Concept Architecture
Concept
Theoretically, Linear Interpolation is a numerical approximation method used to estimate the value of a function between two known data points by assuming that the function changes linearly over the interval. It is one of the simplest interpolation techniques and is founded on analytic geometry and numerical analysis. Linear interpolation provides an efficient approximation when observations are closely spaced or when the underlying function varies approximately linearly.
Mathematically, Linear Interpolation constructs the unique straight line passing through two known points and evaluates the corresponding value at any intermediate point. The method produces a continuous piecewise linear approximation of the underlying function but does not guarantee continuity of the first derivative at successive interpolation intervals. Approximation accuracy depends on the spacing of the data points and the curvature of the true function.
In practice, Linear Interpolation is widely applied in health economics to estimate intermediate costs, utilities, survival probabilities, epidemiological measures and model parameters when observations are available only at discrete intervals. It supports decision modelling, calibration, numerical integration and simulation while providing a computationally efficient approximation technique.
Purpose
Used to estimate unknown values between observed data points, approximate continuous functions, support numerical modelling and improve computational efficiency in health economic analyses.
Mathematical Formulae
Primary Formula
For two points (x?, y?) and (x?, y?),
y = y? + ((x ? x?) � (x? ? x?))(y? ? y?)
where:
- x?, x? = known independent variable values
- y?, y? = corresponding observed values
- x = interpolation point
Supporting Formulae
Slope
m = (y? ? y?) � (x? ? x?)
Linear Equation
y = y? + m(x ? x?)
Related Mathematical Methods
- Polynomial approximation
- Cubic spline interpolation
- Numerical interpolation
- Piecewise linear approximation
- Least squares approximation
- Numerical integration
Example
Suppose healthcare expenditure is observed as:
- Age 50: �2,000
- Age 60: �3,000
Estimate expenditure at age 55.
Slope
= (3,000 ? 2,000) � (60 ? 50)
= 100
Interpolated value
= 2,000 + 100(55 ? 50)
= �2,500
The estimated healthcare expenditure at 55 years of age is therefore �2,500.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Formula | =B2+((D2-A2)/(C2-A2))*(E2-B2) | Calculate a linearly interpolated value. |
| SLOPE | =SLOPE(B2:B3,A2:A3) | Calculate the interpolation slope. |
| FORECAST.LINEAR | =FORECAST.LINEAR(D2,B2:B3,A2:A3) | Estimate intermediate values using linear interpolation. |
| TREND | =TREND(B2:B3,A2:A3,D2) | Produce equivalent linear predictions. |
| IF | =IF(AND(D2>=A2,D2<=C2),"Interpolation","Extrapolation") | Determine whether the estimate is interpolation or extrapolation. |
VBA (Optional)
Automate interpolation of costs, utilities, survival probabilities and model parameters between observed time points while supporting calibration and health economic simulation models.
Sources
- Burden RL, Faires JD. Numerical Analysis.
- Atkinson KE. An Introduction to Numerical Analysis.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing.
- Stoer J, Bulirsch R. Introduction to Numerical Analysis.
- Cheney EW, Kincaid D. Numerical Mathematics and Computing.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (5)
Library
Publications
1
Accuracy and Stability of Numerical Algorithms — Nicholas J. Higham, 2nd Edition ed., 2002 (Society for Industrial and Applied Mathematics (SIAM))
The standard reference on floating-point error, conditioning, stability and the numerical reliability of matrix, polynomial and linear-algebra algorithms.
BookView source →
Frequently Asked Questions (6)
What is linear interpolation?
A method of estimating values between two known data points by assuming a straight-line relationship between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does linear interpolation estimate an intermediate value?
Linear interpolation estimates a value between two known data points by assuming a straight-line relationship between them, so the estimate lies on the line joining the two points. It reads off the point on that straight segment at the position of interest. Assuming a straight line between the known points is how linear interpolation produces an intermediate value Simple and quick, linear interpolation reads a value off the straight line between two points, which serves well when the data changes gently but understates any curvature between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What relationship does linear interpolation assume between data points?
Linear interpolation assumes a straight-line relationship between two known data points, treating the quantity as changing at a constant rate between them. This straight-line assumption is what lets it estimate values in the gap. Relying on a straight line between the points is the defining simplification of linear interpolation Simple and quick, linear interpolation reads a value off the straight line between two points, which serves well when the data changes gently but understates any curvature between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
Between how many points does linear interpolation work?
Linear interpolation works between two known data points, estimating values in the interval separating them by assuming a straight-line relationship. Each estimate draws on the pair of points bracketing the position of interest. This use of two points to define the connecting line is basic to how linear interpolation operates Simple and quick, linear interpolation reads a value off the straight line between two points, which serves well when the data changes gently but understates any curvature between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
What is a limitation of linear interpolation?
A limitation of linear interpolation is that, by assuming a straight-line relationship between two known data points, it cannot capture curvature between them, so it may be inaccurate where the underlying quantity bends. The straight segments give a coarse fit when the data is far from linear. This inability to follow curvature is the cost of the simplicity of linear interpolation Simple and quick, linear interpolation reads a value off the straight line between two points, which serves well when the data changes gently but understates any curvature between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
How does linear interpolation relate to a cubic spline?
Linear interpolation estimates values between two known data points by assuming a straight-line relationship, while a cubic spline fits smooth cubic curves between data points and keeps the function and its derivatives continuous. Linear interpolation gives straight, angular joins, whereas a cubic spline is smooth. The two are connected as interpolation methods differing in the degree of the pieces fitted between the data points Simple and quick, linear interpolation reads a value off the straight line between two points, which serves well when the data changes gently but understates any curvature between them.
Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.
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British health economist
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Verification date: 3 Apr 2026
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