Longi udinal da a analysis wi h SEM
Running Head: Longi udinal da a analysis wi h SEM
Longi udinal da a analysis wi h s uc u al equa ions
Jesús Rosel(1) and Ian Plewis(2)
(1) Depa amen o de Psicología E olu i a, E., S. y Me odología.
Uni e si a Jaume I, Cas ellón, Spain
(2) Cen e o Longi udinal S udies, Ins i u e o Educa ion, Uni e si y o London,
Uni ed Kingdom
D . Jesús Rosel
Depa amen o de Psicología E, E, S y Me odología
Uni e si a Jaume I
Apdo. 224,
12080 Cas ellón,
SPAIN.
Tel: 964 72 93 30.
Fax: 964 72 93 49.
E-mail: [email p o ec ed]
Sup imi : ¶
Longi udinal da a analysis wi h SEM
Longi udinal da a analysis wi h s uc u al equa ions
Abs ac
In his pape we e iew di e en s uc u al equa ion models o he analysis o
longi udinal da a: (a) uni a ia e models o obse able a iables, (b) mul i a ia e models
o obse able a iables, (c) models wi h la en a iables, (d) models ha a e
uncondi ioned o condi ioned o o he a iables (depending on he a iabili y o he
independen a iables: ime- a ying o ime-in a ian , and depending on he ype o
independen a iables: o la en a iables o o obse able a iables), (e) models wi h
in e ac ion o a iables, ( ) models wi h non-linea a iables, (g) models wi h a
cons an , (h) wi h single le el and mul ile el measu emen , and (i) o he ad ances in
SEM o longi udinal da a (la en g ow h cu e model, la en di e ence sco e, e c.).
We ha e paid mo e a en ion o he in e ac ion o a iables and o non-linea
ans o ma ions o a iables because hey a e no equen ly used in empi ical
in es iga ion. They do, howe e , o e in e es ing possibili ies o esea che s who wish
o e i y ela ions be ween he a iables hey ob ain. Po en ial applica ions a e
desc ibed, wi h hei ad an ages and disad an ages.
Sup imi : We
Sup imi : uncondi ioned
Sup imi : and
Sup imi : ¶
Longi udinal da a analysis wi h SEM 1
Longi udinal da a analysis wi h s uc u al equa ions
Since Jö eskog (1969), Keesling (1972) and Wiley (1973) i s de eloped he s a is ical
model o s uc u al equa ions, i has become one o he mos widely used echniques o
analysing longi udinal da a. So much is his he case ha : (a) in mos handbooks abou
longi udinal da a analysis one o se e al chap e s a e de o ed o exempli ying s uc u al
equa ion models (Amsel & Renninge , 1997; Bijle eld & an de Kamp, 1998; Collins
& Ho n, 1991; Collins & Saye , 2001; Dwye , Feinlieb & Ho meis e , 1992;
Fi zmau ice, Lai d & Wa e, 2004; F ees, 2004; Go man, 1995; Li le, Schnabel &
Baume , 2000; Plewis, 1985; Singe & Wille , 2003; on Eye & Clogg, 1994); (b)
cen es ha p o ide aining in longi udinal esea ch o e cou ses in s uc u al equa ion
modelling (SEM); and (c) he e is a s eady inc ease in he numbe o jou nal a icles in
which his me hodology is applied (Ca d & Li le, 2007). An example o he impo ance
o SEM applied o longi udinal s udies is e lec ed by he wo chap e s ha Jö eskog
(1974, 1977) published on his opic 30 yea s ago.
In his a icle we will b ie ly e iew he di e en models applied o he analysis o
longi udinal da a by means o SEM. We keep he s a is ical heo y down o a basic
le el, bu his wo k is also in ended o be a p ac ical guide o esea che s who ha e o
analyse longi udinal da a. Fo his eason, we la gely a oid dwelling on he basics o
SEM and i is he e o e assumed ha he eade has some p io knowledge abou his
subjec ma e .
One impo an poin o be aken in o accoun in any SEM model is ha he esea che ,
on conduc ing his o he esea ch, mus ensu e a p ope in eg a ion o he ollowing
aspec s: (a) a basic heo y wi h he co ec ly o mula ed hypo heses so as o be able o
check whe he he da a ma ch he heo y; (b) a co ec esea ch design, wi h a ho ough
s udy o he a iables o be measu ed, he ime be ween measu emen s, he numbe o
measu emen s, he age o ages o he sample, he ime he esea ch las ed, and so o h;
and (c) he s a is ical model o da a analysis ha is o be used as a me hod o con i ming
(o , should i be he case, ejec ing) he hypo heses ha ha e been posi ed (Collins,
2006; Emb e son, 2007; Li le, Bo ai d & Slege s, 2006; Li le, P eache , Selig & Ca d,
2007; Ram & G imm, 2007).
Be o e beginning any longi udinal s udy i is impo an o pu o wa d hypo heses abou
he s abili y o he obse able and la en a iables, as well as he ela ions be ween
hem, in o de o check whe he : (a) he a iances o he la en and obse able a iables
Sup imi : Un
Sup imi : impo an e aspec o a
ene en cuen a en cualquie
modelo SEM es que el
in es igado , cuando lle a a cabo
su in es igación, ha de ene bien
a iculados en e sí los siguien es
aspec os
Sup imi : una eo ía de base con
las co espondien es hipó esis
co ec amen e o muladas con el
in de comp oba si los da os se
co esponden con la eo ía,
Sup imi : un co ec o diseño de
in es igación
Sup imi : con las a iables a
medi , el iempo en e
mediciones, el núme o de
mediciones, la edad o edades de la
mues a, la du ación de la
in es igación, e c. bien
es udiadas,
Sup imi : el modelo es adís ico
de análisis de da os que si a de
mé odo de comp obación (o, en su
caso, de e u ación) de las
hipó esis plan eadas
Sup imi : An es de comenza
cualquie es udio longi udinal es
impo an e plan ea hipó esis
ace ca de la es abilidad de las
a iables obse ables y la en es
Sup imi : así como de las
elaciones en e las mismas
comp obando si
Sup imi : ¶
Longi udinal da a analysis wi h SEM 2
a e s able o di e o e ime; (b) he same is ue o he measu emen s o bo h he
obse able and la en a iables; (c) he loads, o coe icien s, be ween he la en
a iables and hei co esponding obse able a iables a e equal a each ime o
measu emen ; (d) he in e nal co ela ions be ween la en and/o obse able a iables
emain s able be ween di e en imes o measu emen ; and (e) he e ec s be ween
di e en imes o measu emen a e equal o changing (B own, 2006).
The di e en models we will be looking a a e as ollows: (a) uni a ia e and
mul i a ia e models; (b) obse able a iable models and la en a iable models; (c)
uncondi ioned e sus condi ioned o o he a iables; (d) wi h and wi hou in e ac ion
e ms; (e) wi h linea and non-linea e ec s; ( ) wi h and wi hou cons an s; (g) wi h
measu es a one and mo e han one le el; and (h) di e se ad ances in longi udinal SEM.
In ac , he e can be as many di e en models as he e a e combina ions among he
p eceding condi ions, which is wha makes SEM a highly e sa ile esea ch ool.
Uni a ia e models o obse able a iables
The heo e ical ounda ion o e ec models wi h obse able a iables lies in simple
eg ession (Pea son, 1896) and in pa h analysis (W igh , 1918, 1921). Pa h analyses a e
a u he de elopmen o eg ession, bu wi h he addi ion o in e media e a iables o
p edic he esponse a iable o in e es . SEM encompasses bo h hese obse able
a iable models and hose wi h la en a iables (o ac o s).
Uni a ia e epea ed measu es models a e hose in which he same pe sons a e measu ed
on a single a iable on se e al occasions. Thus, i he a iable V1 is measu ed 4 imes
(e.g. le us suppose we a e dealing wi h a g oup o child en o whom he a iable ‘le el
o knowledge o ma hema ics’ is measu ed a 6, 7, 8 and 9 yea s o age; in panel designs
his is ep esen ed by 4W1V, ha is o say, 4 ‘wa es’ o imes, and 1 a iable), hen he
ep esen a ion o he a iable a he ou imes would be: V1,1, V1,2, V1,3 and V1,4, whe e
he i s subsc ip indica es he a iable (in his example i is always he same: V1, ) and
he second subsc ip indica es he ime o measu emen ( he i s ime, a he age o 6, is
ep esen ed by V1,1; he second ime, a 7 yea s old, by V1,2, and so o h). The simples
model o ep esen ing he na u e o hese da a is shown in Figu e 1.
Inse Figu e 1 abou he e
In he g aphic ep esen a ion o he SEM we ha e ollowed Ben le ’s (1995) sys em o
no a ion, which uses ec angles o ep esen he obse ed a iables (V1,1, V1,2, V1,3 and
V1,4); he e ec s o some a iables on o he s a e ep esen ed by a ows (which show he
Sup imi : las a ianzas de las
a iables la en es y obse ables
son es ables o di ie en a lo la go
del iempo,
Sup imi : lo mismo espec o de
las medias, an o de las a iables
obse ables como de las la en es,
Sup imi : las ca gas, o
coe icien es, en e las a iables
la en es y sus co espondien es
a iables obse ables son iguales
en cada momen o de medición,
Sup imi : las co elaciones
in e nas en e a iables la en es
y/o obse ables se man ienen
es ables en e di e en es
momen os de medición,
Sup imi : los e ec os en e
di e en es momen os de medición
son equi alen es o cambian es
Sup imi : ,
Sup imi : ¶
Longi udinal da a analysis wi h SEM 3
di ec ion o each espec i e e ec ), wi h he co esponding size o he e ec (b1, b2 and
b3); and he a ows ha a e labelled wi h he le e “E” (E1,2, E1,3 and E1,4) a e he
esiduals.
The model in Figu e 1 is called a i s -o de au o eg essi e model (AR(1)), o
‘simplex’ o Ma ko model. The e m ‘au o eg essi e’ e e s o he ac ha each
alue o he a iable V1 a ime depends only on he alue o ha same a iable in he
p e ious measu emen -1 (V1, = (V1,( -1))). This model was pu o wa d by Gu man
(1954), bu has since been de eloped by o he s (Ande son, 1960; Heise, 1969;
Humph eys, 1960). I should be no ed ha he e is no E1,1 in he model because one o
he assump ions o he model is ha he independen (o explana o y) a iables ha e
been measu ed wi hou e o . This model can be ep esen ed algeb aically in a
compac o m:
y = Βy + Γx + ε, (1)
whe e y is he p
×
1-o de ec o o dependen (o esponse) a iables: y’ = [V1,2, V1,3,
V1,4], x is he q
×
1-o de ec o o independen a iables: x=[V1,1], ε is he ec o o
esiduals: ε’ =[E1,2, E1,3, E1,4], while he ela ions be ween he dependen a iables a e
ep esen ed by ma ix Β, o o de p
×
p, and a e as ollows:
Β =
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
00
00
000
3
2
b
b, (2)
Γ is a coe icien ma ix o o de p
×
q ha ela es he independen a iables wi h he
dependen ones, which in ou case will be: Γ’ = [b1, 0, 0].
In o de o es ima e he co esponding pa ame e s, ma ix Ψ = Co (ε), whe e Co is
he co a iance ope a o so ha Co (Y,Y) = Va (Y), and Ψ is a ma ix o o de p
×
p,
mus be added o he p e ious ma ices. In ou case, he esidual a iances (Co (E1,2,
E1,2), Co (E1,3, E1,3), Co (E1,4, E1,4)) will be ep esen ed on he main diagonals o his
ma ix. Ma ix Φ = Co (x), o o de q
×
q, which ep esen s he ma ix o he
co a iances o he independen a iables, would also ha e o be added o he model.
The model in Figu e 1 (which has 3 deg ees o eedom) can be e o mula ed so ha
each measu emen depends on he measu emen immedia ely be o e i bu also on he
measu emen pe o med a he las ime bu one. As a esul , he model in Figu e 2
could be p oposed. Sup imi : ¶

Longi udinal da a analysis wi h SEM 4
Inse Figu e 2 abou he e
The model in Figu e 2 (which would ha e 1 deg ee o eedom) is called second-o de
au o eg essi e (AR(2)) because each alue is a unc ion o he wo measu emen s
immedia ely p eceding i (V1, = (V1,( -1),V1,( -2))), and i he e we e e idence ha he
measu emen s o he di e en imes we e closely ela ed, a hi d-o de au o eg essi e
model could be con empla ed, whe e a new e ec (b6) would ha e o be added o he
model in Figu e 2. This e ec would go om V1,1 o V1,4, and in gene al o any AR(3)
model i would be: V1, = (V1,( -1),V1,( -2),V1,( -3)). To be able o o mula e a p-o de
au o eg essi e model, we need a leas p+1 measu emen s.
The AR(3) model abo e would ha e 0 deg ees o eedom, and he e o e i would no be
possible o es ima e coe icien s o o e all i o he model. Di e en hypo heses abou
he a iables can be es ed. The main pu pose o hese hypo heses (as hey a e shown in
Figu es 1 and 2) a e he equali y o he e ec pa ame e s, o he a iances and be ween
co a iances o he esiduals (B own, 2006):
(a) I i is assumed ha he e ec s among a iables emain cons an o e ime, hey can
be cons ained so ha coe icien s con inue o be equal; hus, in he AR(1) model in
Figu e 1, i could be hypo hesised ha b1 = b2 = b3 and, ollowing on wi h he same
logic, in he AR(2) model in Figu e 2, i could be conside ed ha b1 = b2 = b3, and also
b4 = b5. These assump ions make mo e sense i he ime span be ween a iables is he
same, and i i is assumed ha he p ocess does no change o e ime (which is mo e
likely o occu wi h adul s han wi h small child en).
(b) Since he same a iable V1, is measu ed on se e al occasions, he esidual a iances
could be assumed o be equal; hus, in he models in Figu es 1 and 2, Co (E1,2,
E1,2) = Co (E1,3, E1,3) = Co (E1,4, E1,4) can be included in he syn ax o he cons ain s
pa ag aph o he inpu o he s a is ical so wa e ha is being used (Co is he
co a iance ope a o so ha Co (Y,Y) = Va (Y)). This assump ion makes mo e sense
when he a iances o he a iables ha e oughly he same alue, which is mo e
equen in esea ch conduc ed wi h adul s. This is due o he ac ha when he same
a iable is examined in small child en, he a iance is usually seen o inc ease wi h age
and he p e ious cons ain will no be ul illed empi ically.
(c) Di e en assump ions can be made abou he esiduals. I migh be easonable o
suppose ha , because he same a iable is being measu ed se e al imes, he esiduals a
ime 2 (E1,2) will co a y wi h hose om ime 3 (E1,3), ha is o say, Co (E1,2,E1,3) will
Sup imi : hypo hesis
Sup imi : hypo hesis
Sup imi :
Sup imi : :
Sup imi : :
Sup imi : :
Sup imi : ¶
Longi udinal da a analysis wi h SEM 5
be le ee (i.e. ≠ 0), hose om ime 3 will co a y wi h hose om 4 (Co (E1,3,E1,4)),
which could also be le ee, and so o h, i he e we e mo e measu emen imes
(Co (E1,4, E1,5), …). These pa ame e s will he e o e be included in he co esponding
co a iance sec ion as being ee, since by de aul hey a e assumed o be equal o 0. I
his assump ion we e added o Figu e 1, he esul ing model would be like he one in
Figu e 3. F om Figu e 3 onwa ds, we will ollow he con en ion o ep esen ing
co a iances by means o cu ed lines wi h a owheads a bo h ends.
Inse Figu e 3 abou he e
(d) Bea ing in mind ha i is he same a iable, and ha he same co a iance p ocesses
occu ing be ween e o s can be epea ed be ween consecu i e measu emen s, he
ollowing cons ain can be added: Co (E1,2, E1,3) = Co (E1,3, E1,4) = e c.
(e) The esiduals a each occasion can be in e p e ed as ‘inno a ions’ o he a iable a
he ime o measu emen (Box, Jenkins and Reinsel, 1994), as hey a e a pa o he
same a iable ha is no explained by he p e ious a iable o a iables. Thus, a i s -
o de mo ing a e age model (MA(1)) can be pu o wa d, in which each alue o he
a iable a any gi en ime (V1, ) is a unc ion o he p e ious e o o he same pe son
(E1, -1), ha is V1, = (E1, -1), which esul s in he model in Figu e 4.
Inse Figu e 4 abou he e
In ac , he model in Figu e 4 is a i s -o de au o eg essi e and i s -o de mo ing
a e age model (ARMA(1,1)). I is possible o add he cons ain whe eby he e ec s o
he mo ing a e ages a e in a ian o e ime (as is assumed in Box-Jenkins ime se ies
models), by lea ing b4 = b5. The cons ain s men ioned abo e o he e ec s, a iances
and co a iances o he esiduals can also be applied o he model in Figu e 4.
The o egoing s a is ical hypo hesis mus be based on a heo e ical jus i ica ion o he
model, bu i used wisely hey help o iden i y he unde lying model ha gene a es he
da a. The models ha ha e been ou lined he e cons i u e he basis o SEM, and
unde s anding and being able o apply hem is an almos indispensable condi ion be o e
mo ing on o mo e complex models.
Mul i a ia e models o obse able a iables
Mul i a ia e models o obse able a iables, as we ha e al eady s a ed, a e an ex ension
o uni a ia e models. Fo example, le us suppose ha (as in Figu e 1) he same child en
a e measu ed a 6, 7, 8 and 9 yea s o age on he a iable ‘le el o knowledge o
Sup imi : ,
Sup imi : ,
Sup imi : ing
Sup imi : hypo esis
Sup imi : ¶
Longi udinal da a analysis wi h SEM 6
language’ in addi ion o hei ‘le el o knowledge o ma hema ics’: he design would
now be ep esen ed by 4W2V. As we ha e seen, he a iable ‘le el o knowledge o
ma hema ics’ was ep esen ed a he ou imes by V1,1, V1,2, V1,3 and V1,4 and now he
‘le el o knowledge o language’ is ep esen ed as V2,1, V2,2, V2,3 and V2,4. One
ad an age o measu ing wo a iables is ha i becomes possible o s udy he di ec ional
e ec o one a iable on he o he . Tha is o say, he esea che can o mula e wo
esea ch ques ions, namely, do he wo a iables de elop independen ly o each o he ,
o does he a iable ‘le el o knowledge o ma hema ics’ exe a g ea e in luence o e
ha conce ning ‘le el o knowledge o language’ o ice e sa? This gi es us he model
in Figu e 5.
Inse Figu e 5 abou he e
In Figu e 5 he co a iance be ween he a iables V1,1, V2,1 is ep esen ed by lines wi h
a owheads a bo h ends ha join he wo a iables; i he a iable ‘ma hs abili y’ (V1,1,
V1,2, V1,3 and V1,4) exe ed an in luence on ‘language abili y’ (V2,1, V2,2, V2,3 and V2,4),
hen he e ec s b8, b10 and b12 could be expec ed o be s a is ically signi ican ; i he
opposi e we e ue, hen he signi ican e ec s would be b9, b11 and b13. In each case he
esea che mus ha e some subs an i e hypo heses ha jus i y he speci ica ion; i only
a ew o he p e ious e ec s we e signi ican , he esea che would ha e o es ablish an
explana ion o hese indings and hei meaning.
The ma hema ical ep esen a ion o he model in Figu e 5 would ha e he same o m as
he one in Equa ion 1, bu he y ec o o he dependen a iables is now o o de 6
×
1:
y’ = (V1,2, V1,3, V1,4, V2,2, V2,3, V2,4), he x ec o is now o o de 2
×
1:
x’ = (V1,1, V2,1), while ε is he ec o o esiduals, which in ou case is o o de 6
×
1: ε’ =
(E1,2, E1,3, E1,4, E2,2, E2,3, E2,4), Β being o o de 6
×
6:
Β =
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
0000
0000
0000
0000
000000
000000
612
133
510
112
bb
bb
bb
bb ,
and Γ is a coe icien ma ix o o de 6
×
2 ha ela es he independen a iables wi h he
dependen ones:
Sup imi : ;
Sup imi :
(
,,, , VV VV 22413121
Sup imi : ¶
Longi udinal da a analysis wi h SEM 7
Γ’ = ⎟
⎠
⎞
⎜
⎝
⎛
0000
0000
49
81
bb
bb .
Ma ix Ψ is o o de 6
×
6, and in ou case he a iances o he esiduals (Co (E1,2, E1,2),
Co (E1,3, E1,3), …, Co (E2,4, E2,4)) will be ep esen ed on i s main diagonal; he main
diagonal o ma ix Φ = Co (x), o o de 2
×
2, will con ain he a iances o V1,1 and o
V2,1, while he o he wo elemen s o he ma ix con ain he co a iance be ween V1,1 and
V2,1.
Some imes he c ossed e ec s be ween a iables V1, and V2, a e con empo aneous
(ei he ecip ocal o unidi ec ional) a he han lagged; hus, i can be supposed ha in
he model in Figu e 5 he a iables ‘le el o knowledge o ma hema ics’ and ‘le el o
knowledge o language’ in luence each o he a he same ime. The model would
he e o e be non- ecu si e ( wo o mo e a iables exe an in luence on each o he ); a
easible ep esen a ion o such a model is shown in Figu e 6.
Inse Figu e 6 abou he e
I he a iable ‘le el o knowledge o ma hema ics’ (V1,1, V1,2, V1,3 and V1,4) in luenced
‘le el o knowledge o language’ (V2,1, V2,2, V2,3 and V2,4), hen only e ec s b9, b11 and
b13 in Figu e 6 would be signi ican . I , in con as , he a iable ‘le el o knowledge o
language’ we e he one ha exe ed an in luence on ‘le el o knowledge o
ma hema ics’, hen pa ame e s b8, b10 and b12 would be signi ican . I no c ossed e ec s
be ween he a iables (b8, b9, b10, …, b13) we e signi ican , hen his would indica e ha
he a iables V1, and V2, de elop independen ly, wi h jus he ini ial co a iance
(Co (V1,1, V2,1)) ep esen ing he p elimina y ela ion be ween he wo.
The model in Figu e 6 is non- ecu si e because he e a e one o se e al in e dependence
‘loops’ be ween a iables; no e ha he e is a ecip ocal in luence be ween V1,2 and V2,2,
be ween V1,3 and V2,3, as well as be ween V1,4 and V2,4. These models a e also called
in e dependence and simul aneous equa ion models and can lead o iden i ica ion
di icul ies (Ben le & Rayko , 2000; Bollen, 1989; Hayduk, 1996).
The cons ain s ou lined in he sec ion abou uni a ia e models o obse able a iables
o he e ec s, a iances and co a iances can be added o he models in Figu es 5 and 6,
bu in Figu es 5 and 6 he co a iances o con empo aneous measu emen s can be aken
as being ee: Co (E1,2, E2,2), Co (E1,3, E2,3), Co (E1,4, E2,4). In Figu es 5 and 6, his
could be ep esen ed by lines wi h a owheads a bo h ends ha join he espec i e
measu emen e o s. The cons ain Co (E1,2, E2,2) = Co (E1,3, E2,3) = Co (E1,4, E2,4)
Sup imi : by
Sup imi :
Sup imi : ¶
Longi udinal da a analysis wi h SEM 14
As in he model in Figu e 14, i we p opose ha X1 and X2 p esen a pe manen e ec
o e all he measu emen imes, he co esponding e ec s o X1 and X2 on V1,2, V1,3, V1,4,
V2,2, V2,3 and V2,4 can be added.
The wo ime-in a ian condi ioned models desc ibed he e, ha is, la en a iables (see
Figu e 14) and obse able a iables (see Figu e 15), can be gene alised and applied o
he models o la en a iables men ioned abo e, o example, om Figu e 8 o Figu e
13.
Wi h espec o he ime- a ying condi ional models, le us suppose he model in Figu e
12, in which he cogni i e capaci y o each child has been measu ed a each ime as he
numbe o co ec answe s in an in elligence es (X1, ). This exe s an in luence a each
espec i e measu emen ime and, h ough he espec i e ac o s, would a ec hei
capaci y o language and ma hema ics; he esul would he e o e be he model in
Figu e 16.
Inse Figu e 16 abou he e
No e ha in Figu e 16 we ha e included all he possible co a iances be ween he
independen a iables X1,1, X1,2, X1,3 and X1,4: Co (X1,1, X1,2), Co (X1,1, X1,3), …,
Co (X1,3, X1,4), assuming ha he e is a co ela ion be ween he child en’s sco es a each
ime o measu emen . I we assumed ha he independen a iable X1, was empo ally
dependen on i sel (ins ead o co a ia ion), hen we would ha e o emo e he
co a iances be ween X1,1, X1,2, X1,3 and X1,4 and pu in he e ec s ha co espond om
X1,1 o X1,2, om X1,2 o X1,3 and om X1,3 o X1,4.
Figu e 17 shows a de elopmen model condi ioned o ime- a ying ac o s. I we
assume ha he ollowing a e measu ed in he child en: he a iable ‘le el o knowledge
o ma hema ics’, ep esen ed a he ou imes by V1,1, V1,2, V1,3, and V1,4; he a iable
‘le el o knowledge o language’, ep esen ed as V2,1, V2,2, V2,3, and V2,4; and a gene al
in elligence es (X1, ) and Wechsle ’s block design es (X2, ) a e used as independen
a iables (o co a iables), and ha bo h in elligence es s o m a empo al in elligence
ac o (F5, F6, F7 and F8) and, u he mo e, ha he e is a ime dependence be ween he
ac o s, hen he model would be he one in Figu e 17.
Inse Figu e 17 abou he e
No e ha , s a is ically, he independen a iable o he sys em is ac o F5, which means
i has been measu ed wi h no e o ; i we we e o assume ha he empo al in elligence
ac o (F5, F6, F7 and F8) is co a ying o e ime (ins ead o ha ing a empo al
Sup imi :
Sup imi : ¶

Longi udinal da a analysis wi h SEM 15
dependence), hen we would ha e o emo e e ec s b16, b17 and b18, he e o s o
measu emen o he ac o s D6, D7 and D8, and add he co a iances among ac o s F5,
F6, F7 and F8: Co (F5, F6), Co (F5, F7), …, Co (F7, F8).
The same conside a ions as hose made in he sec ion abou ‘Uni a ia e models o
obse able a iables’ and ‘Mul i a ia e models o obse able a iables’ can be epea ed
he e wi h espec o he measu emen e o s o he a iables V1, and V2, in o de o gain
deg ees o eedom and acili a e he con e gence o he model es ima o s.
As can be seen, including subs an i e (‘ ime-in a ian ’ o ‘ ime- a ying’) independen
a iables in SEM adds new possibili ies o he es ima ion o e ec s be ween a iables,
and o he sea ch o dependence among hem.
Models wi h in e ac ion o a iables
When wo o mo e a iables a e being measu ed, one o hem, o a new dependen
a iable, can be hypo hesised as being a unc ion o he in e ac ion o wo (o mo e)
independen a iables (Aiken & Wes , 1991; Aiken, Wes & Pi s, 2003; Jacca d,
Tu isi & Wan, 2003). In e ac ion o a iables is unde s ood o mean he p oduc o
hem. Thus, i we p opose ha he in e ac ion be ween consuming alcohol and obacco
a ou s he appea ance o cance , hen we ha e o measu e he a e o alcohol and
obacco consump ion o each pe son, and we mus c ea e a hi d a iable ha will be
he p oduc o bo h o each pe son, while a he same ime, as a dependen a iable, i
would se e as an indica o o cance . In in e ac ion o a iables, an o e iding p inciple
is ha o ‘nes ing’: when an in e ac ion o a iables is used, we mus u ilise he
a iables and co esponding in e ac ions ha lie on a lowe le el. Thus, i he
in e ac ion o he a iables X, Y and Z we e used in a model, he a iables and
in e ac ions XYZ, XY, XZ, YZ, X, Y, and Z would ha e o be employed so ha he model
makes subs an i e sense and he o ecas s es ablished by he model can be pe o med
co ec ly.
The e a e h ee in e ac ion p ocedu es in SEM: in e ac ion be ween independen
obse able a iables, in e ac ion when he obse ed a iables a e pa o he same ac o
and in e ac ion be ween ac o s, wi h hei co esponding obse able a iables (Jacca d,
Tu isi & Wan, 1999; Kenny & Judd, 1984; Ping, 1995; Schumacke , 2002;
Schumacke & Ma coulides, 1998; Schumacke & Lomax, 2004).
Sup imi :
Sup imi : ¶
Sup imi : a e
Sup imi : ¶
Longi udinal da a analysis wi h SEM 16
Le us suppose ha in Figu e 5 he in e ac ion o V1, and V2, exe s an in luence on he
de elopmen o he a iables V1, and V2, . Thus, i a new a iable is gene a ed, i should
be he in e ac ion o V1, and V2, o each ime:
V1*2, = V1, *V2, , (1)
In his way we would ha e a new a iable o each measu emen ime and, bea ing in
mind ha i has an in luence on V1, and V2, , we would p opose he model in Figu e 18.
Inse Figu e 18 abou he e
No e ha in Figu e 18: (a) he in e ac ion be ween a iables V1*2,1, V1*2,2 and V1*2,3 ha e
been included in he model; (b) he co a iances be ween all he independen a iables o
he model ha e also been included; (c) he a iable V1,2 is a unc ion o V1,1, o V2,1 and
o he in e ac ion be ween bo h o hem (V1*2,1). The same hing happens wi h he
a iables V1,3 [V1,3 = (V1,2, V2,2, V1*2,2)] and V1,4 [V1,4 = (V1,3, V2,3, V1*2,3)], and he
a iable V2, depends on he in e ac ion o V1*2, -1, ha is o say: V2,2 = (V1,1, V2,1, V1*2,1),
V2,3 = (V1,2, V2,2, V1*2,2)] and V2,4 = (V1,3, V2,3, V1*2,3).
I can also be supposed ha he in e ac ion be ween a iables is ime dependen and, in
ha case, we would need o emo e Co (V1*2,1, V1*2,2, V1*2,3) and add bo h
V1*2,2= (V1*2,1) and V1*2,3= (V1*2,2).
Should he e be an in e ac ion be ween a iables wi hin he same ac o , so ha hey
a ec ano he ac o , we would ha e o c ea e a new ac o esul ing om he
in e ac ion be ween he wo o hem. Thus, i in Figu e 12 i is assumed ha he
in e ac ion o he a iables V1, and V2, wi hin each ac o F a ec s he ac o F +1, we
would ha e o ob ain he p oduc o he a iables om equa ion (6), including he
p oduc wi hin a new ac o , as shown in Figu e 19.
Inse Figu e 19 abou he e
We can hypo hesise ha F5, F6 and F7 a e ime dependen , and he e o e we would need
o emo e Co (F5, F6, F7) and add F6 = (F5) and F7 = (F6).
I is easy o apply he in e ac ion o a iables be ween ac o s and he e e ences ci ed
abo e can be consul ed o u he in o ma ion on his ma e .
In models o in e ac ion o a iables i is also possible o use he same p ocedu e as ha
p oposed in ela ion o he e o s in measu ing he a iables V1, and V2, in he p e ious
sec ions.
Sup imi : in
Sup imi :
Sup imi : ,
Sup imi : ¶
Longi udinal da a analysis wi h SEM 17
Models wi h non-linea a iables
S ic ly speaking, hese ough o be called models wi h ans o ma ion o non-linea
a iables; hus, i becomes possible o p opose a model in which a a iable (Y ) is a
unc ion o he alue o he same a iable a a p e ious ime (Y -1) and o he squa e o
he same a iable a he p e ious ime (Y2 -1). In o he wo ds Y = (Y -1,Y2 -1), which
indica es ha he ela ion be ween Y and Y -1 is quad a ic. This non-linea ela ion can be
o any o he ype (and no only quad a ic): cubic, quad a ic, loga i hmic, squa e oo ed,
and so o h, o o he mo e complex, exponen ial- ype unc ions: logis ic, Gompe z, and
so o h. E en in e ac ion models a e a special case o non-linea ans o ma ion
(E ezadi-Amoli & McDonald, 1983; Jacca d, Tu isi & Wan, 1999; Jö eskog & Yang,
1996; Kenny & Judd, 1984; McA dle & Nessel oade, 2003; Schumacke , 2002;
Schumacke & Ma coulides, 1998; Sebe & Wild, 1989).
I , in he model in Figu e 1, i we e assumed ha Y = (Y -1,Y2 -1), hen he g aphic
ep esen a ion would be ha shown in Figu e 20.
Inse Figu e 20 abou he e
In Figu e 20: (a) he squa es o he espec i e independen a iables ha e been included
in he model; (b) he co a iances be ween all he independen a iables in he model a e
also included, Co (V1,1, V21,1, V21,2, V21,3) meaning all he pai wise co a iances o he
independen a iables, ha is o say, Co (V1,1, V21,1, V21,2, V21,3): Co (V1,1,V21,1),
Co (V1,1,V21,2), …, Co (V21,2,V21,3); and (c) no e ha he a iable V1,2 is a unc ion o
V1,1, and o he squa e o he same a iable [V1,2= (V1,1,V21,1)]. The same hing happens
wi h he a iable V1,3, which is a unc ion o V1,2 and o he squa e o his same a iable
(V21,2) ( ha is: V1,3= (V1,2,V21,2)), and las ly: V1,4= (V1,3,V21,3).
I he quad a ic e ec s we e no signi ican , i would be necessa y o wi hd aw e ec s
b4, b5 and b6, oge he wi h hei co esponding independen a iables (V21,1, V21,2 and
V21,3). By so doing, he model would become he one shown in Figu e 1 again.
Ins ead o he quad a ic model be ween obse able a iables sugges ed he e, any o he
non-linea ans o ma ion o he a iables could also be hypo hesised.
The non-linea model o obse ed a iables can be gene alised o la en a iables and,
hus, a quad a ic model o he e ec o one ac o (F2 ) on ano he ac o (F +1) could
also be p oposed and, hence, F +1= (F , F2 ). The model in Figu e 9 would hen become
he one in Figu e 21.
Sup imi : :
Sup imi : ¶
Longi udinal da a analysis wi h SEM 18
Inse Figu e 21 abou he e
Each quad a ic ac o mus con ain i s espec i e squa ed a iables and he
co esponding in e ac ion o he a iables. I is s ongly ad isable o include o hogonal
coe icien s in he p oduc and he powe o he a iables in o de o a oid colinea i y
(Li le, Bo ai d & Widaman, 2006); his is also alid o in e ac ion be ween a iables,
as we ha e explained in he p e ious sec ion.
The co a iances among he pai wise independen a iables, Co (F1, F5, F6, F7) =
Co (F1, F5), Co (F1, F6), ..., Co (F6, F7), mus be added o Figu e 21.
Supposing he quad a ic ac o s we e no signi ican , hen he co esponding e ec s b12,
b13 and b14 would no be signi ican ei he . I he quad a ic e ec b13 we e signi ican
( ha is o say, om F6 o F3), bu one o he e ec s o F6 on i s indica o a iables we e
no signi ican , hen all he e ec s and hei co esponding indica o a iables would
ha e o be le (due o he p inciple o hie a chy, o he app op ia e o ecas o F3).
We can also assume ha F5, F6 and F7 a e ime dependen and, he e o e, we would
ha e o emo e Co (F5, F6, F7) and add F6 = (F5) and F7 = (F6).
When he ime in e als be ween successi e measu emen s a e di e en and i is also
assumed ha he passage o ime exe s a cons an e ec on he di e en measu es, he
ollowing s a egy can be used o include non-linea e ec s among he a iables: (i)
each measu ing uni is iden i ied by a cons an (e.g. a); (ii) each au o eg essi e
coe icien is made equal o he cons an a aised o he powe o he ime in e al;
addi ionally, (iii) he alue o a mus be less han uni y, so ha he mo e ime has
passed, he lowe he e ec o he passage o ime will be. Thus, le us suppose in
Figu e 21 ha one yea passes be ween he i s and he second measu emen , a yea and
a hal be ween he second and he hi d, and 10 mon hs be ween he hi d and he ou h
measu emen . In he cons ain s pa ag aph o he syn ax o an SEM s a is ical p og am
i could be indica ed (by educing he ime uni o mon hs) ha b1= a12, b2= a18, b3= a10,
and a<1.
The c i e ia o he a iances and co a iances o he e o s ha we e p oposed ea lie can
be applied.
Bo h he p ocedu es o in e ac ion be ween a iables and hose in ol ing hei non-
linea ans o ma ion enable he esea che o op imise he in o ma ion ha is a ailable.
This is because, al hough gene ally speaking da a collec ion ends o be he mos
Sup imi : In ac o F5 ( ac o o
F1 squa ed), he pa ame e s ha e
o be se in acco dance wi h he
ollowing c i e ion: b15=(b4)2;
b16=b4*b5; b17=(b5)2; b18=(b6)2;
b19=b6*b7; b20=(b7)2; b21=(b8)2;
b22=b8*b9; b23=(b9)2; ha is o
say, he a iable V1,1 squa ed
(V21,1) ha ecei es he e ec o
F5 mus ha e he squa e o he
coe icien o F1 on V1,1, he same
being ue o he ac o s F6 and F7
wi h espec o hei a iables.
Sup imi : El
Sup imi : The sys em o
es ima ing pa ame e s o models
ha include in e ac ions o
a iables o non-linea e ec s can
be implemen ed in wo
phasessis ema de es imación de
pa áme os pa a modelos que
incluyan in e acción de a iables
o e ec os no-lineales puede se
implemen ado en dos ases (Ping,
1995), and
Sup imi : e en possible
Sup imi : e incluso pueden
inclui se coe icien es o ogonales
en el p oduc o de pa áme os pa a
e i a la colinealidad
Sup imi : o e
Sup imi : pa a
Sup imi : inclui e ec os no-
lineales en e las
Sup imi : ¶
Longi udinal da a analysis wi h SEM 19
demanding pa , wi h he da a al eady a ailable o he esea che i is possible o es
mo e hypo heses conce ning he same a iables wi h hei co esponding
ans o ma ion.
Models wi h cons an
SEM ini ially ela e co a iances and a iances be ween he a iables. The model
di e en ia es (o cen es) he obse able and la en a iables wi h espec o he mean,
wi h he esul ing loss o in o ma ion abou he cons an s. Ne e heless, i is some imes
e y impo an o ob ain he alue o he cons an s o he model. In longi udinal designs,
his is he case when compa ing he esul s in e ms o an in e en ion in di e en
g oups o o de e mine he change o le el in a a iable.
As an example, le us suppose ha in he case o Figu e 1 we we e in e es ed in
e i ying he cons an a each measu emen ime; his would he e o e gi e us he model
in Figu e 22.
Inse Figu e 22 abou he e
In he ep esen a ion o he cons an s we ollowed he sys em pu o wa d by McA dle
and Eps ein (1987), in which he cons an s a e dis inguished by an a ow wi h a iangle
placed a he o igin wi h a numbe ‘1’ inside i . I mus be bo ne in mind ha he
a iable V1,1 now admi s o ecas e o (E1,1). The subs an i e in e p e a ion o he o he
elemen s and symbols in Figu e 22 is he same as ha o Figu e 1 (excep in he
cons an s ha ha e been added).
The model in Figu e 22 can easily be ex ended by including he ec o o cons an s in
equa ion (1):
y = αy + Βy + Γx + ε, (6)
whe e addi ionally: x = αx + Λxξ + δ.
In e p e a ion o he symbols is he same as ha explained in Equa ion 1. In ou case,
αy’ = [k1 k2 k3 k4], αx, Λx, ξ and δ a e null ma ices, and Ψ’ = [(Co (E1,1, E1,1), (Co (E1,2,
E1,2), Co (E1,3, E1,3), Co (E1,4, E1,4)]. Simila ly, he subs an i e in e p e a ion o he
o he elemen s and symbols in Figu e 22 is he same as ha o Figu e 1 (excep in he
cons an s ha ha e been added).
One impo an aspec o be aken in o accoun in Figu e 22 is ha , as in he o he
eg ession models, he alues o he cons an s do no necessa ily ha e o be hose o he
means (Rosel, A nau & Ja a, 1998; Rosel, Ja a & A nau, 2002). Ins ead, hey a e
Sup imi : )];
Sup imi : simila ly
Sup imi : ¶

Longi udinal da a analysis wi h SEM 20
condi ioned o he alue o he mean o he a iable ha hey ha e as a dependen
a iable, and also ha o he e ec s hey ecei e h ough he independen a iables. Fo
example, in Figu e 1, le us suppose ha he means o each a iable we e 1,1
V= 50, 1,2
V
= 60,
1,3
V= 70 and 1,4
V= 80 and ha he esul s o he coe icien s we e b1 = 0.60, b2
= 0.70 and b3 = 0.85; he equa ion o he a iable V1,1 would be V1,1 = k1 + E1,1, and i s
expec ed alues E(V1,1) = E(k1 + E1,1). As a esul , 1,1
V= k1, and he alue o k1 would
he e o e be he mean o V1,1, ha is, k1 = 50; o V1,2 i would be V1,2 = k2 + b1·V1,1 +
E1,2, and i s expec ed alues would be E(V1,2) = E(k2 + b1·V1,1 + E1,2), which esul s in
1,2
V = k2 + b1·1,1
V, k2 = 1,2
V- b1·1,1
V= 60 - 0.60·50 =30. Hence, he alue o k2 no longe
depends only on he mean o V1,2, bu also on he coe icien b1 and on he mean o V1,1.
Likewise, we would ha e k3 = 28, and k4 = 20.50, which p o es ha he alue o he
cons an s diminishes o e ime, despi e he ac ha he means o he a iables inc ease.
As can be seen wi h his simple example, he e does no necessa ily ha e o be a ela ion
be ween he means and he cons an s, and he e o e he magni udes o he cons an s
canno be aken di ec ly o compa e he means.
In conclusion, he alues o wha mos au ho s call ‘s uc u al means’ a e no hese
means a all bu ins ead cons an s om each espec i e o ecas equa ion. We
ecommend compa ing he means o he obse able a iables aking he co a iances
among a iables (wi h no e ec s among hem) as he e e ence model, which hus
allows he espec i e means o be compa ed di ec ly. On he o he hand, o he la en
a iables i is ad isable o use he p ocedu e p oposed by Li le, Slege s and Ca d
(2006), which cons ains he mean o he e ec s o he la en a iable on i s
co esponding obse able a iables o a alue o one.
Wi h single le el e sus mul ile el measu emen
Up o his poin he models ha e been de eloped unde he assump ion ha he
pa ame e s a e ixed. In o he wo ds, we ha e no conside ed he possibili y ha hese
pa ame e s ( o example: b1, in Figu e 1) migh a y ac oss uni s, pe haps om one
highe le el uni such as a geog aphical a ea o ano he . Suppose we ha e a esponse o
dependen a iable, V, measu ed on ou possibly unequally spaced occasions wi h any
pa e n o missing da a a e he i s occasion. Le us also suppose ha ou sample is
s uc u ed in such a way ha each case is loca ed in one and only one clus e (pupils (i)
wi hin schools (j), o example). Ou obse a ions a e V ij, (measu emen occasion) =
Sup imi : :
Sup imi : :
Sup imi : :
Sup imi : :
Sup imi : Como
Sup imi : conclusión
Sup imi : los alo es de lo que
la mayo ía de au o es denomina
‘medias es uc u ales’, no son
ales medias, sino que son
cons an es de cada espec i a
ecuación de p onós ico
Sup imi : Nues a
ecomendación es compa a las
medias de las a iables
obse ables es ableciendo como
modelo de e e encia las
co a ianzas en e las a iables
(sin ningún e ec o en e las
mismas),
Sup imi : así se compa an
di ec amen e las espec i as
medias;
Sup imi : mien as pa a las
medias de las a iables la en es es
con enien e usa el p ocedimien o
de
Sup imi : que cons iñe igual a
uno la media de los e ec os de la
a iable la en e sob e sus
co espondien es a iables
obse ables
Sup imi : ,
Sup imi : ¶
Longi udinal da a analysis wi h SEM 21
1, …, Tij (max. Tij =4); i (case) = 1, …, nj; j (clus e ) = 1, …, J and we will assume ha V
~ MVN (µ, Ω).
We can app oach he analysis o hese da a in a numbe o ways, depending on ou
unde lying esea ch ques ion. The i s me hod is o model V as a unc ion o one o
mo e ea lie o lagged measu es o V, ha is V -k, k > 0, along wi h o he explana o y
a iables. This app oach is pa icula ly use ul i we seek causal conclusions abou , o
example, he e ec o an in e en ion in he absence o andomisa ion (see, o example,
Plewis, 1985). We can ega d each o he h ee esponses V , > 1 as a mul i a ia e se
(V1 is assumed o be exogenous he e) and so ou model is:
ijijjij eVbbV 2121202 +
+
= (2.a)
ijijijjij eVbVbbV 3232131303
+
+
+
= (2.b)
ijijijijjij eVbVbVbbV 4343242141404
+
+
+
+
= (2.c)
3,2,1,
000 =
+
=kubb kjkjk (2.d)
The in e cep e ms (bk0j , k = 2, 3 ,4) a y andomly om clus e o clus e a ound an
o e all mean bk00 as shown in equa ion (2.d). The eg ession coe icien s bk1 ep esen
he ela ions be ween each esponse and he exogenous V1; bk2, k = 3, 4 link he las wo
esponses o hei alue on he second occasion and b43 links he inal esponse o i s
p e ious o lagged alue.
Equa ions (2.a) o (2.c) de ine a ecu si e model o he cases, equa ion (2.d) de ines he
model a he clus e le el and we assume ha he case esiduals a e unco ela ed wi h
he clus e esiduals. Maximum likelihood es ima ion is ela i ely s aigh o wa d, albei
i e a i e, and can be ca ied ou in mos s a is ical packages as well as wi h he mo e
specialised mul ile el packages such as MLwiN and HLM. P o iding any missing da a
a e ‘missing a andom’, in o he wo ds, any missing V ( > 1) depend only on V -k, k >
0, hen he algo i hms a e ully e icien .
The model gene a es wo co a iance ma ices, one o he cases and he o he o he
clus e s. The mul i a ia e model allows all he esiduals a he case le el o be co ela ed
as migh happen i he e is an omi ed explana o y a iable ha is ela ed o V ( > 1)
a e con olling o V -k, k > 0, in o he wo ds, o he change in V om occasion o
occasion. This pa o he model is essen ially he same as he model o ‘seemingly
un ela ed eg essions’ as in oduced in o he econome ics li e a u e by Zellne (1962).
The model can be simpli ied. Fo example, we migh cons ain he pa ame e s b31, b41
and b42 o be ze o so ha any esponse V depends only on i s immedia ely p eceding
Sup imi : ...,
Sup imi : ...,
Sup imi : ...,
Sup imi : i.e.
Sup imi : ,
Sup imi : exogeneous
Sup imi : exogeneous
Sup imi : a
Sup imi : specialis
Sup imi : ;
Sup imi : o
Sup imi : ¶
Longi udinal da a analysis wi h SEM 22
measu e as in Figu e 1 o a single le el model. In he same ein, we migh also
cons ain some o he elemen s o he co a iance ma ix a bo h le els o be ze o so ha
only he i s -o de au o eg essi e elemen s a e non-ze o. On he o he hand, i will
some imes be app op ia e o make he model mo e complica ed. In pa icula , we migh
wan o allow some o all o he eg ession coe icien s b o a y ac oss clus e s so ha
we eplace bpq by bpqj and he co a iance ma ix a he clus e le el is ex ended o
include bo h he ex a a iances bu also he co a iances be ween he in e cep s (bk0j)
and he slopes (bpqj). We migh also wan o in oduce explana o y a iables measu ed a
he clus e le el and hese migh include clus e a iables c ea ed om he explana o y
a iables a he case le el, such as he clus e mean.
Ra he han ela ing he esponse a one occasion o lagged measu es o he esponse, a
second app oach o analysing epea ed measu es da a comes om modelling he
e olu ion o he esponse o e ime (o age). In o he wo ds, we model V as a unc ion
o ime wi hin a g ow h cu e o uncondi ional amewo k. This can be pa icula ly
use ul in a de elopmen al o ageing con ex . Ou model is now:
∑
=
+= Q
q
ij
q
ijqij ij eaV
0
β
(3.a)
qjjqqij u+
=
0
β
β
(3.b)
qjqjq +
=
000
β
β
(3.c)
He e Q is he o de o he polynomial a age (o ime) a, de e mined by he da a bu wi h
he es ic ion ha Q << max Tj, and βq a e he model coe icien s ha show how he
esponse a ies wi h age. These ( andom) coe icien s can a y om case o case and
om clus e o clus e .
The equa ions (3.a) o (3.c) a e a h ee-le el model wi h a ia ion be ween occasions
wi hin cases (le el one), ep esen ed by 2
e
σ
, a ia ion be ween cases wi hin clus e s
(le el wo) ep esen ed by he andom e ec s u o he in e cep , slope e c., and
a ia ion be ween clus e s (le el h ee) ep esen ed by a second se o andom e ec s .
I can be es ima ed wi h he same algo i hm used o model 3, wi h equa ions (3.a) o
(3.c). In he s uc u al equa ions li e a u e, he andom e ec s u and a e some imes
ea ed as la en a iables ha a e de e mined by he obse ed esponses o e ime
(Me edi h & Tisak, 1984, 1990).
The mul ile el g ow h cu e model is a lexible ool ha can be adap ed and ex ended.
Thus, i can accommoda e he ac ha he e is o en a ia ion in he age o ime o
Sup imi : in
Sup imi : a
Sup imi : h ee
Sup imi : and
Sup imi : ¶
Longi udinal da a analysis wi h SEM 23
measu emen wi hin any one measu emen occasion. I is possible o allow he le el-one
esiduals o be au oco ela ed (Golds ein e al., 1994) and o allow he a iance a le el
one o change wi h age o ime. Explana o y a iables can be in oduced a each le el
( ime- a ying a iables a le el one, case-le el a iables a le el wo and clus e
a iables a le el h ee) and hese a iables can be used o explain a ia ion in any o he
g ow h pa ame e s βq.
The condi ional (o eg ession) and he uncondi ional (o g ow h cu e) app oaches can,
wi h su icien occasions o measu emen , be combined, as shown by Plewis (1996).
Then i is possible o condi ion on he measu e a he i s occasion ( he baseline
measu e), he eby making i possible o compa e g ow h pa ame e s ac oss g oups o
cases s a ing o a he same poin . O he ex ensions a e also possible – o example, o
bina y, uno de ed and o de ed ca ego ical esponses (Plewis e al., 2006), o
mul i a ia e esponses (Plewis, 2005), and o combina ions o hese.
O he ad ances in SEM o longi udinal da a
Al hough we do no in end o go in o g ea dep h on he ma e , o he ad ances in SEM
o longi udinal da a ha e been made in ecen yea s, some o he mos impo an being
he ollowing:
(a) La en g ow h cu e models (Bollen & Cu an, 2006; Duncan, Duncan & S ycke ,
2006; Me edi h & Tisak, 1984, 1990), which use la en a iables and cons an s. One o
he la en a iables would hus ep esen he s a ing le el and he o he la en a iable
would ep esen he a e o g ow h o he g oup, bu now adding indica o cons an s o
each la en a iable. New la en a iables ha ep esen he quad a ic, cubic o some
o he endency o he de elopmen cu e can also be added. In his model he espec i e
a iances o he la en a iables indica e he andom a iabili y o he indi iduals ha
make up he sample, and he e o e each la en a iable becomes a a iable wi h a
andom (mul ile el) coe icien . This la en cu e model is hus di ec ly ela ed o he
mul ile el model. Figu e 23 shows an example o a linea la en g ow h cu e model
wi h a andom cons an and a linea e ec . No e ha all he coe icien s om he la en
a iables owa ds he obse able a iables a e ixed e ec s and ha F1 ep esen s he
le el o he cons an ( ixing he alues: b1 = b2 = b3 = b4 = 1), while F2 is he linea
slope o he model ( ixing he alues: b5 =1, b6 = 2, b7 = 3).
Inse Figu e 23 abou he e
Sup imi : –
Sup imi : –
Sup imi : En
Sup imi : la Figu a
Sup imi : se mues a un
ejemplo de modelo de
Sup imi : con una cons an e
alea o ia y un e ec o lineal
Sup imi : Obsé ese
Sup imi : que odos los
coe icien es desde las a iables
la en es hacia las a iables
obse ables son e ec os ijos
( ixed e ec s) y que
Sup imi : ep esen a
Sup imi : el ni el de la
cons an e
Sup imi : mien as
Sup imi : es
Sup imi : la pendien e lineal del
modelo
Sup imi : ¶
Longi udinal da a analysis wi h SEM 30
S eye , R. (2005). Analyzing indi idual and a e age causal e ec s ia s uc u al
equa ion models. Me hodology, 1, 39-54.
S eye , R., Eid, M. & Schwenkmezge , P. (1997). Modeling ue in aindi idual change:
T ue change as a la en a iable. Me hods o Psychological Resea ch-Online, 2,
21-33.
S eye , R., K ambee , S. & Hannö e , W. (2004). Modeling La en T ai -Change. In: K.
Van Mon o , H. Oud & A. Sa o a (Eds.), Recen de elopmen s on s uc u al
equa ion modeling: heo y and applica ions. Ams e dam: Kluwe .
Tukey, J.W. (1977). Explo a o y da a analysis. Reading, MA: Addison-Wesley.
on Eye, A. & Clogg, C.C. (Eds.) (1994). La en a iables analysis: Applica ions o
de elopmen al esea ch. Thousand Oaks, CA: Sage.
Wiley, D.E. (1973). The iden i ica ion p oblem o s uc u al equa ion models wi h
unmeasu ed a iables. In: A.S. Goldbe ge & O.D. Duncan (Eds.), S uc u al
equa ion models in he social sciences. New Yo k: Semina
W igh , S. (1918). On he na u e o size ac o s. Gene ics, 3, 367-374.
W igh , S. (1921). Co ela ion and causa ion. Jou nal o Ag icul u al Resea ch, 10,
557-585.
Zellne , A. (1962). An e icien me hod o es ima ing seemingly un ela ed eg essions
and es s o agg ega ion bias. Jou nal o he Ame ican S a is ical Associa ion, 57,
348-368.
Sup imi : (www.mp -
online.de).
Sup imi : In
Sup imi :
Sup imi : ¶

Longi udinal da a analysis wi h SEM
1
Figu e 1. Fi s -o de au o eg essi e uni a ia e model (AR(1)), 4W1V.
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
Sup imi : 1
Longi udinal da a analysis wi h SEM
2
Figu e 2. Second-o de au o eg essi e uni a ia e model (AR(2)), 4W1V.
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
b4 b
5
Sup imi : 1
Longi udinal da a analysis wi h SEM
3
Figu e 3. Fi s -o de au o eg essi e model wi h co a iance be ween he measu emen
e o s (4W1V).
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
Co (E1,2, E1,3) Co (E1,3, E1,4)
Sup imi : 1
Longi udinal da a analysis wi h SEM
4
Figu e 4. Fi s -o de mo ing a e age and au o eg essi e model, ARMA(1,1),
design 4W1V.
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b3
b4 b
5
Sup imi : 1
Longi udinal da a analysis wi h SEM
5
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b
5 b
6
Co (V1,1,V2,1)
b8
b9
b10
b11
b12
b13
Figu e 5. AR(1) bi a ia e model wi h c ossed e ec s be ween he wo
a iables, co esponding o he design 4W2V.
Sup imi : 1

Longi udinal da a analysis wi h SEM
6
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b
5 b
6
b8 b
9 b
10 b
11 b
12 b
13
Co (V1,1, V2,1)
Figu e 6. Recu si e model wi h ecip ocal in luences be ween a iables
a he same momen o measu emen . Sup imi : medi ion
Sup imi : 1
Longi udinal da a analysis wi h SEM
7
Figu e 7. Simple ac o model in epea ed measu es.
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3
b3
E1,1
F1
b4
b1
b2
E1,4
Sup imi : 1
Longi udinal da a analysis wi h SEM
8
Figu e 8. Simple ac o model in epea ed measu es, wi h AR(1) e ec s.
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3
b3
E1,1
F1
b4
b1
b2
E1,4
b5 b6 b7
Sup imi : 1
Longi udinal da a analysis wi h SEM
9
Figu e 9. AR(1) uni a ia e model in he ac o .
V1,1 V1,2 V1,4
E1,2 E1,3 E1,4
E1,1
F1 F2 F3 F4
D2 D3 D4
b1 b
2 b
3
b4 b
5 b
6 b
7
V1,3
Sup imi : 1
Longi udinal da a analysis wi h SEM
16
Figu e 16. Time- a ying model condi ioned o he a iable X1, .
Co (X1,1, X1,2, X1,3, X1,4) indica es all he possible co a iances be ween he
a iables X1,1, X1,2, X1,3 and X1,4: Co (X1,1, X1,2), Co (X1,1, X1,3), … ,
Co (X1,3, X1,4).
V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b5 b6
F1 F2 F3 F4
D2 D3 D4
b1 b
2 b
3
b7 b
8 b
9 b
10 b
11
E1,1 E2,1 E1,2 E1,3 E1,4
D1
X1,1 X1,2 X1,3 X1,4
Co (X1,1, X1,2, X1,3, X1,4)
b12 b
13 b
14 b
15
Sup imi : 1

Longi udinal da a analysis wi h SEM
17
Figu e 17. Time- a ying model condi ioned o he in ellec ual capaci y ime
ac o , indica ed om F5 o F8.
V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b5 b6
F1 F2 F3 F4
D2 D
3 D
4
b1 b
2 b
3
b7 b
8 b
9 b
10 b
11
E1,1 E2,1 E1,2 E1,3 E1,4
D1 b12 b
13 b
14 b
15
F5 F6 F7 F8
b16 b
17 b
18
X1,1 X1,2 X1,3 X1,4 X2,1 X2,2 X2,3 X2,4
b19 b20 b21 b
22 b
23 b
24 b
25 b
26
D6 D
7 D
8
Ex2,2 EX2,3 EX2,4
EX1,1 EX2,1 EX1,2 EX1,3 EX1,4
Sup imi : 1
Longi udinal da a analysis wi h SEM
18
Figu e 18. Va iable in e ac ion model, wi h e ec s on he nex ime o measu emen .
Co (V1,1, V2,1, V1*2,1, V1*2,2, V1*2,3) indica es all he possible (pai wise) co a iances
be ween he a iables.
V1*2,2 V1*2,3
b9 b11 b13
b14
b15
b16
b17
b18
b19
Co (V
1,1
, V
2,1
, V
1*2,1
, V
1*2,2
, V
1*2,3
)
b10 b
12
V1,1 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b
5 b
6
b8
V1,2
V1*2,1
Sup imi : 1
Longi udinal da a analysis wi h SEM
19
V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b5 b6
F1 F2 F3 F4
D2 D3 D4
b1 b
2 b
3
b7 b
8 b
9 b
10 b
11
E1,1 E2,1 E1,2 E1,3 E1,4
Figu e 19. Obse ed a iable in e ac ion model wi hin he measu emen ime
ac o s. Double-headed a ows mus be added o he model o ep esen he
co a iances be ween he independen a iables: Co (F1, F5, F6, F7)= Co (F1,
F5), Co (F1, F6), …, Co (F6, F7)
F5 F6 F7
V1*2,1 V1*2,2 V1*2,3
b12 b
13 b
14
Co (F1, F5, F6, F7)
E
1
*
2,1
E
1*2,2 E1*2,3
b15 b
16 b
17
Sup imi : .
Sup imi : 1
Longi udinal da a analysis wi h SEM
20
Figu e 20. Lag 1 quad a ic e ec s model. Co (V1,1, V21,1, V21,2, V21,3) indica es
all he (pai wise) co a iances o he independen a iables: Co (V1,1,V21,1),
Co (V1,1,V21,2), …, Co (V21,2,V21,3).
Co (V1,1, V21,1, V21,2, V21,3)
b4 b
5 b
6
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
V21,1 V21,2 V21,3
Sup imi : 1
Longi udinal da a analysis wi h SEM
21
Figu e 21. Model o quad a ic e ec s o one ac o (F2 ) on ano he ac o
(F +1). Double-headed a ows mus be added o he model o ep esen he
co a iances be ween he independen a iables: Co (F1, F5, F6, F7)= Co (F1,
F5), Co (F1, F6), …, Co (F6, F7).
V1,1 V1,2 V1,3 V1,4 V2,1 V2,2 V2,3 V2,4
E2,2 E2,3 E2,4
b4 b5 b6
F1 F2 F3 F4
D2 D3 D4
b1 b
2 b
3
b7 b
8 b
9 b
10 b
11
E1,1 E2,1 E1,2 E1,3 E1,4
V2
1,1 V2
1,2 V2
1,3 V2
2,1 V2
2,2 V2
2,3
F5 F6 F7
b15 b17 b18 b
20 b21 b23
V1*2,1 V1*2,2 V1*2,3
E4E6
E1 E2 E3 E5
b12 b
13 b
14
b16 b19 b
22
Co (F1, F5, F6, F7)
E1*2,1 E
1*2,2 E
1*2,3
Sup imi : 1

Longi udinal da a analysis wi h SEM
22
Figu e 22. Uni a ia e model AR(1) wi h cons an s in he a iables.
V1,1 V1,2 V1,3 V1,4
E1,2 E1,3 E1,4
b1 b
2 b
3
E1,1
1 1 11
k1 k2 k3 k4
Sup imi : 1
Longi udinal da a analysis wi h SEM
23
Figu e 23. Linea la en g ow h cu e model, wi h a andom cons an and a andom
slope
F1 F2
b1 b2 b3
b4 b5
b6
b7
D1 D2
k1 k2
V1,1
E1,1
V1,2
E1,2
V1,3
E1,3
V1,4
E1,4
1 1
Co (D1,D2)
Sup imi : .
Sup imi : 1
Longi udinal da a analysis wi h SEM
24
Figu e 24. La en g ow h di e ence sco e model, wi h a cons an ini ial le el and an
addi i e coe icien .
k1
b11 b12 b13
b14
b2
V1
,
1 V1
,
2 V1
,
4
E1,2 E1,3 E1,4
E1,1
F
1
F2 F3 F4
b1 b
3
b4 b
5 b
6 b
7
V1
,
3
Δ
F
2
ΔF3 ΔF4
b8 b
9 b
10
F5 F6
1 1
k2
Co (D5,D6)
D5 D6
b15 b
16
b
17
Sup imi : andom
Sup imi : andom
Sup imi : 1