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.

Box 1 for a definition of lifespan, and other demographic terms used throughout this text), and thus be (relatively) short-lived and negligibly senescent (Péron et al., 2019). For example, Baudisch (2011) compared 10 animal species and found that robins Erithacus rubecula rank as having the shortest life expectancy while at the same time having the least senescent survivorship curve (out of this admittedly small sample of 10 animal species). Likewise, an organism can have an increasing but relatively low mortality rate over its entire lifespan, and thus be long-lived and senescent. For example, bamboo (Phyllostachys) stands rapidly die following a period of relatively low mortality lasting 60-100 years (Finch & Rose, 1995;Janzen, 1976).
Similarly, long-lived semelparous plants such as long-lived Puya raimondii (living up to 150 years;Finch (1998)) and Agave americana (which often live decades; Harper and White (1974)) show delayed, but rapid declines in vitality with age.
To disentangle these two dimensions of ageing, namely life expectancy and the shape of the survivorship curve, demographers typically distinguish the two using the pace of ageing and the shape of ageing, respectively (Baudisch, 2011;Keyfitz, 1968Keyfitz, , 1977. The pace of life is often quantified through demographic metrics such as mean life expectancy, reproductive window, or generation time, which tend to be highly correlated. The pace of life behaves intuitively: it is high for short-lived organisms and low for long-lived organisms. The shape of ageing, on the other hand, is determined by the time-standardized shape of the mortality or survival curve. The goal of shape metrics is to classify survival curves by whether the mortality rate mostly increases or decreases with (standardized) time (respectively, senescent versus negative senescent curves), see Keyfitz' entropy is one of the metrics that has been proposed to quantify the shape of ageing (Keyfitz, 1977;Wrycza et al., 2015). Keyfitz' entropy was originally identified as a dimensionless measure of the elasticity of lifespan to a uniform change in age-specific mortality (Leser, 1955). Population entropy was later re-derived and popularized by Keyfitz (1977). A similar measure was introduced through independent proofs by Demetrius, which helped attract interest to the measure (Demetrius, 1974(Demetrius, , 1978. Demetrius (1978) noted the potential use of Keyfitz' entropy for the classification of survivorship curves, pointing out that it has the useful property that H = 1 corresponds to a constant mortality rate (type II curve), H < 1 corresponds to mortality increasing with age (type I curve), and H > 1 corresponds to mortality decreasing with age (type III curve, see Figure 1 for an example of all three types of curves). Salguero-Gómez et al. (2016) introduced a discretized version of Keyfitz entropy that interchanges the integral for summation which has been subsequently used in a number of publications (Beckman et al., 2018;Bernard et al., 2020;Capdevila et al., 2020;Salguero-Gómez, 2017). In this short note, we show that this approach to discretise Keyfitz entropy does not fully capture the expected relationship with increasing, decreasing and constant mortality rates of the continuous-time metric. To resolve this discrepancy, we introduce a different discrete-time version of Keyfitz' entropy based on previous work on matrix formulas for life disparity (Caswell, 2013;Caswell et al., 2018;Keyfitz & Caswell, 2005;Vaupel & Canudas-Romo, 2003).
We then show that our alternative discretisation does preserve the expected relationship between Keyfitz' entropy, and increasing, decreasing, or constant mortality curves in age-structured matrix population models. We analyse the relationship between the two Keyfitz' metrics to test if any consistent biases might exist. We evaluate this relationship empirically using animal and plant matrix population models.
We find that the two metrics classify survivorship with a similar profile across values, with a consistent, strong trend toward underestimating entropy values (suggesting stronger senescence) using the original discrete entropy metric. That is, curves classified as (weakly) negatively senescent by the new discrete entropy metric are likely to be incorrectly classified as senescent by the original discrete entropy metric, and F I G U R E 1 Three example survivorship functions of the three different types (type I: senescence; type II: constant mortality; type III, negative senescence). The two Keyfitz entropy measures given by Equations (5) and (10)) are calculated for these three curves using two different widths of the age classes, Δt, given below in Table 1. That is, we discretized the survivorship curves shown in the Figure using two different sizes of discrete intervals, Δt = 0.01 and Δt = 1 .   Keyfitz (1977) (or the recent edition, Keyfitz and Caswell (2005)) define the measure H as where l(a) is survivorship at age a (see Box 1). Keyfitz and Caswell (2005) note that H has been called entropy of information in other contexts (section 4.3 of Keyfitz & Caswell, 2005). H as defined above, since then generally known as Keyfitz' entropy, is a weighted average of the logarithm of survival, where the weights reflect normalized survivorship. The use of entropies and information theory has a long history in ecology and evolution (e.g. Shannon biodiversity indices, Kullback-Leibler and Jenzen-Shannon divergence, and Maximum Entropy distribution modelling). Note, however, that Keyfitz' entropy does not integrate to one and is therefore not an entropy in the strictest sense (Shannon & Weaver, 1949). Keyfitz and Caswell (2005) derived the formula by calculating the effect of a proportional change in age-specific mortality on the life expectancy at birth, and the measure therefore relates to the similarity of mortality across age-classes. As a result, Keyfitz' entropy is also a measure of the concavity of the survivorship curve. Goldman and Lord (1986), Vaupel (1986), and recently Vaupel and Canudas-Romo (2003) showed that Keyfitz entropy can be decom-  when mortality is constant and show that it approximates one as mortality gets close to zero but is lower than one otherwise.

| An alternative discretisation of Keyfitz entropy
We propose an alternative discrete-time formula for Keyfitz' en- e 0 = 1 Ne 1 , = 1 e 1 , TA B L E 1 Comparison of the two Keyfitz entropy discretisations for the curves in Figure 1 for a discrete time interval of Δt = 0.01 and a discrete time interval of Δt = 1. Original discrete entropy is given by Equation (5), and new discrete entropy is given by Equation (10) Keyfitz and Caswell (2005)). We did not derive our formula from this starting point, and instead used existing discrete-time expressions for the numerator and denominator in the continous-time expression derived by Keyfitz. Therefore it remains to be shown whether our new expression for H, H N in Equation (10), can also be derived by following Keyfitz' proof and considering a proportional change in mortality at all ages in a discrete-time model.

| Comparing the metrics in real species using the COM(P)ADRE databases
To compare how the two discrete-time entropy measures perform in the context of biologically realistic models, we calculated entropies We initially screened the matrix population models in COMPADRE and COMADRE based on their inclusion in previous publications that used Keyfitz' entropy (Bernard et al., 2020). These models were selected based on duration of study, and whether the population monitored was subject to experimental manipula-  (2001)). We only retained stage-based models where differences between stage and their age-equivalent matrices were within 5% of one another along the above demographic metrics. Nineteen matrices were dropped from COMADRE; 117 matrices were dropped from COMPADRE. The 5% cut-off was an arbitrary threshold that allowed us to retain only those models that in the age-from-stage conversion remain similar to the original models. In most cases for the matrices satisfying the 5% threshold, the difference in demographic metrics from the corresponding stage matrix were < 1 %.

| RE SULTS: B IA S OF THE E XIS TING ME TRIC CORREL ATE S WITH LONG E VIT Y
To compare the original and new discrete-time entropy measures, in Table 1 we calculate the entropy using both measures for a few example survivorship curves shown in Figure 1. Both metrics change as the step size is changed, as is generally the case for demographic outcomes in matrix models (Enright et al., 1995;Picard et al., 2010;Torres et al., 2008). However, the value given by the New Discrete-  which required a stage-to-age conversion as described in the methods. We excluded models if the demographic quantities such as population growth rate of the converted stage-to-age model differed from the original stage-structured models by more than 5%, which led to many exclusions and therefore a sparser plot for COMPADRE than for COMADRE (117 exclusions versus 19 exclusions, respectively; bottom two panels versus top two panels in Figure 2). As a consequence of these exclusions, virtually no models from the orig- short-lived species, therefore introducing a spurious correlation between shape and pace.
In Supporting Information 1 we show why the original discretisation, H lx , classifies constant mortality curves as negatively senescent curves, and why it does so more strongly for species with shorter lifespans. We find that for constant mortality , H lx = exp( − ) 1 − exp( − ) . This function is smaller than one whenever the mortality rate is nonzero, that is, when is larger than 0. Furthermore, exp( − ) 1 − exp( − ) is a decreasing function of such that the Original Discrete Keyfitz metric gets closer to zero as the constant mortality gets larger and life expectancy gets shorter, leading to the correlation seen in Figure 2b. In the limit of infinitesimally small time steps, survival approaches one (or approaches zero), and the sum approximates the continuous time formula well in this limit.

| D ISCUSS I ON AND CON CLUS I ON
We have shown that the commonly used time-discretized formula for Keyfitz' entropy (referred to here as the Original Discrete Entropy measure) does not preserve the relationship between Keyfitz' entropy and the shape of the survivorship curve that exists for a continuous-time definition of survivorship entropy. That is, constant mortality curves do not yield a Keyfitz' entropy of one, and life histories with decreasing mortality will not always yield values above one (e.g. see Table 1). Specifically, nearly 40% of classified survivorship curves (126 of 336) changed classification when using the new discrete metric from senescent to negatively senescent, or vice versa. Furthermore, the distance between the original and the new discrete Keyfitz' entropy metric correlates with life expectancy ( Figure 2b). As a consequence, any correlations obtained between pace and shape of life in previous publications using the existing Keyfitz metric may need to be reevaluated.
We propose a different formula for the discretisation of Keyfitz' entropy (referred to here as the new entropy measure), based on life disparity and life expectancy in Equation 10. We show in Supporting Information 2 that this new formula does preserve the relationship between the shape of the survivorship curve and Keyfitz entropy (that is, H N > 1 when mortality is a decreasing function of age, H N < 1 when mortality is an increasing function of age, and H N = 1 when mortality is constant). However, a major downside of the formula we have proposed is that it is only a measure of the shape of ageing for age-structured survival matrices (Leslie matrices). If the survival and population matrix are stage-structured, then the New Discrete Entropy measure quantifies whether mortality rate increases or decreases with stage. Stage-to-age conversion methods can offer one way around this limitation to the method (for more information on stage-to-age conversion methods see section 5.3 in Caswell, 2001, and for an implementation of the methods in R see Jones et al., 2022).
Besides the new shape metric proposed here, there are many other shape metrics that have been proposed and can be used to classify survivorship curves. For example, other life table statistics that have been used to quantify the age-specific decline in survival include Hayley's median (Hailey, 1874); the age-dependent mortality parameter of mortality distributions (e.g. Gompertz, Weibull, Siler, Logistic, etc.; Ricklefs & Scheuerlein, 2002); the age at the onset of senescence (Jones et al., 2008) and the integration of the remaining lifespan and survival function (Wrycza et al., 2015). Wrycza et al. (2015) highlight a number of other potential candidates, such as a modified Gini coefficient (reviewing 7 possible metrics), and highlight the value of the entropy as a measure of the shape of life (see also Aburto et al. (2022) for a recent discussion of measures of lifespan inequality).

AUTH O R CO NTR I B UTI O N S
Charlotte de Vries conceptualized the article with input from Connor Bernard and Roberto Salguero-Gómez. Charlotte de Vries performed analytical mathematical analyses, Connor Bernard performed analyses in R. Charlotte de Vries wrote first draft: all authors contributed to subsequent drafts.