t_test.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": [],
      "authorship_tag": "ABX9TyPSzg/6guADLqRDWd4Zhlw4"
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# t検定\n",
        "２群間の平均値の差の検定"
      ],
      "metadata": {
        "id": "yLpoRGaAZ8J4"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# ライブラリの読み込み\n",
        "import numpy as np\n",
        "import pandas as pd\n",
        "from scipy import stats"
      ],
      "metadata": {
        "id": "xU3THRUnmuIn"
      },
      "execution_count": 50,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## サンプルデータの生成\n",
        "numpy の randomモジュールを使って、２グループ分の「正規乱数」を生成します。\n",
        "\n",
        "\n",
        "ｔ検定の仮定として、データは以下のようなものとします\n",
        "* 母集団が正規分布に従っている（母集団の正規性の仮定）\n",
        "* 二群の分散が等しい（分散の均一性の仮定）\n",
        "* サンプルが独立に抽出されている（サンプルの独立性の仮定）"
      ],
      "metadata": {
        "id": "_XxQHOxraFDH"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 54,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 283
        },
        "id": "CAGRhlX3Xbv0",
        "outputId": "eeb034d8-88de-42f9-f002-bf487adf5678"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<AxesSubplot:>"
            ]
          },
          "metadata": {},
          "execution_count": 54
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "# Group A のデータ生成\n",
        "a = np.random.normal(\n",
        "    loc = 75, #平均\n",
        "    scale = 10, #標準偏差\n",
        "    size = 30, #データサイズ\n",
        "    )\n",
        "pd.Series(a).hist()\n",
        "\n",
        "# Group B のデータ生成\n",
        "b = np.random.normal(\n",
        "    loc = 68, #平均\n",
        "    scale = 12, #標準偏差\n",
        "    size = 30, #データサイズ\n",
        "    )\n",
        "pd.Series(b).hist()"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 実質的な差の評価\n",
        "ここで差がなければ、検定をする意味がないので・・"
      ],
      "metadata": {
        "id": "sodc3abWci2-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 各グループの平均を確認\n",
        "print(f'Group A の平均 = {a.mean():.2f}')\n",
        "print(f'Group B の平均 = {b.mean():.2f}')\n",
        "print(f'2群の平均の差 = {(abs(a.mean()-b.mean())):.2f}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "s5CZEaitcdYn",
        "outputId": "d40cef55-44f1-4d80-88e9-95db920b4dc7"
      },
      "execution_count": 55,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Group A の平均 = 73.48\n",
            "Group B の平均 = 64.21\n",
            "2群の平均の差 = 9.28\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 対応のある t 検定\n",
        "Group A と Group B には対応があるという前提で検定します。  \n",
        "つまり、同じメンバーに対して異なる条件で行なった実験結果と考えます。"
      ],
      "metadata": {
        "id": "6SJaS73fgiH-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 対応のある検定は ttest_rel\n",
        "t, p = stats.ttest_rel(a, b)\n",
        "\n",
        "print('Studentのt検定｜対応あり')\n",
        "print(f'p値 = {p:.3f}')\n",
        "print(f't値 = {t:.2f}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3NwvysbmeDFb",
        "outputId": "be4aef7b-b5e2-40d3-890d-d9bdaa603a00"
      },
      "execution_count": 56,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Studentのt検定｜対応あり\n",
            "p値 = 0.010\n",
            "t値 = 2.77\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "実行例の場合、p<0.05 で「有意差あり」（帰無仮説は棄却）と判定されます。"
      ],
      "metadata": {
        "id": "yRQwFQJAns1p"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 対応のない t 検定\n",
        "\n",
        "Group A と Group B には対応がないという前提で検定します。  *斜体テキスト*\n",
        "つまり、異なるメンバーにそれぞれ異なる条件で行なった実験結果と考えます。"
      ],
      "metadata": {
        "id": "qHDLnYMshnku"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 対応のない検定は ttest_ind\n",
        "t, p = stats.ttest_ind(a, b, equal_var=True)\n",
        "\n",
        "print('Studentのt検定｜対応なし')\n",
        "print(f'p値 = {p:.3f}')\n",
        "print(f't値 = {t:.2f}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "aNnV32J3hsdn",
        "outputId": "3279d3ed-a745-467d-ac4e-6f1b42de8359"
      },
      "execution_count": 57,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Studentのt検定｜対応なし\n",
            "p値 = 0.009\n",
            "t値 = 2.69\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "実行例の場合、p<0.05 で「有意差あり」（帰無仮説は棄却）と判定されます。"
      ],
      "metadata": {
        "id": "iZ7YV33XogOD"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 対応のない t 検定（等分散性なし）\n",
        "等分散性がないと仮定した場合には、Welchの t検定はを利用します。  \n",
        "ttest_ind 関数の引数 equal_var を False とすることで Welch の方法が採用されます。"
      ],
      "metadata": {
        "id": "NVaDSgirfe7K"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# 等分散性が仮定できない場合 ttest_ind で equal_var=False\n",
        "t, p = stats.ttest_ind(a, b, equal_var=False)\n",
        "\n",
        "print('Welchのt検定｜対応なし｜等分散性なし')\n",
        "print(f'p値 = {p:.3f}')\n",
        "print(f't値 = {t:.2f}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "iXDXS_66gYU3",
        "outputId": "f2c72833-9733-4f9f-9eb7-85588af49f50"
      },
      "execution_count": 58,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Welchのt検定｜対応なし｜等分散性なし\n",
            "p値 = 0.010\n",
            "t値 = 2.69\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "実行例の場合、p<0.05 で「有意差あり」（帰無仮説は棄却）と判定されます。"
      ],
      "metadata": {
        "id": "ISflb9Iyoh7S"
      }
    }
  ]
}