2.42.3.2 Model Fitting and Selection

After an experimental design has been conducted and response data have been collected, a statistical model can be fitted to describe the relationship between the experimental factors and the response.

Model fitting and selection help determine which factor effects and interactions should be included in the model and whether the selected model adequately describes the experimental data.

Origin provides several tools for evaluating and selecting a model, including model term selection, Stepwise Regression, and the Lack of Fit Test.

Model Fitting

A DOE model describes how the response changes as a function of the experimental factors. Depending on the design and the selected model, the model may include:

Main effects, which describe the individual effects of factors. Interaction effects, which describe how the effect of one factor depends on another factor. Higher-order terms, such as quadratic terms, which can describe curvature in the response. Other design-specific terms, depending on the type of experimental design.

For example, a model for two quantitative factors \(X_1\) and \(X_2\) may be written as:

\[Y=\beta_0+\beta_1X_1+\beta_2X_2+\beta_{12}X_1X_2+\beta_{11}X_1^2+\beta_{22}X_2^2+\epsilon\]

where:

• \(Y\) is the response.
• \(beta_0\) is the intercept.
• \(\beta_1\) and \(\beta_2\) represent the main effects.
• \(\beta_{12}\) represents the interaction between \(X_1\) and \(X_2\).
• \(\beta_{11}\) and \(\beta_{22}\) represent quadratic effects.
• \(\epsilon\) represents experimental error.

The appropriate model terms depend on the experimental design and the objective of the analysis.

Model Selection

Including unnecessary terms can make a model more complicated than necessary, while omitting important terms can result in an inadequate model. Model selection therefore aims to identify a model that adequately describes the response while avoiding unnecessary terms.

When selecting a model, consider:

• Model fit statistics
• Lack of fit.
• Residual diagnostics
• Plots for analyze design

Stepwise Regression

Stepwise Regression provides an automated approach for selecting model terms based on specified statistical criteria.

During stepwise regression, terms can be added to or removed from the model according to the selected entry and removal criteria. This can help reduce a model containing many candidate terms to a smaller set of potentially important terms.

See Stepwise Regression for Analyze Design for details.

Note: Stepwise regression is a model-selection procedure and should not replace consideration of the experimental design, model hierarchy, subject-matter knowledge, and diagnostic results.

Model Prediction

The Find Y from X feature calculates predicted response values for selected factor settings using the fitted DOE model. It can be used to examine the predicted response at specific combinations of factor values without manually calculating the model equation.

See Find Y from X for details