Multiple+linear+regression+and+Adjusted+R_squared.ipynb

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      "metadata": {
        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/Nargis45/Regression-techniques/blob/master/Multiple%2Blinear%2Bregression%2Band%2BAdjusted%2BR_squared.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Glmw9yu-8PUM"
      },
      "source": [
        "# Multiple linear regression and adjusted R-squared"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "bf8b0stl8PUf"
      },
      "source": [
        "## Import the relevant libraries"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "-LOrlrHu8PUg"
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n",
        "import statsmodels.api as sm\n",
        "import seaborn\n",
        "seaborn.set()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XQPlxT4V8PUl"
      },
      "source": [
        "## Load the data"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "id": "7BKxtAdm8PUm"
      },
      "outputs": [],
      "source": [
        "# Load the data from a .csv in the same folder\n",
        "data = pd.read_csv('/content/1.02.+Multiple+linear+regression.csv')"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "scrolled": true,
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          "height": 424
        },
        "id": "mZpJ7Zzd8PUn",
        "outputId": "91b6fa81-03a6-4576-8741-6c3c445ba1fa"
      },
      "outputs": [
        {
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              "     SAT   GPA  Rand 1,2,3\n",
              "0   1714  2.40           1\n",
              "1   1664  2.52           3\n",
              "2   1760  2.54           3\n",
              "3   1685  2.74           3\n",
              "4   1693  2.83           2\n",
              "..   ...   ...         ...\n",
              "79  1936  3.71           3\n",
              "80  1810  3.71           1\n",
              "81  1987  3.73           3\n",
              "82  1962  3.76           1\n",
              "83  2050  3.81           2\n",
              "\n",
              "[84 rows x 3 columns]"
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              "      <th>SAT</th>\n",
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              "      <th>3</th>\n",
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              "      <th>81</th>\n",
              "      <td>1987</td>\n",
              "      <td>3.73</td>\n",
              "      <td>3</td>\n",
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              "      <th>82</th>\n",
              "      <td>1962</td>\n",
              "      <td>3.76</td>\n",
              "      <td>1</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>83</th>\n",
              "      <td>2050</td>\n",
              "      <td>3.81</td>\n",
              "      <td>2</td>\n",
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              "  </tbody>\n",
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              "<p>84 rows × 3 columns</p>\n",
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              "                                                     [key], {});\n",
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              "\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",
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              "  "
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          },
          "metadata": {},
          "execution_count": 3
        }
      ],
      "source": [
        "# Let's check what's inside this data frame\n",
        "data"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
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      "outputs": [
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          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "               SAT        GPA  Rand 1,2,3\n",
              "count    84.000000  84.000000   84.000000\n",
              "mean   1845.273810   3.330238    2.059524\n",
              "std     104.530661   0.271617    0.855192\n",
              "min    1634.000000   2.400000    1.000000\n",
              "25%    1772.000000   3.190000    1.000000\n",
              "50%    1846.000000   3.380000    2.000000\n",
              "75%    1934.000000   3.502500    3.000000\n",
              "max    2050.000000   3.810000    3.000000"
            ],
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              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>SAT</th>\n",
              "      <th>GPA</th>\n",
              "      <th>Rand 1,2,3</th>\n",
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              "  </thead>\n",
              "  <tbody>\n",
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              "      <th>count</th>\n",
              "      <td>84.000000</td>\n",
              "      <td>84.000000</td>\n",
              "      <td>84.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>mean</th>\n",
              "      <td>1845.273810</td>\n",
              "      <td>3.330238</td>\n",
              "      <td>2.059524</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>std</th>\n",
              "      <td>104.530661</td>\n",
              "      <td>0.271617</td>\n",
              "      <td>0.855192</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>min</th>\n",
              "      <td>1634.000000</td>\n",
              "      <td>2.400000</td>\n",
              "      <td>1.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25%</th>\n",
              "      <td>1772.000000</td>\n",
              "      <td>3.190000</td>\n",
              "      <td>1.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>50%</th>\n",
              "      <td>1846.000000</td>\n",
              "      <td>3.380000</td>\n",
              "      <td>2.000000</td>\n",
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              "    <tr>\n",
              "      <th>75%</th>\n",
              "      <td>1934.000000</td>\n",
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              "    <tr>\n",
              "      <th>max</th>\n",
              "      <td>2050.000000</td>\n",
              "      <td>3.810000</td>\n",
              "      <td>3.000000</td>\n",
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              "    .colab-df-convert:hover {\n",
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              "\n",
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              "\n",
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              "          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",
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              "  </div>\n",
              "  "
            ]
          },
          "metadata": {},
          "execution_count": 4
        }
      ],
      "source": [
        "# This method gives us very nice descriptive statistics. We don't need this as of now, but will later on!\n",
        "data.describe()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ghW_nNkE8PUr"
      },
      "source": [
        "## Create your first multiple regression"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "id": "CJ9A0kbR8PUt"
      },
      "outputs": [],
      "source": [
        "# Following the regression equation, our dependent variable (y) is the GPA\n",
        "y = data ['GPA']\n",
        "# Similarly, our independent variable (x) is the SAT score\n",
        "x1 = data [['SAT','Rand 1,2,3']]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "B4eVCXNF8PUv",
        "outputId": "59bac564-354e-4ac3-df69-4a0177a5a7d8"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/statsmodels/tsa/tsatools.py:142: FutureWarning: In a future version of pandas all arguments of concat except for the argument 'objs' will be keyword-only\n",
            "  x = pd.concat(x[::order], 1)\n"
          ]
        }
      ],
      "source": [
        "# Add a constant. Esentially, we are adding a new column (equal in lenght to x), which consists only of 1s\n",
        "x = sm.add_constant(x1)\n",
        "# Fit the model, according to the OLS (ordinary least squares) method with a dependent variable y and an idependent x\n",
        "results = sm.OLS(y,x).fit()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 499
        },
        "id": "yp2leTAm8PUx",
        "outputId": "fd71be0e-c030-40f7-cb8d-401c3bbb3ac5"
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      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<class 'statsmodels.iolib.summary.Summary'>\n",
              "\"\"\"\n",
              "                            OLS Regression Results                            \n",
              "==============================================================================\n",
              "Dep. Variable:                    GPA   R-squared:                       0.407\n",
              "Model:                            OLS   Adj. R-squared:                  0.392\n",
              "Method:                 Least Squares   F-statistic:                     27.76\n",
              "Date:                Thu, 10 Nov 2022   Prob (F-statistic):           6.58e-10\n",
              "Time:                        09:49:38   Log-Likelihood:                 12.720\n",
              "No. Observations:                  84   AIC:                            -19.44\n",
              "Df Residuals:                      81   BIC:                            -12.15\n",
              "Df Model:                           2                                         \n",
              "Covariance Type:            nonrobust                                         \n",
              "==============================================================================\n",
              "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
              "------------------------------------------------------------------------------\n",
              "const          0.2960      0.417      0.710      0.480      -0.533       1.125\n",
              "SAT            0.0017      0.000      7.432      0.000       0.001       0.002\n",
              "Rand 1,2,3    -0.0083      0.027     -0.304      0.762      -0.062       0.046\n",
              "==============================================================================\n",
              "Omnibus:                       12.992   Durbin-Watson:                   0.948\n",
              "Prob(Omnibus):                  0.002   Jarque-Bera (JB):               16.364\n",
              "Skew:                          -0.731   Prob(JB):                     0.000280\n",
              "Kurtosis:                       4.594   Cond. No.                     3.33e+04\n",
              "==============================================================================\n",
              "\n",
              "Notes:\n",
              "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
              "[2] The condition number is large, 3.33e+04. This might indicate that there are\n",
              "strong multicollinearity or other numerical problems.\n",
              "\"\"\""
            ],
            "text/html": [
              "<table class=\"simpletable\">\n",
              "<caption>OLS Regression Results</caption>\n",
              "<tr>\n",
              "  <th>Dep. Variable:</th>           <td>GPA</td>       <th>  R-squared:         </th> <td>   0.407</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.392</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   27.76</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Date:</th>             <td>Thu, 10 Nov 2022</td> <th>  Prob (F-statistic):</th> <td>6.58e-10</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Time:</th>                 <td>09:49:38</td>     <th>  Log-Likelihood:    </th> <td>  12.720</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>No. Observations:</th>      <td>    84</td>      <th>  AIC:               </th> <td>  -19.44</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Df Residuals:</th>          <td>    81</td>      <th>  BIC:               </th> <td>  -12.15</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Df Model:</th>              <td>     2</td>      <th>                     </th>     <td> </td>   \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
              "</tr>\n",
              "</table>\n",
              "<table class=\"simpletable\">\n",
              "<tr>\n",
              "       <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
              "</tr>\n",
              "<tr>\n",
              "  <th>const</th>      <td>    0.2960</td> <td>    0.417</td> <td>    0.710</td> <td> 0.480</td> <td>   -0.533</td> <td>    1.125</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>SAT</th>        <td>    0.0017</td> <td>    0.000</td> <td>    7.432</td> <td> 0.000</td> <td>    0.001</td> <td>    0.002</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Rand 1,2,3</th> <td>   -0.0083</td> <td>    0.027</td> <td>   -0.304</td> <td> 0.762</td> <td>   -0.062</td> <td>    0.046</td>\n",
              "</tr>\n",
              "</table>\n",
              "<table class=\"simpletable\">\n",
              "<tr>\n",
              "  <th>Omnibus:</th>       <td>12.992</td> <th>  Durbin-Watson:     </th> <td>   0.948</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Prob(Omnibus):</th> <td> 0.002</td> <th>  Jarque-Bera (JB):  </th> <td>  16.364</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Skew:</th>          <td>-0.731</td> <th>  Prob(JB):          </th> <td>0.000280</td>\n",
              "</tr>\n",
              "<tr>\n",
              "  <th>Kurtosis:</th>      <td> 4.594</td> <th>  Cond. No.          </th> <td>3.33e+04</td>\n",
              "</tr>\n",
              "</table><br/><br/>Notes:<br/>[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.<br/>[2] The condition number is large, 3.33e+04. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
            ]
          },
          "metadata": {},
          "execution_count": 7
        }
      ],
      "source": [
        "# Print a nice summary of the regression.\n",
        "results.summary()"
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "GarlpL7I8dxu"
      },
      "execution_count": 7,
      "outputs": []
    }
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