Linear & Quadratic Mathematical Models

Mathematical models are the heart of designed experiments. They translate experimental results into equations that describe how factors influence the response. Linear models capture straight‑line relationships, while quadratic models capture curvature. Understanding both is essential for designing effective improvements. 

A linear model includes main effects and interactions but assumes that the relationship between each factor and the response is straight. Linear models are appropriate for screening and characterization experiments where the goal is to identify key drivers and understand interactions. 

A quadratic model includes squared terms and, in some cases, cross‑product terms. These models capture curvature—situations where the response increases up to a point and then decreases, or vice versa. Quadratic models are essential for optimization because they allow you to identify peaks, valleys, and optimal settings. 

Quadratic models are typically built using response surface methods such as central composite or Box‑Behnken designs. These designs include additional levels that allow you to estimate curvature accurately. 

In the Improve phase, linear and quadratic models help you understand the structure of the process and design improvements that align with its true behavior. They provide a mathematical foundation for optimization and support confident, data‑driven decision‑making.