Multiple Regression Analysis
Fit multiple linear regression model with diagnostics. Performs multiple linear regression with R², adjusted R², coefficient tests, VIF for multicollinearity…
Performs multiple linear regression with R², adjusted R², coefficient tests, VIF for multicollinearity, and residual diagnostics. Last column is treated as the response variable.
What Is Multiple Linear Regression?
Multiple linear regression extends simple regression to model the relationship between a response variable (y) and two or more predictor variables (x₁, x₂, …, xₚ). The model y = β₀ + β₁x₁ + β₂x₂ + ⋯ + βₚxₚ + ε is fitted using Ordinary Least Squares, minimizing the sum of squared residuals.
Key diagnostics include R² (variance explained), Adjusted R² (penalized for number of predictors), F-test p-value (overall model significance), individual coefficient p-values, and the Variance Inflation Factor (VIF) for detecting multicollinearity — correlation among predictors that inflates standard errors.
A VIF above 5-10 indicates problematic multicollinearity. Remedies include removing redundant variables, combining correlated predictors, or using regularization techniques (Ridge, Lasso). The residual plot should show random scatter; patterns indicate model misspecification.
Formula: y = β₀ + β₁x₁ + β₂x₂ + ⋯ + βₚxₚ + ε β = (XᵀX)⁻¹Xᵀy R² = 1 − SS_res/SS_tot Adj R² = 1 − (1−R²)(n−1)/(n−p−1) VIF_j = 1/(1 − R²_j)
Example Calculation
Predicting house price (y) from area x₁ (sq ft) and age x₂ (years). Five houses: area 1,300/1,800/1,900/2,000/2,400 sq ft, age 10/5/12/25/15 years, price $150,000/$233,000/$216,000/$208,000/$249,000. The least-squares fit gives β₀ = 50,000, β₁ = 100 (each sq ft adds $100), β₂ = −2,000 (each year reduces $2,000). R² = 0.92, Adj R² = 0.84. VIF(x₁) = 1.2, VIF(x₂) = 1.2 — no multicollinearity.
When to Use This Calculator
- A researcher modeling the relationship between multiple independent variables and an outcome to identify significant predictors
- A business analyst building a predictive model for sales based on multiple factors like price, advertising spend, and seasonality
- A quality engineer determining which process variables jointly affect product quality using multi-factor analysis
- A data scientist checking for multicollinearity (VIF) before using regression coefficients for feature interpretation
Common Mistakes to Avoid
- Including highly correlated predictors without checking VIF — multicollinearity makes individual coefficients unreliable and sign-flips common; remove or combine redundant variables
- Adding too many predictors relative to sample size — the rule of thumb is at least 10-20 observations per predictor; overfitting occurs when this ratio is too low
- Interpreting coefficient magnitudes without standardizing — a coefficient of 100 for a variable measured in millimeters is not 'more important' than a coefficient of 0.5 for a variable in meters; compare standardized coefficients
- Ignoring the residual plot — a good R-squared with patterned residuals indicates model misspecification; always check that residuals are randomly scattered
How to Interpret Results
- Adjusted R-squared below 0.5 means the model explains less than half the variation — consider adding more relevant predictors or transforming variables
- F p-value below 0.05 means the model as a whole is statistically significant — at least one predictor contributes meaningfully
- Individual coefficient p-values above 0.05 suggest that predictor may not contribute significantly given the other variables — consider removing it to simplify the model
Related Standards & References
- Draper & Smith, Applied Regression Analysis (Wiley) — classic OLS reference including F-tests and residual analysis
- Kutner, Nachtsheim & Neter, Applied Linear Statistical Models (McGraw-Hill) — textbook treatment of multiple regression, adjusted R², and VIF
- Belsley, Kuh & Welsch (1980), Regression Diagnostics (Wiley) — standard reference on detecting multicollinearity and influential observations
Frequently Asked Questions
What is Adjusted R² and why is it better than R²?
R² always increases (or stays the same) when you add more predictors, even irrelevant ones. Adjusted R² penalizes for the number of predictors, decreasing when a new variable does not improve the model enough to justify the added complexity. Use Adjusted R² for model comparison when the number of predictors differs.
What does a high VIF value mean?
VIF (Variance Inflation Factor) measures how much the variance of a coefficient is inflated due to correlation with other predictors. VIF = 1 means no collinearity; VIF > 5 suggests moderate collinearity; VIF > 10 indicates severe multicollinearity. High VIF makes individual coefficient estimates unreliable, though overall predictions may still be accurate.