P ep in
Bi mann, F. (2025): Ad ancing he Age-Happiness Deba e: Reconciling G aphical and Nume ical App oaches.
doi:10.5281/zenodo.17285803
A icle
Ad ancing he Age-Happiness Deba e: Reconciling
G aphical and Nume ical App oaches
Felix Bi mann1*
1Leibniz Ins i u e o Educa ional T ajec o ies; 96047 Bambe g; Ge many
Abs ac
While in he pas , especially nume ical app oaches such as eg ession models and p- alues we e u ilized o in es iga e whe he he unc-
ional o m be ween age and happiness is linea o U-shaped in a gi en coun y, ecen ad ances ha e shown ha such app oaches can be
misleading, and g aphical analyses should amend he analyses. Howe e , hese ha e he downside ha hey ely o some ex en on subjec i e
in e p e a ions and a e ha dly quan i iable. I applied ca elessly, hey can be misleading as well. We sugges wo easily compu ed s a is ics o
combine g aphical and nume ical app oaches. We demons a e hei usage wi h ESS da a (N > 440,000) and show how hey enable a mo e
nuanced in es iga ion o unc ional o ms. Fu he mo e, we discuss how s a is ical unce ain y can be handled in he age-happiness deba e
and emphasize p ac ical signi icance, which needs o be kep in mind.
Keywo ds:
happiness, ageing, unc ional o m, g aphical analysis, p ac ical signi icance
Acknowledgemen s:
Special hanks o Da id Ba am, who ead an ea ly e sion o he d a and ga e help ul commen s.
1. In oduc ion
How does happiness a y by age, and does his unc ional
o m di e by coun y? These seemingly simple ques ions
a e he co e o a decade-old esea ch deba e. Despi e dozens
o published s udies, including hund eds o housands o su -
eyed indi iduals and hund eds o coun ies, a consensus has
ye o eme ge. While some schola s claim ha he e is a
U-shape in almos all coun ies (Beja,2018;Blanch lowe
e al.,2023;Blanch lowe and Oswald,2009;Cheng e al.,
2017;G aham and Ruiz Pozuelo,2017), o he s disag ee (Ba -
am,2021,2022;Becke and T au mann,2022;Bi mann,
2021;K a z and B üde l,2025). In e es ingly, he da a used
o answe hese ques ions is o en common o mul iple s udies,
ye s a is ical and me hodological ques ions, such as he mod-
eling s a egy o he inclusions o omission o con ol a i-
ables, in luence he esul s and in e p e a ion. A ecen pub-
lica ion has demons a ed ha an o e - eliance on eg ession
coefficien s and hei espec i e p- alues can dis o he e-
sul s and lead o ques ionable esul s, especially when g aph-
ical and desc ip i e me hods suppo di e en in e p e a ions
(Ba am,2024). A his poin , we would like o con ibu e o
his me hodological discussion ega ding using g aphical and
desc ip i e app oaches. They ha e he ad an age o equi ing
ewe (s a is ical) assump ions and suppo ing a lexible in e -
p e a ion, ega dless o sample sizes o model speci ica ions
used (in con as o eg ession- ype app oaches). Howe e ,
g aphical app oaches a e no gene ally accep ed by e e yone
as hey come wi h a mo e subjec i e pa , lea ing he in e p e-
a ion up o each esea che , which di e s om he classical
and “ha d” nume ical s a is ics ha appea o be p ecise and
s ingen .
A his poin , we sugges econciling g aphical and nu-
me ical app oaches o some ex en and ou lining how hey
can amend each o he . We sugges wo common and eas-
ily compu ed s a is ics ha help esea che s decide lexibly
whe he he unc ional o m be ween age and happiness is
a he linea o no . These s a is ics ha e he ad an age o
being independen o sample size and no using ( eg ession-
compu ed) p- alues. They a e mo e nuanced and a oid a bi-
na y classi ica ion and hinking. We will discuss he p ob-
lem o bina y conclusions in empi ical esea ch in mo e de-
ail and sugges al e na i es. In addi ion, we would like o
emphasizep ac icalsigni icance (in con as o s a is ical sig-
ni icance), which is o en o go en in bo h g aphical and nu-
me ical analyses (Mohaje i e al.,2020). We will demons a e
* Co esponding Au ho :
Felix Bi mann, Leibniz Ins i u e o Educa ional T ajec o ies
[email p o ec ed] 1
© 2025 Copy igh by he Au ho s.
Licensed as an open access a icle using a CC BY 4.0 license.
doi:10.5281/zenodo.17285803
ha some coun ies ha e a a he dis inc unc ional o m ye
p obably lack any p ac ical ele ance ega ding he in luence
o age. This conclusion aligns wi h o he s udies ha es
he speci ic in luence o a ious p edic o s on (un)happiness
(Bi mann,2024). When age’s o e all and o al e ec on hap-
piness is iny, he ques ion abou unc ional o ms becomes
a he academic as hey ha e li le eal-li e implica ions. We
sugges discussing hese aspec s openly so ha esea ch is no
disconnec ed om eali y. This is no a p oposal o shu down
he deba e o unc ional o ms bu a he he con a y. I ap-
pea s o be highly ele an o unde s and in which coun ies
age has los i s in luence on happiness, why his has happened,
and which mechanisms a e ele an .
Summa ized, he ollowing pape has h ee esea ch
ques ions: Fi s , how can we assess he unc ional o m be-
ween age and happiness g aphically and nume ically? Sec-
ond, wha is he o e all in luence o age on happiness (p ac i-
cal signi icance o age)? Thi d and inal, how a e unc ional
o ms and he o e all in luence o age on happiness ela ed
o each o he ? Using a la ge-scale Eu opean da ase (N >
440,000), we will answe hese ques ions empi ically and
make sugges ions o ad ancing he age-happiness deba e in
gene al.
2. Theo e ical amewo k
This sec ion has wo main pu poses. Fi s , we gi e an o e iew
o ecen de elopmen s in he age-happiness deba e o see
wha has been esea ch and how in he pas . This is ele an
o see whe e esea ch gaps and po en ial p oblems lie and how
hey can be amended. Second, we discuss in mo e de ail he
implica ions o ou esea ch app oach and how bina y con-
clusion aps can be a oided and we sugges a way o handle
unce ain y in his esea ch deba e.
2.1. Recen de elopmen s and esea ch
agenda
Thanks o he ongoing esea ch on he in luence o age on hap-
piness, he e ha e been majo de elopmen s. He e, we will
b ie ly summa ize he main ad ances in he pas ew yea s.
• Con ol a iables’ unc ion has been explained in much
mo e de ail (Ba am,2021;K a z and B üde l,2025).
While in he pas , he addi ion o con ol a iables such
as gende , educa ion le el, heal h, o ma i al s a us was
a he common in eg ession models o he main p e-
dic o s (age o highe -o de e ms such as age-squa ed),
con incing a gumen s ha e been pu o wa d o e-
mo e mos such con ols om he models. As has
been shown, no classical con ols a e equi ed o es-
ima e he o al and causal e ec o age on happiness as
he e a e no an eceden s o age. Adding mo e a iables
o he model can be ele an o es ima ing medi a ion
pa hways and explain how and why age in luences hap-
piness; howe e , i he gene al unc ional o m is o be
es ima ed, hese should be emo ed o a oid o e con-
ol bias (Elwe and Winship,2014). Howe e , some
a iables can s ill be help ul o accoun o pe iod and
su ey e ec s.
• I is much clea e now how ca eless in e p e a ion o
eg ession models can be misleading when es ima ing
unc ional o ms. Especially only elying on he s a is i-
cal signi icance o some coefficien s is no a alid way
o p o e such o ms. To s a wi h, explo a i e a emp s
ha do no ely on s a is ical signi icance a all bu on
clus e app oaches ha e shown ha unc ional o ms
can be di e se and ha a leas h ee majo unc ional
o ms a e p esen , disp o ing olde claims ha he U-
shape is gene al and alid e e ywhe e (Bi mann,2021).
Follow-up s udies ha e also shown ha he app oach o
explain unc ional o ms solely based on model coeffi-
cien s can be misleading as e en o highly linea o ms,
squa ed e ms o age can s ill be s a is ically signi ican
(Ba am,2024). The ole o sample sizes has also been
conside ed o a oid w ong conclusions. This connec s
o he deba e abou he p oblema ic usage o p- alues in
s a is ics and ha mo e nuanced app oaches a e highly
encou aged (Wasse s ein e al.,2019). Mo e de ails a e
ou lined below in sec ion 2.2.
• Based on hese de elopmen s, a gumen s ha e been
made o a oid emphasizing he classical eg ession-
es ing con ex whe e only coefficien s a e e alua ed
nume ically. Ins ead, g aphical in e p e a ions a e lex-
ible and accessible as a highly ele an addi ion o nu-
me ical app oaches. They can demons a e di ec ly,
o example, ha highe -o de e ms’ s a is ical signi i-
cance does no gua an ee a non-linea unc ional o m.
Fu he mo e, hey encou age a mo e nuanced discus-
sion and hey a oid bina y classi ica ions. While hese
de elopmen s a e welcome and ad ancing he ield, we
would like o make a ew addi ions and econcile nume -
ical and g aphical app oaches o some ex en . While
bene icial, we would like o demons a e how g aphical
me hods can also be misleading i applied ca elessly
and o show how nume ical analyses a e s ill bene icial.
Fu he mo e, we would like o emphasize p ac ical sig-
ni icance (compa ed o s a is ical signi icance). E en
coefficien s wi h e y small p- alues can be meaning-
less in he eal wo ld, which should be pa o he gen-
e al discou se. We should make i mo e salien why
and how much he age-happiness ela ionship ma e s
(and how his di e s be ween coun ies).
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2.2. Mo ing o a wo ld beyond bina y
conclusions
In he pas ew yea s, a shi has happened in s a is ics. Mo e
and mo e esea che s ag ee ha bina y conclusions, which
a e mos ly acili a ed by an unheal hy ocus on p- alues and
o he “ha d” s a is ics ha suppo such conclusions (“ he p-
alue is smalle han 0.05; hence I conclude ha my indings
a e ue…”) a e p oblema ic and do mo e ha m han good.
This shi in hinking abou unce ain y and s a is ics ega ds
he ype o s a is ics we should compu e, epo , and in e p e ,
such as p- alues, con idence in e als, o measu es o e ec
size, and how unce ain y in gene al should be add essed. Un-
o una ely, his hinking has no ye qui e eached he age-
happiness deba e. Too many p e ious s udies easily each
bina y conclusions, such as ha he s a is ical signi icance
o a squa ed eg ession e m leads o he conclusion ha a
unc ional o m mus be ( e e sed) u-shaped (Ba am,2024)
o ha coun ies can be easily g ouped in clus e s (Bi mann,
2021), esul ing in ei he -o classi ica ions wi h no oom o
in-be ween. We a gue ha such bina y hinking is ha m ul
when discussing complex and messy aspec s such as unc-
ional o ms, as, in eali y, hese o ms a e ne e uly “linea ”
o any hing else. Fo example, linea i y is a ma hema ically
de ined e m ha da a om eal da a can ne e each, only ap-
p oach o a ce ain deg ee. When in doub , esea che s should
epo his anspa en ly and discuss he deg ee o simila i y,
po en ial biases, and especially he unce ain y a ound hei
es ima es. Only his acili a es an open and hones discussion
abou phenomena in he eal wo ld ha we, as esea che s,
would like o desc ibe and assess s a is ically.
In oducing g aphical ways o in e p e a ion o he de-
ba e is a s ep in he igh di ec ion as “a pic u e is wo h a hou-
sand wo ds”, which enables nuanced and mo e complex con-
clusions han a ew numbe s e e can. Howe e , as explained
be o e, such an app oach is also limi ed. We wan o amend
hese g aphical app oaches wi h some s a is ics ye no all
in o bina y decision aps again. Howe e , his endea o also
comes wi h cos s. When discussing s a is ics in he ollowing,
he e a e no clea -cu guidelines o when an e ec o change
is “small” o “la ge”. Resea che s a e so used o s ic guide-
lines (“a p- alue below 0.05 shows a eal e ec ” / “an e ec
size a ound 0.50 is o medium size”, e c…) ha i seems ou
o place o make such a s a emen ha epo ed numbe s can-
no be easily e alua ed o classi ied. Fo he cu en analyses
and p obably la ge pa s o he o e all deba e, we ecommend
a oiding absolu e classi ica ions bu compa ing numbe s and
e ec sizes o each o he . Fo example, when mul iple coun-
ies a e compa ed wi hin one s udy, one can see ha some
coun ies ha e a mo e linea shape han o he s. When alk-
ing abou he o al e ec o age on happiness, one can po en-
ially conclude ha age ma e s mo e in some coun ies han
o he s. These ela i e compa isons a oid bina y conclusions
and a e mo e nuanced as hey all on o a con inuous spec um,
which enables ine-g ained conclusions. When a s udy only
includes a single coun y, one can e e ence p e ious esea ch
esul s and see whe he e ec sizes a e smalle o la ge bu
no “small” o “la ge”. Leading a discussion in such a way
p o ides ele an in o ma ion o eade s ye is nuanced and
akes he complexi y o he eal wo ld in o accoun . While i
may ake a while o expe ienced esea che s o ge used o
alking abou da a and s a is ics in such a way, we a gue ha
unce ain y and ela i e compa isons should be emb aced o
a oid pas mis akes and mo e on as a esea ch communi y.
3. Da a, a iables, and me hods
3.1. Da a, sample, and a iables
In he ollowing, we use ESS da a om he las en ounds
(ESS 1, edi ion 6.7, o ESS 10, edi ion 3.1). The ESS p o-
ides high-quali y da a o mo e han hi y Eu opean coun-
ies. While adding e en mo e coun ies om o he con i-
nen s and egions was possible wi h u he da ase s o ou line
ou poin s, he ESS is adequa e. Especially since g aphical in-
e p e a ions a e ele an , showing e en mo e coun ies migh
be o e whelming. Fu he mo e, as olde s udies ha e shown,
he e a e di e se unc ional o ms o age and happiness in
Eu ope. Due o he long- unning na u e o he ESS, many
coun ies a e included mul iple imes in he da ase , which
p o ides u he insigh and s a is ical powe . To be conc e e,
only coun ies ha ha e a leas wo non-consecu i e pa ic-
ipa ions a e included. Fu he mo e, only indi iduals aged
om 18 o 80 a e included. This ensu es ha he e is also
enough e idence in he oldes coho s, as a e y low numbe
o e y old pa icipan s could dis o unc ional o ms due o
ou lie s. The esul ing sample o all analyses hence includes
33 coun ies wi h a o al o 440,160 pa icipan s. To a oid
he in luence o pe iod e ec s and accoun o di e ences in
su ey me hodology, he ESS ound will be included as he
sole con ol a iable in he ollowing analyses. The depen-
den a iable is happiness (“How happy a e you?”), which is
measu ed on an 11-poin Like scale wi h alues om 0 (“ex-
emely unhappy”) o 10 (“Ex emely happy”). This measu e-
men is ha monized o e all wa es and coun ies and p o ides
a well-es ablished measu emen o happiness o li e sa is ac-
ion. The main independen a iable is age in yea s. Indi id-
uals wi h missing alues a e emo ed om he analyses (lis -
wise dele ion). Howe e , he sha e o missingness o bo h
a iables is ex emely low (less han 1% each).
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3.2. S a egy o analysis
3.2.1. G aphical app oach: local polynomial
smoo hing
The goal o his analysis is o show how g aphical app oaches
can be bo h bene icial and misleading. Fu he mo e, i a -
emp s o demons a e how g aphical and nume ical app oaches
can complemen each o he . To be mos lexible, one could
s a wi h a simple sca e plo be ween happiness and age, as
his is pu ely desc ip i e and does no make any assump ions.
Howe e , gi en ha bo h a iables only con ain a limi ed
ange (especially happiness wi h only 11 possible alues) and
a e y la ge numbe o indi iduals in each coun y (N a ies
be ween 3,706 and 31,996), e e y po en ial poin in he g aph
would be occupied, making a g aphical in e p e a ion close
o u ile. To esol e his issue, local polynomial smoo hing is
a po en ial solu ion (Cle eland and Loade ,1996;Fan,1992).
Fo as many poin s as speci ied by he esea che ( ypically
25 o 100) o e he ange o he x- a iable (age, in ou case),
a eg ession model is es ima ed a each poin , and a ke nel
unc ion weighs he in luence o each obse a ion. The in e -
cep o he compu ed eg ession is hen used as he esul a
his x- alue. By epea ing his s ep o all speci ied poin s,
he immense in o ma ion o he sca e plo is educed o a
a he small numbe o poin s, which can be used o i a line,
a i ing a a simple line g aph ha is easy o in e p e as he
da a is condensed in o his i ed line, showing he unc ional
o m be ween x and y a iable. A simila app oach is LOESS
o LOWESS, which could also ha e been used. While his
app oach is sensible and use ul, i can s ill be misleading.
Fo emos , as wi h any g aph, scaling ma e s. Should a g aph
always s a wi h ze o o no ? When di e en coun ies a e
compa ed o each o he , is i sensible o pu hem all on he
same scaling o is i be e o use lexible scales as o un-
de sco e smalle di e ences be ween coun ies? Below, we
demons a e how di e en answe s o hese ques ions can in-
luence how g aphs a e ead and in e p e ed and some ad ice
is gi en. Fu he mo e, when using polynomial smoo hing,
he e a e se e al pa ame e s ha he esea che has o se
(which is some imes done au oma ically by he so wa e). Fo
example, he numbe o poin s o es ima e he local unc ion
has o be speci ied, as well as he ype o ke nel algo i hm o
he deg ee o he eg essions used in he compu a ions. How-
e e , hese speci ica ions a e usually a he echnical and ha e
no la ge in luence (as long as no ex eme alues a e chosen).
In any case, we ecommend o be anspa en and discuss such
decisions when discussing esul s so eade s ha e he change
o hink abou his and con empla e al e na i es, which migh
lead o di e en conclusions.
3.2.2. Nume ical app oach: absolu e and changing
R²
Fo he nume ical app oach, we would like o ocus on h ee
dis inc aspec s: i s , wha is he unc ional o m be ween age
and happiness, o which we use he ela i e change o R².
Second, we would like o see he o al and absolu e ele ance
o age o explain happiness, o which we use he R² alue o a
well- i ing eg ession model. Thi d and inal, we would like
o discuss how hese wo s a is ics ela e o each o he and
wha his can ell us abou di e en coun ies and con ex s.
Since we wo k wi h nume ical alues, i is also c ucial o alk
abou some kind o “e ec ” size when compa ing numbe s,
which is also aken up a he end o his sec ion.
We s a wi h he unc ional o m ( ela i e change o R²).
We would like o ha e a nume ical app oach ha does no ely
on p- alues o solely on he coefficien s in a eg ession model
and a oid coa se o bina y classi ica ions. We compa e he
empi ical unc ional- o m wi h a ange o heo e ical ones and
selec he one wi h he bes i . This goes back o he p oblem
o s a is ical i in a eg ession model, o which a wide ange
o s a is ics has been de eloped. As ou models ha e only one
main p edic o (age, o de i ed highe -o de e ms), we would
like o keep hings simple and wo k wi h R² (while o he i
measu es migh wo k simila ly). R², o he coefficien o de-
e mina ion, is easily compu ed in an OLS eg ession model
as one minus he esidual sum o he squa es di ided by he
o al sum o squa es Guja a i and Po e (2010). When he
model has exac ly one p edic o , his can also be shown g aph-
ically on pape . R² is bound be ween 0 and 1 and adding addi-
ional p edic o s o he eg ession models can only inc ease i s
alue bu ne e dec ease i . The e o e, adding “useless” p e-
dic o s wi h li le p edic i e powe can ne e make he model
i wo se, as R² does no conside model pa simony. Fo ou
means, his is comple ely ine. Since we a e mos ly in e es ed
in he dis inc ion “linea -shape” s “non-linea shape”, we
compa e he linea model o a mo e lexible one. In ou case,
we decided o use he cubic model1as i can model ei he a
u-shaped o m o e en mo e complex o ms. By compa ing
he model i s be ween he wo models, we can assess, nume -
ically, whe he a non-linea unc ional o m is mo e plausible
han he linea one. I wo ks as ollows: Imagine he mos sim-
ple case, a highly linea ela ionship wi h exac ly one indepen-
den a iable. This p edic o pe ec ly explains he unc ional
o m (excep o andom e o s, which will always be p esen
in eal-li e da ase s). Adding highe o de e ms o his model
( he squa ed o cubed e m o he p edic o ) will no change
R² in any signi ican o m. Howe e , keep in mind ha “sig-
1This model con ains age, age² and age³ as independen a iables. How-
e e , no e ha his decision is somewha a bi a y and e en mo e complex
models migh be bene icial o o he da a o esea ch ques ions.
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ni ican ” does no mean “s a is ically signi ican ” bu needs o
be discussed and assessed by he esea che s! We call his he
ela i e change o R² (which can be exp essed as a pe cen age
change om he base). Hence, adding a highe -o de e m
will ha dly inc ease R² i he ela ionship is uly linea . This
is a he di e en i he ela ionship is (in e sely) U-shaped
o e en mo e complica ed. While he linea model speci ica-
ion will usually ha e some explana o y powe , adding one
o mo e highe -o de e ms will g ea ly imp o e he model i
(again, in he iew o he esea che s). Hence, he ela i e
pe cen age change o R² will be la ge . We a gue ha his
ela i e change o R² is a simple way o es ima e whe he he
unc ional o m is linea o no . I he change is a he small-
ish, he i is linea ; i he change is la ge , i is no . This
is he main claim we a e es ing empi ically u he below.
We will demons a e p ac ically how assessing “smalle ” o
“la ge ” wo ks in a mul i-coun y compa ison when absolu e
judgmen s a e o be a oided. Since hese pe cen age changes
can be huge, especially when base le els a e e y small, log-
a i hmising hem can be bene icial o a oid la ge numbe s.
Fo he app oach as jus explained, he e is a ca ea ,
which is he second main a gumen o his pape , which is
abou p ac icalsigni icance. The idea is o s udy bo h he el-
a i e changes in R² and he absolu e R² ( om he bes - i ing
model a ailable). This means, ega dless o how complex
and well- i ed he eg ession model is (po en ially adding
squa ed, cubed, and e en u he e ms), i he inal R² alue
is low, he e is an o e all e y limi ed in luence o age on
happiness. This is ele an o p ac ical ques ions. E en
i he unc ional o m be ween happiness and age was pe -
ec ly U-shaped, i he o al explained a iance o happiness
by age is e y low, why should one ca e much abou his ac ?
The happiness an indi idual epo s is, o a e y small ex-
en , dependen on age, and o he in luences migh be much
mo e ele an . This connec s he age-happiness deba e o a
b oade ques ion: I we a e in e es ed in s udying happiness
and which ac o s explain and p edic i , age migh be negli-
gible (a leas in some coun ies). This ac mus always be
kep in mind; o he wise, his en i e deba e and li e a u e can
degene a e in o an o e ly academic discussion de ached om
eal li e. Tha is why we a e also in e es ed in he o al R²
ha a well- i ing model has. As explained abo e, since R²
can ne e ge lowe , e en i highe -o de e ms a e added o a
pe ec ly linea model, we decided o use he R² alue o he
cubic model as a measu emen o absolu e i and ele ance
o happiness. We would like o emphasize ha his second
nume ical alue is also a he meaningless wi hou con ex .
We sugges o ei he compa e hese alues o each o he in
mul i-coun y compa isons o compa e hem o he in luence
o o he p edic o s o happiness ( o example, gende , heal h,
o ma iage s a us). By doing so, mo e nuanced insigh s a e
possible, as o he s udies ha e a emp ed o es a la ge se
o p edic o s o explain happiness (Bi mann,2024;Dohe y
and Kelly,2010;Halle and Hadle ,2006).
Las ly, we would like o es how ela i e changes in R²
( unc ional o ms) and absolu e alues o R² ( o al in luence
o age on happiness) a e ela ed. One p e ious publica ion
indica ed g aphically ha he e migh be a ela ion as mo e
de eloped coun ies, also wi h a highe HDI, usually display
non-linea unc ional o ms, while coun ies wi h lowe HDI
ha e mo e linea o ms (Ba am,2024). Ano he publica-
ion ha a emp ed o so unc ional o ms in o clus e s also
demons a ed his nume ically (Bi mann,2021). This gi es
a i s hin ha he e migh be a deepe ela ion o why some
coun ies ha e linea shapes while o he s ha e no . As in mo e
de eloped coun ies, he p ac ical signi icance o age o hap-
piness us lowe due o social wel a e sys ems, such as heal h-
insu ance o pension schemes, we would also expec a ela ion
be ween o al explained R² by happiness and he ela i e ex-
plana o y powe . This esea ch ques ion is es ed nume ically
and g aphically. Nume ically, he co ela ion coefficien is
compu ed o absolu e explained a iance in he cubic model
and he ela i e imp o emen o R² om he linea o cubic
model. G aphically, his is done by plo ing hese wo alues
agains each o he in a sca e plo , sepa a ely o each coun y.
I he e is a ela ionship ound, his could hin o s uc u al,
social, o cul u al di e ences be ween coun ies ha dese e
a en ion. Fu he mo e, one can see which coun ies a e sim-
ila o each o he and which a e a he di e en in his ela-
ionship. Finally, his g aphical o m o p esen a ion, again,
unde lines how g aphical app oaches can be bene icial o nu-
anced in e p e a ions, especially when mul iple coun ies o
da a poin s a e o be compa ed. All analyses a e conduc ed
using S a a 16.1, and do- iles a e a ailable upon eques .
4. Resul s
4.1. G aphical app oach: local polynomial
smoo hing
As desc ibed be o e, he i s pa o he analysis is a g aphical
in e p e a ion. Fo compa ison, he unc ional o m be ween
happiness and age has been compu ed o all coun ies in he
ESS da a using local polynomial smoo hing. These esul s
a e shown in he appendix (Figu es 4 and 5). Since his has
been done be o e in o he publica ions, i is only depic ed as
a e e ence so one can la e compa e he nume ical esul s o
all coun ies o he g aphs. Wha is o g ea es in e es o us
is o see how g aphs can s ill be misleading i applied ca e-
lessly. This p oblem is no he me hod i sel bu a he he
scaling o he axes (B yan,1995). To demons a e his, wo
coun ies, Po ugal and Ge many, ha e been selec ed om he
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da ase . On pape , hese coun ies a e compa able. They ha e
pa icipa ed in e e y ESS ound since he s a , and many ob-
se a ions a e a ailable (31,224 o Ge many and 16,350 o
Po ugal). Bo h ha e been membe s o he Eu opean Union
o mo e han 35 yea s and ha e a high Human De elopmen
Index (0.942 / 0.866 in 2021). Howe e , when he unc ional
o m o happiness and age is in es iga ed, he g aphical ap-
p oach yields wo di e en conclusions. We s a wi h Figu e
1, he uppe panel (Flexible y-scales). This scaling app oach
uses di e en y-scales o bo h coun ies. The bene i is ha
wi hin-coun y di e ences a e emphasized and de ails a e e-
ealed, howe e , a c oss-coun y compa ison becomes diffi-
cul , po en ially misleading.
Fo Po ugal, we obse e a highly linea and mono onic
decline in happiness o e he age cou se. The olde people
become, he less happiness hey epo . Based on his g aph,
one would ha dly e e assume ha a U-shaped ela ionship is
p esen in Po ugal. Howe e , o Ge many, he conclusions
a e di e en . In hei you h, indi iduals a e a he happy, bu
he olde hey become, he less happy hey a e. The e is min-
imum alue o indi iduals aged a ound 50 yea s. A e wa d,
he e is a u n, and happiness ises o he age o abou 70.
Then, happiness s a s o decline again. This is an ideal case
o he classical U-shape, especially in he middle pa o li e.
Howe e , i should be ou lined how his in e p e a ion can
change as soon as we change he scaling o he y-axis, mea-
su ing happiness. As shown in he uppe panel, he axes a e
di e en be ween bo h coun ies. In Po ugal, i anges om
7.5 o 6.0 bu in Ge many only om 7.4 o 7.1. I we look a
he global maximum and minimum o he dis ibu ion in he
g aphs (compa e also Table 2 in he appendix), we see ha
hese alues a e 7.48 and 6.33 o Po ugal ( ange 1.15) bu
7.38 and 7.20 o Ge many ( ange 0.18). When we o ce he
same scaling in he lowe panel o Figu e 1, ou conclusions
a e a he di e en . Po ugal displays he wide ange o hap-
piness and emains unchanged. Howe e , i he same scaling
is o ced on Ge many, he dis inc U-shape has mos ly an-
ished. I looking only a his new g aph, i becomes a he
difficul o see he U-shape as i is a he la . Wha a e he
implica ions o his? While i has been well-known ha one
can c ea e misleading g aphs when changing scales in g aphs,
he e is no clea -cu ecommenda ion. In his example, bo h
scalings ha e bene i s and disad an ages.
We sugges : when compa ing di e en coun ies o
each o he , using he same scaling appea s help ul o enable
a ai compa ison on he same scale. I only a single coun-
y is o in e es , a lexible scaling can help e eal nuances
and de ails. In any case, esea che s should hink ca e ully
abou his issue and handling i anspa en ly, ou lining why
hey ha e chosen a ce ain scaling and how he in e p e a ion
o conclusions changes when he scaling is changed. The
p oblem o scaling also ela es o he nex sec ions whe e
nume ical app oaches a e discussed. When conside ing he
iden ical scaling, one could conclude ha he o e all ange
he agg ega ed happiness alls in o is much la ge in Po ugal
han in Ge many. In o he wo ds, he a ia ion o e he a -
e age li e-cou se is much la ge . This can also be quan i ied
nume ically, as we ha e al eady done abo e. In he appendix
(Table 2), we epo he ange o agg ega ed happiness o all
coun ies in he ESS. To assess whe he hese wi hin-coun y
a ia ions a e a he la ge o no , he e a e some hin s in he
li e a u e. Jebb e al. (2020) a gue: “Fo ou Can il ladde
scale, esponden s epo ed (and p obably hough ) in e ms
o he nea es whole scale poin om 1 o 10. The e o e,
i seemed ha di e ences below 1.00 should be conside ed
qui e small.” (p.296). Gi en his e e ence, he a ia ion is
small in Ge many bu no so in Po ugal. Howe e , o ela e
o wha we ha e w i en in sec ion 2.2, we sugges a oiding
absolu e compa isons o judgmen s bu keep a ela i e one.
Wha can be said wi hou a doub is ha he a ia ion is much
la ge in Po ugal han in Ge many. As discussed in mo e de-
ail below, his sugges s ha age ma e s mo e in his coun y
o explain happiness han in he o he .
4.2. Nume ical analyses: absolu e and
changing R²
As ou lined in sec ion 3.2, we ha e p oposed some ele an
s a is ics o judge nume ically how ele an he in luence
o age is on happiness. We show ou esul s o all coun-
ies in he ESS in Figu e 2 o a con enien o e iew. Two
main s a is ics a e p esen ed and discussed; o a comple e
o e iew, e e o Table 2 in he appendix. Fi s , we would
like o alk abou p ac ical signi icance and he o e all in-
luence o age on happiness. No e ha his is no abou he
unc ional o m be ween happiness and age bu only on he
ele ance o age o happiness in a gi en coun y. To achie e
his we concen a e on he absolu e R² explained by he cubic
model.2The la ge his numbe , he be e age can explain
he a ia ion in happiness ha is due o he in luence o age.
No e ha he ESS ound is included as a con ol a iable.
Howe e , his speci ic in luence has been sub ac ed om he
esul , so he epo ed R² alues a e he pu e in luence o age
and i s de i ed e ms (ne in luences).3Resul s a e p esen ed
in Figu e 2 (lowe panel). Wha we see is ha he highes al-
ues a e ound in Bulga ia, Li huania, and Uk aine. In hese
coun ies, mo e han 5% o he o al a ia ion in happiness
is explained by age. This is e y di e en in o he coun ies,
2This model con ains age, age² and age³ as independen a iables. How-
e e , no e ha his decision is somewha a bi a y and e en mo e complex
models migh be bene icial o o he da a o esea ch ques ions.
3This ne ing ou has been done o all nume ical analyses p esen ed in
his pape .
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Figu e 1: G aphical compa ison o Po ugal and Ge many using local polynomial smoo hing. Two di e en y-scales applied Sou ce: ESS1-
10. Pos -s a i ica ion weigh s applied. 95% CIs included.
such as he Ne he lands, Ge many, o Finland, he e no e en
1% o his a ia ion is explained by age. This also ela es
o wha has been discussed be o e in he g aphical app oach,
ha is he o e all a ia ion wi hin a coun y. The conclusion
is clea , age ma e s mo e o happiness in some coun ies
han in o he s. In he ESS se ing, no he absolu e alues
o R² can be in e p e ed bu mo e so he ela i e compa i-
son be ween coun ies. Fo example, as he alue is 1.1 in
he Uni ed Kingdom bu abou 7.7 in Bulga ia, one can con-
clude ha age ma e s abou se en imes mo e o happiness
in Bulga ia han in he UK, which ou lines a huge di e ence
be ween hese wo coun ies.
The nex ques ion we would like o answe is he unc-
ional o m be ween happiness and age. He e we u ilize he
change o R² as soon as he assump ion o a linea unc ional
o m is abandoned (by adding he squa ed and cubed e m o
age o he model). This is done as ollows: R² is compu ed
o he linea and cubic model, and bo h R² alues a e s o ed
sepa a ely o each coun y (again, ESS ound is included
as he sole con ol a iable o accoun o pe iod-e ec s o
o he unwan ed in luences, such as changes o su ey quali y
o me hodology o e ime). The pe cen age change om he
linea o he cubic model is compu ed, and he loga i hm is
aken. This is necessa y since, o a ew coun ies, he pe -
cen age change is huge as he i s alue is e y small and
close o ze o (hence, a minuscule change can s ill esul in
a la ge pe cen age change, which could dis o he esul s).
This s a is ic is ele an o check whe he a unc ional o m
is linea o no , ega dless o he absolu e in luence o age on
happiness. No e ha absolu e changes below 1 pe cen be-
come nega i e numbe s due o he loga i hm aken. Based on
his s a is ic (Figu e 2, uppe panel), he coun ies ha ha e
he leas p onounced linea shapes a e Ge many, Tu key, and
Belgium. The mos linea ones a e Hunga y, Uk aine, and
Li huania. When compa ing his in o ma ion in Figu e 2 o
he unc ional o ms in he appendix, we can see a good con-
g uence be ween g aphical and nume ical esul s when only
he unc ional o m is o in e es , ega dless o he o al in lu-
ence. No e ha his always is a ela i e compa ison, meaning
ha he unc ional o ms a e less-linea in some coun ies han
in o he s. We do no belie e ha an absolu e classi ica ion is
a ge -o ien ed. We would a gue ha his is a clea bene i
o he app oach we ha e jus in oduced. Ins ead o elying
on p- alues, which o en lead esea che s o bina y conclu-
sions, ou app oach is mo e nuanced and p o ides a con in-
uum o ine-g ained compa isons wi h an inhe en meaning.
This can be used o be ween-coun y compa isons (such as in
ou case) bu also o wi hin-coun y compa isons ( o exam-
ple, when da a om mul iple poin s in ime in a ailable o a
single coun y. A inal bene i o ou app oach is ha con ol
a iables can be included like in any o he eg ession model,
which means ha he s a is ical analysis pa is easy o handle
and well-known o mos esea che s.
Nex , we would like o ela e hese h ee s a is ics o
each o he : log. R² changes om he linea o he cubic model
(1), absolu e R² alues o he cubic model (2) and he ange
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Figu e 2: O e iew o key s a is ics by coun y Sou ce: ESS1-10.
o happiness o agg ega ed da a (3, his las s a is ic is e-
po ed in Table 2 in he appendix). To do his, Pea son’s
is compu ed o measu e he co ela ion be ween hese h ee
a iables; esul s a e epo ed in Table 1.
Figu e 2 al eady indica es ha coun ies wi h a high pe -
cen age change show a low o al explana o y powe o age,
and indeed, he e is a mode a e nega i e co ela ion (Pea -
son’s = -.559). Rega ding he ange o happiness and he
absolu e R² alue, we obse e a a he s ong co ela ion (.94).
We can conclude om his ha in coun ies wi h a la ge a ia-
ion o happiness o e agg ega ed li e-cou ses, age in gene al
explains mo e happiness han in coun ies wi h low a ia ion.
Wha we can do in a inal s ep is plo hese s a is ics agains
each o he o each coun y using a sca e plo . Bu i s , we
ha e also a emp ed o classi y each coun y in o linea and
non-linea shapes based on Figu e 4 and 5. Appa en ly, his
is a subjec i e classi ica ion ha is no w i en in s one. And
again, bina y classi ica ions as his one can hide a ious nu-
ances and a e, in gene al, no encou aged, as hey undo he
bene i s o he g aphical app oach. Howe e , o his demon-
s a ion we would like o use his app oach as i is qui e ele-
an o he ollowing sca e plo (Figu e 3). Coun ies classi-
ied as ha ing a linea ela ionship a e BG, CY, CZ, DK, EE,
ES, GR, HR, HU, IL, IS, IT, LT, LV, PL, PT, RS, RU, SI, SK,
UA. All o he s a e classi ied as ei he ha ing a non-linea , U-
shaped, o e en mo e complex unc ional o m. By doing so
we can explo a i ely check whe he any special pa e ns a e
p esen , which migh bene i conclusions o u he esea ch
s udies.
Fi s , no e ha he x-axis is loga i hmic due o he wide
ange o numbe s p esen ed he e. In con as o Figu e 2 ( op
panel), we ha e decided o plo he o iginal alues wi h a log-
scale ins ead o loga i hmising he alues i s o he sake o
a clea e p esen a ion. We see ha coun ies classi ied wi h
a linea shape ha e a e age alues be ween 1 and 10% im-
p o emen in R² om he linea o he cubic model. Fo he
non-linea coun ies, hese imp o emen s a e usually be ween
10 and 100%, wi h wo ou lie s wi h ex emely la ge alues.
While his migh seem imp essi e i s , i is a he i ial as he
base o he change is iny and e y close o ze o. All coun ies
classi ied as non-linea ha e absolu e R² alues be ween e y
close o ze o up o abou 1.8%. They all lie closely oge he a
he bo om o he igu e. Howe e , his is a he di e en o
he o he coun ies as hey ha e a wide ange, up o almos
8% o o al explana o y powe . Rega dless o he g aphical
classi ica ion, we belie e ha Figu e 3 has much bene i s as
i enables a e mo e nuanced in e p e a ion. As we see, he e
is a wide ange o alues and mos coun ies all be ween 1
and 100% imp o emen by swi ching om he linea o he
cubic eg ession model. This highligh s nea ly ha he ques-
ion o linea i y in he unc ional o m be ween age and hap-
piness is no so much bina y as a he a con inui y. While a
pu ely g aphical in e p e a ion as in Figu e 4 and 5 is bene i-
cial, Figu e 3 summa izes he in o ma ion o many coun ies.
Especially when compa ing coun ies, esea che s should be
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Table 1:Co ela ion ma ix (Pea son’s ) Sou ce: ESS1-10. N = 33. *** p < 0.001
Log. R2change R2(absolu e) Range o happiness
Log. R2change 1
R2(absolu e, cubic) -.559∗∗∗ 1
Range o happiness -.597∗∗∗ .941∗∗∗ 1
Figu e 3: Sca e plo o o al R² and change in R² om a linea o a cubic model by coun y Sou ce: ESS1-10. No e he loga i hmic scaling
o he x-axis. G aphical shape classi ica ion is subjec i e and based on Figu es 4 and 5 in he appendix. The highe a nume ical alue on he
x-axis, he less linea he unc ional o m be ween age and happiness is. The highe a nume ical alue on he y-axis, he mo e impo an age
is o explain happiness in gene al.
awa e o such g aphical o ms o in e p e a ion and conside
hem as al e na i es o amendmen s o nume ical app oaches.
5. Discussion
The main pu pose o he cu en s udy is o econcile g aph-
ical and nume ical app oaches when s udying he unc ional
o m be ween happiness and age. In he i s pa , we demon-
s a ed ha g aphical app oaches, such as local polynomial
smoo hing, a e highly bene icial o isualize unc ional o ms
lexibly. We sugges his is always he i s s ep in an analysis
since g aphs can quickly summa ize a la ge da ase . Howe e ,
we ha e also demons a ed ha eaching a conclusion h ough
a g aph can some imes be challenging, especially when di e -
en popula ions a e o be compa ed. Fu he mo e, hey ha e
he downside o p oducing a la ge numbe o g aphs and com-
pa ing many coun ies can be difficul . As explained abo e,
we sugges esea che s o s ay anspa en and explain and jus-
i y a ious decisions, such as he choice o he axes scaling.
Consequen ly, we ha e dedica ed he second pa o he
pape o sugges ing some s a is ics ha help cha ac e ize unc-
ional o ms. These measu es a e quickly compu ed in any
mode n s a is ical so wa e package, well-de ined, and easily
unde s ood by a b oad ange o esea che s. They do no de-
pend on sample size o p- alues, which s ill oo o en encou -
age bina y conclusions. Using he s a is ics oge he , one can
cha ac e ize whe he a unc ional o m is a he linea o no
and whe he he independen a iable in luences he ou come
in a way ha has any p ac ical signi icance. We wan o un-
de sco e ha his las poin has been neglec ed by esea ch
in he age-happiness deba e. Ou esul s indica e ha he o-
al amoun o a ia ion in happiness explained by age is ex-
emely low and, o many coun ies, e en below 1%. The e-
o e, he unc ional o m be ween age and happiness is wi h-
ou any p ac ical signi icance. In hese coun ies, he a ia-
ion o happiness due o a di e en age is so low ha no in e -
en ion can make any p ac ical di e ence, and he ques ion
o unc ional o ms becomes a he academic. This inding
is in line wi h ano he s udy ha es s he in luence o age on
happiness in Ge many, using high-quali y panel da a. E en in
a bi a ia e model wi hou u he con ols, age explains only
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