Filling a design table with response values is not the end. Only after calculating effects, expressing a model, and checking whether residuals support that candidate model can you explore conditions.

Effects and regression coefficients express the same information on different scales

For a 2² design, the coded model is ŷ=b₀+bAA+bBB+bABAB. An effect is high mean minus low mean, while a ±1 coded coefficient is effect/2. In a balanced design, b₀ is the grand mean.

ANOVA partitions SS explained by the model and residual SS. With replication, residuals can be divided further into pure error and lack of fit. Before scanning each effect p-value, inspect the complete candidate model, degrees of freedom, and residual information.

Model hierarchy preserves an interpretable coordinate system

When including AB interaction, retain A and B main effects; when including A², retain A. This is the hierarchy principle. Removing lower-order terms because their p-values are large changes the model's meaning when the coding origin changes and breaks interaction interpretation.

In a small DOE, fitting every term leaves almost no residual degrees of freedom. Effect plots, a prior sparsity assumption, and scientific knowledge can help, but repeatedly deleting terms through automatic stepwise p-value rules creates selection bias. A pooled term is not declared exactly zero; it is a modelling judgment to combine it into error estimation.

Coded and natural-unit equations have different roles

Coded −1/+1 units make effect comparisons and orthogonal structure clear. A natural-unit equation in real °C, pH, or rpm is needed for operational interpretation and communication.

For low X_L and high X_H, coded x is (X−center)/half-range. Transforming to natural units changes coefficient numbers and units, but must give the same predictions within the same range. Preserve the transformation, units, and valid range together.

The Profiler moves a cross-section of the fitted model

Prediction Profiler shows a prediction while moving one factor and holding the others at their current values. With interaction, the slope for A changes with B's setting, so panels are not independent. A contour shows a two-factor combination at once.

Desirability turns “maximize,” “target,” or “range” into 0–1 functions and can combine responses. Maximum desirability is optimal only for the supplied goals; it does not automatically define biological importance or manufacturing robustness. Changing function shapes or weights changes the candidate.

In-Silico Lab: turn effects into an equation and move coordinates

1. Compare B-dependence of the A profiler at AB=0 and AB=6.
2. Move A and B from −1 to +1 and read the current prediction.
3. Explain why values outside the design range are unavailable.
4. List range information needed to convert coded coefficients to natural units.

Read JMP analysis in judgment order, not output order

Recommended reasoning order:

1. verify actual runs, missingness, and deviations;
2. inspect effect directions and interaction plots;
3. define candidate model and hierarchy;
4. review ANOVA and effect uncertainty;
5. diagnose residuals, influence, and run order;
6. reconcile coded and natural equations;
7. explore candidates with Profiler and contour;
8. record confirmation experiments and prediction error.

An optimum is a candidate for the next experiment

The model is estimated from finite design points and error. At a predicted maximum, perform an independent confirmation run and compare observed values with the prediction and CI or PI. A boundary optimum may suggest a wider range is useful, but never extend automatically beyond safety or physical limits.

Example result statement

We hierarchically fitted the pre-specified candidate model A+B+AB. Its coded equation was 70+8A+5B+6AB, with no time trend in residuals by run order. A=+1 and B=+1 maximized prediction in the Profiler, but this was a design boundary, not a declared final optimum. We planned an independent confirmation batch to check prediction error.

Takeaways

• A factorial effect and a ±1 coded coefficient differ by a factor of two.
• Retain related lower-order terms with interactions and squared terms.
• Pooling and automatic term deletion require modelling judgment.
• Residuals and run order must not trail behind the effect table.
• Attach units and a valid range to natural-unit equations.
• A Profiler optimum is a candidate requiring confirmation.

The next unit is U20. It examines which effects become aliased and indistinguishable when runs from a full design are reduced, then stops automatically.

Official supplementary resources

• NIST/SEMATECH · How to model DOE data
• NIST/SEMATECH · Check of assumptions

The model and Lab in this article are educational synthetic material.