16
0
0
0
16
0
0
0
The Multiple Regression Calculator fits a linear model with two independent variables (X₁ and X₂) to predict a dependent variable (Y). Multiple regression extends simple linear regression by allowing you to examine how two or more factors simultaneously influence an outcome, while controlling for each other's effects. This is one of the most powerful and commonly applied techniques in statistics, used extensively in econometrics, epidemiology, psychology, and engineering.
This simplified calculator uses Cramer's rule to solve the normal equations for a 2-predictor model. Enter up to 5 data points with two predictor values and one response value each. The calculator returns the intercept (b₀), both predictor coefficients (b₁, b₂), and the coefficient of determination (R²).
The multiple regression model with two predictors is:
$$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$
The ordinary least squares estimates minimize the sum of squared residuals. The normal equations in matrix form are:
$$\begin{bmatrix} n & \sum X_1 & \sum X_2 \\ \sum X_1 & \sum X_1^2 & \sum X_1 X_2 \\ \sum X_2 & \sum X_1 X_2 & \sum X_2^2 \end{bmatrix} \begin{bmatrix} b_0 \\ b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} \sum Y \\ \sum X_1 Y \\ \sum X_2 Y \end{bmatrix}$$
This calculator solves the 3×3 system using Cramer's rule, computing the determinant of the coefficient matrix and replacing each column with the right-hand side vector to obtain each coefficient. The determinant formula ensures an exact algebraic solution without iterative methods.
Each coefficient b₁ and b₂ represents the partial effect of that predictor, holding the other predictor constant. For example, b₁ = 2.5 means that Y increases by 2.5 units for each one-unit increase in X₁, assuming X₂ remains unchanged. This “controlling for” interpretation is what makes multiple regression so valuable for isolating individual effects in observational data.
The R² is calculated as \(R^2 = 1 - SSE/SST\), where SSE is the sum of squared residuals and SST is the total sum of squares around the mean of Y.
Multiple regression results require careful interpretation:
Be cautious about multicollinearity — when X₁ and X₂ are highly correlated with each other, the individual coefficients become unstable and difficult to interpret, even though the overall model fit may still be good. With only 5 data points and 3 parameters to estimate, the model has very few degrees of freedom (n - p - 1 = 2), so results should be considered exploratory.
Inputs
Results
House prices ($K) are modeled by size (sq ft) and age (years). The model shows each additional sq ft adds $168 to the price, while each year of age decreases value by $1,586 (holding size constant). R² = 0.999 indicates an excellent fit.
Inputs
Results
Exam scores are predicted by study hours and sleep hours. Each study hour adds 5.2 points, and each sleep hour adds 4.1 points. Both predictors contribute positively, with R² = 0.996.
Running two separate simple regressions gives you the marginal effect of each predictor, ignoring the other. Multiple regression gives you the partial effect of each predictor, controlling for the other. These can differ substantially when the predictors are correlated. For example, both hours studied and IQ might predict exam scores, but their simple regression coefficients would be inflated if high-IQ students also study more. Multiple regression disentangles their independent contributions.
Multicollinearity occurs when two or more predictor variables are highly correlated with each other. This makes it difficult to isolate the individual effect of each predictor, leading to unstable coefficients (large standard errors), coefficients that may have unexpected signs, and sensitivity to small data changes. The overall model fit (R²) is not affected, but interpreting individual coefficients becomes unreliable. Variance Inflation Factor (VIF) is commonly used to detect multicollinearity.
As a minimum, you need more observations than parameters. For 2 predictors plus an intercept (3 parameters), you need at least 4 data points, but this leaves almost no degrees of freedom. A common rule of thumb is 10-20 observations per predictor. This calculator supports up to 5 points with 2 predictors (3 parameters, 2 degrees of freedom), which is adequate for demonstration but insufficient for reliable inference.
Adjusted R² penalizes R² for the number of predictors: \(R^2_{adj} = 1 - \frac{(1-R^2)(n-1)}{n-p-1}\), where p is the number of predictors. Unlike regular R², adjusted R² can decrease when adding irrelevant predictors, making it better for comparing models with different numbers of predictors. This calculator reports standard R²; for small samples with multiple predictors, always check adjusted R² in practice.
Yes. In multiple regression, a predictor's coefficient reflects its effect after controlling for other predictors. This can lead to Simpson's paradox-like situations where a variable that positively correlates with Y in simple regression has a negative coefficient in multiple regression (or vice versa). This occurs when predictors are correlated. The partial coefficient is often more informative about the true relationship than the simple correlation.
Cramer's rule provides an exact algebraic solution to the system of normal equations without requiring iterative algorithms or matrix inversion libraries. For a 3×3 system (intercept + 2 predictors), the formulas are manageable and can be computed using basic arithmetic operations. This makes it suitable for an AST-based calculator engine that evaluates mathematical expressions directly. For larger systems (3+ predictors), Cramer's rule becomes computationally expensive and methods like Gaussian elimination or QR decomposition are preferred.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Linear Regression Calculator
Regression & Correlation Analysis
Simple Linear Regression Calculator
Regression & Correlation Analysis
Polynomial Regression Calculator
Regression & Correlation Analysis
Exponential Regression Calculator
Regression & Correlation Analysis
Logarithmic Regression Calculator
Regression & Correlation Analysis
Power Regression Calculator
Regression & Correlation Analysis
