What Is Multiple Regression Analysis?
Multiple linear regression predicts one outcome y from two or more predictors x₁, x₂, … using the equation y = b₀ + b₁x₁ + b₂x₂ + …. The coefficients are chosen by least squares, b = (XᵀX)⁻¹Xᵀy, to minimise the sum of squared residuals.
Paste your data with y in the first column and the predictors after it. The calculator returns the regression equation, a coefficient table with standard errors, t statistics, p-values and 95 % confidence intervals, plus R², adjusted R², the overall F test, the residual standard error and charts of actual versus predicted values and of residuals.
The article below explains the formulas, how to read each number, and answers to common questions about adjusted R² and multicollinearity.
How to Use the Multiple Regression Analysis Calculator
- Paste your data with y in the first column and the predictors after it.
- Add a header row if you want named variables.
- Read the equation, R², adjusted R² and the overall F test.
- Check each predictor's p-value and the residual plots.
Multiple Regression Analysis Formula
Model
Least squares solution
Adjusted R²
Overall F
Multiple Regression Analysis Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| bⱼ | Coefficient of predictor j | change in y per unit of xⱼ, others fixed |
| R² | Share of variation in y explained | |
| Adj. R² | R² corrected for the number of predictors | |
| F | Overall significance statistic | df = (k, n − k − 1) |
| k | Number of predictors |
Multiple Regression Analysis Worked Example
Each coefficient is the expected change in y for a one-unit rise in that predictor, holding the others fixed. Adjusted R² penalises useless predictors, so it can fall when you add a variable that does not help. The overall F test asks whether at least one predictor is useful; the t-tests ask about each predictor on its own.
With the example values loaded in the calculator above, the steps are:
- Build the design matrix X (a column of 1s plus 3 predictor columns) and solve the normal equations b = (XᵀX)⁻¹ Xᵀy.
- Coefficients: Intercept = 80.8361, Size = 0.151829, Rooms = -7.47076, Age = -1.27027.
- SSE = Σ(y − ŷ)² = 408.975, SST = 32,600.5, so R² = 1 − SSE ÷ SST = 0.987455.
- Residual variance MSE = SSE ÷ (n − k − 1) = 68.1626; standard errors come from the diagonal of MSE × (XᵀX)⁻¹.
- Overall F = (SSR ÷ k) ÷ MSE = 157.425.
Result: Regression equation = ŷ = 80.836 + 0.15183·Size − 7.4708·Rooms − 1.2703·Age; R² = 0.987455; Adjusted R² = 0.981182.
Where Multiple Regression Analysis Is Used
Prediction
Predicting price, sales or scores from several factors.
Understanding drivers
Seeing which predictors matter when others are held fixed.
Research projects
Reporting coefficients, p-values and R² for coursework.
Model comparison
Comparing adjusted R² with and without a predictor.
Common Multiple Regression Analysis Mistakes to Avoid
- Do not read a coefficient as causal.
- Avoid including two nearly identical predictors.
- Do not extrapolate far beyond your data.
Multiple Regression Analysis Terms and Definitions
- Adjusted R²
- R² corrected for the number of predictors.
- Multicollinearity
- Predictors that are strongly correlated with each other.
- Residual
- The difference between an observed and a predicted y.
Multiple Regression Analysis FAQs
What is multiple linear regression?
A model that predicts one outcome from two or more predictor variables using a linear equation.
What is the difference between R² and adjusted R²?
Adjusted R² penalises extra predictors, so it only rises if a new variable improves the model more than chance would.
How many predictors can I use?
This calculator allows up to 8, but you should have many more observations than predictors.
What does a p-value for a coefficient mean?
It tests whether that coefficient is zero given the other predictors are in the model.