Concept Architecture
Concept
Theoretically, Response Surface Methodology (RSM) is a collection of statistical and mathematical techniques used to model, analyse and optimise the relationship between one or more response variables and a set of quantitative explanatory variables. It is founded on regression modelling and the design of experiments, with the objective of identifying combinations of input variables that optimise a specified response. In health economics, it is used to approximate complex model outputs, calibrate simulation models and optimise resource allocation or intervention strategies.
Mathematically, response surface methodology represents the response variable using a low-order polynomial regression model, most commonly a second-order quadratic function. The fitted response surface estimates how changes in input variables affect the outcome and is subsequently analysed to identify stationary points corresponding to local maxima, minima or saddle points. Optimisation is performed using calculus-based or numerical methods applied to the estimated response surface.
In practice, response surface methodology is implemented by selecting an experimental design, estimating the regression model, assessing model adequacy, and identifying optimal operating conditions. In health economics it is applied to calibrating disease models, approximating computationally expensive simulation models, evaluating uncertainty in model parameters, optimising screening strategies and improving the efficiency of probabilistic analyses.
Purpose
Used to model and optimise the relationship between multiple input variables and health economic outcomes, enabling efficient calibration, approximation and optimisation of complex decision models.
Mathematical Formulae
Primary Formula
Second-order response surface model:
y = ?? + ????? ??x? + ????? ???x?� + ???<?? ???x?x? + �
where:
- y = response variable
- x? = explanatory variables
- ? = regression coefficients
- � = random error
Supporting Formulae
Stationary point:
?y = 0
Estimated response:
? = X??
Related Mathematical Methods
- Design of experiments
- Multiple regression
- Least-squares estimation
- Quadratic regression
- Numerical optimisation
- Model calibration
- Sensitivity analysis
Example
A health economist investigates the effect of screening interval (x?) and screening uptake (x?) on the incremental net monetary benefit of a screening programme.
A quadratic response surface is fitted using simulation outputs:
? = 250 + 18x? + 35x? ? 2.4x?� ? 3.1x?� + 1.2x?x?
Optimising the fitted response surface identifies the combination of screening interval and uptake that maximises expected net monetary benefit without repeatedly evaluating the full simulation model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(Y2:Y30,B2:E30,TRUE,TRUE) | Estimates regression coefficients for the response surface. |
| SUMPRODUCT | =SUMPRODUCT(Coefficients,Variables) | Calculates predicted response values. |
| Solver Add-in | Maximise predicted response by changing input variables | Identifies optimal intervention or model parameter values. |
| RSQ | =RSQ(Predicted,Observed) | Assesses goodness of fit of the response surface model. |
VBA (Optional)
A VBA procedure can automate repeated fitting of response surface models and optimise the estimated response across alternative parameter combinations.
Sources
- Box GEP, Wilson KB. On the Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society Series B. 1951;13(1):1?45.
- Myers RH, Montgomery DC, Anderson-Cook CM. Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Wiley.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports on decision-analytic modelling.
Related Concepts (2)
Library
Publications
1
Decision Modelling for Health Economic Evaluation — Briggs, Claxton & Sculpher, 1st Edition ed., 2006 (Oxford University Press)
Foundational textbook on decision-analytic modelling for economic evaluation, covering decision trees, Markov models, handling parameter and structural uncertainty, probabilistic sensitivity analysis, and value of information. Volume 1 in the Handbooks in Health Economic Evaluation series.
BookView source →
Frequently Asked Questions (6)
What is response surface methodology?
Statistical techniques for modelling and optimising the relationship between several inputs and an output by fitting a smooth surface to limited results.
Source: Box & Wilson 1951
What does response surface methodology fit to a set of results?
Response surface methodology fits a smooth mathematical surface to the results of running a system at a limited set of input combinations, so that the output can be predicted and explored between the points actually tried. The fitted surface stands in for the full system, allowing the analyst to see how the output changes with the inputs and to locate the combination that optimises it. It turns a scatter of individual runs into a continuous picture of the input-output relationship. Myers and colleagues (2016) set out the method.
Source: Myers et al. 2016
How does response surface methodology work?
Response surface methodology works by conducting a set of experiments, chosen through a statistical design to sample the input space efficiently, measuring the output at each, and fitting a smooth model, often a low-order polynomial, to the results. This fitted surface approximates how the output depends on the inputs. The surface is then used to locate the region of the optimum and to estimate the input values that maximise or minimise the output, sometimes iteratively refining the search near the optimum.
Source: Box & Wilson 1951
What is the role of experimental design in response surface methodology?
Experimental design is central to response surface methodology because it determines which input combinations are evaluated, aiming to gather the most information about the response with a limited number of runs. Designs such as factorial and central composite designs sample the input space efficiently, allowing a surface to be fitted reliably from few results. Good design ensures the fitted surface accurately represents the relationship, so the optimisation is sound, which matters when each evaluation is costly.
Source: Box & Wilson 1951
Where is response surface methodology used?
Response surface methodology is used to optimise processes and systems with several inputs, originally in industrial and chemical experimentation, and more broadly wherever an output must be optimised over several factors using limited, possibly costly, evaluations. In modelling, it can serve to approximate and optimise the behaviour of a complex or expensive model, fitting a surface to a sample of its runs. It is one approach to building a metamodel and finding good input values efficiently when full evaluation is impractical.
Source: Box & Wilson 1951
What are the limitations of response surface methodology?
Response surface methodology relies on the fitted surface, usually a low-order polynomial, adequately approximating the true relationship, which may fail if the relationship is highly non-linear or irregular, so the surface can mislead away from the sampled region. It works best locally, near the region explored, and may need iterative refinement to reach an optimum. Its accuracy depends on the experimental design and the model form chosen. These limitations mean it suits smooth relationships and local optimisation rather than complex global search.
Source: Box & Wilson 1951
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 3 Oct 2025
Content version: 1.0.0
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