We filled the CCD table with response values and fitted a full quadratic model. If R² is 0.98, can we now find the optimal conditions? A high proportion of explained variance is just a starting point; it does not check for missing curvature, sequence effects, extrapolation, or selection bias.
The question for this section is:
How do you determine if a quadratic model adequately describes the data?
First, link candidate models to the design
A two-factor full quadratic model has six terms.
ŷ = β₀ + β₁x + β₂y + β₁₁x² + β₂₂y² + β₁₂xy
The list of terms is not invented after looking at the results to find the best combination of p-values. It is pre-specified based on the design objective and scientific question, and then we check whether the design can estimate those terms. The first thing to check is whether the row with the attached response values deviates from the original design rows.
Maintain the quadratic hierarchy
If x² is kept in the model, then x should also be kept; if xy is kept, then the main effects of x and y should also be kept. Deleting terms simply because the p-value of the lower-order term is large can change the meaning of the higher-order term depending on the coordinate origin and coding.
Compare three candidate models on the same data
Suppose we created a composite response as follows:
Y = 82 + 5x − 3y − 6x² − 4.2y² + 3xy + error
| Candidate Model | Included Terms | Expected Issues |
|---|---|---|
| linear | 1, x, y | Curvature and xy will remain in the residuals |
| quadratic without interaction | 1, x, y, x², y² | The xy structure may remain |
| full quadratic | All six terms | Fits the current generating function, but check for overfitting and range in real data |
In a teaching fixture with zero noise, the full quadratic recovers the true coefficients. However, in real data, each coefficient estimate has an SE and CI, and the information content is not uniform across the design region.
ANOVA partitions variance but does not automatically approve a model
The Model F and p-value of ANOVA look at whether "this candidate model explains the signal better than a model with only an intercept." Each Parameter Estimate represents the coded effect with other terms fixed. The following questions remain to be answered separately:
- Is there a selection bias in choosing the terms to fit the data?
- Was pure error estimated using repeat points?
- Have blocks, run order, and measurement batches been included in the model?
- Does the variance and shape of the residuals vary depending on the conditions?
- Are the locations for which you want to make predictions actually within the design region?
R² and adjusted R² summarize the proportion of variance explained, but they do not answer these questions.
Lack of fit and residuals provide different warnings
If there are repeated conditions, the residual SS can be partitioned into pure error and lack-of-fit SS. If the LOF is larger than the pure error, it is possible that the current model has missed the structure beyond the repeated variation. Conversely, a large LOF p-value does not prove that the model is correct. In small samples, the power may be low.
In residual plots, look for the following:
- Does a U-shape or S-shape remain depending on the predicted value?
- Does the spread widen in a fan shape as the response increases?
- Is there a drift along the run order?
- Does one point excessively pull the model?
Iteratively deleting from the largest p-value does not reflect the selection process in the final p-value and prediction uncertainty. Maintain the prior model, hierarchy, and domain importance, and review the reduced model by comparing it with the full model and with new confirmation data.
Actual by Predicted reads the identity diagonal
The closer the points are to the diagonal, the closer the observed and fitted predictions are. However, fitting well at the design points is different from predicting new conditions well. If the repeated center points fall in the same direction, or if only the boundary points deviate significantly, consider the structure separately.
In-Silico Lab: Re-examine missing terms with residuals
- In the full quadratic model, examine the number of coefficients, R², RMSE, and LOF SS.
- Change to a linear model and increase the curvature to see how the residuals and SSE change.
- In
quadratic-no-interaction, increase the true xy coefficients. - Change the noise and seed to see how much the small sample diagnostics fluctuate.
Change the candidate model and look at the structure of the residuals.
Compare SSE, LOF, and residuals by fitting linear, square term, and full quadratic models to the same CCD synthetic data.
If this is your first time: What should I press?
- 1. Read the question firstIn the Lab title, check the one thing you will compare this time.
- 2. Change just one conditionInitially, change only one of the inputs: n, effect, or spread.
- 3. new composite specimen pressureNew synthetic data is created. The same conditions may vary depending on the sample.
- 4. Pictures and calculation results CompareWrite in one sentence what moves and what stays the same before and after the change.
If it gets stuckresetGo back to see the default results and change just one condition. This Lab is not a correct answer tester but a pattern observation tool.
Synthetic observations of the same settings
calculation result
Even if R² is high, check the residual curve/order pattern and lack of fit. Without loop points, pure error and LOF cannot be separated.
educational synthetic modelbjs-response-surface-sequence-v1. One row is one independent simulation or design run unless otherwise indicated. It cannot be used for actual research, quality, or regulatory decisions.
The Lab's 13 rows comprise 8 CCF coordinates and 5 center points. Model fitting is calculated using least squares and is not a substitute for using the menu or actual JMP values.
Read JMP output as a set
The terms of the prior quadratic model and the overall uncertainty are viewed together.
The remaining structure is checked based on the repetition point pure error.
Missing curvature, heteroscedasticity, order, and outliers are not compressed into one screen.
ANOVA and Parameter Estimates show the signal and coefficients, Lack of Fit shows the discrepancy compared to the replicate points, and Residual and Actual by Predicted show the shape. None of these are a single gate. If you have made a model change, leave a receipt with the initial model, the reason for the change, and the diagnostics before and after.
Example of a result statement
A pre-specified full quadratic model was fitted to the 13-run CCF data. The hierarchy was maintained, and pure error was estimated using the replicate center points. There were no obvious curvature or fan patterns in the Actual by Predicted and residual-by-predicted plots, and LOF was not significant in the current sample. This supports the adequacy of the candidate model within the study range, but does not replace a confirmation experiment under new conditions.
End of the section
- The second-order model connects the design to the prior questions.
- It adheres to the quadratic hierarchy.
- Neither R², p-value, nor lack of fit alone is a criterion for acceptance.
- Residuals serve as a map for identifying curvature, heteroscedasticity, order, and points of influence.
- Record the reduction and selection process, as well as the confirmation data.
In the next section, we will read the candidate model that has passed the diagnosis using contour plots, Profiler, and desirability, and then proceed to the confirmation experiment.
Official Supplementary Materials
- NIST/SEMATECH · Assessing the Model
- NIST/SEMATECH · Response Surface Model Example
- JMP Help · Fitting Linear Models
This article and the Lab are synthetic materials for educational purposes and should not be used as evidence for actual research, process, quality, or regulatory decisions.