場合の**理論度数(または期待度数)**$(np_1, np_2, ..., np_k$)と観測度数と比べる必要があります。\n",
        "\n",
        "ここで、適合度基準は\n",
        "\n",
        "$\\chi^2=\\sum_{i=1}^{k}\\dfrac{(f_i-np_i)^2}{np_i}$\n",
        "\n",
        "と定義されます。\n",
        "\n",
        "この適合度の$\\chi^2$統計量は$n$が大きいとき、自由度$k-1$の$\\chi^2$分布$\\chi^2(k-1)$に従うことが知られています。\n",
        "\n",
        "得られた観察結果が理論値と近ければ$\\chi^2$は小さい値となるので、適合度検定ではその大小を$\\chi^2$分布から調べる形になります。\n",
        "\n",
        "帰無仮説を\n",
        "\n",
        "$H_0:P(A_1)=p_1, P(A_2)=p_2..., P(A_k)=p_k$とすると、\n",
        "\n",
        "$\\chi^2 > \\chi_{\\alpha}^2(k-1)$であれば、帰無仮説(観測度数は理論上の値と適合している)は棄却されることになります。\n",
        "\n",
        "<br>\n",
        "\n",
        "メンデルの例で計算してみると、\n",
        "\n",
        "\\begin{array}{lrr} \\hline\n",
        "   表現型 & 黄・丸 & 黄・しわ & 緑・丸 & 緑・しわ & 計 \\\\ \\hline\n",
        "   観測度数 & 315 & 101 & 108 & 32 & 556 \\\\\n",
        "   確率 & 9/16 & 3/16 & 3/16 & 1/16 & 1 \\\\\n",
        "   理論度数 & 312.75 & 104.25 & 104.25 & 34.75 & 556 \\\\ \\hline\n",
        "   度数の差 & 2.25 & -3.25 & 3.75 & -2.75 & 0 \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "$\\chi^2 = \\dfrac{(2.25)^2}{312.75}+\\dfrac{(-3.25)^2}{104.25}+\\dfrac{(3.75)^2}{104.25}+\\dfrac{(-2.75)^2}{34.75}$\n",
        "\n",
        "$=0.470$\n",
        "\n",
        "自由度$4-1=3$の$\\chi^2$分布のパーセント点$\\chi_{0.05}^2(3)$をRの`qchisq`関数で見てみると\n",
        "\n",
        "```\n",
        "qchisq(パーセント, 自由度)\n",
        "```"
      ],
      "metadata": {
        "id": "E38388vvJa50"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度3のカイ二乗分布におけるα=0.05のパーセント点を求める\n",
        "qchisq(0.95, 3)"
      ],
      "metadata": {
        "id": "z-F2g20EQq4S",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "ba029be1-2017-4661-cd89-991f7e6fdaeb"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "7.81472790325118"
            ],
            "text/markdown": "7.81472790325118",
            "text/latex": "7.81472790325118",
            "text/plain": [
              "[1] 7.814728"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "ということで、$\\chi^2 > \\chi_{\\alpha}^2(k-1)$ではないので、帰無仮説は棄却されない。\n",
        "\n",
        "つまり、理論上の度数と大きく矛盾はしていないことになります。\n",
        "\n",
        "適合度の検定はしばしば\n",
        "\n",
        "$\\chi^2=\\sum(O-E)^2/E$\n",
        "\n",
        "と表されます。$O$はObservedで観測度数を表し、$E$はExpectedで期待度数を表します。"
      ],
      "metadata": {
        "id": "MFCI1iZYQ85_"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### 独立性検定\n",
        "\n",
        "$\\chi^2$検定はあるデータの持つ2つの属性の独立性を調べることもできます。\n",
        "\n",
        "$n$個の標本について、2つの属性$A, B$を測定したとします。\n",
        "\n",
        "例えば属性$A$が葉の色という属性、属性$B$が病気にかかりやすいかどうか、という属性を測定した場合\n",
        "\n",
        "下の様な、記述統計学の際にも触れた分割表を得ることができます。\n",
        "\n",
        "\\begin{array}{c|ccc|c}  \\hline\n",
        "  属性 & 病気に強い & 中くらい & 病気に弱い &　計\\\\ \\hline \\hline\n",
        "  葉の色・緑 & 4 & 2 & 3 & 9 \\\\\n",
        "  葉の色・黄 & 8 & 4 & 6 & 18 \\\\\n",
        "  葉の色・茶 & 6 & 3 & 6 & 15 \\\\ \\hline\n",
        "  計 & 18 & 9 & 15 & 42 \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "この分割表において**独立**というのがどういう意味かというと、\n",
        "\n",
        "病気の強さと葉の色には関連が無い、ということを指します。\n",
        "\n",
        "一般化すると、下の様な一般的な分割表において\n",
        "\n",
        "\\begin{array}{c|cccc|c}  \\hline\n",
        "  属性 & B_1 & B_2 & ... & B_c & 計\\\\ \\hline \\hline\n",
        "  A_1 & f_{11} & f_{12} & ... & f_{1c} & \\sum\\limits_{j}f_{1j} \\\\\n",
        "  A_2 & f_{21} & f_{22} & ... & f_{2c} & \\sum\\limits_{j}f_{2j}\\\\\n",
        "  ... & ... & ... & ... & ... & ... \\\\\n",
        "  A_r & f_{r1} & f_{r2} & ... & f_{rc} & \\sum\\limits_{j}f_{rj}\\\\ \\hline\n",
        "  計 & \\sum\\limits_{i}f_{i1} & \\sum\\limits_{i}f_{i2} & ... & \\sum\\limits_{i}f_{ic} & n \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "帰無仮説$H_0:すべてのi, jに対し、\n",
        "P(A_i\\cap B_j)=P(A_i)P(B_j)$\n",
        "\n",
        "であることを指します。\n",
        "\n",
        "つまり、$A_1, A_2, ..., A_r$になる確率はどの$B$に対しても変わらないことを意味します。\n",
        "\n",
        "当然、$B_1, B_2, ..., B_c$になる確率はどの$A$に対しても変わらないことも意味します。\n",
        "\n",
        "<br>\n",
        "\n",
        "この独立性を検定するには、\n",
        "\n",
        "まず、$P(A_i)=p_{i\\cdot}, P(B_j)=p_{\\cdot j}$とすると、相対度数から$p_{i\\cdot},p_{\\cdot j}$を推定し\n",
        "\n",
        "$\\hat{p_{i\\cdot}}=f_{i\\cdot}/n$　　$\\hat{p_{\\cdot j}}=f_{\\cdot j}/n$\n",
        "\n",
        "と計算することができます。\n",
        "\n",
        "帰無仮説$H_0$が成立している場合、$A_i$かつ$B_j$の属性の理論度数(期待度数)は、\n",
        "\n",
        "$E_{ij} = n\\hat{p_{i\\cdot}}\\hat{p_{\\cdot j}}=f_{i\\cdot}f_{\\cdot j}/n$\n",
        "\n",
        "と計算できます。\n",
        "\n",
        "\\begin{array}{c|ccc|c}  \\hline\n",
        "  属性 & B_1 & ... & B_c & 確率\\\\ \\hline \\hline\n",
        "  A_1 & E_{11}=n\\hat{p_{1\\cdot}}\\hat{p_{\\cdot 1}} & ... & E_{1c}=n\\hat{p_{1\\cdot}}\\hat{p_{\\cdot c}} & \\hat{p_{1\\cdot}} = \\sum\\limits_{j}f_{1j}/n \\\\\n",
        "  ... & ... & ... & ... & ... \\\\\n",
        "  A_r & E_{r1}=n\\hat{p_{r\\cdot}}\\hat{p_{\\cdot 1}} & ... & E_{rc}=n\\hat{p_{r\\cdot}}\\hat{p_{\\cdot c}} & \\hat{p_{r\\cdot}} = \\sum\\limits_{j}f_{rj}/n \\\\ \\hline\n",
        "  確率 & \\hat{p_{\\cdot 1}} = \\sum\\limits_{i}f_{i1}/n & ... & \\hat{p_{\\cdot c}} = \\sum\\limits_{i}f_{ic}/n &  \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "ここで、適合度の検定(独立している状況に適合しているか)を用いることで、\n",
        "\n",
        "**独立性の$\\chi^2$検定**を行うことができます。\n",
        "\n",
        "この時統計量は\n",
        "\n",
        "$\\chi^2=\\sum\\limits_{i}\\sum\\limits_{j}\\dfrac{(f_{ij}-f_{i\\cdot}f_{\\cdot j}/n)^2}{f_{i\\cdot}f_{\\cdot j}/n}$\n",
        "\n",
        "$=\\sum\\limits_{i}\\sum\\limits_{j}\\dfrac{(nf_{ij}-f_{i\\cdot}f_{\\cdot j})^2}{nf_{i\\cdot}f_{\\cdot j}}$\n",
        "\n",
        "として計算されます。\n",
        "\n",
        "この$\\chi^2$が従う$\\chi^2$分布の自由度は、$(r-1)(c-1)$となります。\n",
        "\n",
        "<br>\n",
        "\n",
        "葉の色・病気の分割表の例でみてみると、\n",
        "\n",
        "\\begin{array}{c|ccc|c}  \\hline\n",
        "  期待度数 & 病気に強い & 中くらい & 病気に弱い & 確率 \\\\ \\hline \\hline\n",
        "  葉の色・緑 & 9\\times18/42 & 9\\times9/42 & 9\\times15/42 & 9/42 \\\\\n",
        "  葉の色・黄 & 18\\times18/42 & 18\\times9/42 & 18\\times15/42 & 18/42 \\\\\n",
        "  葉の色・茶 & 15\\times18/42 & 15\\times9/42 & 15\\times15/42 & 15/42 \\\\ \\hline\n",
        "  確率 & 18/42 & 9/42 & 15/42 & n=42 \\\\ \\hline\n",
        "\\end{array}\n",
        "\n",
        "$\\chi^2=\\dfrac{(4-9\\cdot18/42)^2}{9\\cdot18/42}+\\dfrac{(2-9\\cdot9/42)^2}{9\\cdot9/42}+...+\\dfrac{(6-15\\cdot15/42)^2}{15\\cdot15/42}=0.19$\n",
        "\n",
        "自由度は$(3-1)(3-1)=4$なので、$\\chi^2(4)$のパーセント点を`qchisq`関数で求めると、\n",
        "\n",
        "```\n",
        "qchisq(確率, 自由度)\n",
        "```"
      ],
      "metadata": {
        "id": "jz45TDonGtoS"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 自由度4のカイ2乗分布のパーセント点を求める\n",
        "qchisq(0.95, 4)"
      ],
      "metadata": {
        "id": "B1S7oku4zEJ5",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        },
        "outputId": "b2a4a24b-7b64-4e91-c732-2f4487a539ef"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "9.48772903678115"
            ],
            "text/markdown": "9.48772903678115",
            "text/latex": "9.48772903678115",
            "text/plain": [
              "[1] 9.487729"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "となり、有意水準5%で葉の色と病気のなりやすさの独立性の仮説は棄却されず、\n",
        "\n",
        "両者に関係があるとはいえない、と評価される。"
      ],
      "metadata": {
        "id": "q9w5AbWmZjcc"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "#### Rによる$\\chi^2$検定\n",
        "\n",
        "適合度検定や独立性検定はRの`chisq.test`という関数を使用する事でも実施できます。\n",
        "\n",
        "例えば、先ほどの黄丸・黄しわ・緑丸・緑しわの観測値が$315, 101, 108, 32$であったとき、\n",
        "\n",
        "これが$9:3:3:1$に適合しているかを検定する場合は\n",
        "\n",
        "```\n",
        "chisq.test(x=観測値,p=各観測値の確率)\n",
        "```\n",
        "\n",
        "という形で実施できます。"
      ],
      "metadata": {
        "id": "QKpiQhuzpbsD"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 適合度検定をchisq.testで実施する\n",
        "chisq.test(x=c(315, 101, 108, 32), p=c(9/16, 3/16, 3/16, 1/16))"
      ],
      "metadata": {
        "id": "0Vn0Cy2qqVQw"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "と、$\\chi^2$値と$p$値が求まるので、検定を実施できます。\n",
        "\n",
        "また、独立性の検定の場合は`matrix`関数でデータの行列を作成し、`chisq.test`関数に与えます。\n",
        "\n",
        "```\n",
        "chisq.test(matrixデータ)\n",
        "```"
      ],
      "metadata": {
        "id": "abRbKClHqnDA"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 独立性の検定をchisq.testで実施する\n",
        "x <- matrix(c(4, 2, 3, 8, 4, 6, 6, 3, 6),nrow=3,byrow=T)\n",
        "chisq.test(x)"
      ],
      "metadata": {
        "id": "RVTOASYMqfkx",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 148
        },
        "outputId": "1e675a7e-dc55-4f17-d1ad-c535f668ee0e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "Warning message in chisq.test(x):\n",
            "“Chi-squared approximation may be incorrect”\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "\n",
              "\tPearson's Chi-squared test\n",
              "\n",
              "data:  x\n",
              "X-squared = 0.18667, df = 4, p-value = 0.9959\n"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "こちらも同様に、$\\chi^2$値と$p$値が求まるので、検定を実施できます。\n"
      ],
      "metadata": {
        "id": "gdluVXSRrW8H"
      }
    }
  ]
}