 5. Process Improvement
5.5.9. An EDA approach to experimental design
5.5.9.9. Cumulative residual standard deviation plot

## Motivation: What are the Advantages of the Linear Combinatoric Model?

Advantages: perfect fit and comparable coefficients The linear model consisting of main effects and all interactions has two advantages:
1. Perfect Fit: If we choose to include in the model all of the main effects and all interactions (of all orders), then the resulting least squares fitted model will have the property that the predicted values will be identical to the raw response values Y. We will illustrate this in the next section.

2. Comparable Coefficients: Since the model fit has been carried out in the coded factor (-1, +1) units rather than the units of the original factor (temperature, time, pressure, catalyst concentration, etc.), the factor coefficients immediately become comparable to one another, which serves as an immediate mechanism for the scale-free ranking of the relative importance of the factors.
Example To illustrate in detail the above latter point, suppose the (-1, +1) factor X1 is really a coding of temperature T with the original temperature ranging from 300 to 350 degrees and the (-1, +1) factor X2 is really a coding of time t with the original time ranging from 20 to 30 minutes. Given that, a linear model in the original temperature T and time t would yield coefficients whose magnitude depends on the magnitude of T (300 to 350) and t (20 to 30), and whose value would change if we decided to change the units of T (e.g., from Fahrenheit degrees to Celsius degrees) and t (e.g., from minutes to seconds). All of this is avoided by carrying out the fit not in the original units for T (300,350) and t (20, 30), but in the coded units of X1 (-1, +1) and X2 (-1, +1). The resulting coefficients are unit-invariant, and thus the coefficient magnitudes reflect the true contribution of the factors and interactions without regard to the unit of measurement.
Coding does not lead to loss of generality Such coding leads to no loss of generality since the coded factor may be expressed as a simple linear relation of the original factor (X1 to T, X2 to t). The unit-invariant coded coefficients may be easily transformed to unit-sensitive original coefficients if so desired. 