Multiple Linear Regression

Regression and Prediction Models

This tool performs multiple linear regression to model the relationship between a dependent variable and one or more independent variables.

Practical Examples

Explore these examples to understand how to use the calculator for different scenarios.

House Price Prediction

Real Estate

Predicting house prices (Y) based on square footage (X1) and number of bedrooms (X2).

Y: 300000, 450000, 500000, 620000

X: 

1500, 2
2000, 3
2200, 3
2800, 4

Predict for: 2500, 3

Sales Performance Analysis

Marketing

Analyzing product sales (Y) based on advertising spend (X1) and website traffic (X2).

Y: 250, 320, 400, 500, 550

X: 

1000, 5000
1500, 6000
2000, 7500
2500, 9000
3000, 10000

Predict for: 2200, 8000

Crop Yield Estimation

Agriculture

Estimating crop yield (Y, in tons per acre) based on rainfall (X1, in inches) and fertilizer used (X2, in kg per acre).

Y: 3.5, 4.2, 4.0, 5.1, 4.8

X: 

20, 100
25, 120
22, 110
30, 150
28, 140

Predict for: 26, 130

Student Exam Score Prediction

Education

Predicting a student's final exam score (Y) based on hours studied (X1) and attendance rate (X2, as a percentage).

Y: 65, 72, 78, 85, 92

X: 

5, 80
8, 85
10, 90
12, 95
15, 98

Predict for: 11, 92

Other Titles
Understanding Multiple Linear Regression: A Comprehensive Guide
An in-depth look at the principles, applications, and mathematics behind multiple linear regression analysis.

What is Multiple Linear Regression?

  • Defining the Model
  • The Core Equation
  • Key Assumptions
Multiple Linear Regression (MLR) is a statistical technique used to model the relationship between a single dependent (or response) variable and two or more independent (or predictor) variables. It is an extension of simple linear regression, which only considers one predictor. The goal of MLR is to find a linear equation that best predicts the value of the dependent variable based on the values of the independent variables.
The Core Equation
The fundamental equation for a multiple linear regression model is: Y = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε. Where: Y is the dependent variable, X₁, X₂, ..., Xₖ are the independent variables, β₀ is the y-intercept (the value of Y when all X's are 0), β₁, β₂, ..., βₖ are the regression coefficients representing the change in Y for a one-unit change in the respective X, and ε is the model error or residual.
Key Assumptions
For the model to be valid and reliable, several assumptions must be met: 1. Linearity: A linear relationship exists between the dependent variable and the independent variables. 2. Independence: The residuals (errors) are independent of each other. 3. Homoscedasticity: The variance of the residuals is constant for all observations. 4. Normality: The residuals are normally distributed. 5. No Multicollinearity: The independent variables are not highly correlated with each other.

Step-by-Step Guide to Using the Calculator

  • Inputting Your Data
  • Making Predictions
  • Interpreting the Results
Inputting Your Data
In the 'Dependent Variable (Y)' field, enter the values of the variable you want to predict. In the 'Independent Variables (X)' field, enter the data for your predictors. Each row should correspond to an observation, and each column to a different variable. Ensure that the number of rows in the X data matches the number of entries in the Y data.
Making Predictions
To predict a new Y value, enter the corresponding values for each independent variable into the 'Predict Y for new X values' field, separated by commas. The number of values must match the number of independent variables used in the model.
Interpreting the Results
The calculator provides several key outputs: the regression equation, the coefficients (including the intercept), R-squared, Adjusted R-squared, and the standard error. These values help you understand the strength, direction, and significance of the relationships between your variables.

Real-World Applications of Multiple Linear Regression

  • Economics and Finance
  • Medical Research
  • Marketing and Sales
Economics and Finance
MLR is widely used to predict asset prices, forecast GDP growth based on inflation and unemployment rates, or model a company's stock price based on its earnings, debt, and other market factors.
Medical Research
In healthcare, it can be used to predict a patient's blood pressure based on factors like age, weight, and cholesterol levels, or to identify risk factors for diseases.
Marketing and Sales
Companies use MLR to predict product sales based on advertising expenditure, promotional activities, and competitor pricing. It helps in optimizing marketing strategies and resource allocation.

Common Pitfalls and How to Avoid Them

  • Overfitting the Model
  • Ignoring Multicollinearity
  • Misinterpreting Causation
Overfitting the Model
Overfitting occurs when the model performs well on the training data but poorly on new, unseen data. This can happen if you include too many independent variables. Use Adjusted R-squared and techniques like cross-validation to check for overfitting.
Ignoring Multicollinearity
When independent variables are highly correlated, it becomes difficult to determine the individual effect of each predictor on the dependent variable. This can lead to unstable and unreliable coefficient estimates. Check correlation matrices or Variance Inflation Factor (VIF) to detect multicollinearity.
Misinterpreting Causation
Regression analysis reveals relationships, not causation. A strong relationship between X and Y doesn't automatically mean X causes Y. There could be a lurking variable influencing both. Always apply domain knowledge to interpret the results correctly.

Mathematical Derivation and Formulas

  • The Matrix Formulation
  • Calculating Coefficients
  • Measuring Model Fit
The Matrix Formulation
The MLR model can be expressed concisely using matrix algebra: y = Xβ + ε. Here, 'y' is a vector of the dependent variable observations, 'X' is the design matrix (with a leading column of ones for the intercept), 'β' is the vector of coefficients, and 'ε' is the vector of errors.
Calculating Coefficients
The coefficients (β) are estimated using the method of least squares, which minimizes the sum of the squared residuals. The formula to calculate the coefficient vector is: β = (XᵀX)⁻¹Xᵀy, where Xᵀ is the transpose of X and (XᵀX)⁻¹ is the inverse of the matrix product XᵀX.
Measuring Model Fit
R-squared is calculated as R² = 1 - (SSR / SST), where SSR is the sum of squared residuals (Σ(yᵢ - ŷᵢ)²) and SST is the total sum of squares (Σ(yᵢ - ȳ)²). Adjusted R-squared is calculated as 1 - [(1 - R²)(n - 1) / (n - k - 1)], where 'n' is the number of observations and 'k' is the number of predictors.