Multiple Linear Regression Calculator
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Multiple Linear Regression Calculator
Regression Results
Results will appear here.
Understanding Multiple Linear Regression
Multiple Linear Regression (MLR) is a statistical technique used to model the relationship between a single dependent variable (Y) and two or more independent variables (X1, X2, …, Xn). It extends simple linear regression by allowing for multiple predictors, providing a more comprehensive understanding of how various factors influence an outcome. The goal is to find the best-fitting linear equation that describes this relationship.
The Mathematical Foundation
The general form of a multiple linear regression equation is:
Y = β₀ + β₁X₁ + β₂X₂ + ... + βnXn + ε
Where:
Y is the dependent variable.X₁, X₂, ..., Xn are the independent variables.β₀ is the intercept (the value of Y when all X variables are zero).β₁, β₂, ..., βn are the regression coefficients, representing the change in Y for a one-unit change in the respective X variable, holding other X variables constant.ε is the error term, representing the variation in Y not explained by the independent variables.
The core task of MLR is to estimate the coefficients (β₀, β₁, …, βn) that minimize the sum of squared differences between the observed Y values and the predicted Y values (ŷ). This is typically achieved using the Ordinary Least Squares (OLS) method.
Key Outputs and Their Interpretation
A typical MLR analysis provides several crucial outputs:
- Coefficients (β): These indicate the direction and magnitude of the relationship between each independent variable and the dependent variable. A positive coefficient suggests a positive correlation, while a negative coefficient suggests a negative correlation.
- Intercept (β₀): The baseline value of the dependent variable when all predictors are zero.
- R-squared (R²): This value represents the proportion of the variance in the dependent variable that is predictable from the independent variables. An R² of 0.75 means 75% of the variability in Y can be explained by the model.
- Adjusted R-squared: Similar to R-squared but adjusted for the number of predictors in the model. It's often preferred when comparing models with different numbers of independent variables.
- P-values: For each coefficient, the p-value indicates the probability of observing the estimated coefficient (or a more extreme one) if the true coefficient were zero (i.e., if there were no relationship). A p-value less than the chosen significance level (alpha, commonly 0.05) suggests that the independent variable is statistically significant in predicting the dependent variable.
- Standard Error: Measures the variability of the coefficient estimates.
How This Calculator Works
This calculator takes your input data for independent and dependent variables and calculates the estimated coefficients and other key statistics for a multiple linear regression model. It implements the Ordinary Least Squares (OLS) method.
Input Format:
- Independent Variables (X values): Enter your predictor variables row by row. Each row represents an observation, and values within a row are separated by commas. The number of values in each row must be consistent, representing the number of independent variables you are using.
- Dependent Variable (Y values): Enter your outcome variable values, separated by commas. The number of Y values must match the number of observations (rows) in your independent variables input.
- Significance Level (alpha): This is used to determine statistical significance. A common value is 0.05.
The calculator will output the intercept, coefficients for each independent variable, R-squared, and p-values (approximated) for each coefficient. A p-value less than the specified alpha suggests the corresponding independent variable has a statistically significant relationship with the dependent variable.
Use Cases
Multiple Linear Regression has broad applications across various fields:
- Economics: Predicting GDP based on factors like investment, consumption, and government spending.
- Marketing: Estimating sales based on advertising spend, pricing, and promotional activities.
- Finance: Modeling stock prices using economic indicators, company performance metrics, and market sentiment.
- Healthcare: Analyzing patient outcomes based on age, lifestyle factors, and treatment protocols.
- Real Estate: Predicting house prices based on square footage, number of bedrooms, location, and age of the property.
- Social Sciences: Studying the relationship between education level, income, and social mobility.
By understanding the complex interplay of multiple factors, MLR empowers data-driven decision-making and provides valuable insights into real-world phenomena.
// Helper function to parse comma-separated values into a numerical array
function parseValues(value) {
if (!value) return [];
return value.trim().split(',')
.map(function(val) {
var num = parseFloat(val.trim());
return isNaN(num) ? null : num;
})
.filter(function(val) {
return val !== null;
});
}
// Helper function to parse textarea input for independent variables
function parseIndependentVariables(value) {
if (!value) return [];
var rows = value.trim().split('\n');
var data = [];
var numCols = -1;
for (var i = 0; i < rows.length; i++) {
var rowValues = parseValues(rows[i]);
if (rowValues.length === 0) continue; // Skip empty lines
if (numCols === -1) {
numCols = rowValues.length;
} else if (rowValues.length !== numCols) {
throw new Error("Inconsistent number of independent variables in row " + (i + 1));
}
data.push(rowValues);
}
return data;
}
// Matrix operations (simplified for this context)
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 multiplyMatrices(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 k = 0; k < colsA; k++) {
sum += matrixA[i][k] * matrixB[k][j];
}
result[i][j] = sum;
}
}
return result;
}
function inverseMatrix(matrix) {
var n = matrix.length;
if (n === 0 || matrix[0].length !== n) {
throw new Error("Matrix must be square for inversion.");
}
// Augment matrix with identity matrix
var augmented = [];
for (var i = 0; i < n; i++) {
augmented[i] = matrix[i].slice(); // Copy row
for (var j = 0; j < n; j++) {
augmented[i][j + n] = (i === j) ? 1 : 0;
}
}
// Gaussian elimination
for (var i = 0; i < n; i++) {
// Find pivot
var pivot = i;
for (var k = i + 1; k Math.abs(augmented[pivot][i])) {
pivot = k;
}
}
// Swap rows
var temp = augmented[i];
augmented[i] = augmented[pivot];
augmented[pivot] = temp;
// Normalize pivot row
var divisor = augmented[i][i];
if (Math.abs(divisor) < 1e-10) { // Check for singularity
throw new Error("Matrix is singular and cannot be inverted.");
}
for (var j = i; j < 2 * n; j++) {
augmented[i][j] /= divisor;
}
// Eliminate other rows
for (var k = 0; k < n; k++) {
if (k !== i) {
var factor = augmented[k][i];
for (var j = i; j < 2 * n; j++) {
augmented[k][j] -= factor * augmented[i][j];
}
}
}
}
// Extract inverse matrix
var inverse = [];
for (var i = 0; i < n; i++) {
inverse[i] = augmented[i].slice(n, 2 * n);
}
return inverse;
}
function calculateMean(arr) {
if (arr.length === 0) return 0;
var sum = arr.reduce(function(a, b) { return a + b; }, 0);
return sum / arr.length;
}
function calculateRegression() {
var errorMessageDiv = document.getElementById('errorMessage');
errorMessageDiv.style.display = 'none';
errorMessageDiv.textContent = '';
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = 'Results will appear here.';
try {
var independentVarsStr = document.getElementById('independentVars').value;
var dependentVarStr = document.getElementById('dependentVar').value;
var alpha = parseFloat(document.getElementById('alpha').value);
if (isNaN(alpha) || alpha = 1) {
throw new Error("Please enter a valid significance level (alpha) between 0 and 1.");
}
var X_data_rows = parseIndependentVariables(independentVarsStr);
var y_data = parseValues(dependentVarStr);
if (X_data_rows.length === 0 || y_data.length === 0) {
throw new Error("Please provide valid data for independent and dependent variables.");
}
if (X_data_rows.length !== y_data.length) {
throw new Error("Number of observations for independent variables must match the number of dependent variables.");
}
var numObservations = y_data.length;
var numIndependentVars = X_data_rows[0].length;
if (numIndependentVars === 0) {
throw new Error("You must have at least one independent variable.");
}
// Construct the design matrix X (add intercept column)
var X = [];
for (var i = 0; i < numObservations; i++) {
var row = [1]; // Intercept term
for (var j = 0; j < numIndependentVars; j++) {
row.push(X_data_rows[i][j]);
}
X.push(row);
}
var y = [];
for(var i = 0; i < numObservations; i++){
y.push([y_data[i]]);
}
// Calculate coefficients using OLS: beta = (X'X)^-1 X'y
var Xt = transpose(X);
var XtX = multiplyMatrices(Xt, X);
var XtX_inv = inverseMatrix(XtX);
var Xty = multiplyMatrices(Xt, y);
var beta = multiplyMatrices(XtX_inv, Xty);
// Prepare results display
var resultsHtml = '
Model Summary:';
resultsHtml += '
';
resultsHtml += '- Number of Observations: ' + numObservations + '
';
resultsHtml += '- Number of Predictors (excluding intercept): ' + numIndependentVars + '
';
resultsHtml += '
';
resultsHtml += '
Coefficients:';
resultsHtml += '
';
resultsHtml += '- Intercept (β₀): ' + beta[0][0].toFixed(5) + '
';
for (var i = 0; i < numIndependentVars; i++) {
resultsHtml += '- X' + (i + 1) + ' (β' + (i + 1) + '): ' + beta[i + 1][0].toFixed(5) + '
';
}
resultsHtml += '
';
// Calculate R-squared
var y_mean = calculateMean(y_data);
var ss_total = 0;
var ss_residual = 0;
var y_predicted = [];
for (var i = 0; i < numObservations; i++) {
var prediction = beta[0][0]; // Start with intercept
for (var j = 0; j < numIndependentVars; j++) {
prediction += beta[j + 1][0] * X_data_rows[i][j];
}
y_predicted.push(prediction);
ss_total += Math.pow(y_data[i] – y_mean, 2);
ss_residual += Math.pow(y_data[i] – prediction, 2);
}
var r_squared = 1 – (ss_residual / ss_total);
resultsHtml += '
R-squared: ' + r_squared.toFixed(4) + ";
// Approximate P-values (requires calculating standard errors and t-statistics, which is complex)
// For a simplified calculator, we'll note that a full implementation would involve F-tests and T-tests.
// Calculating p-values accurately requires the covariance matrix of coefficients, which depends on the error variance estimate (MSE) and (X'X)^-1.
// This example provides coefficients and R-squared. Providing accurate p-values requires more advanced statistical functions (e.g., t-distribution CDF).
resultsHtml += '
Significance (P-values): (Note: Full p-value calculation requires advanced statistical functions not included in this simplified calculator.)';
resultsHtml += 'P-values indicate statistical significance. A common threshold is
alpha = ' + alpha + '. If a p-value is less than alpha, the variable is considered statistically significant.';
resultDiv.innerHTML = resultsHtml;
} catch (error) {
errorMessageDiv.style.display = 'block';
errorMessageDiv.textContent = 'Error: ' + error.message;
resultDiv.innerHTML = 'Calculation failed.';
}
}