Post provided by Lotte de Vries

Animals and plants exhibit a wide range of patterns of longevity, growth, and reproduction but the general drivers of this enormous variation in life history are poorly understood. Comparative demography uses large demographic databases to attempt to identify patterns in life-history strategies across the tree of life (e.g. this PNAS paper, and this one). In this paper, we show that a formula widely used in this comparative demographic literature in recent years (e.g. here, here, and here) does not behave mathematically as had been assumed. As a consequence, nearly 40% of survivorship curves were getting erroneously classified (for example, as senescent when they were in fact non-senescent). 

Human mortality rate increases after puberty (or in biological terms: reproductive maturation). However, not all creatures across the tree of life show such a decrease in vitality after maturity. Hydra, for example, do not appear to suffer such an increase in mortality (aka senescence), and there are many other examples in this paper. 

What does senescence mean and what is this classification? 

Getting older cannot be avoided (in fact it should be celebrated!), but getting more fragile with age is not as unavoidable as it might seem from a human perspective. Once we have gone through puberty, we as humans as well as most of our pets have a higher chance of dying with every passing year. This increasing mortality rate with age is referred to as (actuarial) senescence. It turns out that not all species suffer from senescence. Instead, a wide variety of patterns is found when we plot age-specific mortality for different species. These curves can be roughly classified into 3 types: 1) increasing mortality rate with age (senescent, type I in figure below), 2) constant mortality with age (type II in figure below), or 3) decreasing mortality rate with age (negatively senescent, type III in figure below). 

The three types of survivorship curves that Keyfitz’ entropy can distinguish:  type I, an increasing mortality rate with age (senescent), type II, a constant mortality with age, and type III, a decreasing mortality rate with age (negatively senescent).  

Continuous versus discrete-time

In continuous time, a formula referred to as Keyfitz’ entropy can be used to classify survivorship curves into those three categories. However, most demographic data and most demographic models use discrete time steps since, in most cases, ecologist do not monitor individuals continuously in our experiments or field work. Therefore, a discrete-time version of Keyfitz’ entropy was used to classify survivorship curves, but we found that this version was actually not correctly classifying curves. 

Consequences:

We found that, as a consequence, nearly 40% of survivorship curves were getting erroneously classified (for example, as senescent when they were in fact non-senescent).  In addition, we show that the error induced by the old metric correlates strongly with life expectancy (in Figure  2b of our paper).   Any  correlations  obtained  between Keyfitz’ entropy  and measures of the so-called pace of life, such as life expectancy, in  these previous therefore need to be reevaluated, and conclusions drawn from these comparative demographic analyses need to be carefully reconsidered. 

Solutions

Borrowing from existing discrete-time formulae (e.g. see formulae in this paper), we proposed a new formula for discrete-time Keyfitz’ entropy in this paper that we proved correctly classifies survivorship curves. However, our formula only works for age-classified models. Luckily, since then, Stefano Giaimo have extended the formula to work for both age and stage-classified models (article in press). 

You can read the full article here