Enter your data points. Each row represents an observation. The first column is your dependent variable (Y), and subsequent columns are your independent variables (X1, X2, …).
Example: For two independent variables, you would have 3 columns. For three, 4 columns.
Enter data separated by commas for each row.
Results
Enter data and click "Calculate Coefficients".
Understanding Multiple Regression
Multiple regression is a powerful statistical technique used to model the relationship between a single dependent variable (Y) and two or more independent variables (X1, X2, …, Xk). Unlike simple linear regression which examines the relationship between two variables, multiple regression allows us to understand how multiple factors collectively influence an outcome.
The Mathematical Model
The general form of a multiple linear regression model is:
Y = β₀ + β₁X₁ + β₂X₂ + … + βkXk + ε
Y: The dependent variable (the outcome we want to predict or explain).
X₁, X₂, …, Xk: The independent variables (the factors that may influence Y).
β₀: The intercept, representing the expected value of Y when all independent variables are zero.
β₁, β₂, …, βk: The regression coefficients for each independent variable. Each coefficient represents the expected change in Y for a one-unit increase in the corresponding independent variable, holding all other independent variables constant.
ε: The error term, representing the variation in Y that cannot be explained by the independent variables.
How the Calculator Works
This calculator estimates the coefficients (β₀, β₁, …, βk) using the Ordinary Least Squares (OLS) method. OLS aims to find the coefficients that minimize the sum of the squared differences between the observed values of Y and the values predicted by the regression model.
Mathematically, this involves solving the normal equations:
(XᵀX)β = Xᵀy
Where:
X is the design matrix, which includes a column of ones (for the intercept), and columns for each independent variable.
Xᵀ is the transpose of the design matrix.
y is the vector of observed dependent variable values.
β is the vector of coefficients (β₀, β₁, …, βk) that we want to estimate.
The solution for β is:
β = (XᵀX)⁻¹ Xᵀy
This calculator performs these matrix operations to compute the estimated coefficients.
Use Cases
Multiple regression has wide-ranging applications across various fields:
Economics: Predicting GDP based on factors like interest rates, inflation, and consumer spending.
Finance: Estimating stock prices based on market indices, company performance, and economic indicators.
Marketing: Determining the impact of advertising spend, promotions, and pricing on sales revenue.
Healthcare: Analyzing patient outcomes based on treatment, age, lifestyle factors, and genetic predispositions.
Social Sciences: Understanding factors influencing educational attainment, such as parental income, school quality, and student motivation.
Environmental Science: Modeling pollution levels based on industrial output, traffic volume, and weather patterns.
By understanding the relationships between variables, businesses and researchers can make more informed decisions, forecast future trends, and identify key drivers of outcomes.
function calculateRegression() {
var n = parseInt(document.getElementById('numObservations').value);
var k = parseInt(document.getElementById('numIndependentVars').value);
var dataTextArea = document.getElementById('dataInput').value.trim();
var resultsDiv = document.getElementById('results');
if (isNaN(n) || isNaN(k) || n <= 0 || k < 0) {
resultsDiv.innerHTML = "Please enter valid numbers for observations and independent variables.";
return;
}
var dataRows = dataTextArea.split('\n');
if (dataRows.length !== n) {
resultsDiv.innerHTML = "The number of data rows entered does not match the specified number of observations (n).";
return;
}
var Y = []; // Dependent variable
var X = []; // Independent variables matrix
var maxYCols = k + 1; // Max columns per row (Y + k Xs)
for (var i = 0; i < dataRows.length; i++) {
var rowValues = dataRows[i].split(',').map(function(val) { return parseFloat(val.trim()); });
if (rowValues.length !== maxYCols) {
resultsDiv.innerHTML = "Row " + (i + 1) + " has an incorrect number of values. Expected " + maxYCols + " (Y + " + k + " X's).";
return;
}
for (var j = 0; j < rowValues.length; j++) {
if (isNaN(rowValues[j])) {
resultsDiv.innerHTML = "Invalid number found in row " + (i + 1) + ", column " + (j + 1) + ".";
return;
}
}
Y.push(rowValues[0]);
X.push(rowValues.slice(1)); // X values for this observation
}
// Prepare the design matrix X_design (including intercept)
// X_design will have n rows and k+1 columns
var X_design = [];
for (var i = 0; i < n; i++) {
var row = [1]; // Add intercept term (X0 = 1)
row = row.concat(X[i]); // Add independent variables
X_design.push(row);
}
// — Matrix Operations —
// Helper functions for matrix operations
function transpose(matrix) {
var rows = matrix.length;
var cols = matrix[0].length;
var transposed = [];
for (var j = 0; j < cols; j++) {
transposed[j] = [];
for (var i = 0; i < rows; i++) {
transposed[j][i] = matrix[i][j];
}
}
return transposed;
}
function multiply(matrixA, matrixB) {
var rowsA = matrixA.length;
var colsA = matrixA[0].length;
var rowsB = matrixB.length;
var colsB = matrixB[0].length;
if (colsA !== rowsB) {
throw new Error("Matrices cannot be multiplied: incompatible dimensions.");
}
var result = [];
for (var i = 0; i < rowsA; i++) {
result[i] = [];
for (var j = 0; j < colsB; j++) {
var sum = 0;
for (var l = 0; l < colsA; l++) {
sum += matrixA[i][l] * matrixB[l][j];
}
result[i][j] = sum;
}
}
return result;
}
// Function to calculate the inverse of a matrix (requires a library for general case,
// but for small matrices like X'X, we can implement specific logic or use a simple inversion method if needed).
// For simplicity, this example will assume a basic inverse calculation is needed,
// but a robust solution would use a dedicated linear algebra library.
// NOTE: A full matrix inversion is complex. For this example, we'll use a placeholder
// and focus on the structure. A real-world implementation would need a robust library.
// Let's implement a 2×2 and 3×3 inverse for demonstration purposes,
// as typical basic regression examples might use this.
// For larger matrices, a proper library is essential.
function inverse(matrix) {
var n = matrix.length;
if (n === 0 || matrix[0].length !== n) {
throw new Error("Matrix must be square to find inverse.");
}
// Simplified inversion for small matrices
if (n === 2) {
var a = matrix[0][0], b = matrix[0][1], c = matrix[1][0], d = matrix[1][1];
var det = a * d – b * c;
if (det === 0) throw new Error("Matrix is singular, cannot invert.");
return [[d / det, -b / det], [-c / det, a / det]];
} else if (n === 3) {
// Using adjugate method for 3×3
var m = matrix;
var det = m[0][0] * (m[1][1] * m[2][2] – m[1][2] * m[2][1]) –
m[0][1] * (m[1][0] * m[2][2] – m[1][2] * m[2][0]) +
m[0][2] * (m[1][0] * m[2][1] – m[1][1] * m[2][0]);
if (det === 0) throw new Error("Matrix is singular, cannot invert.");
var invDet = 1 / det;
var inv = [
[(m[1][1] * m[2][2] – m[1][2] * m[2][1]) * invDet, (m[0][2] * m[2][1] – m[0][1] * m[2][2]) * invDet, (m[0][1] * m[1][2] – m[0][2] * m[1][1]) * invDet],
[(m[1][2] * m[2][0] – m[1][0] * m[2][2]) * invDet, (m[0][0] * m[2][2] – m[0][2] * m[2][0]) * invDet, (m[0][2] * m[1][0] – m[0][0] * m[1][2]) * invDet],
[(m[1][0] * m[2][1] – m[1][1] * m[2][0]) * invDet, (m[0][1] * m[2][0] – m[0][0] * m[2][1]) * invDet, (m[0][0] * m[1][1] – m[0][1] * m[1][0]) * invDet]
];
return inv;
} else {
// For matrices larger than 3×3, a more complex algorithm (like Gaussian elimination)
// or a library is required. This is a limitation of this basic implementation.
throw new Error("Matrix inversion for size " + n + "x" + n + " is not supported in this basic calculator. Use a dedicated library for larger systems.");
}
}
// Convert Y to a column vector matrix
var y_vector = [];
for (var i = 0; i < Y.length; i++) {
y_vector.push([Y[i]]);
}
try {
var Xt = transpose(X_design);
var XtX = multiply(Xt, X_design);
var XtX_inv = inverse(XtX);
var XtY = multiply(Xt, y_vector);
var beta_vector = multiply(XtX_inv, XtY);
var coefficients = [];
coefficients.push(beta_vector[0][0]); // Intercept (beta_0)
for (var i = 1; i < beta_vector.length; i++) {
coefficients.push(beta_vector[i][0]); // Other coefficients (beta_1, beta_2, …)
}
var resultHtml = "