Multiple Linear Regression Calculator
Calculate Multiple Linear Regression goodness-of-fit ($R^2$), Adjusted $R^2$, overall model $F$-statistic ($F = \frac{R^2/k}{(1-R^2)/(N-k-1)}$), and ANOVA $df$.
Calculate Multiple Regression Model Fit
Enter model R-squared (R²), number of predictors (k), and sample size (N).
Calculation Results
Calculated using Multiple Linear Regression ANOVA equations: F = \frac{R^2 / k}{(1 - R^2) / (N - k - 1)}, \quad \bar{R}^2 = 1 - \frac{(1 - R^2)(N - 1)}{N - k - 1}
Quick Summary
The Multiple Linear Regression Calculator evaluates overall model significance ($F$-statistic), unadjusted $R^2$, Adjusted $R^2$, and residual degrees of freedom.
Formula Explanation
\hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_k x_k
F = \frac{R^2 / k}{(1 - R^2) / (N - k - 1)}, \quad \bar{R}^2 = 1 - \frac{(1 - R^2)(N - 1)}{N - k - 1}How It Works
Multiple Linear Regression extends simple regression to incorporate multiple explanatory variables simultaneously. The Multiple Linear Regression Calculator evaluates overall model fit metrics including Adjusted $R^2$ and $F$-ratio.
Step-by-Step Worked Example
Practical Problem: A multiple regression model with $k = 3$ predictors across $N = 50$ observations yields $R^2 = 0.75$. Calculate Adjusted $R^2$ and overall model $F$-statistic.
- Step 1: Calculate model degrees of freedom ($df_1$): $df_1 = k = \mathbf{3}$.
- Step 2: Calculate residual degrees of freedom ($df_2$): $df_2 = 50 - 3 - 1 = \mathbf{46}$.
- Step 3: Calculate Adjusted R²: $\bar{R}^2 = 1 - \frac{(1 - 0.75)(49)}{46} = 1 - \frac{0.25 \times 49}{46} = 1 - \frac{12.25}{46} = 1 - 0.2663 = \mathbf{0.7337\text{ (73.37\%)}}.$
- Step 4: Calculate Mean Square Regression / Error numerator & denominator: Numerator $= 0.75 / 3 = 0.25$; Denominator $= (1 - 0.75) / 46 = 0.25 / 46 = 0.005435$.
- Step 5: Calculate F-statistic: $F = 0.25 / 0.005435 = \mathbf{46.0000}$ ($p < 0.0001$, model is overall statistically significant).
Real-World Calculation Examples
Scenario 1: 3-Predictor Model (R² = 0.75, k = 3, N = 50)
Parameters: R² = 0.75, k = 3, N = 50
Result: Adj R² = 73.37%, F = 46.00 (Significant).
Scenario 2: Real Estate Price Valuation Model
Parameters: Predictors: Sq Ft, Bedrooms, Zipcode
Result: Multiple linear regression fit assessment.
Scenario 3: Employee Salary Predictor Model
Parameters: Years Experience, Education Level, Age
Result: Adjusted R² penalty assessment.
Scenario 4: Marketing Mix Attribution Model
Parameters: TV, Search Ads, Social Media Spend
Result: Model F-test evaluation.
Key Benefits of Using This Calculator
Adjusted R-Squared Calculation
Penalizes unnecessary predictors to prevent overfitting error bias.
Overall Model F-Test
Tests global null hypothesis $H_0: \beta_1 = \beta_2 = \dots = \beta_k = 0$.
Residual df Tracking
Computes residual degrees of freedom $N - k - 1$.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is Multiple Linear Regression?
Multiple Linear Regression predicts a continuous dependent variable $Y$ using two or more independent predictor variables ($X_1, X_2, \dots, X_k$).
Why is Adjusted R-Squared necessary?
Unadjusted $R^2$ ALWAYS increases when adding predictors (even useless ones); Adjusted $R^2$ penalizes adding irrelevant predictors.
What is Multicollinearity?
Multicollinearity occurs when two or more independent predictors are highly correlated, inflating variance of regression coefficients.
How do I detect Multicollinearity?
Use Variance Inflation Factor (VIF); VIF $> 5$ or $10$ indicates problematic multicollinearity.
How do I run Multiple Regression in Excel?
Use Data Analysis Toolpak: Regression, selecting multiple contiguous columns for Input X Range.
What does global model F-test evaluate?
The $F$-test checks if AT LEAST ONE predictor has a non-zero slope ($H_0: \beta_1 = \beta_2 = \dots = \beta_k = 0$).
What is partial regression coefficient ($\beta_i$)?
$\beta_i$ measures the change in $Y$ per unit change in $X_i$, HOLDING ALL OTHER PREDICTORS CONSTANT.
What is overfitting in multiple regression?
Overfitting occurs when a complex model fits noise in sample data well ($R^2$ high) but fails on new unseen data (low Adjusted $R^2$).
What is dummy coding for categorical predictors?
Categorical variables with $c$ categories are converted into $c - 1$ binary (0/1) indicator dummy variables.
What is homoscedasticity assumption?
The assumption that error residuals $\epsilon_i$ have constant variance across all predicted values $\hat{y}$.