In the real world, outcomes rarely depend on a single variable. Multiple linear regression (MLR) lets us examine how several factors simultaneously affect a response variable. This is important for making accurate predictions and understanding complex systems.
MLR is used in engineering, business, healthcare, and social sciences to:
Single-variable models oversimplify reality. For example, predicting a student's GPA using only study hours ignores other important factors like sleep, stress, and class attendance. MLR accounts for multiple influences simultaneously.
Suppose we want to predict a student’s GPA (Y). Possible predictors:
This MLR model helps quantify how each behavior affects GPA while holding others constant.
Multiple linear regression involves more than one predictor variable. The general form of the model is:
This equation describes a hyperplane in (k+1)-dimensional space. Each coefficient βⱼ indicates the effect of one predictor holding the others constant.
Vehicle mileage depends on weight (x₁) and engine displacement (x₂):
Example model: E(Y) = 50 + 10x₁ + 7x₂
Higher-order polynomial regression can also be expressed in linear regression format:
Rewriting using new variables x₁ = x, x₂ = x², x₃ = x³:
Interaction effects are represented by cross-product terms. For example:
This model accounts for effects that vary depending on the level of another predictor.
Flexible model that includes quadratic and interaction terms:
Can be rewritten with new variables:
Where: x₃ = x₁², x₄ = x₂², x₅ = x₁x₂
This section can include regression plane plots and contour plots using 3D graphics or Chart.js projections (coming soon).