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.
How to Read a Multiple Regression
Each coefficient is the expected change in y for a one-unit rise in that predictor, holding the other predictors fixed. R² says how much of the variation in y the whole model explains, and adjusted R² penalises predictors that add little, so it can fall when you add a useless variable.
The overall F test asks whether any predictor is useful. Then look at each coefficient’s p-value. Predictors that are strongly correlated with each other make the coefficients unstable, even when the overall fit is good.
Where Multiple Regression Analysis Is Used
Several predictors
Model a response with more than one explanatory variable.
Model comparison
Compare R² and adjusted R² as you add or drop predictors.
Significance
Read the overall F test and the coefficient table.
Residual spread
Use the residual standard error to judge typical prediction error.
Common Multiple Regression Analysis Mistakes to Avoid
- You need many more observations than predictors.
- Highly correlated predictors make coefficients unstable.
- Do not read a coefficient on its own. It depends on which other predictors are in the model.
- Check residuals for patterns before you trust the tests.
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?
It is a model y = b₀ + b₁x₁ + b₂x₂ + … fitted by least squares, with more than one predictor.
What is the difference between R² and adjusted R²?
R² never falls when you add a predictor. Adjusted R² accounts for the number of predictors and falls if a new one adds too little.
How many predictors can I use?
There must be fewer than the number of observations, and in practice you want many more observations than predictors, as a common guide, around 10 or more per predictor.
What does a p-value for a coefficient mean?
It tests whether that coefficient could be zero given the other predictors in the model. A small p-value is evidence that the predictor adds information.