2024_two_sided_ab_test.ipynb

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  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# **<font color='crimson'>Алгоритм получения желаемых метрик A/B-теста для двусторонней гипотезы, проверяющей монету на \"честность\"</font>**\n",
        "\n",
        "---\n",
        "\n",
        "**Выполнил**: Юмаев Егор\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "lp7UQIboH6K8"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## <font color='green'>**1 Моделирование поведения \"честной\" монеты с оценкой False Positive Rate**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "sZV-nhKFOt8o"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "YkLhK7KaHqNp"
      },
      "outputs": [],
      "source": [
        "import pandas as pd\n",
        "import numpy as np\n",
        "from tqdm.notebook import tqdm"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### <font color='green'>**1.1 Оценка False Positive Rate (1 - Specificity)**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "y-9KOGYsX6sk"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Поведение \"честной\" монеты можно описать с помощью биномиального распределения:"
      ],
      "metadata": {
        "id": "Nossjr62O9qQ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.binomial(1, 0.5, size=10)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "hHRm87GvHsN9",
        "outputId": "61d8f127-4d22-4697-b778-3a29b494fb7a"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([1, 1, 1, 0, 0, 1, 1, 0, 0, 0])"
            ]
          },
          "metadata": {},
          "execution_count": 2
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "В качестве базового сценария рассмотрим подход, когда одному эксперименту будет соответствовать 10 бросков монеты."
      ],
      "metadata": {
        "id": "BFjUhzdsl8pZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Если убрать параметр size, переместив количество повторений на место единицы (1), то функция np.random.binomial будет возвращать количество положительных исходов из серии запусков (количество \"орлов\" в случае \"честной\" монеты):"
      ],
      "metadata": {
        "id": "LRmeN61vPhiC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "np.random.binomial(10, 0.5)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "43mdCo7GHsQn",
        "outputId": "b3633424-79c4-4c09-e82c-9d18b2ea8ad1"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "5"
            ]
          },
          "metadata": {},
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Сформулируем **гипотезы** для исследуемой монеты:\n",
        "\n",
        "* **Н0**: вероятность выпадения \"орла\" равна 50% (монета \"честная\")\n",
        "\n",
        "* **Н1**: вероятность выпадения \"орла\" **не** равна 50% (монета **не** честная)"
      ],
      "metadata": {
        "id": "cnmF_NdNQtaO"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\"Честная\" монета или \"не честная\" определяется тем, насколько, например, при 10 бросках, количество положительных исходов отличается от 5 (50%).\n",
        "\n",
        "Сделаем предположение, что если по итогам 10 бросков количество положительных исходов равно 5 (отклонение 0), 4 или 6 (отклонение 1), 3 или 7 (отклонение 2), 2 или 8 (отклонение 3), - все такие запуски будут свидетельствовать в пользу **Н0**.\n",
        "\n",
        "Если по итогам 10 бросков количество положительных исходов равно 1 или 9 (отклонение 4), 0 или 10 (отклонение 5), - такие запуски будут свидетельствовать в пользу **Н1**.\n"
      ],
      "metadata": {
        "id": "fuFj64fYRrMq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Смоделируем 100_000 экспериментов, во время каждого эксперимента \"честная\" монета будет \"подбрасываться\" 10 раз с подсчетом количества положительных исходов. Затем оценим отклонения полученных исходов от 5. Вычислим процент ситуаций, когда отклонение от 5 больше или равно 4."
      ],
      "metadata": {
        "id": "80it0BvqQQwt"
      }
    },
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      "cell_type": "code",
      "source": [
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(10, 0.5))"
      ],
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        "outputId": "f574ddd7-4fad-4159-951a-609f857f4e20"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
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    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "t37QR46THsW2"
      },
      "execution_count": 5,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Каждая строчка датафрейма аккумулирует сведения о количестве выпавших \"орлов\" из 10 подбрасываний монеты."
      ],
      "metadata": {
        "id": "0DTE5aTUUOGj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# выведем последние 5 строк датафрейма\n",
        "df.tail()"
      ],
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        "colab": {
          "base_uri": "https://localhost:8080/",
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              "      display:flex;\n",
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              "      buttonEl.style.display =\n",
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              "        const element = document.querySelector('#df-721971e0-0153-446e-922f-736bc4939cc1');\n",
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              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
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              "  </button>\n",
              "\n",
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              "      --disabled-bg-color: #3B4455;\n",
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              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
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              "    display: none;\n",
              "    fill: var(--fill-color);\n",
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              "    width: 32px;\n",
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              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
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              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
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              "    background-color: var(--disabled-bg-color);\n",
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              "\n",
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              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
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              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-d06ac899-4b81-44a8-809f-842f89617d3d button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "dataframe",
              "summary": "{\n  \"name\": \"df\",\n  \"rows\": 5,\n  \"fields\": [\n    {\n      \"column\": 0,\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2,\n        \"min\": 3,\n        \"max\": 8,\n        \"num_unique_values\": 4,\n        \"samples\": [\n          7,\n          4,\n          8\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# вычисляем число отклонений для каждого эксперимента\n",
        "df['deviation'] = abs(5 - df[0])\n",
        "df.tail()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 206
        },
        "id": "zOmW6F8uHscz",
        "outputId": "31c1ee03-cbd4-491d-eab8-faba080be862"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "       0  deviation\n",
              "99995  8          3\n",
              "99996  7          2\n",
              "99997  7          2\n",
              "99998  3          2\n",
              "99999  4          1"
            ],
            "text/html": [
              "\n",
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              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
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              "\n",
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              "        vertical-align: top;\n",
              "    }\n",
              "\n",
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              "        text-align: right;\n",
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              "</style>\n",
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              "      <th></th>\n",
              "      <th>0</th>\n",
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              "\n",
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              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
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              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-3bb362a6-60ae-45e1-be6a-be98a47bf11c button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-3bb362a6-60ae-45e1-be6a-be98a47bf11c');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
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              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
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              "\n",
              "\n",
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              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
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              "\n",
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              "    width: 32px;\n",
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              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
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              "    fill: var(--button-hover-fill-color);\n",
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              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-19782b80-c83b-499d-b159-04ba41a3cddc button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "dataframe",
              "summary": "{\n  \"name\": \"df\",\n  \"rows\": 5,\n  \"fields\": [\n    {\n      \"column\": 0,\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 2,\n        \"min\": 3,\n        \"max\": 8,\n        \"num_unique_values\": 4,\n        \"samples\": [\n          7,\n          4,\n          8\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    },\n    {\n      \"column\": \"deviation\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0,\n        \"min\": 1,\n        \"max\": 3,\n        \"num_unique_values\": 3,\n        \"samples\": [\n          3,\n          2,\n          1\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно 4\n",
        "(df['deviation'] >= 4).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BcwlgEazHsfe",
        "outputId": "3aa2e87f-bdfd-4726-f0d2-5627127adda4"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.02179"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "В 2.2% случаев фиксируются значительные отклонения от 5.\n",
        "\n",
        "С самого начала эксперимента мы знаем, что монета является \"честной\", - это отражено в параметрах заданного биномиального распределения. Однако в 2.2% случаев \"честная\" в реальности монета может быть ошибочно признана \"нечестной\". Это и будет оценкой **False Positive Rate**."
      ],
      "metadata": {
        "id": "7JY6wV9sWy1G"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### <font color='green'>**1.2 Оценка True Positive Rate (Sensitivity)**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "1y_EEDdPYJF4"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Чтобы оценить True Positive Rate, необходимо, чтобы монета была \"немного нечестной\". \"Нечестность\" можно определить в фунции np.random.binomial, задав вероятность выпадения положительного исхода, например, на уровне 0.6."
      ],
      "metadata": {
        "id": "CXlaG4idR6md"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# зададим размер MDE, при добавлении которого к вероятности 0.5\n",
        "# монета перестанет быть честной\n",
        "# при заданном теоретическом распределении\n",
        "mde = 0.1"
      ],
      "metadata": {
        "id": "U2sv7EP5TT9D"
      },
      "execution_count": 9,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(10, 0.5 + mde))"
      ],
      "metadata": {
        "id": "Ck34aSrSWb4l",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "8537f0e6733944e2942cbc731ac7d337",
            "0402beb8b53e4410af4283753fc95537",
            "583af66ae32a41ce9a02605630d7f013",
            "9e42e4c89d80432db6a22f318b7033d5",
            "45d9bfee88034ba898445c0d1c1a8c68",
            "4ceb5e7f57b84249a942033bb66f31f6",
            "995748975b62466eab50ed0bc670aea0",
            "4012d7e792264d55b64d3b4ffd5f2370",
            "b5d43fba09a14e77b4390ebe5037c27b",
            "cc7ad83affb04446bc387eedca2b2483",
            "75fccd722afe419091c5642d1611b3a9"
          ]
        },
        "outputId": "4f5d0d9a-9f84-409d-d4d8-cb22ea5fe55f"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "8537f0e6733944e2942cbc731ac7d337"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "6If2X8xdWb7Q"
      },
      "execution_count": 11,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# считаем отклонения\n",
        "df['deviation'] = abs(5 - df[0])"
      ],
      "metadata": {
        "id": "o0UCUJJPWb-b"
      },
      "execution_count": 12,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "В предыдущем пункте, когда вычислили долю экстремальных отклонений от значения 5, мы получили оценку **False Positive Rate**, т.к., моделируя эксперимент, изначально знали, что монета честная. Для этого вероятнотсь выпадения успешного исхода в функции np.random.binomial() была установлена на уровне 0.5.\n",
        "\n",
        "Теперь, на этапе подготовки эксперимента, мы указали эту вероятность равной 0.6. С самого начала знаем, что монета нечестная. Поэтому вычисление доли экстремальных отклонений от значения 5 дает оценку **True Positive Rate**."
      ],
      "metadata": {
        "id": "B8_lOykPUvgX"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно 4\n",
        "(df['deviation'] >= 4).mean()"
      ],
      "metadata": {
        "id": "0Kh_SfXZWcBe",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "3aac4c4b-ad80-4943-9ae4-272a2af18cf5"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.04894"
            ]
          },
          "metadata": {},
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Только лишь в 5% случаев мы смогли нечестную в реальности монету назвать нечестной. **Вывод**: при столь небольшой выборке, которую задали в функции np.random.binomial(), выявить маленький эффект (MDE) правильно не получится. Минимально необходимая мощность в соответствии со сложившейся практикой должна быть 80% (на 75% больше полученного результата)."
      ],
      "metadata": {
        "id": "aZ5WMgQOWPFS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Повысим значение MDE и повторно рассчитаем мощность."
      ],
      "metadata": {
        "id": "WkON05O2XGJG"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "mde = 0.2\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(10, 0.5 + mde))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "e1e538c5829d43bb89d199c13dafe574",
            "6937c98b7da04098b2530729cbdb147e",
            "91ae2e7396f24ddfb35c9253d146c087",
            "826d7943c7b54c7eba377d4bda12cc18",
            "59966f50d69c4761b3db1d907a46aedf",
            "3fa6e6fd33604de3b789323bca048f46",
            "977611a2ff454bc6b1ddbcf02486d3dd",
            "76ca00da78144469bcafe215c8e61b5c",
            "029d025ab0e34fabb257bf8755d74487",
            "4b505ba92a86462191f6171c8520f5d4",
            "6dba952480004386bbee414fb30c2c9d"
          ]
        },
        "outputId": "3090660c-430b-4fee-cf84-4ce0bf652eb5",
        "id": "FY7jtIKrXSta"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "e1e538c5829d43bb89d199c13dafe574"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "AUWxwvBhXStg"
      },
      "execution_count": 15,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# считаем отклонения\n",
        "df['deviation'] = abs(5 - df[0])"
      ],
      "metadata": {
        "id": "JytsIl7jXStg"
      },
      "execution_count": 16,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно 4\n",
        "(df['deviation'] >= 4).mean()"
      ],
      "metadata": {
        "id": "R3BZzFaFWcE1",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "f8cccd1d-56de-433d-d773-bdf43e86192a"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.15009"
            ]
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Увеличение MDE без увеличения размера выборки в функции np.random.binomial() позволило повысить мощность сразу в три раза."
      ],
      "metadata": {
        "id": "1xoc3hQiXlR9"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Увеличить **True Positive Rate** можно, понизив **False Positive Rate**. Для этого необходимо понизить порог, после которого разница 5 и среднего значения каждого отдельного эксперимента признается экстремальной. Понизим порог с 4 до 3."
      ],
      "metadata": {
        "id": "czQCgtg5YmEV"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно 3 при MDE = 0.2\n",
        "(df['deviation'] >= 3).mean()"
      ],
      "metadata": {
        "id": "2evFxjAlHsiT",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "b4c0b89e-4182-4d22-9e5f-61c2389de174"
      },
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.38332"
            ]
          },
          "metadata": {},
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "С пониженным порогом мощность выросла более чем в два раза, но все еще не дотягивает до общепринятого рубежа в 80%."
      ],
      "metadata": {
        "id": "2Jk7lh3QZYFK"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Однако, мы не можем только для **True Positive Rate** изменить порог. Порог для **False Positive Rate** тоже изменится."
      ],
      "metadata": {
        "id": "W8QnAurDZnh3"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Посчитаем, как изменится **False Positive Rate** с новым порогом."
      ],
      "metadata": {
        "id": "oukOfVI1aELC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# для False Positive Rate\n",
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(10, 0.5))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "fe628fec18304aa3b27fa2d1c41bacdc",
            "b218215b413b4c8a849ececdb8ababb6",
            "51bee7ab1d42476aa91502ed10258ff0",
            "904fcbffd27d4969876a792537c75e2a",
            "9e956ac4909b423ea6274960e08c2afd",
            "497fe5b9262c4ea48b5e6508467d9fc7",
            "5d63c5972a7448ffb77d5f813a7908f6",
            "bdd3c9b9c7124cf2ad375ffaaf6fc66e",
            "95728bc032164135906c8f4895fe505a",
            "2d4233bed3944978a263158cee21f990",
            "7a064e9067f5453bbe5fe2fe74678c50"
          ]
        },
        "outputId": "162ab2c7-0ec0-4fb4-d19d-82457f06fcc5",
        "id": "J3nOQtgqaQPm"
      },
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "fe628fec18304aa3b27fa2d1c41bacdc"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "0ddC949iaQPn"
      },
      "execution_count": 20,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычисляем число отклонений для каждого эксперимента\n",
        "df['deviation'] = abs(5 - df[0])"
      ],
      "metadata": {
        "id": "oRKBAe4xaQPn"
      },
      "execution_count": 21,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно 3\n",
        "(df['deviation'] >= 3).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "d310920d-a9ef-4e4b-9349-5cec3df5bd77",
        "id": "qPaE9zFPaQPn"
      },
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.11145"
            ]
          },
          "metadata": {},
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Количество случаев, когда честная монета будет признана нечестной (**False Positive Rate**), выросло с 2.2% до 11.1%. Таким образом, мы можем пожертвовать точностью статистического теста для получения желаемого размера мощности."
      ],
      "metadata": {
        "id": "vQUgry6Ca2jz"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Снова рассчитаем **True Positive Rate**. На этот раз MDE установим еще выше: 0.3."
      ],
      "metadata": {
        "id": "dCuNNeyEbnAF"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "mde = 0.3\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(10, 0.5 + mde))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "f908b6f65a2b42efae6ab08eb9caed20",
            "5c57dc507ca7435ba6f0cb26003a1b25",
            "b08886e5d9fd4f3bb076577ddc110a4d",
            "9eb582b8fbc7435d89fc2f63b9010a84",
            "409042e6f2ae427f915f106d4020f34f",
            "83508890ee6b46c49f8365407530757b",
            "03f0e9a6fd5245a290b5723c2578c523",
            "418232c7be47478eab111bf6a7eb03d7",
            "3d28ad41737e495bbf441cac7372ed67",
            "ca83711f9fc04a67bc8bf6580f5955b4",
            "aea8e3de4169493695ea2f31b1e37282"
          ]
        },
        "outputId": "53ab546d-3d3c-49a2-f834-b142a4d2096f",
        "id": "Cmiqz_GLb6do"
      },
      "execution_count": 23,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "f908b6f65a2b42efae6ab08eb9caed20"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "AWaVnHw0b6dp"
      },
      "execution_count": 24,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# считаем отклонения\n",
        "df['deviation'] = abs(5 - df[0])"
      ],
      "metadata": {
        "id": "8PeUNkSzb6dp"
      },
      "execution_count": 25,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно 3\n",
        "(df['deviation'] >= 3).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "de3ff62a-6162-4a09-fdcd-4489f84a0831",
        "id": "lB6Fonjdb6dp"
      },
      "execution_count": 26,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.67735"
            ]
          },
          "metadata": {},
          "execution_count": 26
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Одновременное увеличение MDE и снижение порога позволило увеличить мощность теста до 67%. В 67% случаев нечестная монета называется нечестной. Однако размер MDE оставляет желать лучшего, но это лучший результат, какой мы можем получить при выборке size = 10 в функции np.random.binomial().\n",
        "\n",
        "Для улучшения результатов эксперимента необходимо увеличить размер выборки.\n",
        "\n",
        "Разработаем алгоритм, которая позволит получить стандартные alpha, beta и  power при MDE = 0.1 (10%)."
      ],
      "metadata": {
        "id": "ZbRhUPM2cW0O"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## <font color='green'>**2 Алгоритм получения желаемых метрик A/B-теста, проверяющего монету на \"честность\"**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "TyQp6op8jEZK"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Зафиксируем желаемые значения метрик A/B-теста (процедуры проверки монеты на \"честность\"):\n",
        "\n",
        "* **MDE**: 10%\n",
        "\n",
        "* **Power (1 - beta)**: 80%\n",
        "\n",
        "* **Significance (alpha)**: 5%"
      ],
      "metadata": {
        "id": "nPrTDCU3jXLq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Требуется подобрать:\n",
        "\n",
        "* **размер выборки**\n",
        "\n",
        "* **отклонение от нормы**."
      ],
      "metadata": {
        "id": "1mlZwH3NkCad"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Начнем с подбора отклонения от нормы. Для этого зафиксируем первоначальный размер выборки на уровне 100 (по итогам ранее проведенных экспериментов мы знаем, что размер выборки 10 точно недостаточен)."
      ],
      "metadata": {
        "id": "2XwjifiekQau"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### <font color='green'>**2.1 Подбор FPR (Significance) (1-я итерация)**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "NL9TepPalMpp"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# размер выборки\n",
        "sample_size = 100"
      ],
      "metadata": {
        "id": "R4XiHi11HsrL"
      },
      "execution_count": 27,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# для False Positive Rate\n",
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(sample_size, 0.5))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "bfa675fd8ac44c3180d293c5a623c19a",
            "1434c187635842d39bb67b6c23a6a3b2",
            "e2a28c694ebb4c0f89d9f038b8231516",
            "0ed5a12e204a41aabf5d9b62c08e2a50",
            "eaa406463c7045879a595a30ed690432",
            "18918008143747f789c44f1ba1c05572",
            "1bbb145f0ab645199d477aa689af21ec",
            "68d002c8b3ae4fa1bfe72eec40845594",
            "e74aa98e6a7546ed82c1f6ebdd5d028f",
            "5f625080728e4c3c8273924c52d3f26f",
            "72578fa0b79e4c45a4c0ad83463b6db1"
          ]
        },
        "outputId": "cfc4ee3d-33cf-44dc-e282-82e9cd94b30d",
        "id": "inkTOSFFlir7"
      },
      "execution_count": 28,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "bfa675fd8ac44c3180d293c5a623c19a"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "NoScgvwplisD"
      },
      "execution_count": 29,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Т.к. размер выборки изменился, теперь величину 10 необходимо заменить на размер sample_size, умноженный на необходимый процент."
      ],
      "metadata": {
        "id": "9Ac9vFVlmF52"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# вычисляем число отклонений для каждого эксперимента\n",
        "df['deviation'] = abs(sample_size * 0.5 - df[0])"
      ],
      "metadata": {
        "id": "HxuWPCfblisD"
      },
      "execution_count": 30,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно вручную подобранному порогу\n",
        "(df['deviation'] >= sample_size * 0.11).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "828717f7-1713-47b5-a108-a54a0924fcea",
        "id": "pmmLVONUlisD"
      },
      "execution_count": 31,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.03379"
            ]
          },
          "metadata": {},
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "При размере выборки, равной 100 и пороге отклонения больше либо равном 11 **FPR (Significance)** равен **3.4%**."
      ],
      "metadata": {
        "id": "vtZGN_sCnGVd"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### <font color='green'>**2.2 Подбор TPR  и MDE (1-я итерация)**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "6wHJ5xE1nmpT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Подбирать TPR и MDE будем одновременно, т.к. они связаны."
      ],
      "metadata": {
        "id": "vTLW8zEqnuoN"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Порог отклонения зададим равным значению, подобранному в предыдущем пункте: sample_size * 0.11."
      ],
      "metadata": {
        "id": "Jm3e8ZrPoUZ5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "mde = 0.1\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(sample_size, 0.5 + mde))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "39027b1906444924ba1ea31f5484b5fa",
            "8fa336e8b2fc49678ef08a54a2d029df",
            "20106c1484ea41f881b93433f1da6f45",
            "51a4ef22ad9b4495af19f52ada5996f9",
            "e8528cba67ad41b48fbce21a54ff9b54",
            "32604ed0e6e441898ffd98210b15e345",
            "9a582591987d4bd18d3d0adfe7a62336",
            "302a6f5ee7ea4267af6257da2eb4feab",
            "b831cb2dd5f046c287468c1ff2dce117",
            "c67d71914d17449b9bd7cbf43c290431",
            "719b420071f3417d90ebb70022a08a49"
          ]
        },
        "outputId": "7862136e-f96d-4793-e02b-5b411c88b476",
        "id": "QuNw6hjXn8aN"
      },
      "execution_count": 32,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "39027b1906444924ba1ea31f5484b5fa"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "MdGhAuFcn8aU"
      },
      "execution_count": 33,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# считаем отклонения\n",
        "df['deviation'] = abs(sample_size * 0.5 - df[0])"
      ],
      "metadata": {
        "id": "-k1o6W9an8aU"
      },
      "execution_count": 34,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от 5\n",
        "# больше или равно вручную подобранному порогу\n",
        "(df['deviation'] >= sample_size * 0.11).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "03b70cd0-ec9a-48cb-e0a0-32b094d7bac2",
        "id": "ZkOIC0XOn8aV"
      },
      "execution_count": 35,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.46202"
            ]
          },
          "metadata": {},
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Получена мощность, равная 46%. Этого недостаточно, т.к. запланировано значение мощности в 80%.\n",
        "\n",
        "Очевидно, необходимо увеличить размер выборки. Релаизуем вторую итерацию подбора желаемых значений метрик A/B-теста."
      ],
      "metadata": {
        "id": "1oSJ9-2AoxPL"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### <font color='green'>**2.3 Подбор FPR (Significance) (2-я итерация)**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "qS_FSqxHpkP7"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# размер выборки увеличиваем со 100 до 200\n",
        "sample_size = 200"
      ],
      "metadata": {
        "id": "qDrwscjIpkP8"
      },
      "execution_count": 36,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# для False Positive Rate\n",
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(sample_size, 0.5))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "0f5c1acf782440bd83f1506bbd60e0b4",
            "a7f08d9f11084d8ab7d792168d1d0572",
            "5920901b93b44751b5cf7d4df110a1cf",
            "c942d6a002a6442787614f1f2d5c24ea",
            "d21cc423616444fbbb5e60a23707c51f",
            "b8281012ddaf4ec2b2d4e7d708822bb3",
            "f2b425c5fa3a4ce9879ebe08ceffc192",
            "44cb7d9ab6814b7da4605a893a85dcbf",
            "6748683db49546f99b1af808bbb6f715",
            "450be80f215b4969957229df8538057c",
            "cb2a2b7370534180bbf061d30983ec23"
          ]
        },
        "outputId": "12cd5a88-04cb-4091-d2be-7eec032e3c8c",
        "id": "f_VYqgNNpkP9"
      },
      "execution_count": 37,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "0f5c1acf782440bd83f1506bbd60e0b4"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "Qa232HNQpkP9"
      },
      "execution_count": 38,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Т.к. размер выборки изменился, теперь величину 10 необходимо заменить на размер sample_size, умноженный на необходимый процент."
      ],
      "metadata": {
        "id": "Tn7tfmtlpkP9"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# вычисляем число отклонений для каждого эксперимента\n",
        "df['deviation'] = abs(sample_size * 0.5 - df[0])"
      ],
      "metadata": {
        "id": "rqVKXFjgpkP-"
      },
      "execution_count": 39,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим отклонение от нормы\n",
        "threshold = sample_size * 0.07\n",
        "threshold"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "jdnhfoPJqIRb",
        "outputId": "ea955ad9-ca80-4b27-8865-09ca1b211ae5"
      },
      "execution_count": 40,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "14.000000000000002"
            ]
          },
          "metadata": {},
          "execution_count": 40
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от sample_size * 0.5\n",
        "# больше или равно threshold\n",
        "(df['deviation'] >= threshold).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "0f88ba87-6a56-4dcd-ced7-4e683ca012c8",
        "id": "OEpDT9lvpkP-"
      },
      "execution_count": 41,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.04011"
            ]
          },
          "metadata": {},
          "execution_count": 41
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "При размере выборки, равной 200 и пороге отклонения больше либо равном 14 **FPR (Significance)** равен **4%**."
      ],
      "metadata": {
        "id": "UwKEg6UZpkP-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### <font color='green'>**2.4 Подбор TPR  и MDE (2-я итерация)**</font>\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "_LhBtJZlqreR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Порог отклонения зададим равным значению, подобранному в предыдущем пункте: threshold."
      ],
      "metadata": {
        "id": "aOaEvnhZqreT"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# моделируем 100_000 экспериментов\n",
        "n = 100_000\n",
        "mde = 0.1\n",
        "result = []\n",
        "\n",
        "for i in tqdm(range(n)):\n",
        "    np.random.seed(i) # воспроизводимость результатов\n",
        "    result.append(np.random.binomial(sample_size, 0.5 + mde))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "81d062489b5c4ad092a40fb6b9209f13",
            "d8f929dd86214c4e908209b1a250250c",
            "ca7c9183aec4411db786d5b6c35347d1",
            "ac3346304dae44a59a476fbda619bdb4",
            "72479e8f688a4aeaad300aac5b2dd790",
            "5777d3749403401abab2e934d2258543",
            "8dbbaef22b104259a9205b54a19f3bfa",
            "84d72c1bda0143c786107f9570ab0c08",
            "cfcb18b6b91043b49546265e4d506323",
            "8248bf84b2b740e1b8f52423e300bb71",
            "784ae9f94d8146ee9f1b93a6780d5f02"
          ]
        },
        "outputId": "fd05fb20-2224-4cea-82c3-3ee675e86016",
        "id": "Phi9U6MzqreT"
      },
      "execution_count": 42,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/100000 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "81d062489b5c4ad092a40fb6b9209f13"
            }
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# сохраняем результаты в датафрейм\n",
        "df = pd.DataFrame(result)"
      ],
      "metadata": {
        "id": "tw-675h4qreU"
      },
      "execution_count": 43,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# считаем отклонения\n",
        "df['deviation'] = abs(sample_size * 0.5 - df[0])"
      ],
      "metadata": {
        "id": "ns46GIKmqreU"
      },
      "execution_count": 44,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# вычислим процент ситуаций, когда отклонение от sample_size * 0.5\n",
        "# больше или равно threshold\n",
        "(df['deviation'] >= threshold).mean()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "38efc604-db92-44d6-f7d3-55ecb7fabf86",
        "id": "Jc5SajjxqreU"
      },
      "execution_count": 45,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.78681"
            ]
          },
          "metadata": {},
          "execution_count": 45
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Получена мощность, равная 78%. Этого недостаточно, т.к. запланировано значение мощности в 80%.\n",
        "\n",
        "Однако, на 2-й итерации получено значение мощности, достаточно близкое к желаемому. Добиться мощности в 80% можно, регулируя объем выборки, но менять ее теперь необходимо незначительно, т.к. мы вплотную подошли к желаемой мощности, а значения Significance и MDE соответствуют запланированным."
      ],
      "metadata": {
        "id": "FyqwGYz1qreV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# **<font color='royalblue'>Выводы</font>**\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "Q5UByJHYf2r0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "(1) С целью подбора желаемых метрик A/B-теста для **двусторонней** гипотезы были зафиксированы следующие значения метрик:\n",
        "\n",
        "* **MDE**: 10%\n",
        "\n",
        "* **Power (1 - beta)**: 80%\n",
        "\n",
        "* **Significance (alpha)**: 5%"
      ],
      "metadata": {
        "id": "B_Egu8EDf3_c"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "(2) В результате проведения ряда итераций подобраны:\n",
        "\n",
        "* **размер выборки**\n",
        "\n",
        "* **отклонение от нормы**\n"
      ],
      "metadata": {
        "id": "KbpdGbPnf4DJ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "(3) Алгоритм подбора параметров определен следующим образом:\n",
        "\n",
        "* **Отклонение от нормы** (пороговое значение) на каждой итерации корректируется таким образом, чтобы значение **Significance (alpha)** было максимально приближено к 0.05 (5%).\n",
        "\n",
        "* Оптимальный, наиболее эффективный **размер выборки** определялся таким образом, чтобы подобрать наименьший размер выборки, позволяющий получить желаемые размеры метрик A\\B-теста, - для снижения стоимости и длительности проведения теста.\n"
      ],
      "metadata": {
        "id": "O3Vh3EjWf4GZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "(4) В отчете представлена схема алгоритма получения желаемых метрик A/B-теста на этапе планирования эксперимента."
      ],
      "metadata": {
        "id": "ef365nejf4Jb"
      }
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "2Yk-mUz_HtBC"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}