Linear Regression and Logistic Regression
Regression analysis is a core statistical technique used to model the relationship between a dependent variable (the outcome) and one or more independent variables (the predictors). The choice between Linear and Logistic regression depends entirely on the nature of your outcome variable.
1. Linear Regression
Linear Regression is used when the dependent variable is continuous (numerical). It models the relationship by fitting a straight line through the observed data points.
- Goal: To predict a specific numerical value (e.g., predicting the price of a house based on its square footage).
- Mathematical Equation: Represented as Y = β0 + β_1X + epsilon, where Y is the dependent variable, X is the predictor, β1 is the slope (the change in Y per unit of X), and β0 is the intercept.
- Key Assumption: There is a linear relationship between the independent and dependent variables.
2. Logistic Regression
Despite its name, Logistic Regression is used for classification tasks. It is employed when the dependent variable is categorical (binary, such as Yes/No, True/False, or 0/1).
- Goal: To predict the probability that an observation belongs to a particular category (e.g., predicting the probability that a patient has a disease based on their symptoms).
- The Logit Function: Because linear regression could predict values below 0 or above 1 (which are impossible for probabilities), Logistic Regression uses the Sigmoid function (or Logistic function) to map any real-valued number into a range between 0 and 1. P(Y=1) = frac{1}{1 + e^{-(beta_0 + beta_1X)}}
- Result: The output is a probability score between 0 and 1. A threshold (usually 0.5) is then applied to classify the outcome into a category.
3. Comparative Overview
| Feature | Linear Regression | Logistic Regression |
| Dependent Variable Type | Continuous (e.g., Height, Price) | Categorical/Binary (e.g., Win/Loss, Spam/Not) |
| Output | A continuous numerical value | A probability between 0 and 1 |
| Relationship | Linear | Sigmoidal (S-curve) |
| Primary Use | Forecasting/Estimation | Classification |
4. Which One to Use?
- Use Linear Regression when: Your question asks “How much?” or “How many?” (e.g., “How much will this stock price change?”).
- Use Logistic Regression when: Your question asks “Which one?” or “Is it true?” (e.g., “Will the customer buy the product? Yes or No?”).
Statistical Fact
In Linear Regression, we minimize the Residual Sum of Squares (RSS) to fit the line. In Logistic Regression, we cannot use this method because the relationship isn’t linear. Instead, we use Maximum Likelihood Estimation (MLE), which is an iterative statistical method that finds the parameter values that maximize the likelihood of observing the actual data.
