Concept Architecture
Concept
Theoretically, Interpolation is the process of estimating unknown values that lie between known observations. It assumes that the underlying relationship between adjacent data points can be approximated by a specified mathematical function. In health economics, interpolation is used to estimate values between observed time points, including survival probabilities, costs, utilities and epidemiological parameters, when complete data are unavailable.
Mathematically, Interpolation is represented by constructing a function that passes through known data points and evaluating that function at intermediate values. The choice of interpolation method depends on the characteristics of the data and the required level of accuracy. Linear interpolation is the most commonly applied approach in health economic modelling because of its simplicity and transparency, although spline and polynomial interpolation may be appropriate for smoother relationships.
In practice, Interpolation is applied when converting published survival curves into model inputs, estimating intermediate utility values between follow-up visits, deriving transition probabilities for alternative cycle lengths and completing incomplete parameter tables. Appropriate interpolation methods are selected according to the underlying data structure, recognising that interpolation estimates values only within the observed range and should not be used for extrapolation beyond the available data.
Purpose
Used to estimate intermediate values between observed data points, enabling complete parameterisation of health economic models when measurements are available only at discrete intervals.
Mathematical Formulae
Primary Formula
Linear interpolation:
y = y? + ((x ? x?)/(x? ? x?))(y? ? y?)
where:
- x?, x? are known values of the independent variable
- y?, y? are the corresponding observed values
- x is the value at which interpolation is required
- y is the interpolated estimate.
Supporting Formulae
There is no universally recognised canonical mathematical formula.
Related Mathematical Methods
- Linear interpolation
- Polynomial interpolation
- Cubic spline interpolation
- Piecewise linear modelling
- Survival curve reconstruction
- Numerical analysis
Example
A health utility study reports:
| Time (months) | Utility |
|---|---|
| 6 | 0.78 |
| 12 | 0.70 |
The utility at 9 months is estimated using linear interpolation:
y = 0.78 + ((9 ? 6)/(12 ? 6))(0.70 ? 0.78)
y = 0.78 + (3/6)(?0.08)
y = 0.74
The interpolated utility of 0.74 is used in calculating quality-adjusted life-years for the model cycle.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| FORECAST.LINEAR | =FORECAST.LINEAR(9,{0.78,0.70},{6,12}) | Estimate intermediate values assuming a linear relationship. |
| INDEX | =INDEX(ValueRange,MATCH(Time,TimeRange,1)) | Retrieve the lower interpolation point. |
| MATCH | =MATCH(TargetTime,TimeRange,1) | Identify adjacent observed values for interpolation. |
| LET | =LET(x,9,x1,6,x2,12,y1,0.78,y2,0.70,y1+((x-x1)/(x2-x1))*(y2-y1)) | Perform linear interpolation within a health economic model. |
| XLOOKUP | =XLOOKUP(TargetTime,TimeRange,ValueRange,,1) | Retrieve neighbouring observations before interpolation. |
VBA (Optional)
Automate interpolation of costs, utilities, transition probabilities or survival estimates between observed time points throughout a health economic model.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.
- Burden RL, Faires JD. Numerical Analysis. 10th ed. Cengage Learning; 2015.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing. 3rd ed. Cambridge University Press; 2007.
- National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.
Related Concepts (6)
Library
Publications
1
Parameter Estimation and Uncertainty: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-6 — Briggs, Weinstein, Fenwick, Karnon, Sculpher & Paltiel, Task Force Report 6 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on parameter estimation and the characterisation of uncertainty in decision models, covering probabilistic sensitivity analysis, distributional choices, and correlation between parameters.
Journal ArticleView source →
Frequently Asked Questions (6)
What is interpolation?
A mathematical technique estimating an unknown value falling between two known data points, based on an assumed relationship between them.
Source: Press et al. 2007
What assumption does interpolation rely on?
Interpolation estimates a value between known data points by assuming the quantity behaves in a particular way between them, most simply that it changes at a steady rate so the points can be joined by a straight line. More elaborate versions assume a smooth curve instead. The estimate is only as good as that assumption, so interpolation is reliable when the underlying quantity really does vary smoothly and gently between the known points. Where it jumps or bends sharply, the assumption fails. Press and colleagues (2007) describe these methods.
Source: Press et al. 2007
How does interpolation work?
Interpolation works by fitting a function through the known data points and using it to estimate values between them. Linear interpolation assumes a straight line between two adjacent points, estimating an intermediate value proportionally. Higher-order methods, such as polynomial or spline interpolation, fit smooth curves through several points for greater accuracy where the relationship is not linear. The chosen method determines the assumed shape between points, so the estimate reflects how the values are assumed to vary in the gap.
Source: Press et al. 2007
How does interpolation differ from extrapolation?
Interpolation estimates values within the range of the known data points, between points where data exist, whereas extrapolation estimates values outside that range, beyond the observed data. Interpolation is generally more reliable, since it is constrained by data on both sides, while extrapolation is riskier, relying on assumptions about behaviour where no data exist. The distinction matters because extrapolation, common in projecting model outcomes beyond trial follow-up, carries greater uncertainty than interpolation between observed points.
Source: Press et al. 2007
Where is interpolation used in modelling?
In modelling, interpolation is used to estimate values between known data points, such as filling in costs, probabilities, or utility values for ages or times where only some values are tabulated, or reading values from a function known only at certain points. It is also used within numerical methods. By providing estimates between known points, interpolation lets models use continuous or finely spaced inputs derived from coarser data, though the assumed relationship between points affects the estimates.
Source: Press et al. 2007
What are the limitations of interpolation?
Interpolation depends on the assumed relationship between the known points, so it can be inaccurate if the true behaviour differs from the assumption, for instance if a straight line is assumed where the relationship curves. Sparse or noisy data points give less reliable interpolation, and higher-order methods can oscillate between points. Interpolation is only valid within the range of the data; estimating beyond it is extrapolation, which is riskier. These limitations mean the method and its assumptions are chosen to suit the data.
Source: Press et al. 2007
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Oct 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/interpolation
- Term code
- HE-EM-MP-018
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