.. _dpa-logistic-regression-modeling-binary-outcomes-via-odds-and-log-odds:

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Logistic Regression: Modeling Binary Outcomes via Odds and Log-Odds
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**Stage 6 · 🎯 Classification & Logistic Regression**  ·  Lesson 38 of 56  ·  *advanced*

:doc:`◀ Previous · How Shapley Values Work <37-how-shapley-values-work>`   ·   :doc:`Next · Maximum Likelihood (MLE): Fitting a Distribution to Observed Data ▶ <39-maximum-likelihood-mle-fitting-a-distribution-to-observed-data>`   ·   :doc:`↑ Section <index>`


.. important::

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When the outcome is yes or no
-------------------------------

Regression so far has predicted a **number** — a fare, a price. But many outcomes are **binary**: will
a customer **churn** or not? will a student **return** next year? Ordinary linear regression fails
here, because a straight line runs off to :math:`\pm\infty` and would predict "probabilities" **below
0 or above 1**. **Logistic regression** is the standard model for a **yes/no** outcome, and it works by
predicting a **probability** instead.

Odds and log-odds
-------------------

The trick is to transform the probability so a linear model fits. Start with the **odds** — the ratio
of the probability of the event to its complement, :math:`p / (1 - p)` — which stretches :math:`[0, 1]`
out to :math:`[0, \infty)`. Then take the logarithm, giving the **log-odds** or **logit**, which spans
**all** real numbers. Logistic regression makes *this* linear in the features:

.. math::

   \ln\!\left(\frac{p}{1 - p}\right) = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p.

The model is an ordinary linear equation — just on the **log-odds scale** rather than the probability
scale.

The logistic curve
--------------------

To get a probability back, invert the transform. Solving for :math:`p` gives the **logistic**
(sigmoid) function:

.. math::

   p = \frac{1}{1 + e^{-z}}, \qquad z = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p.

This S-shaped curve takes the linear combination :math:`z` — any real number — and squashes it
smoothly into a valid probability between 0 and 1. Large positive :math:`z` gives :math:`p` near 1,
large negative near 0, and :math:`z = 0` gives :math:`p = 0.5`.

Reading coefficients
----------------------

The coefficients read on the **odds** scale. A one-unit rise in :math:`x_j` adds :math:`\beta_j` to the
log-odds, which **multiplies** the odds by :math:`e^{\beta_j}` — the **odds ratio**. So
:math:`e^{\beta_j} > 1` means the feature raises the odds of the event, :math:`< 1` lowers them. In
Python it is ``LogisticRegression`` in scikit-learn, or ``Logit`` in statsmodels for the full
coefficient table. Unlike least squares, its coefficients have **no closed form** — they are found by
**maximum likelihood**, the subject of the next lesson.

.. hint::

   **Related lessons:** :doc:`Maximum Likelihood (MLE): Fitting a Distribution to Observed Data <39-maximum-likelihood-mle-fitting-a-distribution-to-observed-data>`  ·  :doc:`Assessing Model Fit in Logistic Regression <40-assessing-model-fit-in-logistic-regression>`  ·  :doc:`Binary Classification Models – Conceptual Framework and Evaluation Metrics <51-binary-classification-models-conceptual-framework-and-evaluation-metrics>`  ·  :doc:`Complete and Quasi-Complete Separation in Logistic Regression <41-complete-and-quasi-complete-separation-in-logistic-regression>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2026/01/16/logistic-regression-modeling-binary-outcomes-via-odds-and-log-odds/ <https://insightful-data-lab.com/2026/01/16/logistic-regression-modeling-binary-outcomes-via-odds-and-log-odds/>`__ (insightful-data-lab.com).

.. tags:: purpose: reference, topic: data analysis, topic: data preparation, level: advanced