Using Normal Probability Plots to Determine Parameters for Higher‐level Factorial Experiments with Orthogonal and Orthonormal Bases

ABSTRACTIn chemical engineering applications such as optimizing plant operations and product quality, factorial experiments are often conducted to obtain empirical models for systems. While the mathematical underpinning for two‐level factorial designs is well understood, a methodology for analyzing higher‐level experiments is not readily available. Often an orthogonal or orthonormal basis is selected for a factorial design matrix. In factorial design, an orthonormal basis is defined as an orthogonal matrix where the Euclidean two‐norms of the column vectors are equal. This investigation examines, for full factorial design, the selection of parameters using normal probability plots and the effect that the design basis has on parameter determination. When using normal probability plots to determine parameter significance, the traditional orthogonal basis for higher‐level experiments may result in erroneous conclusions. A Monte‐Carlo experiment was developed to simulate 3‐level and mixed‐level factorial experiments with different types of measurement error. The basis chosen is shown to affect the shape of probability plots, and measurement errors from a given normal distribution are shown to result in a constant standard deviation for all parameter estimates only when using an orthonormal basis.