Multiple Linear Regression Calculator

Multiple Linear Regression

Dependent Variable (Y) Separate observations with commas, spaces, or newlines

Please enter valid numeric data for Dependent Variable Y.

Predictor X₁

Predictor X₂

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Multivariable Predictive Modeling

What Is Multiple Linear Regression?

Multiple Linear Regression (MLR) models the linear relationship between a single continuous dependent variable ($Y$) and two or more independent predictor variables ($X_1, X_2, \dots, X_k$).

It extends Simple Linear Regression by isolating the unique partial contribution of each predictor while holding all other explanatory variables constant, providing comprehensive omnibus model testing ($F$-test) and individual slope coefficient $t$-tests.

When to Use It

Multivariable Forecasting: Predicting an outcome variable from multiple explanatory features.
Hypothesis Testing: Evaluating whether specific predictors exert statistically significant partial effects ($\beta_j \neq 0$).
Controlling for Confounders: Estimating an intervention effect while mathematically adjusting for baseline covariates.
Variance Explanation: Measuring the proportion of total variance explained via the coefficient of determination ($R^2$).

The Matrix OLS Formulas

$$\mathbf{\hat{\beta}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}$$ $$R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}, \quad R^2_{\text{adj}} = 1 - \frac{SS_{\text{res}}/(n - k - 1)}{SS_{\text{tot}}/(n - 1)}$$ $$F = \frac{MS_{\text{reg}}}{MS_{\text{res}}} = \frac{SS_{\text{reg}} / k}{SS_{\text{res}} / (n - k - 1)}$$ $$SE(\hat{\beta}_j) = \sqrt{MS_{\text{res}} \cdot [(\mathbf{X}^T \mathbf{X})^{-1}]_{jj}}, \quad t_j = \frac{\hat{\beta}_j}{SE(\hat{\beta}_j)}$$

Where $\mathbf{X}$ is the $n \times (k+1)$ design matrix with leading ones for the intercept, $\mathbf{y}$ is the observation vector, and $MS_{\text{res}} = s_e^2$ is the residual mean square error with $df_{\text{res}} = n - k - 1$.

Worked Example

Scenario: Regressing $Y = [15, 20, 24, 28, 33, 38]$ on $X_1 = [2, 4, 5, 7, 8, 10]$ and $X_2 = [1, 2, 3, 2, 4, 5]$ ($n=6, k=2$) at $\alpha=0.05$.

Step 1: Compute OLS Estimates ($\mathbf{\hat{\beta}}$):
$\hat{\beta}_0 = 8.7908, \quad \hat{\beta}_1 = 2.4716, \quad \hat{\beta}_2 = 0.9575$.
Model: $\hat{Y} = 8.7908 + 2.4716 X_1 + 0.9575 X_2$.

Step 2: Partition Sums of Squares ($SS$):
$SS_{\text{tot}} = 357.3333, \quad SS_{\text{res}} = 0.9007, \quad SS_{\text{reg}} = 356.4326$.

Step 3: Goodness of Fit & Overall Model Significance:
$R^2 = \frac{356.4326}{357.3333} = 0.9975, \quad R^2_{\text{adj}} = 0.9958$.
$F = \frac{356.4326 / 2}{0.9007 / 3} = \frac{178.2163}{0.3002} = 593.60 \implies p < 0.0001$.

Conclusion: The regression model explains **99.75% of variance** and is statistically significant overall ($p < 0.0001$).

Regression Fit Complete

Multiple Regression Results

Estimated Regression Equation

Ŷ = 8.7908 + 2.4716 X₁ + 0.9575 X₂

Overall Model Is Statistically Significant (F = 593.60, p < 0.0001)

R-Squared (R²) 0.9975

Adjusted R² 0.9958

F-Statistic 593.60

Model p-value < 0.0001

Residual Std Error (sₑ) 0.5479

Sample Size (n) 6

Regression Coefficients & Parameter Estimates

Predictor	Coefficient (β̂)	Std Error	t-Statistic	p-value	95% Confidence Interval	Sig? (α = 0.05)

ANOVA Summary Table (Overall Model Fit)

Source of Variation	Sum of Squares (SS)	df	Mean Square (MS)	F-Statistic	p-value

Multiple Linear Regression Calculator