How is logistic regression different from linear regression?

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Logistic regression and linear regression are both statistical models used for prediction, but they differ fundamentally in their purpose, output, and underlying assumptions:

1. Purpose

• Linear Regression:
  Used for predicting a continuous numeric outcome based on one or more predictor variables. For example, predicting house prices or temperature.

• Logistic Regression:
  Used for predicting a categorical outcome, typically binary (yes/no, 0/1). For example, predicting whether an email is spam or not.

2. Output

• Linear Regression:
  Produces a continuous value as output. The model fits a straight line (or hyperplane) to the data by minimizing the squared differences between actual and predicted values.

• Logistic Regression:
  Produces a probability between 0 and 1 using the logistic (sigmoid) function. The predicted probability is then mapped to classes (e.g., probability > 0.5 → class 1).

3. Model Equation

• Linear Regression:

  $$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \ldots + \epsilon$$

  where $y$ is continuous.

• Logistic Regression:

  $$P(y=1) = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \ldots)}}$$

  The output is the probability of class 1.

4. Assumptions

• Linear Regression:
  Assumes a linear relationship between predictors and the outcome, residuals are normally distributed, and homoscedasticity (constant variance).

• Logistic Regression:
  Does not assume linearity of outcome but assumes the log-odds of the outcome is linear in the predictors.

5. Loss Function

• Linear Regression: Uses Mean Squared Error (MSE).

• Logistic Regression: Uses Log Loss (cross-entropy) to measure the error in probability predictions.

Summary:

• Linear regression is for continuous outcomes; logistic regression is for classification (categorical outcomes).

• Logistic regression outputs probabilities via a sigmoid function, whereas linear regression outputs direct numeric predictions.

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