Regression analysis is a foundational tool in Business Statistics. It allows us to understand relationships between variables and make predictions. As UGC NET Commerce aspirants, it is essential to not just understand individual types of regression but also know when and why to use each. This article provides an in-depth comparison of three major types of regression:
- Linear Regression
- Logistic Regression
- Multiple Regression
We'll explore their assumptions, mathematical models, applications, and interpretations laying out the full spectrum of regression tools.
Dependent Variable Type
Linear Regression is used when the dependent variable is continuous and numeric (e.g., sales, income, height).
Logistic Regression is used when the dependent variable is binary or categorical (e.g., Yes/No, Pass/Fail, 0/1).
Multiple Regression is an extension of linear regression but includes two or more independent variables to predict a continuous dependent variable.
Model Form & Estimation Method
1. Linear Regression
Model Form: Y = β₀ + β₁X + ε
Estimation: Ordinary Least Squares (OLS)
2. Logistic Regression
Model Form: log(p/(1−p)) = β₀ + β₁X
This is called the logit function. The output is the log-odds of the probability of an event.
Estimation: Maximum Likelihood Estimation (MLE)
3. Multiple Regression
Model Form: Y = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε
Estimation: Ordinary Least Squares (OLS)
Goodness-of-Fit Metrics
| Model | Main Metric | Interpretation |
|---|---|---|
| Linear Regression | R² (Coefficient of Determination) | Proportion of variance explained by the model. |
| Logistic Regression | Log-likelihood, Pseudo R², AUC-ROC | Model’s accuracy in predicting categories. |
| Multiple Regression | Adjusted R² | Like R² but adjusted for number of predictors. |
Use-Cases & Limitations
1. Linear Regression
Use-Case: Forecasting demand, estimating cost, predicting performance scores.
Limitations: Assumes linearity and normality. Not suitable for categorical outcomes.
2. Logistic Regression
Use-Case: Predicting likelihood of loan default, churn analysis, disease detection.
Limitations: Only for binary/categorical outcomes. Interpretation of coefficients is in terms of odds.
3. Multiple Regression
Use-Case: When the outcome is influenced by multiple variables—like sales influenced by pricing, advertising, and seasonality.
Limitations: Multicollinearity may affect reliability of coefficients. Interpretation becomes complex with more predictors.
Example
1. Linear Regression
Data: Predict sales (Y) based on advertising spend (X)
Y = 50 + 4X
If X = ₹10,000, then Y = 50 + 4(10) = ₹90
2. Logistic Regression
Data: Predict whether a student will pass (1) or fail (0) based on study hours (X)
log(p/1-p) = -1 + 0.8X → p = 1 / (1 + e^(-(-1+0.8X)))
If X = 2 hours, p = 1 / (1 + e^(-0.6)) ≈ 0.645 (i.e., 64.5% chance of passing)
3. Multiple Regression
Data: Predict salary (Y) based on experience (X₁) and education level (X₂)
Y = 30 + 5X₁ + 10X₂
If X₁ = 4 years, X₂ = 2 (postgraduate), then Y = 30 + 20 + 20 = ₹70
Interpretation of Results
Linear Regression: A one-unit increase in X increases Y by β₁ units.
Logistic Regression: A one-unit increase in X increases the log-odds of the outcome by β₁. Odds ratio = e^β₁.
Multiple Regression: Coefficients are interpreted while holding other variables constant.
Application Cases (Qualitative Insights)
- Marketing: Linear regression is used to forecast ad impact on sales. Logistic regression helps identify customers likely to churn.
- Healthcare: Logistic regression helps predict disease probability. Multiple regression helps understand what combination of diet, exercise, and medication impact health scores.
- Finance: Multiple regression helps understand investment returns based on economic indicators.
Quick Comparison Table
| Aspect | Linear Regression | Logistic Regression | Multiple Regression |
|---|---|---|---|
| Dependent Variable | Continuous | Binary / Categorical | Continuous |
| Estimation Method | OLS | MLE | OLS |
| Equation Type | Linear | Log-odds | Linear (with multiple Xs) |
| Evaluation | R² | AUC, Pseudo R² | Adjusted R² |
| When to Use | Simple prediction with one numeric X | Probability prediction | Multifactor prediction |
Summary
Each regression method has its strengths and intended usage. Linear regression is best for simple numeric predictions. Logistic regression is ideal for classification problems, especially binary outcomes. Multiple regression is powerful when you need to analyze the influence of several variables at once.
For UGC NET, understanding not only how to apply these tools but also their theoretical foundations, assumptions, and interpretive frameworks is essential. Always check data type, problem type, and output needs before selecting a regression method.