Response-surface modeling
Model curvature in a response and find promising settings within a studied factor range.
Concepts and reasoning
Moving from screening to local optimization
A screening experiment identifies factors that matter. A response-surface experiment asks how the response changes within a promising region and where useful settings might lie. A quadratic model can represent diminishing returns, a peak, a trough, or a saddle where different directions behave differently.
Two-level data alone cannot identify a separate square term for each coded factor: x² equals 1 at both −1 and +1. Additional levels are needed. Central composite designs add axial and center points; Box–Behnken designs use another structured collection of settings. Which design is suitable depends on feasible combinations, desired prediction precision, and run constraints.
Response-surface modeling is local. A polynomial is a convenient approximation around the studied region, not a physical law valid at all temperatures, concentrations, or pressures. A numerical optimizer will always seek an optimum of the fitted function within its constraints, even when that function is poorly supported.
Multiple meanings of “best”
The setting with the highest predicted mean may have unacceptable variability, cost, safety constraints, or performance on another response. Distinguish optimization of one mean from robust process design or multiresponse optimization. Scientific constraints belong in the problem definition, not as an afterthought once an attractive peak appears.
NIST's response-surface design guidance explains why a design must support the model being fitted. Merely collecting many observations does not ensure that quadratic coefficients can be separately estimated.
Mathematical foundations
A full quadratic model
For k coded factors, the model is
It has 1 + 2k + k(k − 1)/2 coefficients: six for two factors and ten for three. More observations than coefficients are generally needed to estimate residual variation, and the design matrix must have full rank. Repeatedly measuring only a few settings cannot resolve missing directions.
In matrix form, write the surface as β₀ + bᵀx + xᵀBx, where B has square-term coefficients on its diagonal and half the interaction coefficients off the diagonal. A stationary point solves
when B is invertible. If B is negative definite, the stationary point is a maximum; positive definite gives a minimum; mixed-sign eigenvalues give a saddle. A stationary point outside the feasible region is not an admissible operating recommendation.
Worked example
Worked example: a peak inside the experimental region
Suppose a fitted two-factor response is
over coded settings −2 to +2. Setting the derivatives to zero gives 4 − 4x₁ = 0 and −2 − 2x₂ = 0, so the stationary point is (1, −1). Both square coefficients are negative and there is no interaction, so this is a maximum. The predicted response is 80 + 4 + 2 − 2 − 1 = 83.
| Coded setting (x₁, x₂) | Predicted response |
|---|---|
| (0, 0) | 80 |
| (1, 0) | 82 |
| (0, −1) | 81 |
| (1, −1) | 83 |
| (2, −1) | 81 |
If x₁ = (temperature − 50)/10 and x₂ = (speed − 100)/20, the predicted optimum corresponds to temperature 60 and speed 80 in their respective physical units. Coding must be documented to make a recommendation usable.
Now add an interaction of 6x₁x₂ while leaving the other coefficients unchanged. The curvature matrix has diagonal entries −2 and −1 and off-diagonal entries 3. Its determinant is 2 − 9 = −7, indicating mixed curvature. Negative square coefficients alone no longer establish a maximum. Interactions can change the geometry substantially.
Even for the original surface, an observed mean of 81 in confirmation runs might be compatible with prediction uncertainty or might reveal model inadequacy. Compare confirmation results with an appropriate prediction interval rather than expecting the fitted peak exactly.
Interpreting the results
Interpret coefficients in the coded coordinate system
A quadratic response surface includes linear terms, squared terms, and pairwise interactions. A linear coefficient describes a slope at the coded origin when other factors are zero; it is not a constant slope everywhere once curvature or interactions are present. Squared terms describe curvature, and interactions allow a slope to depend on another factor.
The predicted optimum is the setting with the best fitted response under the optimization constraints. It is a model prediction, not a confirmed operating condition or a confidence region for the optimum. A boundary optimum may indicate that the best setting lies outside the studied range, or simply that the fitted surface increases toward that boundary.
A high R squared measures fit to the observed response variation. It does not establish useful prediction at untested combinations. Translate coded coordinates back to physical units and report the experimental range whenever presenting the proposed setting.
Assumptions and limitations
Enough distinct settings are required for curvature
A full quadratic model with k factors has an intercept, k linear terms, k squared terms, and k(k−1)/2 interactions. The design must distinguish all those columns and leave information to estimate residual variation. Repeating the same few settings can increase the number of rows without resolving rank deficiency.
The surface assumes that a quadratic approximation is adequate over the chosen region. Independent errors with suitable variance behavior are needed for ordinary least-squares uncertainty calculations. Changes in batches, operators, or measurement precision can violate that simple model.
Optimization within each factor's minimum and maximum creates a rectangular search region. Corners of that rectangle may never have been studied and may violate physical constraints. The model must not be treated as permission to use an infeasible combination.
Diagnostics and sensitivity analysis
Diagnose fit before optimizing the process
Inspect residuals against fitted values for curvature or changing spread and against run order for drift. Review unusual observations against experimental records. Large leverage at boundary settings can strongly change the fitted optimum; investigate sensitivity without automatically discarding valid runs.
Assess lack of fit using replicated settings when the design permits it, and compare predictions at scientifically relevant combinations. A two-level design plus centers can detect some curvature without separating all quadratic terms; NIST explains this limitation.
Run confirmation experiments at the proposed optimum and nearby feasible settings. Compare observed responses with predicted uncertainty and practical tolerances. If the optimum shifts sharply under small data changes, collect additional informative runs rather than reporting excessive numerical precision.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- NIST handbook: response surface designs — Designs that support estimation of curvature.