What is the clinical evidence from this study?

Study design: Other. Population: Aortic valve disease. Intervention: Aortic valve repair vs. Matched general population. Primary outcome: Excess mortality (HR 2.70).

What question did this study set out to answer?

This research presents a method to estimate excess mortality using aggregated data instead of patient-level data.

June 17, 2026Open Access

Estimating the hazard ratio of excess mortality using a microsimulation framework with golden section search calibration: a methodological primer

RRReda RhellabErasmus MC MNMaximiliaan L. NotenboomErasmus MC JEJonathan R.G. EtnelErasmus MC

Key Result

In a methodological case study of non-elderly adults undergoing aortic valve repair, the estimated excess mortality HR was 2.70 (2.43 when accounting for modeled events) compared to the general population.

Key Points

This research presents a method to estimate excess mortality using aggregated data instead of patient-level data.
Developed a microsimulation framework utilizing the golden section search algorithm for calibration.
Calibrated microsimulation-derived survival curves to observed patient-sample survival curves.
Applied the method to a case study of aortic valve repair in non-elderly adults.
Estimated excess mortality hazard ratio (HR) of 2.70 without modelled events and 2.43 with modelled events.
The HR increased over time: 1.23 at 0–60 months, 2.32 at 61–120 months, and 5.50 at 121–188 months.
Provided a method for estimating time-dependent excess mortality with residual excess mortality estimation.

Structured PICO

Population

An aggregate data case study of non-elderly adults undergoing aortic valve repair, followed for up to 15 years, used to demonstrate a methodological framework.

Exposure

Microsimulation framework with golden section search calibration

Comparator

Incremental search method

Outcome

Excess mortality hazard ratio (overall and time-dependent)

The golden section search algorithm within a microsimulation framework offers an efficient, accessible method to estimate time-varying and residual excess mortality hazard ratios using only aggregated data.

Main Result

Hazard Ratio: 2.7

Limitations

The method does not jointly model excess mortality and background mortality correction parameters (unlike Touraine models)
The practical example focused on point estimates and left probabilistic sensitivity analysis out of scope

Abstract

Excess mortality in a population with a disease can be defined as extra mortality observed compared to the matched-general population, and expressed as a hazard ratio (HR). While excess hazard models and cure models exist, these commonly need patient-level data. In this paper, we show a method for estimation of excess mortality with aggregated data. We detail the estimation of excess mortality in a microsimulation framework using the golden section search algorithm. This iterative algorithm identifies a local minimum within a predefined search interval. The overall excess mortality and residual excess (remaining mortality after accounting for modelled events) is estimated by calibrating microsimulation-derived survival curves to observed patient-sample survival curves. R code is provided. We apply the method to a study of aortic valve repair in non-elderly adults. The method enabled efficient estimation of excess mortality, expressed as overall and time-dependent hazard ratios through a piecewise function. The estimated excess mortality HR in the case-study was 2.70 without accounting from modelled events and 2.43 while accounting for modelled events, and the HR increased over time (1.23; 2.32; 5.50 at 0–60 months, 61–120 months and 121–188 months, respectively). This microsimulation framework provides an accessible, efficient method for estimating time-varying excess mortality hazard ratios, allowing for estimation of residual excess mortality, which excludes mortality due to modelled events, using only aggregated input data.

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