Academic journal article Demographic Research

Taylor's Power Law in Human Mortality

Academic journal article Demographic Research

Taylor's Power Law in Human Mortality

Article excerpt

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Taylor (1961) established a power law that describes a pattern in ecology regarding the (spatial or temporal) variability of populations. For many species, it describes a linear relationship between the logarithms of the variance (Var) of population size or density (P) and its mean (E):

... (1)

Although the interpretation of the two parameters a and b is controversial, Taylor (1961: 735) calls the constant a a (less relevant) computing factor and suggests that the slope b is a species-specific index of aggregation. Kilpatrick and Ives (2003) report that many empirical analyses identify values between 1 and 2 for the slope b, due to environmental and demographic stochasticity as well as competitive interactions between species. Kendal (2004b) gives a detailed overview of the history of TL. He shows that TL has mostly been found for population densities in ecology, but that power laws have also been identified in other contexts, such as outbreaks of infectious diseases (Anderson and May 1988; Rhodes and Anderson 1996) and for physical distributions of gene structures within chromosomes (Kendal 2004a). In these non-ecological realizations of TL, Taylor's interpretation of the exponent is obviously not viable.

In human demography, TL has been applied by Cohen, Xu, and Brunborg (2013), who verify a log-linear variance to mean relationship for Norway's population (disaggregated in 19 counties) from 1978 to 2010; they suggest using TL as an evaluation criterion for population forecasts, i.e., to determine if such a linear relationship can be found in both observed population data and in forecasts.

Vaupel, Zhang, and van Raalte (2011) analyzed variation in the age at death by e[dagger], which is the average number of life-years lost in a population (Vaupel and Canudas Romo 2003). They showed that populations with very low variation typically also had the lowest mean, as measured by life expectancy at birth. In this article, we quantify further the relationship between the variation and the mean of human mortality by testing the application of TL to human death rates and to rates of improvement in mortality: We compute variances and means of human mortality (change) for single ages over time for different countries to analyze their temporal variability across ages and across countries.

1.1 Hypothesis 1

In our so-called cross-age-scenarios, we analyze temporal variance to temporal mean relationships of mortality in multiple countries on the logarithmic scale across ages. We hypothesize that these relationships are almost log-linear for all ages in each country. Such a finding might indicate that TL could be used to evaluate mortality forecasts and to justify linear assumptions in mortality forecasts on a logarithmic scale.

If TL were confirmed in observed mortality data, then TL could be tested in mortality forecasts, and the results of this consistency test would be available immediately after generating a forecast. In contrast, forecast errors can typically only be computed after the mortality of forecast years occurs. Of course, in the long run, the empirical usefulness of using TL as a consistency test would have to be evaluated post hoc.

Justifying linear assumptions in mortality forecasting with TL would be beneficial, since many models (Lee and Carter 1992; Renshaw and Haberman 2003, 2006) rely on linear predictors for death rates on a logarithmic scale. Tuljapurkar, Li, and Boe (2000) use the observed long-term linear mortality decline in the G7 countries on the logarithmic scale to justify the modeling of log-linear mortality forecasts. If we found support for our hypothesis of (almost) linear variance to mean relationships for death rates and their rates of change, TL might also justify linear assumptions for forecasting models relying on the change of mortality (Mitchell et al. 2013; Haberman and Renshaw 2012; Bohk and Rau 2014). …

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