Multiple Linear Regression Calculator

Multiple linear regression is a statistical technique that models the relationship between one dependent variable (Y) and two or more independent predictor variables (X1, X2, …). It fits a linear equation of the form Y = β₀ + β₁X1 + β₂X2 + ε, where β values are the estimated coefficients and ε is the error term.

R² (the coefficient of determination) measures what proportion of the variance in Y is explained by the predictors. A value of 0.85 means 85% of the variability in Y is accounted for by X1 and X2. Higher values indicate a better fit, but adding more predictors will always increase R² even if they are not meaningful — use Adjusted R² to account for this.

Adjusted R² penalizes the model for adding predictors that do not improve explanatory power. Unlike plain R², it can decrease if a new variable adds noise rather than signal. It is generally a more honest measure of model fit when comparing models with different numbers of predictors.

The F-statistic tests whether the regression model as a whole is statistically significant — i.e., whether at least one predictor is meaningfully related to Y. If the model p-value is below your chosen significance level α (e.g., 0.05), you reject the null hypothesis that all coefficients are zero and conclude the model has explanatory value.

Each coefficient's p-value tests whether that specific predictor has a statistically significant relationship with Y, holding all other predictors constant. A p-value below α suggests the predictor contributes meaningfully to the model. A high p-value suggests the predictor may not be necessary.

Transformations are useful when your data violates the linearity or normality assumptions. For example, if Y grows exponentially with X, a log transformation of Y can linearize the relationship. If data is right-skewed, a square root transformation can reduce skewness and stabilize variance.

When you force zero intercept, the regression line is constrained to pass through the origin (0, 0), meaning β₀ = 0. This is appropriate only when you have theoretical or physical justification that Y must be zero when all predictors are zero. In most practical cases, you should include the intercept.

When 'Exclude Outliers' is selected, data points whose standardized residuals exceed ±2.5 standard deviations from the mean are removed before fitting the model. This can improve model fit when extreme values are due to data entry errors or anomalies, but should be used carefully so legitimate data is not discarded.