.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "packages/statistics/auto_examples/plot_regression.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_packages_statistics_auto_examples_plot_regression.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_packages_statistics_auto_examples_plot_regression.py:


Simple Regression
====================

Fit a simple linear regression using 'statsmodels', compute corresponding
p-values.

.. GENERATED FROM PYTHON SOURCE LINES 8-21

.. code-block:: Python


    # Original author: Thomas Haslwanter

    import numpy as np
    import matplotlib.pyplot as plt
    import pandas

    # For statistics. Requires statsmodels 5.0 or more
    from statsmodels.formula.api import ols

    # Analysis of Variance (ANOVA) on linear models
    from statsmodels.stats.anova import anova_lm








.. GENERATED FROM PYTHON SOURCE LINES 22-23

Generate and show the data

.. GENERATED FROM PYTHON SOURCE LINES 23-34

.. code-block:: Python

    x = np.linspace(-5, 5, 20)

    # To get reproducible values, provide a seed value
    rng = np.random.default_rng(27446968)

    y = -5 + 3 * x + 4 * np.random.normal(size=x.shape)

    # Plot the data
    plt.figure(figsize=(5, 4))
    plt.plot(x, y, "o")




.. image-sg:: /packages/statistics/auto_examples/images/sphx_glr_plot_regression_001.png
   :alt: plot regression
   :srcset: /packages/statistics/auto_examples/images/sphx_glr_plot_regression_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    [<matplotlib.lines.Line2D object at 0x7b207328ef60>]



.. GENERATED FROM PYTHON SOURCE LINES 35-37

Multilinear regression model, calculating fit, P-values, confidence
intervals etc.

.. GENERATED FROM PYTHON SOURCE LINES 37-54

.. code-block:: Python


    # Convert the data into a Pandas DataFrame to use the formulas framework
    # in statsmodels
    data = pandas.DataFrame({"x": x, "y": y})

    # Fit the model
    model = ols("y ~ x", data).fit()

    # Print the summary
    print(model.summary())

    # Perform analysis of variance on fitted linear model
    anova_results = anova_lm(model)

    print("\nANOVA results")
    print(anova_results)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

                                OLS Regression Results                            
    ==============================================================================
    Dep. Variable:                      y   R-squared:                       0.845
    Model:                            OLS   Adj. R-squared:                  0.836
    Method:                 Least Squares   F-statistic:                     97.76
    Date:                Mon, 17 Nov 2025   Prob (F-statistic):           1.06e-08
    Time:                        00:19:37   Log-Likelihood:                -53.560
    No. Observations:                  20   AIC:                             111.1
    Df Residuals:                      18   BIC:                             113.1
    Df Model:                           1                                         
    Covariance Type:            nonrobust                                         
    ==============================================================================
                     coef    std err          t      P>|t|      [0.025      0.975]
    ------------------------------------------------------------------------------
    Intercept     -4.1877      0.830     -5.044      0.000      -5.932      -2.444
    x              2.7046      0.274      9.887      0.000       2.130       3.279
    ==============================================================================
    Omnibus:                        1.871   Durbin-Watson:                   1.930
    Prob(Omnibus):                  0.392   Jarque-Bera (JB):                0.597
    Skew:                           0.337   Prob(JB):                        0.742
    Kurtosis:                       3.512   Cond. No.                         3.03
    ==============================================================================

    Notes:
    [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

    ANOVA results
                df       sum_sq      mean_sq          F        PR(>F)
    x          1.0  1347.476043  1347.476043  97.760281  1.062847e-08
    Residual  18.0   248.102486    13.783471        NaN           NaN




.. GENERATED FROM PYTHON SOURCE LINES 55-56

Plot the fitted model

.. GENERATED FROM PYTHON SOURCE LINES 56-64

.. code-block:: Python


    # Retrieve the parameter estimates
    offset, coef = model._results.params
    plt.plot(x, x * coef + offset)
    plt.xlabel("x")
    plt.ylabel("y")

    plt.show()



.. image-sg:: /packages/statistics/auto_examples/images/sphx_glr_plot_regression_002.png
   :alt: plot regression
   :srcset: /packages/statistics/auto_examples/images/sphx_glr_plot_regression_002.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_packages_statistics_auto_examples_plot_regression.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_regression.ipynb <plot_regression.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_regression.py <plot_regression.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_regression.zip <plot_regression.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_