Discretising Keyfitz' entropy for studies of actuarial senescence and comparative demography

Keyfitz' entropy is a widely used metric to quantify the shape of the survivorship curve of populations, from plants to animals and microbes. Keyfitz' entropy values <1 correspond to life histories with an increasing mortality rate with age (i.e. actuarial senescence), whereas values >1 correspond to species with a decreasing mortality rate with age (negative senescence), and a Keyfitz entropy of exactly 1 corresponds to a constant mortality rate with age. Keyfitz' entropy was originally defined using a continuous-time model, and has since been discretised to facilitate its calculation from discrete-time demographic data. Here, we show that the previously used discretisation of the continuous-time metric does not preserve the relationship with increasing, decreasing or constant mortality rates. To resolve this discrepancy, we propose a new discrete-time formula for Keyfitz' entropy for age-classified life histories. We show that this new method of discretisation preserves the relationship with increasing, decreasing, or constant mortality rates. We analyse the relationship between the original and the new discretisation, and we find that the existing metric tends to underestimate Keyfitz' entropy for both short-lived species and long-lived species, thereby introducing a consistent bias. To conclude, to avoid biases when classifying life histories as (non-)senescent, we suggest researchers use either the new metric proposed here, or one of the many previously suggested survivorship shape metrics applicable to discrete-time demographic data such as Gini coefficient or Hayley's median.