Discretising Keyfitz' entropy for studies of actuarial senescence and comparative demography
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Disc e ising Key i z' en opy o s udies o ac ua ial senescence and compa a i e
demog aphy
© 2023 The Au ho s.
Published e sion
de V ies, Cha lo e; Be na d, Conno ; Salgue o‐Gómez, Robe o
de V ies, C., Be na d, C., & Salgue o‐Gómez, R. (2023). Disc e ising Key i z' en opy o s udies o
ac ua ial senescence and compa a i e demog aphy. Me hods in Ecology and E olu ion, 14(5),
1312-1319. h ps://doi.o g/10.1111/2041-210x.14083
2023
Me hods Ecol E ol. 2023;00:1–8.
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1wileyonlinelib a y.com/jou nal/mee3
1 | INTRODUCTION
Ac ua ial senescence is de ined as he inc eased isk o dying as an
indi idual ge s olde (Hamil on, 1966; Medawa , 1952). Ge ing olde
canno be a oided in ha i is a na u al consequence o su i ing, bu
some species seem o be able o a oid senescing (Jones e al., 2014;
Rope e al., 2021; Vaupel e al., 2004). I is emp ing o assume ha
long- li ed o ganisms su e om ac ua ial senescence less han
sho - li ed o ganisms, bu o cou se i is possible o an o ganism o
ha e a cons an bu high mo ali y a e o e i s en i e li espan (see
Recei ed: 20 Sep embe 2022
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Accep ed: 13 Feb ua y 2023
DOI: 10.1111/2041-210X.14083
RESEARCH ARTICLE
Disc e ising Key i z' en opy o s udies o ac ua ial senescence
and compa a i e demog aphy
Cha lo e de V ies1,2,3 | Conno Be na d4 | Robe o Salgue o- Gómez4
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium,
p o ided he o iginal wo k is p ope ly ci ed.
© 2023 The Au ho s. Me hods in Ecology and E olu ion published by John Wiley & Sons L d on behal o B i ish Ecological Socie y.
1Ins i u e o Biodi e si y and Ecosys em
Dynamics, Uni e si y o Ams e dam,
Ams e dam, The Ne he lands
2Depa men o Biological and
En i onmen al Science, Uni e si y o
Jy äskylä, Jy äskylä, Finland
3Depa men o E olu iona y Biology
and En i onmen al S udies, Uni e si y o
Zu ich, Zu ich, Swi ze land
4Depa men o Biology, Uni e si y o
Ox o d, Ox o d, UK
Co espondence
Cha lo e de V ies
Email: c.de ies@u a.nl
Funding in o ma ion
Academy o Finland, G an /Awa d
Numbe : 340130; H2020 Eu opean
Resea ch Council, G an /Awa d
Numbe : 788195; Na u al En i onmen
Resea ch Council, G an /Awa d Numbe :
NE/M018458/1; Schweize ische
Na ional onds zu Fö de ung de
Wissenscha lichen Fo schung, G an /
Awa d Numbe : 310030B_182836 and
P500PB_211003
Handling Edi o : Res Al wegg
Abs ac
1. Key i z' en opy is a widely used me ic o quan i y he shape o he su i o ship
cu e o popula ions, om plan s o animals and mic obes. Key i z' en opy al-
ues <1 co espond o li e his o ies wi h an inc easing mo ali y a e wi h age (i.e.
ac ua ial senescence), whe eas alues >1 co espond o species wi h a dec easing
mo ali y a e wi h age (nega i e senescence), and a Key i z en opy o exac ly 1
co esponds o a cons an mo ali y a e wi h age. Key i z' en opy was o iginally
de ined using a con inuous- ime model, and has since been disc e ised o acili-
a e i s calcula ion om disc e e- ime demog aphic da a.
2. He e, we show ha he p e iously used disc e isa ion o he con inuous- ime
me ic does no p ese e he ela ionship wi h inc easing, dec easing o cons an
mo ali y a es. To esol e his disc epancy, we p opose a new disc e e- ime o -
mula o Key i z' en opy o age- classi ied li e his o ies.
3. We show ha his new me hod o disc e isa ion p ese es he ela ionship wi h
inc easing, dec easing, o cons an mo ali y a es. We analyse he ela ionship
be ween he o iginal and he new disc e isa ion, and we ind ha he exis ing
me ic ends o unde es ima e Key i z' en opy o bo h sho - li ed species and
long- li ed species, he eby in oducing a consis en bias.
4. To conclude, o a oid biases when classi ying li e his o ies as (non- )senescen , we
sugges esea che s use ei he he new me ic p oposed he e, o one o he many
p e iously sugges ed su i o ship shape me ics applicable o disc e e- ime de-
mog aphic da a such as Gini coe icien o Hayley's median.
KEYWORDS
demog aphy, Key i z en opy, mo ali y, senescence, shape o li e measu es
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Me hods in Ecology and E oluon
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Box 1 o a de ini ion o li espan, and o he demog aphic e ms used
h oughou his ex ), and hus be ( ela i ely) sho - li ed and negli-
gibly senescen (Pé on e al., 2019). Fo example, Baudisch (2011)
compa ed 10 animal species and ound ha obins E i hacus ubecula
ank as ha ing he sho es li e expec ancy while a he same ime
ha ing he leas senescen su i o ship cu e (ou o his admi edly
small sample o 10 animal species). Likewise, an o ganism can ha e
an inc easing bu ela i ely low mo ali y a e o e i s en i e li es-
pan, and hus be long- li ed and senescen . Fo example, bamboo
(Phyllos achys) s ands apidly die ollowing a pe iod o ela i ely low
mo ali y las ing 60– 100 yea s (Finch & Rose, 1995; Janzen, 1976).
Simila ly, long- li ed semelpa ous plan s such as long- li ed Puya ai-
mondii (li ing up o 150 yea s; Finch (1998)) and Aga e ame icana
(which o en li e decades; Ha pe and Whi e (1974)) show delayed,
bu apid declines in i ali y wi h age.
To disen angle hese wo dimensions o ageing, namely li e ex-
pec ancy and he shape o he su i o ship cu e, demog aphe s
ypically dis inguish he wo using he pace o ageing and he shape
o ageing, espec i ely (Baudisch, 2011; Key i z, 1968, 1977). The
pace o li e is o en quan i ied h ough demog aphic me ics such
as mean li e expec ancy, ep oduc i e window, o gene a ion ime,
which end o be highly co ela ed. The pace o li e beha es in u-
i i ely: i is high o sho - li ed o ganisms and low o long- li ed
o ganisms. The shape o ageing, on he o he hand, is de e mined
by he ime- s anda dized shape o he mo ali y o su i al cu e.
The goal o shape me ics is o classi y su i al cu es by whe he
he mo ali y a e mos ly inc eases o dec eases wi h (s anda dized)
ime ( espec i ely, senescen e sus nega i e senescen cu es), see
Figu e 1 o some examples o di e en su i o ship cu e shapes.
Key i z’ en opy is one o he me ics ha has been p oposed
o quan i y he shape o ageing (Key i z, 1977; W ycza e al., 2015).
Key i z' en opy was o iginally iden i ied as a dimensionless measu e
o he elas ici y o li espan o a uni o m change in age- speci ic mo -
ali y (Lese , 1955). Popula ion en opy was la e e- de i ed and pop-
ula ized by Key i z (1977). A simila measu e was in oduced h ough
independen p oo s by Deme ius, which helped a ac in e es o
he measu e (Deme ius, 1974, 1978). Deme ius (1978) no ed he
po en ial use o Key i z' en opy o he classi ica ion o su i o ship
cu es, poin ing ou ha i has he use ul p ope y ha
H=1
co e-
sponds o a cons an mo ali y a e ( ype II cu e),
H<1
co esponds
o mo ali y inc easing wi h age ( ype I cu e), and
H>1
co esponds
o mo ali y dec easing wi h age ( ype III cu e, see Figu e 1 o an
example o all h ee ypes o cu es).
Salgue o- Gómez e al. (2016) in oduced a disc e ized e sion o
Key i z en opy ha in e changes he in eg al o summa ion which
has been subsequen ly used in a numbe o publica ions (Beckman
e al., 2018; Be na d e al., 2020; Capde ila e al., 2020; Salgue o-
Gómez, 2017). In his sho no e, we show ha his app oach o
disc e ise Key i z en opy does no ully cap u e he expec ed ela-
ionship wi h inc easing, dec easing and cons an mo ali y a es o
he con inuous- ime me ic. To esol e his disc epancy, we in oduce
a di e en disc e e- ime e sion o Key i z' en opy based on p e i-
ous wo k on ma ix o mulas o li e dispa i y (Caswell, 2013; Caswell
e al., 2018; Key i z & Caswell, 2005; Vaupel & Canudas- Romo, 2003).
We hen show ha ou al e na i e disc e isa ion does p ese e
he expec ed ela ionship be ween Key i z' en opy, and inc easing,
dec easing, o cons an mo ali y cu es in age- s uc u ed ma ix pop-
ula ion models. We analyse he ela ionship be ween he wo Key i z'
me ics o es i any consis en biases migh exis . We e alua e his e-
la ionship empi ically using animal and plan ma ix popula ion models.
We ind ha he wo me ics classi y su i o ship wi h a simila p o ile
ac oss alues, wi h a consis en , s ong end owa d unde es ima ing
en opy alues (sugges ing s onge senescence) using he o iginal dis-
c e e en opy me ic. Tha is, cu es classi ied as (weakly) nega i ely
senescen by he new disc e e en opy me ic a e likely o be inco -
ec ly classi ied as senescen by he o iginal disc e e en opy me ic, and
FIGURE 1 Th ee example su i o ship unc ions o he h ee
di e en ypes ( ype I: senescence; ype II: cons an mo ali y; ype
III, nega i e senescence). The wo Key i z en opy measu es gi en
by Equa ions (5) and (10)) a e calcula ed o hese h ee cu es
using wo di e en wid hs o he age classes,
Δ
, gi en below in
Table 1. Tha is, we disc e ized he su i o ship cu es shown in
he Figu e using wo di e en sizes o disc e e in e als,
Δ =0.01
and
Δ =1
.
0246810
Age
10-20
10-10
10
0
Su i o ship
Type I
Type II
Type III
BOX 1 De ini ions o common demog aphic
e ms.
Li e expec ancy: he a e age numbe o addi ional yea s
ha an indi idual o age
x
can expec o li e, gi en a se
o age- speci ic mo ali y a es. Li e expec ancy a bi h is
he e o e he mean numbe o yea s a newbo n indi idual
can expec o li e.
Li e span: he amoun o ime (e.g. days, weeks, yea s) ha
an indi idual li es.
Su i o ship (
l(x)
): he p obabili y ha an indi idual su -
i es om age
0
o age
x
.
Longe i y: gene ally used as a synonym o li e
expec ancy.
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3
Me hods in Ecology and E oluon
de VRIES e al.
cons an mo ali y cu es a e egula ly inco ec ly classi ied as senes-
cen cu es by he o iginal disc e e- ime en opy me ic. In addi ion, li e
expec ancy s ongly co ela es wi h de ia ion be ween he wo en opy
me ics, wi h long- li ed species showing li le bias and sho - li ed spe-
cies showing a s onge bias in he o iginal disc e e- ime Key i z me ic.
2 | METHODS: KEYFITZ ENTROPY
2.1 | Con inuous- ime o mulas o Key i z en opy
Key i z (1977) (o he ecen edi ion, Key i z and Caswell (2005)) de-
ine he measu e H as
whe e
l(a)
is su i o ship a age
a
(see Box 1). Key i z and Caswell (2005)
no e ha H has been called en opy o in o ma ion in o he con ex s (sec-
ion 4.3 o Key i z & Caswell, 2005). H as de ined abo e, since hen gen-
e ally known as Key i z' en opy, is a weigh ed a e age o he loga i hm o
su i al, whe e he weigh s e lec no malized su i o ship. The use o en-
opies and in o ma ion heo y has a long his o y in ecology and e olu ion
(e.g. Shannon biodi e si y indices, Kullback– Leible and Jenzen- Shannon
di e gence, and Maximum En opy dis ibu ion modelling). No e, how-
e e , ha Key i z' en opy does no in eg a e o one and is he e o e no
an en opy in he s ic es sense (Shannon & Wea e , 1949).
Key i z and Caswell (2005) de i ed he o mula by calcula ing
he e ec o a p opo ional change in age- speci ic mo ali y on he
li e expec ancy a bi h, and he measu e he e o e ela es o he
simila i y o mo ali y ac oss age- classes. As a esul , Key i z' en-
opy is also a measu e o he conca i y o he su i o ship cu e.
Goldman and Lo d (1986), Vaupel (1986), and ecen ly Vaupel and
Canudas- Romo (2003) showed ha Key i z en opy can be decom-
posed in o wo cons i uen measu es ha demog aphe s use: Li e
dispa i y,
e†
, which is de ined as he a e age emaining li e expec-
ancy a he ages when dea h occu s, and measu es he numbe o
li e yea s los due o dea h. A e age li espan o an indi idual a he
ime o bi h,
e0
, which is calcula ed by in eg a ing he su i al den-
si y unc ion. The a io o he a o emen ioned indices ep esen s an
equi alen o mula ion o Key i z' en opy:
whe eby
2.2 | Disc e isa ion o Key i z en opy
Salgue o- Gómez e al. (2016) in oduced a disc e ised e sion o
Key i z en opy,
see Table S2 om he Supplemen a y Ma e ials o Salgue o- Gómez
e al. (2016). A ew lines o algeb a will show ha he con inuous- ime
e sion o Key i z’ en opy, Equa ion (1) is equal o 1 when he mo -
ali y a e,
𝜇
, is cons an such ha
l(x)=exp(−𝜇x)
. The disc e iza ion
in Equa ion 5 (which we e e o as he O iginal Disc e e- ime En opy
measu e) no longe sums o one o cons an mo ali y cu es as can
be seen om he Key i z en opies in Table 1. The Key i z en opies
in Table 1 a e calcula ed o he h ee example su i o ship cu es in
Figu e 1. In Supplemen a y Ma e ials 1, we calcula e he alue o
Hlx
when mo ali y is cons an and show ha i app oxima es one as mo -
ali y ge s close o ze o bu is lowe han one o he wise.
2.3 | An al e na i e disc e isa ion o
Key i z en opy
We p opose an al e na i e disc e e- ime o mula o Key i z' en-
opy, de i ed om he de ini ion o Key i z' en opy as he a io
o li e dispa i y o li e expec ancy a bi h (which we e e o as
he New Disc e e- ime En opy measu e). S a ing om he un-
damen al ma ix
N
(Caswell (2001), p. 112), li e expec ancy a bi h
is de ined as
whe e
𝜼1
is he ec o o li e expec ancies a each age,
1T
is a ec-
o o ones and
e1
is a ec o wi h ze os in all en ies excep he
i s en y which is one (see o example Caswell (2013) o Caswell
(1)
H
=−
∫𝜔
0
[
logl(a)
]
l(a)da
∫
𝜔
0
l(a)da
,
(2)
H
=
e†
e
0
,
(3)
e
†=−
∫𝜔
0
[logl(a)]l(a)da
,
(4)
e
0=
∫𝜔
0
l(a)da
.
(5)
H
lx =−∑
∞
x=0log
�
lx
�
lx
∑
∞
x=0
l
x
,
(6)
e0
=1
𝚃
Ne
1,
(7)
=
𝜼
𝚃
1
e
1,
TABLE 1 Compa ison o he wo Key i z en opy disc e isa ions o he cu es in Figu e 1 o a disc e e ime in e al o
Δ =0.01
and a
disc e e ime in e al o
Δ =1
. O iginal disc e e en opy is gi en by Equa ion (5), and new disc e e en opy is gi en by Equa ion (10).
O iginal disc e e en opy New disc e e en opy
Δ = 0.01 Δ = 1 Δ = 0.01 Δ = 1
Su i o ship cu e ype I 0.33 0.30 0.33 0.44
Su i o ship cu e ype II 0.96 0.75 1.00 1.00
Su i o ship cu e ype III 1.85 0.76 1.94 1.18
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4
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Me hods in Ecology and E oluon
de VRIES e al.
e al. (2018)). Simila ly, li e dispa i y a bi h can be calcula ed om
he undamen al ma ix
N
and he mo ali y ma ix
M
as
whe e he ma ix
B
con ains he dis ibu ion o age a dea h o an in-
di idual o each age, and he ec o
Be1
selec s he dis ibu ion o age
a dea h o a newbo n indi idual, see also equa ion (57) in Caswell
e al. (2018) o sec ion 3 in Caswell (2013). Key i z' en opy is hen
gi en by
o equi alen ly by
In Suppo ing In o ma ion 2, we show ha Equa ion (11) yields a alue
g ea e han 1 when mo ali y a e
𝜇
is a dec easing unc ion o age, a
alue less han 1 when mo ali y is an inc easing unc ion o age, and
exac ly 1 when mo ali y is cons an .
No e ha Key i z de i ed his con inuous- ime measu e by con-
side ing a p opo ional change in mo ali y a all ages (sec ion 4.3 in
Key i z and Caswell (2005)). We did no de i e ou o mula om his
s a ing poin , and ins ead used exis ing disc e e- ime exp essions
o he nume a o and denomina o in he con inous- ime exp es-
sion de i ed by Key i z. The e o e i emains o be shown whe he
ou new exp ession o H,
HN
in Equa ion (10), can also be de i ed
by ollowing Key i z' p oo and conside ing a p opo ional change in
mo ali y a all ages in a disc e e- ime model.
2.4 | Compa ing he me ics in eal species
using he COM(P)ADRE da abases
To compa e how he wo disc e e- ime en opy measu es pe o m in
he con ex o biologically ealis ic models, we calcula ed en opies
o age- s uc u ed ma ix popula ion models. We e alua ed bo h
animal and plan popula ion models om he COMADRE ( e sion
4.21.8.0) and COMPADRE ( e sion 6.22.5.0) ma ix popula ion da-
abases (bo h a ailable om h ps://www.compa d e- db.o g). The
empi ical compa ison included 401 species animal ma ix popula ion
models and 34 species plan ma ix popula ion models.
We ini ially sc eened he ma ix popula ion models in
COMPADRE and COMADRE based on hei inclusion in p e ious
publica ions ha used Key i z' en opy (Be na d e al., 2020). These
models we e selec ed based on du a ion o s udy, and whe he
he popula ion moni o ed was subjec o expe imen al manipula-
ion, among o he c i e ia. Wi hin he abo emen ioned subse , we
selec ed unique eco ds whe e duplica es exis ed based on maxi-
mizing s udy du a ion. We e alua ed whe he models we e age o
s age classi ied and emo ed any eco ds ha included NA alues in
ep oduc i e elemen s o s age- speci ic su i al alues g ea e han
uni y ( h ee ma ices we e emo ed om COMADRE, none om
COMPADRE).
Fo models om p e ious analyses ha we e s age- based (80
o 400 in COMADRE; 148 o 150 in COMPADRE), we con e ed
hem o age- based ma ices. The s age- o- age ans o ma ion was
based on li e able p ojec ions o he s age ma ix using he mpm o
able unc ion om he package Rage (Coch an & Ellne , 1992; Jones
e al., 2022). We cons uc ed Leslie ma ices by pu ing su i al
ec o s (px) on he subdiagonal and e ili y ec o s (mx) in he op
ow o ages wi hin 90% o he s a ing popula ion size based on he
su i o ship cu e (lx). We compa ed highe and lowe h esholds o
he coho size cu - o and ound li le a ia ion in he numbe o i-
able age- speci ic analogues o s age ma ices. The di e ence in he
numbe o ma ices using 90%, 95%, and 98% h esholds was h ee
and i e ma ices o COMADRE and COMPADRE, espec i ely.
Age- speci ic models con e ed om s age- speci ic models we e
alida ed o see i he in insic popula ion g ow h a e, ep oduc i e
a e, and gene a ion ime we e consis en wi h hose o he s age
ma ix ha gene a e he li e able om which he age- models we e
calcula ed. Di e ences a ise be ween demog aphic me ics om
he s age ma ix and om he age- om- s age ma ix because some
in o ma ion is ine i ably los in he con e sion ( o mo e in o ma ion,
see sec ion 5.3 in Caswell (2001)). We only e ained s age- based
models whe e di e ences be ween s age and hei age- equi alen
ma ices we e wi hin 5% o one ano he along he abo e demo-
g aphic me ics. Nine een ma ices we e d opped om COMADRE;
117 ma ices we e d opped om COMPADRE. The 5% cu - o was
an a bi a y h eshold ha allowed us o e ain only hose models
ha in he age- om- s age con e sion emain simila o he o iginal
models. In mos cases o he ma ices sa is ying he 5% h eshold,
he di e ence in demog aphic me ics om he co esponding s age
ma ix we e
<1%
.
3 | RESULTS: BIAS OF THE EXISTING
METRIC CORRELATES WITH LONGEVITY
To compa e he o iginal and new disc e e- ime en opy measu es,
in Table 1 we calcula e he en opy using bo h measu es o a ew
example su i o ship cu es shown in Figu e 1. Bo h me ics change
as he s ep size is changed, as is gene ally he case o demog aphic
ou comes in ma ix models (En igh e al., 1995; Pica d e al., 2010;
To es e al., 2008). Howe e , he alue gi en by he New Disc e e-
ime En opy me ic does no c oss he classi ica ion h eshold o
one, and he e o e i s classi ica ion o he su i o ship cu e as
senescen , non- senescen , o nega i e senescen (sensu Vaupel
e al. (2004)) does no change. The O iginal Disc e e- ime En opy
me ic, on he o he hand, changes om abo e one o below one o
he ype III cu e.
Figu e 2a,c show how he O iginal and he New Disc e e En opy
me ics a e co ela ed o ma ix models om COMADRE and
COMPADRE, espec i ely. The spa seness o da a in he bo om
(8)
e†
=1
𝚃
NMNe
1,
(9)
=
𝜼
𝚃
1
Be
1,
(10)
H
N=
1
𝚃
NMNe1
1𝚃Ne
1
,
(11)
H
N=𝜼
𝚃
1Be1
𝜼𝚃
1
e
1
.
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Me hods in Ecology and E oluon
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panels C and D o plan s (COMPADRE) a e a consequence o he
ac ha COMPADRE con ains la gely s age- s uc u ed ma ices
which equi ed a s age- o- age con e sion as desc ibed in he me h-
ods. We excluded models i he demog aphic quan i ies such as pop-
ula ion g ow h a e o he con e ed s age- o- age model di e ed
om he o iginal s age- s uc u ed models by mo e han 5%, which
led o many exclusions and he e o e a spa se plo o COMPADRE
han o COMADRE (117 exclusions e sus 19 exclusions, espec-
i ely; bo om wo panels e sus op wo panels in Figu e 2). As a
consequence o hese exclusions, i ually no models om he o ig-
inal da ase s analysed in Be na d e al. (2020) we e le (one o 143
ma ices con e ed). The e o e we included s age- o- age con e ed
models om COMPADRE ou side o he se analysed by Be na d
e al. (2020), shown in g ey in panels C and D (19 ma ices).
En opy es ima es we e co ela ed be ween he O iginal
Disc e e- ime En opy measu e and he New Disc e e- ime En opy
o mula ion, bu exp essed high a iabili y. The O iginal Disc e e
En opy measu e in oduced a consis en bias o o e - es ima ing
senescence (78.6% o models (264/336) had a g ea e New Disc e e
En opy alue han O iginal Disc e e En opy alue). Bias was
weake a he ex emes (colinea ly high and low alues o en opy)
wi h he s onges o e es ima ion o senescence occu ing unde
he O iginal Disc e e En opy cen e ed a he h eshold alue whe e
New Disc e e En opy = 1 (Figu e 2a).
In COMADRE, senescence was s ongly unde es ima ed when
using he O iginal Disc e e- ime En opy me ic (Figu e 2a), and
also changed sign in a subs an ial numbe o cases (126 o 334
models; wo blue su aces in Figu e 2a). Sign changed be ween he
wo shape measu es in COMADRE almos exclusi ely in he di ec-
ion o weak nega i e senescence (small su i o ship inc ease wi h
age) in e p e ed as posi i e senescence by he O iginal Disc e e
En opy me ic (dec easing su i o ship wi h age; bo om igh
quad an o Figu e 2a). Nea ly 40% o models in COMADRE (126
o 334) in e ed sign be ween he wo en opy measu es. A ound
34% o he models we e mo e han 0.25 uni s en opy in absolu e
e o (i.e. esidual di e ence om he 1:1 equi alency line), and
FIGURE 2 Va ia ion be ween he O iginal Disc e e En opy (
Hlx
) and New Disc e e En opy (
HN
) measu es om ma ix popula ion models
o animals in he COMADRE da abase (panels a and b), and o plan s in he COMPADRE da abase (panels c and d). The blue shaded a eas in
(a and c) ep esen egions whe e he wo en opy me ics ha e gi en a di e en classi ica ion o a su i o ship cu e ( op blue squa e: New
Disc e e me ic classi ied he cu e as senescen whe eas he O iginal Disc e e me ic classi ied he cu e as nega i ely senescen ; bo om
blue squa e: ice e sa). (a) Compa ing Key i z' en opy o animal ma ix models om he COMADRE da abase using he O iginal Disc e e
and he New Disc e e me ic. Poin s in da k blue a e ma ices whe e en opy shi ed om nega i e alues (posi i e senescence) o posi i e
alues (nega i e senescence); poin s in ligh blue do no ha e ha shi . Ma ices con e ed om s age o age a e shown in o ange. (b) Plo
o he di e ence be ween he new and he exis ing me ic
(
HN−Hlx
)
as a unc ion o he li e expec ancy o animal species om COMADRE.
(c) Compa ing Key i z' en opy o plan ma ix models om he COMPADRE da abase using he O iginal Disc e e and New Disc e e En opy
me ic. Poin s in g ey a e con e ed s age- o- age ma ix models ha we e no included in he analyses in Be na d e al. (2020). (d) Plo o he
di e ence be ween he new and he exis ing me ic
(
HN−Hlx
)
as a unc ion o he li e expec ancy o plan om COMPADRE.
(a) (b)
(c) (d)
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Me hods in Ecology and E oluon
de VRIES e al.
18% o models we e mo e han 0.50 uni s en opy in e o om
he ue alue.
In COMPADRE, senescence was also unde es ima ed when
using he O iginal Disc e e- ime En opy me ic (Figu e 2c), changing
sign in a modes numbe o cases (8 o 33 models). Sign changed be-
ween he wo shape measu es in COMPADRE in bo h di ec ions (2
o e es ima ed; 6 unde es ima ed; blue su ace in Figu e 2a). A ound
25% o models (8 o 33) in e ed sign be ween he wo measu es.
The mean en opy di e ence be ween he me ics was 0.35 uni s
in COMPADRE. De ia ions om he 1:1 line occu ed in bo h di-
ec ions in COMPADRE wi h 16 models o e es ima ing en opy and
10 models unde es ima ing en opy wi h he O iginal Disc e e- ime
me ic.
The dis ance be ween he wo disc e e en opy me ics was co -
ela ed wi h li e expec ancy (Figu e 2b,d). Fo da a om COMADRE
(panel B), he mean e o o models wi h li e expec ancy <2 was
−0.63, he mean e o wi h li e expec ancy be ween 3– 5 yea s was
−0.15, and o 5– 10 yea s i was −0.06.
These indings ha e impo an implica ions o he ecen and
u u e wa es o compa a i e demog aphic esea ch e alua ing he
numbe o species escaping o unde going ac ua ial senescence (e.g.
Beckman e al., 2018; Be na d e al., 2020; Capde ila e al., 2020;
Salgue o- Gómez, 2017) because Figu e 2a implies ha hese s ud-
ies ha e likely unde es ima ed he numbe o species wi h negligible
ac ua ial senescence using his O iginal Disc e e Key i z me ic, and
Figu e 2b implies ha his unde es ima ion was especially s ong o
sho - li ed species, he e o e in oducing a spu ious co ela ion be-
ween shape and pace.
In Suppo ing In o ma ion 1 we show why he o iginal disc e isa-
ion,
Hlx
, classi ies cons an mo ali y cu es as nega i ely senescen
cu es, and why i does so mo e s ongly o species wi h sho e
li espans. We ind ha o cons an mo ali y
𝜇
,
H
lx =
𝜇exp(−𝜇)
1−exp(−𝜇)
. This
unc ion is smalle han one whene e he mo ali y a e is nonze o,
ha is, when
𝜇
is la ge han 0. Fu he mo e,
𝜇exp(−𝜇)
1−exp(−𝜇)
is a dec eas-
ing unc ion o
𝜇
such ha he O iginal Disc e e Key i z me ic ge s
close o ze o as he cons an mo ali y ge s la ge and li e expec-
ancy ge s sho e , leading o he co ela ion seen in Figu e 2b. In he
limi o in ini esimally small ime s eps, su i al app oaches one (o
𝜇
app oaches ze o), and he sum app oxima es he con inuous ime
o mula well in his limi .
4 | DISCUSSION AND CONCLUSION
We ha e shown ha he commonly used ime- disc e ized o -
mula o Key i z’ en opy ( e e ed o he e as he O iginal Disc e e
En opy measu e) does no p ese e he ela ionship be ween
Key i z’ en opy and he shape o he su i o ship cu e ha ex-
is s o a con inuous- ime de ini ion o su i o ship en opy. Tha is,
cons an mo ali y cu es do no yield a Key i z’ en opy o one, and
li e his o ies wi h dec easing mo ali y will no always yield alues
abo e one (e.g. see Table 1). Speci ically, nea ly 40% o classi ied
su i o ship cu es (126 o 336) changed classi ica ion when using
he new disc e e me ic om senescen o nega i ely senescen , o
ice e sa. Fu he mo e, he dis ance be ween he o iginal and he
new disc e e Key i z' en opy me ic co ela es wi h li e expec ancy
(Figu e 2b). As a consequence, any co ela ions ob ained be ween
pace and shape o li e in p e ious publica ions using he exis ing
Key i z me ic may need o be ee alua ed.
We p opose a di e en o mula o he disc e isa ion o Key i z'
en opy ( e e ed o he e as he new en opy measu e), based
on li e dispa i y and li e expec ancy in Equa ion 10. We show in
Suppo ing In o ma ion 2 ha his new o mula does p ese e he e-
la ionship be ween he shape o he su i o ship cu e and Key i z
en opy ( ha is,
HN>1
when mo ali y is a dec easing unc ion o age,
HN<1
when mo ali y is an inc easing unc ion o age, and
HN=1
when mo ali y is cons an ). Howe e , a majo downside o he o -
mula we ha e p oposed is ha i is only a measu e o he shape o
ageing o age- s uc u ed su i al ma ices (Leslie ma ices). I he
su i al and popula ion ma ix a e s age- s uc u ed, hen he New
Disc e e En opy measu e quan i ies whe he mo ali y a e inc eases
o dec eases wi h s age. S age- o- age con e sion me hods can o e
one way a ound his limi a ion o he me hod ( o mo e in o ma ion
on s age- o- age con e sion me hods see sec ion 5.3 in Caswell, 2001,
and o an implemen a ion o he me hods in R see Jones e al., 2022).
Besides he new shape me ic p oposed he e, he e a e many
o he shape me ics ha ha e been p oposed and can be used o clas-
si y su i o ship cu es. Fo example, o he li e able s a is ics ha
ha e been used o quan i y he age- speci ic decline in su i al include
Hayley's median (Hailey, 1874); he age- dependen mo ali y pa am-
e e o mo ali y dis ibu ions (e.g. Gompe z, Weibull, Sile , Logis ic,
e c.; Rickle s & Scheue lein, 2002); he age a he onse o senescence
(Jones e al., 2008) and he in eg a ion o he emaining li espan and
su i al unc ion (W ycza e al., 2015). W ycza e al. (2015) highligh
a numbe o o he po en ial candida es, such as a modi ied Gini coe -
icien ( e iewing 7 possible me ics), and highligh he alue o he
en opy as a measu e o he shape o li e (see also Abu o e al. (2022)
o a ecen discussion o measu es o li espan inequali y).
AUTHOR CONTRIBUTIONS
Cha lo e de V ies concep ualized he a icle wi h inpu om
Conno Be na d and Robe o Salgue o- Gómez. Cha lo e de V ies
pe o med analy ical ma hema ical analyses, Conno Be na d pe -
o med analyses in R. Cha lo e de V ies w o e i s d a : all au ho s
con ibu ed o subsequen d a s.
ACKNO WLE DGE MENTS
Cha lo e de V ies hanks Hal Caswell o e y help ul discus-
sions in he ea ly s ages o his p ojec . Robe o Salgue o- Gómez
was suppo ed by a NERC Independen Resea ch Fellowship (NE/
M018458/1). Cha lo e de V ies was suppo ed by Swiss Na ional
Science Founda ion g an numbe 310030B_182836 awa ded
o Hanna Kokko, Swiss Na ional Science Founda ion g an
P500PB_211003 awa ded o Cha lo e de V ies, Academy o Finland
g an numbe 340130 awa ded o Jussi Leh onen, and Eu opean
Resea ch Council Ad anced G an 788195 awa ded o Hal Caswell.
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7
Me hods in Ecology and E oluon
de VRIES e al.
CONFLICT OF INTEREST STATEMENT
The au ho s decla e no con lic s o in e es .
PEER REVIEW
The pee e iew his o y o his a icle is a ailable a h ps://
www.webo scien ce.com/api/ga ew ay/wos/pee - e ie
w/10.1111/2041- 210X.14083.
DATA AVAILABILITY STATEMENT
Code used in his pape can be ound a Zenodo wi h h ps://zenodo.
o g/badge/ la es doi/53287 1458, and a h ps://gi hub.com/Lo e
- biolo gy/Key i z.
ORCID
Cha lo e de V ies h ps://o cid.o g/0000-0001-8955-0479
Conno Be na d h ps://o cid.o g/0000-0003-2175-5712
Robe o Salgue o- Gómez h ps://o cid.
o g/0000-0002-6085-4433
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SUPPORTING INFORMATION
Addi ional suppo ing in o ma ion can be ound online in he
Suppo ing In o ma ion sec ion a he end o his a icle.
Suppo ing In o ma ion S1. P ope ies o he O iginal Disc e e- ime
En opy me ic.
Suppo ing In o ma ion S2. P ope ies o New Disc e e Key i z
En opy me ic.
How o ci e his a icle: de V ies, C., Be na d, C., & Salgue o-
Gómez, R. (2023). Disc e ising Key i z' en opy o s udies o
ac ua ial senescence and compa a i e demog aphy. Me hods in
Ecology and E olu ion, 00, 1–8. h ps://doi.o g/10.1111/2041-
210X.14083
2041210x, 0, Downloaded om h ps://besjou nals.onlinelib a y.wiley.com/doi/10.1111/2041-210X.14083 by Uni e si y O Jy äskylä Lib a y, Wiley Online Lib a y on [29/03/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License