Response Surface

Concept

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Theoretically, a Response Surface is a mathematical function that represents the relationship between one or more input variables and an output of interest. It is founded on response surface methodology, regression modelling and approximation theory and is used to characterise complex systems through a smooth mathematical approximation. In health economics, response surfaces are used to emulate computationally expensive decision models, approximate model outputs and support value of information analysis and model optimisation.

Mathematically, a response surface expresses the expected output as a function of multiple input variables. The surface is commonly estimated using polynomial regression, Gaussian process regression or other surrogate modelling techniques. Once fitted, the response surface approximates the underlying model, enabling rapid prediction of outcomes without repeated execution of the original computationally intensive model.

In practice, response surfaces are estimated by evaluating a health economic model at selected combinations of input parameters and fitting an approximation model to the resulting outputs. Health economists use response surfaces to accelerate probabilistic sensitivity analysis, Expected Value of Partial Perfect Information estimation, calibration, optimisation and uncertainty analysis while maintaining high predictive accuracy.

Purpose

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Used to approximate complex health economic models, reduce computational burden and support efficient uncertainty analysis, optimisation and value of information calculations.

Mathematical Formulae

Primary Formula

y = f(x?, x?, ?, x?)

where:

y = predicted model output

x? ? x? = model input variables

f = response surface function

Supporting Formulae

Second-order polynomial response surface:

y = ?? + ???x? + ????x?� + ?????x?x? + �

Gaussian process representation:

f(x) ? GP(m(x), k(x, x?))

where:

? = regression coefficients

� = random error term

GP = Gaussian Process

Related Mathematical Methods

• Response Surface Methodology
• Gaussian Process
• Gaussian Process Regression
• Regression Analysis
• Surrogate Modelling
• Bayesian Optimisation
• Expected Value of Partial Perfect Information
• Model Calibration

Example

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A health economic microsimulation requires several hours to evaluate a single parameter combination. The model is run for 1,000 carefully selected combinations of treatment effect, disease progression and cost parameters. A Gaussian process response surface is then fitted to these outputs, allowing predicted net monetary benefits to be generated almost instantly for new parameter combinations. The response surface is subsequently used to estimate EVPPI without repeatedly executing the original simulation model.

Excel Implementation

Function	Example Formula	Health Economics Application
LINEST	=LINEST(YRange,XRange,TRUE,TRUE)	Estimate regression coefficients for a polynomial response surface.
TREND	=TREND(YRange,XRange,NewXRange)	Predict model outputs for new combinations of input variables.
MMULT	=MMULT(InputVector,CoefficientVector)	Calculate predicted responses using matrix algebra.
SUMPRODUCT	=SUMPRODUCT(InputRange,CoefficientRange)	Compute fitted values for linear response surface models.

VBA (Optional)

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VBA can automate response surface fitting, generate predictions and evaluate surrogate models across large probabilistic sensitivity analyses.

Sources

• Myers RH, Montgomery DC, Anderson-Cook CM. Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Wiley.
• Rasmussen CE, Williams CKI. Gaussian Processes for Machine Learning. MIT Press.
• Oakley JE, O'Hagan A. Probabilistic sensitivity analysis of complex models: a Bayesian approach. Journal of the Royal Statistical Society: Series B. 2004;66(3):751?769.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR Value of Information Good Practice Reports.