Statistics & Probability

Multiple Regression Calculator | Coefficients, R², F-Test & Residuals

Fit a multiple linear regression model with 2–5 predictor variables. Computes regression coefficients, standard errors, t-statistics, p-values, R², adjusted R², and F-statistic. Includes variance inflation factor (VIF) to diagnose multicollinearity and residual analysis.

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Number of Predictors

Each row: y x₁ x₂ (space or comma separated)

What Is the Multiple Regression Calculator | Coefficients, R², F-Test & Residuals?

Multiple linear regression extends simple OLS to model a response variable y using two or more predictor variables. The model assumes a linear relationship: y = β₀ + β₁x₁ + β₂x₂ + … + βₚxₚ + ε, where ε ~ N(0, σ²).

The OLS estimator minimises the sum of squared residuals by solving the normal equations. R² quantifies the proportion of variance explained; the adjusted R² penalises for additional predictors. The F-test checks whether the model explains significantly more variance than an intercept-only model. VIF detects multicollinearity between predictors.

Formula

OLS normal equations: β = (XᵀX)⁻¹Xᵀy

SE(βⱼ) = √(s² · [(XᵀX)⁻¹]ⱼⱼ) | s² = SSₑ/(n−p−1)

t-stat = βⱼ/SE(βⱼ) | R² = 1 − SSₑ/SS_tot | Adj.R² = 1−(1−R²)(n−1)/(n−p−1)

F = (R²/p) / ((1−R²)/(n−p−1)) | VIFⱼ = 1/(1−R²ⱼ)

How to Use

1
Select the number of predictors (1–5) using the dropdown.
2
Type or paste your data in the textarea — one observation per line, values separated by spaces or commas.
3
Ensure each row starts with the response variable y followed by predictor values.
4
Click Run Regression to compute OLS coefficients.
5
Read R², adjusted R², and the F-test p-value from the summary cards.
6
Check the Coefficient Table for t-statistics and p-values for each predictor.
7
Review the VIF table for multicollinearity — values above 10 indicate serious concern.

Enter your data as one row per observation — first column is the response y, followed by predictor values x₁, x₂, etc. Select the number of predictors, then click Run Regression.

Example Calculation

Example 1 — Height/weight prediction: Data with y = weight (kg), x₁ = height (cm), x₂ = height of father (cm) for 10 people. After fitting, R² = 0.94 indicates 94% of weight variance is explained. VIF ≈ 1.1 confirms low multicollinearity between the predictors.

Example 2 — Multicollinearity demo: If x₁ and x₂ are nearly identical (e.g., temperature in °C and °F), VIF will approach ∞ and the matrix becomes near-singular. The calculator will flag this and return an error, prompting you to remove one of the redundant predictors.

Understanding Multiple Regression | Coefficients, R², F-Test & Residuals

OLS Regression Assumptions

Assumption	Description	Diagnostic
Linearity	y is linearly related to each xⱼ	Residual vs fitted plot
Independence	Observations are independent of each other	Durbin-Watson test
Homoscedasticity	Residual variance is constant across fitted values	Scale-location plot
Normality	Residuals are normally distributed	Q-Q plot, Shapiro-Wilk
No multicollinearity	Predictors are not highly correlated with each other	VIF < 10

R² and Model Fit Benchmarks

R² range	Interpretation	Typical domain
0.00 – 0.30	Weak fit	Social sciences, noisy data
0.30 – 0.60	Moderate fit	Psychology, economics
0.60 – 0.80	Good fit	Engineering, biology
0.80 – 1.00	Strong fit	Physical sciences, controlled experiments

Interpreting Regression Coefficients

▸β₀ (intercept): predicted value of y when all predictors equal zero. Often not meaningful by itself.
▸βⱼ (slope): for a one-unit increase in xⱼ, y changes by βⱼ units, holding all other predictors constant.
▸Standardised β: divide β by (SD of xⱼ / SD of y) to compare importance across predictors on different scales.
▸A predictor with p > 0.05 does not mean it is unimportant — collinearity, sample size, and confounding all affect significance.
▸RMSE (root mean square error) is in the same units as y and estimates the typical prediction error.

Frequently Asked Questions

What is R² and adjusted R²?

R² measures the proportion of variance in y explained by the model. Adjusted R² penalises for additional predictors using the formula 1−(1−R²)(n−1)/(n−p−1). Always prefer adjusted R² when comparing models with different numbers of predictors, as R² always increases when you add variables regardless of their usefulness.

What does the F-test tell me?

The F-statistic tests the null hypothesis that all regression coefficients (except the intercept) are simultaneously zero. A significant F p-value (<0.05) means the model as a whole explains a statistically meaningful amount of variance in y. A significant F does not imply all individual predictors are significant.

What is VIF and when should I be worried?

VIF (Variance Inflation Factor) measures how much the variance of a coefficient is inflated due to correlation with other predictors. VIF = 1/(1−R²ⱼ) where R²ⱼ is from regressing predictor j on all other predictors. VIF < 5 is acceptable; 5–10 is moderate; >10 indicates serious multicollinearity that inflates standard errors and makes individual t-tests unreliable.

How many observations do I need?

As a rule of thumb, you need at least 10–20 observations per predictor variable for reliable estimates. The calculator requires n ≥ p+2 to have at least one degree of freedom for residuals. With fewer observations than predictors, the model is underdetermined and cannot be fitted.

Why are some p-values showing as N/A?

N/A appears when the matrix (XᵀX) is singular or near-singular — this happens with perfect multicollinearity (e.g., one predictor is a linear combination of others), insufficient data, or all values in a column are identical. Remove redundant predictors or add more varied data to resolve this.

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