Concept Architecture
Concept
Theoretically, Model Averaging is a statistical approach that accounts for model uncertainty by combining estimates from multiple candidate models rather than selecting a single best model. It recognises that several plausible models may adequately represent the available evidence and reduces the risk of biased inference arising from model selection. In health economics, model averaging is used to incorporate structural uncertainty into economic evaluations, survival extrapolation and evidence synthesis.
Mathematically, model averaging represents the overall estimate as a weighted average of estimates obtained from individual models. The weights are typically derived from model probabilities, information criteria or Bayesian posterior probabilities, ensuring that models with greater empirical support contribute more heavily to the final estimate while still acknowledging uncertainty across competing model structures.
In practice, model averaging is implemented by fitting multiple plausible models, calculating model weights and combining predicted costs, health outcomes or survival estimates using these weights. It is commonly applied in survival analysis, Bayesian model averaging, health technology assessment and decision modelling to improve the robustness of cost-effectiveness estimates and reduce the impact of structural model uncertainty.
Purpose
Used to account for structural model uncertainty by combining estimates from multiple plausible models, improving the robustness and reliability of health economic evaluations.
Mathematical Formulae
Primary Formula
? = ?w???
where:
? = model-averaged estimate
?? = estimate from model i
w? = weight assigned to model i
?w? = 1
Supporting Formulae
Akaike weights:
w? = exp(??AIC? / 2) � ?exp(??AIC? / 2)
Bayesian model averaging:
P(M?|D) = (P(D|M?) ? P(M?)) � ?(P(D|M?) ? P(M?))
Related Mathematical Methods
- Bayesian Model Averaging
- Akaike Information Criterion
- Bayesian Information Criterion
- Survival Model Averaging
- Model Selection
- Structural Uncertainty
Example
Five parametric survival models are fitted to oncology trial data. Their Akaike weights are estimated as 0.35, 0.28, 0.18, 0.12 and 0.07. Rather than selecting only the best-fitting model, projected survival and QALYs are calculated as the weighted average of predictions from all five models, thereby incorporating uncertainty associated with model choice into the economic evaluation.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(WeightRange,EstimateRange) | Calculate the model-averaged estimate from weighted model predictions. |
| EXP | =EXP(-A2/2) | Calculate the numerator for Akaike weights from ?AIC values. |
| SUM | =SUM(B2:B6) | Normalise model weights so they sum to one. |
| IF | =IF(A2=MIN(A$2:A$6),"Best","Alternative") | Identify the best-fitting model while retaining alternative models for averaging. |
VBA (Optional)
VBA can automate fitting multiple candidate models, calculate model weights and produce model-averaged predictions for probabilistic health economic analyses.
Sources
- Burnham KP, Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer.
- Hoeting JA, Madigan D, Raftery AE, Volinsky CT. Bayesian model averaging: A tutorial. Statistical Science. 1999;14(4):382?401.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practices for Outcomes Research.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is model averaging?
A statistical technique combining predictions from several plausible candidate models, weighted by their relative statistical support, rather than relying on one selected model.
Source: Hoeting et al. 1999
What problem does model averaging address?
Selecting a single model and reporting its results as if it were certainly correct ignores the uncertainty about which model is right, when several fit the data almost equally well yet give different answers. Model averaging addresses this by combining the predictions of the candidate models, weighting each by its statistical support, so the final result reflects the range of plausible structures rather than one chosen form. This carries structural uncertainty through to the conclusion instead of hiding it. It hedges across models. Jackson and colleagues (2011) describe this approach.
Source: Jackson et al. 2011
How does model averaging work?
Model averaging works by fitting several candidate models, assigning each a weight reflecting its relative support from the data, and combining their predictions as a weighted average using those weights. In the Bayesian form, the weights are the posterior probabilities of the models, derived from their fit and prior probabilities; other forms use weights based on information criteria. The averaged prediction, and its uncertainty, then incorporate both the uncertainty within each model and the uncertainty about which model is correct.
Source: Hoeting et al. 1999
How are the weights determined in model averaging?
The weights in model averaging are determined by each candidate model's relative support from the data. In Bayesian model averaging, they are the posterior model probabilities, combining each model's marginal likelihood with its prior probability, so better-fitting, plausible models receive more weight. Alternatively, weights can be derived from information criteria, such as those approximated by the Bayesian information criterion, which balance fit against complexity. Either way, the weights reflect how well each model is supported, giving greater influence to models the data favour.
Source: Schwarz 1978
Why is model averaging used?
Model averaging is used to account for uncertainty about which model is correct, since choosing a single model and ignoring the others understates uncertainty and can give overconfident or biased predictions if the chosen model is wrong. Averaging over plausible models incorporates structural uncertainty, often yielding better-calibrated uncertainty and predictive performance. In settings where several models fit comparably, such as survival extrapolation with competing distributions, model averaging combines them rather than forcing a single choice, reflecting genuine uncertainty about the model form.
Source: Hoeting et al. 1999
What are the limitations of model averaging?
Model averaging depends on the set of candidate models considered, so it accounts only for uncertainty among those included and can miss a better model outside the set, and computing the weights, especially Bayesian posterior model probabilities, can be demanding and sensitive to the priors. The averaged model may be harder to interpret than a single one. Choosing the candidate set and weighting scheme involves judgement. These limitations mean model averaging is applied with a well-considered candidate set and attention to how the weights are derived.
Source: Hoeting et al. 1999
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Verified by Dr Darrin Baines
British health economist
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Verification date: 28 Oct 2025
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