# Propagation of uncertainty through economic models using Taylor series approximations: the Delta-PSA (Δ-PSA) method

Purpose: To demonstrate the feasibility of using Taylor series approximations in health economic models for propagation of uncertainty and probabilistic sensitivity analysis (PSA) as an alternative to Monte Carlo. Method: At a high level, the method treats the economic model as a function from inputs (parameters) to outputs (results), and then finds the derivatives of that function with respect to its inputs and uses this to determine how uncertainty propagates through the model. The first-order Taylor series approximation for variance (Delta method) of model outputs given the joint distribution for parameter uncertainty, and the second-order Taylor series approximation for expected value, are combined to enable an analytic PSA (∆-PSA). This is demonstrated with a moment-generating function model and a Markov model. The ∆-PSA method also facilitates value of information analyses. Result: The joint distribution of outputs (e.g., life years gained, discounted QALYs, incremental net benefit) in the examples closely match those obtained through standard Monte Carlo PSA (MC-PSA) and the resulting cost-effectiveness acceptability curves closely match (see Figure). Conclusion: The ∆-PSA method is an alternative to Monte Carlo methods for PSA and value of information analyses.

## Citation

```
@inproceedings{tristan2020,
author = {Snowsill, Tristan},
title = {Propagation of Uncertainty Through Economic Models Using
{Taylor} Series Approximations: The {Delta-PSA} {(Δ-PSA)} Method},
booktitle = {42nd Annual North American Meeting of the Society for
Medical Decision Making},
date = {2020-10-06},
eventdate = {2020-10-06},
url = {https://smdm.confex.com/smdm/2020/meetingapp.cgi/Paper/13470},
langid = {en},
abstract = {Purpose: To demonstrate the feasibility of using Taylor
series approximations in health economic models for propagation of
uncertainty and probabilistic sensitivity analysis (PSA) as an
alternative to Monte Carlo. Method: At a high level, the method
treats the economic model as a function from inputs (parameters) to
outputs (results), and then finds the derivatives of that function
with respect to its inputs and uses this to determine how
uncertainty propagates through the model. The first-order Taylor
series approximation for variance (Delta method) of model outputs
given the joint distribution for parameter uncertainty, and the
second-order Taylor series approximation for expected value, are
combined to enable an analytic PSA (∆-PSA). This is demonstrated
with a moment-generating function model and a Markov model. The
∆-PSA method also facilitates value of information analyses. Result:
The joint distribution of outputs (e.g., life years gained,
discounted QALYs, incremental net benefit) in the examples closely
match those obtained through standard Monte Carlo PSA (MC-PSA) and
the resulting cost-effectiveness acceptability curves closely match
(see Figure). Conclusion: The ∆-PSA method is an alternative to
Monte Carlo methods for PSA and value of information analyses.}
}
```

*42nd Annual North American Meeting of the Society for Medical Decision Making*. https://smdm.confex.com/smdm/2020/meetingapp.cgi/Paper/13470.