A two-parameter hazard function to describe age patterns of mortality in ancient Northwestern Europe

Hugo J P La Poutré,Fanny Janssen

doi:10.1186/s41118-021-00122-w

Hugo J P La Poutré, Fanny Janssen

Open Access

https://doi.org/10.1186/s41118-021-00122-w

Copy DOI

Abstract

When seeking to describe the age patterns of mortality for ancient populations, researchers are often confronted with small sample sizes or with missing data for several age groups. The traditional approach to dealing with these challenges is to smooth or complement such patterns by matching them to a model life table, either directly or through the Brass logit transformation. This procedure requires an appropriate model life table, which may not be available. We propose a hazard model that is both flexible enough to accurately describe an age pattern of mortality in ancient Northwestern Europe and restrictive enough to complement incomplete data. This paper presents a hazard function that contains four free-to-choose parameters. Tested against a large collection of life tables for northwestern European countries from the 17th to the 21st century, the number of free-to-choose parameters is stepwise reduced from four to only two. Compared with the Brass logit transformation with the Princeton Model West as its reference, the presented two-parameter hazard model is shown to fit the abovementioned dataset much better. The mean fitting error is found to be half the size. Moreover, this model is shown to fit a 13th-century mortality age pattern much better. The proposed two-parameter hazard model is capable of fitting a wide range of age patterns of mortality more closely than the traditional approach can. We therefore conclude that the proposed model facilitates the smoothing and the completion of age patterns of mortality in ancient Northwestern Europe even if they deviate substantially from well-documented patterns.

Highlights

IntroductionEver since Gompertz formulated his law of mortality in the 1820s, new mathematical hazard functions have been suggested to adequately describe age patterns of mortality through forces of mortality (Thiele and Sprague 1871; Perks 1932; Siler1983; Thatcher 1999; Turner and Hanley 2010: p.488), mortality probabilities (Heligman and Pollard 1980; De Beer and Janssen 2016) or survivorship (Wong and Tsui 2015).Poutré and Janssen Genus (2021) 77:12The underlying concept of a hazard rate, which was originally known as the intensity, or force, of mortality, plays a vital role in various scientific disciplines and has been applied in fields as diverse as demography, actuarial science, epidemiology, biology and engineering (Weibull 1951; Siler 1979; Hoem 1983; Kizilersü et al 2018)
We find that for the Human Mortality Database (HMD) dataset, the two-parameter model performs worse than the Siler model, which contains as many as 5 parameters
The two-parameter model has been found to be a simple, restrictive, hazard model, that is suitable for use in cases of limited data to fit mortality on a country-wide scale within north-western Europe from the 19th century onwards

Summary

Introduction

Ever since Gompertz formulated his law of mortality in the 1820s, new mathematical hazard functions have been suggested to adequately describe age patterns of mortality through forces of mortality (Thiele and Sprague 1871; Perks 1932; Siler1983; Thatcher 1999; Turner and Hanley 2010: p.488), mortality probabilities (Heligman and Pollard 1980; De Beer and Janssen 2016) or survivorship (Wong and Tsui 2015).Poutré and Janssen Genus (2021) 77:12The underlying concept of a hazard rate, which was originally known as the intensity, or force, of mortality, plays a vital role in various scientific disciplines and has been applied in fields as diverse as demography, actuarial science, epidemiology, biology and engineering (Weibull 1951; Siler 1979; Hoem 1983; Kizilersü et al 2018). In order to adequately capture the hazard rates due to juvenility, senescence, the so-called 'accident hump' and maternal mortality, these descriptions often require sophisticated mathematical functions containing many parameters (Heligman and Pollard 1980; Wong and Tsui 2015; De Beer and Janssen 2016). A reconstructed mortality pattern is based on limited data, it might be quite irregular or only partly known In such a case, the application of a sophisticated model may not work well. When the data are limited, a simpler and much more restrictive model is needed to ensure reliable parameter fitting Such a model would be able to correct for irregularities, it would be able to fill in the gaps. This kind of model is essential when describing mortality patterns of ancient populations, like those from the Early Modern Period or the Middle Ages

Objectives

Methods

Findings

Conclusion