.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statistics/hypothesis_tests/plot_04_one_way_anova.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_statistics_hypothesis_tests_plot_04_one_way_anova.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statistics_hypothesis_tests_plot_04_one_way_anova.py:


Fisher's one-way analysis of variance
======================================

Compares three fertilizer treatments' effect on plant growth, as in
Fisher's Rothamsted field trials. ANOVA splits the total variability
into a between-groups and a within-groups sum of squares; their ratio,
the F-statistic, tests all group means for equality in one test.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import stats

    from mathematicskit.statistics import one_way_anova








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Simulated data: three treatment groups
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-29

.. code-block:: Python


    rng = np.random.default_rng(0)
    groups = {
        "control": rng.normal(loc=10.0, scale=2.0, size=30),
        "treatment A": rng.normal(loc=11.5, scale=2.0, size=30),
        "treatment B": rng.normal(loc=13.0, scale=2.0, size=30),
    }








.. GENERATED FROM PYTHON SOURCE LINES 30-32

Partitioning the sum of squares
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-46

.. code-block:: Python


    values = list(groups.values())
    grand_mean = np.concatenate(values).mean()
    ss_total = sum(np.sum((g - grand_mean) ** 2) for g in values)
    ss_between = sum(g.size * (g.mean() - grand_mean) ** 2 for g in values)
    ss_within = sum(np.sum((g - g.mean()) ** 2) for g in values)
    print(f"SS_total = {ss_total:.2f} = SS_between {ss_between:.2f} + SS_within {ss_within:.2f}")

    result = one_way_anova(*values)
    df_between, df_within = result.df, result.extra["df_within"]
    f_manual = (ss_between / df_between) / (ss_within / df_within)
    print(f"F = {result.statistic:.3f} (by hand {f_manual:.3f}) on ({df_between:.0f}, {df_within:.0f}) df, p = {result.p_value:.2e}")
    print("group means:", np.round(result.extra["group_means"], 3))





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

 .. code-block:: none

    SS_total = 522.68 = SS_between 198.54 + SS_within 324.14
    F = 26.645 (by hand 26.645) on (2, 87) df, p = 9.41e-10
    group means: [ 9.757 12.052 13.349]




.. GENERATED FROM PYTHON SOURCE LINES 47-49

Groups around the grand mean, and the F reference distribution
----------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-66

.. code-block:: Python


    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    for i, g in enumerate(values):
        ax1.plot(np.full(g.size, i) + rng.uniform(-0.1, 0.1, g.size), g, "o", alpha=0.5)
        ax1.hlines(g.mean(), i - 0.3, i + 0.3, color="k")
    ax1.axhline(grand_mean, color="gray", ls="--", label="grand mean")
    ax1.set_xticks(range(len(groups)), list(groups))
    ax1.set_ylabel("growth")
    ax1.legend()

    f_grid = np.linspace(0.0, max(12.0, result.statistic * 1.1), 400)
    ax2.plot(f_grid, stats.f.pdf(f_grid, df_between, df_within), label=f"F({df_between:.0f}, {df_within:.0f}) under equal means")
    ax2.axvline(result.statistic, color="C3", ls="--", label="observed F")
    ax2.set_xlabel("F")
    ax2.set_ylabel("density")
    ax2.legend()
    fig.tight_layout()



.. image-sg:: /api/gallery/statistics/hypothesis_tests/images/sphx_glr_plot_04_one_way_anova_001.png
   :alt: plot 04 one way anova
   :srcset: /api/gallery/statistics/hypothesis_tests/images/sphx_glr_plot_04_one_way_anova_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.050 seconds)


.. _sphx_glr_download_api_gallery_statistics_hypothesis_tests_plot_04_one_way_anova.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_one_way_anova.ipynb <plot_04_one_way_anova.ipynb>`

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

      :download:`Download Python source code: plot_04_one_way_anova.py <plot_04_one_way_anova.py>`

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

      :download:`Download zipped: plot_04_one_way_anova.zip <plot_04_one_way_anova.zip>`


.. only:: html

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

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