Piecewise Model

Concept

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Theoretically, Piecewise Model is a mathematical modelling approach in which different mathematical functions are defined over distinct regions of the domain, with each function representing behaviour within a specified interval or condition. Piecewise modelling exists because many biological, clinical and economic processes exhibit structural changes that cannot be adequately represented by a single continuous function. In health economics, piecewise models are commonly applied to survival analysis, hazard functions, cost trajectories and disease progression where different phases require different mathematical representations.

Mathematically, a piecewise model is represented as a function composed of multiple sub-functions, each applying within a defined interval. The mathematical framework partitions the domain into mutually exclusive regions and specifies the corresponding functional form for each region. Parameters may differ between segments while continuity may or may not be imposed at transition points depending on the modelling assumptions.

In practice, piecewise models are estimated by identifying clinically or statistically justified breakpoints and fitting separate models within each interval. In health economic evaluation they are frequently implemented as piecewise exponential survival models, piecewise linear cost functions or segmented treatment-effect models to improve model fit where hazards or costs change over time.

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Purpose

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Used to represent processes whose mathematical behaviour changes across predefined intervals, improving the accuracy of survival models, economic evaluations and disease progression models when a single functional form is inadequate.

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Mathematical Formulae

Primary Formula

f(x) = { f?(x), x < c? f?(x), c? � x < c? ? f?(x), x � c??? }

Supporting Formulae

Piecewise linear model:

y = { ?? + ??x, x � c (?? + ??c) + ??(x ? c), x > c }

Piecewise exponential hazard:

h(t) = ??, �??? � t < �?

Survival function:

S(t) = exp(???(t ? �???))

Related Mathematical Methods

• Piecewise Exponential Model
• Segmented Regression
• Change-Point Analysis
• Survival Analysis
• Maximum Likelihood Estimation
• Spline Regression

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Example

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A health economic evaluation models overall survival after cancer treatment using a piecewise exponential model. The estimated hazard is 0.03 per month during the first 12 months and 0.08 per month thereafter.

For month 8:

h(t) = 0.03

For month 18:

h(t) = 0.08

The piecewise model captures the increase in mortality risk after the first year and provides a better fit than a model assuming a constant hazard throughout follow-up.

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Excel Implementation

Function	Example Formula	Health Economics Application
IF	=IF(A2<=12,0.03,0.08)	Apply interval-specific hazard rates
IFS	=IFS(A2<=6,Rate1,A2<=12,Rate2,TRUE,Rate3)	Implement multiple piecewise intervals
INDEX	=INDEX($B$2:$B$5,MATCH(A2,$A$2:$A$5,1))	Retrieve parameters for each interval
MATCH	=MATCH(A2,$A$2:$A$5,1)	Identify the appropriate interval
EXP	=EXP(-B2*C2)	Calculate interval-specific survival probabilities

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VBA (Optional)

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A VBA procedure can automatically allocate observations to predefined intervals and apply the appropriate model parameters throughout a health economic simulation.

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Sources

• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• Collett D. Modelling Survival Data in Medical Research.
• ISPOR-SMDM Modeling Good Research Practices Task Force Reports.
• NICE. Health Technology Evaluation Manual.