Model Browser User's Guide

Linear Regression

This introduction to linear regression in the Model-Based Calibration Toolbox is divided into the following sections:

• Stepwise Regression
• Definitions
• Prediction Error Variance

Stepwise Regression

Building a regression model that includes only a subset of the total number of available terms involves a tradeoff between two conflicting objectives:

• Increasing the number of model terms always reduces the Sum Squared Error.
• However, you do not want so many model terms that you overfit by chasing points and trying to fit the model to signal noise. This reduces the predictive value of your model.

The best regression equation is the one that provides a satisfactory tradeoff between these conflicting goals, at least in the mind of the analyst. It is well known that there is no unique definition of best. Different model building criteria (for example, forward selection, backward selection, PRESS search, stepwise search, Mallows C_p Statistic...) yield different models. In addition, even if the optimal value of the model building statistic is found, there is no guarantee that the resulting model will be optimal in any other of the accepted senses.

Principally the purpose of building the regression model for calibration is for predicting future observations of the mean value of the response feature. Therefore the aim is to select the subset of regression terms such that PRESS, defined below, is minimized. Minimizing PRESS is consistent with the goal of obtaining a regression model that provides good predictive capability over the experimental factor space. This approach can be applied to both polynomial and spline models. In either case the model building process is identical.

1. The regression matrix can be viewed in the Design Evaluation Tool. Terms in this matrix define the full model. In general, the stepwise model is a subset of this full term set.
2. All regressions are carried out with the factors represented on their coded scales (-1,1).
3. All factors and response features are identified by appropriate symbols defined by the test plan. These symbols are automatically carried across to the model building tool.

Technical Documents

Definitions