In This Article
Introduction
Regression analysis is a statistical technique for modeling the relationship between a dependent variable (Y) and one or more independent variables (X). It's one of the most widely used tools in business analytics for prediction and understanding relationships.
Simple Linear Regression
Y = β₀ + β₁X + ε
Where: Y = dependent variable, X = independent variable
β₀ = intercept (Y when X=0), β₁ = slope (change in Y per unit X)
ε = error term
Ordinary Least Squares (OLS)
OLS finds the line that minimizes the sum of squared residuals (differences between actual and predicted values).
Example
If Sales = 100 + 5×Advertising, then:
• Base sales (no advertising) = ₹100
• Each ₹1 in advertising adds ₹5 in sales
Multiple Linear Regression
Y = β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ + ε
Multiple regression includes several independent variables, allowing you to:
- Control for other factors
- Understand relative importance of variables
- Make more accurate predictions
Variable Types
- Continuous: Price, age, income
- Dummy (0/1): Gender, region, yes/no
- Interaction: Combined effect of two variables
Key Assumptions
| Assumption | Description | Violation Consequence |
|---|---|---|
| Linearity | Relationship is linear | Biased estimates |
| Independence | Errors are independent | Inefficient estimates |
| Homoscedasticity | Constant error variance | Unreliable standard errors |
| Normality | Errors normally distributed | Invalid hypothesis tests |
| No multicollinearity | IVs not highly correlated | Unstable coefficients |
Interpreting Results
Key Metrics
- R² (R-squared): % of variance in Y explained by X (0-1, higher is better)
- Adjusted R²: R² adjusted for number of predictors
- p-value: Statistical significance (typically < 0.05)
- Coefficients: Effect size—change in Y per unit change in X
- Standard Error: Precision of coefficient estimate
- F-statistic: Overall model significance
Business Applications
- Sales forecasting: Predict sales from marketing spend, seasonality
- Pricing: Estimate price elasticity of demand
- HR: Predict employee turnover from satisfaction scores
- Finance: Model stock returns, credit risk
- Marketing: Attribution modeling, customer lifetime value
- Operations: Demand planning, capacity optimization
Conclusion
Key Takeaways
- Regression models relationship between Y and X variables
- Simple regression: One predictor; Multiple: Several predictors
- OLS minimizes sum of squared errors
- Check assumptions: Linearity, independence, homoscedasticity, normality
- R² shows variance explained; p-value shows significance
- Coefficients show effect size and direction
- Correlation ≠ causation—be careful with interpretation