Hai , Joseph F.; Babin, Ba y J.; Ringle, Ch is ian M.; Sa s ed , Ma ko; Becke , Jan-
Michael
A icle — Published Ve sion
Co a iance-based s uc u al equa ion modeling (CB-SEM):
a Sma PLS 4 so wa e u o ial
Jou nal o Ma ke ing Analy ics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Hai , Joseph F.; Babin, Ba y J.; Ringle, Ch is ian M.; Sa s ed , Ma ko; Becke , Jan-
Michael (2025) : Co a iance-based s uc u al equa ion modeling (CB-SEM): a Sma PLS 4 so wa e
u o ial, Jou nal o Ma ke ing Analy ics, ISSN 2050-3326, Palg a e Macmillan, London, Vol. 13, Iss. 3,
pp. 709-724,
h ps://doi.o g/10.1057/s41270-025-00414-6
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SOFTWARE REVIEW
Co a iance‑based s uc u al equa ion modeling (CB‑SEM): aSma PLS
4 so wa e u o ial
JosephF.Hai 1 · Ba yJ.Babin2 · Ch is ianM.Ringle3,4 · Ma koSa s ed 5,6 · Jan‑MichaelBecke 7
Re ised: 7 May 2025 / Accep ed: 8 May 2025 / Published online: 7 June 2025
© The Au ho (s) 2025
Abs ac
Co a iance-based s uc u al equa ion modeling (CB-SEM) enables esea che s o es ima e models wi h hypo hesized cause-
e ec ela ionships be ween la en a iables (i.e., cons uc s), each o which is ope a ionalized by se e al i ems (i.e., indi-
ca o s). To conduc CB-SEM analyses, esea che s can ely on a ange o so wa e applica ions. Howe e , many o hese
applica ions equi e esea che s o engage in some imes complica ed and e o -p one p og amming asks. While IBM SPSS
AMOS p o ides a g aphical use in e ace(GUI), i does no ully mee he expec a ions o con empo a y so wa e. In o de
o add ess hese challenges, he s a is ical Sma PLS 4 so wa e has ecen ly in oduced a new CB-SEM module, which
imp o es he use expe ience h ough a mode n and in ui i e g aphical in e ace and comp ehensi e esul epo s. This
u o ial desc ibes he key CB-SEM analysis s eps (i.e., model se up, es ima ion, and esul s e alua ion) using he Sma PLS
so wa e.
Keywo ds CB-SEM· CFA· Con i ma o y ac o analysis· Co a iance-based s uc u al equa ion modeling· SEM·
Sma PLS
In oduc ion
S uc u al equa ion modeling (SEM) is a gene al mul i a i-
a e analysis amewo k ha allows esea che s o empi i-
cally es heo e ically es ablished models wi h ela ionships
be ween cons uc s, ypically ope a ionalized by mul iple
indica o s (Sa s ed e al. 2016). To es ima e s uc u al equa-
ion models, esea che s can d aw on a ious es ima o s,
which di e in e ms o how hey s a is ically app oxima e
cons uc s and model ela ionships (Cho e al. 2022). A gu-
ably he mos p ominen and widely de eloped es ima o is
co a iance-based SEM (CB-SEM) ia maximum likelihood
es ima ion (MLE; Jö eskog 1978, 1993). Nume ous a icles
p o ide in oduc ions o CB-SEM, documen i s widesp ead
impac on he managemen and ma ke ing li e a u e, and
o e guidance on bes p ac ices (e.g., Bagozzi and Yi 1988;
Rigdon 1998; Iacobucci 2010; Diaman opoulos and Rie -
le 2011; Hai e al. 2017; Baumga ne and Weij e s 2020;
Zyphu e al. 2023). Addi ionally, se e al ex books o e
a comp ehensi e unde s anding o how o use CB-SEM in
esea ch and p ac ice (e.g., Diaman opoulos and Siguaw
2000; Rayko and Ma coulides 2006; By ne 2016; Whi -
ake and Schumacke 2022; Kline 2023).
While he o igins o SEM go back o he ea ly wen i-
e h cen u y, i was only wi h he ad en o la e wen ie h
cen u y compu ing powe and he de elopmen o mul iple
comme cial so wa e applica ions, such as EQS, LISREL,
and Mplus, ha SEM p ocedu es became p ominen among
academic esea che s. These so wa e applica ions elied
p edominan ly on manual speci ica ion o he model in some
* Ch is ian M. Ringle
[email p o ec ed]
1 Cle e don Chai o Business, Mi chell College o Business,
Uni e si y o Sou h Alabama, Mobile, AL, USA
2 Chai o heDepa men o Ma ke ing, Analy ics,
andP o essional Sales (MAPS), Ole Miss Business School,
The Uni e si y o Mississippi, Ox o d, MS, USA
3 Ins i u e o Managemen andDecision Sciences, Hambu g
Uni e si y o Technology, Hambu g, Ge many
4 College o Business, Law andGo e nance, James Cook
Uni e si y, Towns ille, Aus alia
5 LMU Munich School o Managemen ,
Ludwig-Maximilians-Uni e si y Munich, Munich, Ge many
6 Facul y o Economics andBusiness Adminis a ion,
Babes-Bolyai Uni e si y, Cluj-Napoca, Romania
7 Depa men o Ma ke ing, BI No wegian Business School,
Oslo, No way
710 J.F.Hai e al.
o m o coded syn ax. Today, he la aan package o he R
s a is ical so wa e (R Co e Team 2022) o e s a ee and
open-sou ce al e na i e ha is widely applied by esea ch-
e s o pe o m CB-SEM analyses, bu s ill equi ing use s
o speci y he model using syn ax. An app oach o a oid
coding he model in a syn ax eme ged wi h he in oduc ion
o AMOS: Analysis o Momen S uc u es (A buckle 1989).
A ew yea s a e AMOS’s in oduc ion as a s andalone
op ion ha elied p edominan ly on a g aphical use in e -
ace(GUI), IBM acqui ed he SPSS so wa e and i became
a widely used SEM applica ion. Academics and p ac i ion-
e s app ecia ed he ease o using he AMOS poin -and-click
in e ace o build hei models and o isualize he esul s
(Hai e al. 2014).
IBM SPSS AMOS was de eloped mo e han 30yea s
ago, howe e , and has changed e y li le in e ms o he
GUI. The so wa e’s use in e ace and o e all usabili y
he e o e do no ully align wi h cu en s anda ds. As an
al e na i e, he s a is ical Sma PLS so wa e (h ps:// www.
sma pls. com; Ringle e al. 2024), which has been de el-
oped o SEM analyses using pa ial leas squa es (Wold
1982; Lohmölle 1989; Chin 1998), now includes a CB-
SEM module wi h a mode n GUI—among o he me hods
such as mul iple eg ession analysis, logis ic eg ession,
necessa y condi ion analysis, pa h analysis, and gene alized
s uc u ed componen analysis (GSCA; Hwang and Takane
2004). P io e iews o he Sma PLS so wa e emphasize
he so wa e’s in ui i e design and ease o use (e.g., Sa s ed
and Cheah 2019). Se e al ex books (e.g., Hai e al. 2022,
2024; Chua 2024), a icles (Ma hews e al. 2016; Sa s ed
e al. 2024b; Me kle 2025), and so wa e e iews (e.g.,
Memon e al. 2021; Cheah e al. 2024; Sa s ed e al. 2024b)
ha e highligh ed he po en ials o he Sma PLS so wa e
and p o ided guidance on i s use o a wide ange o s a is-
ical analyses. Howe e , he applica ion o he Sma PLS
so wa e in a CB-SEM con ex has no ye been documen ed.
In ligh o he abo e, his u o ial a icle demons a es
how o conduc a CB-SEM analysis using he Sma PLS 4
so wa e. To do so, we d aw on he case s udy used in Hai
e al. (2019; i.e., Chaps. 9 o 12), which anks among he
mos widely used ex book on mul i a ia e da a analysis in
he social sciences (e.g., Black and Babin 2019). Ou illus-
a ions aim a helping esea che s o eliably un CB-SEM
analyses, he eby acili a ing he use o he ull spec um o
SEM es ima o s in o de o sa egua d he esul s’ obus -
ness when using me hods wi h di e en assump ions (e.g.,
Sa s ed e al. 2024a).
In he ollowing, we i s desc ibe he case s udy and p in-
ciples o CB-SEM es ima ion, ollowed by a s ep-by-s ep
desc ip ion o model se up, es ima ion, and esul s e alua-
ion using Sma PLS4. This so wa e u o ial a icle con-
cludes wi h addi ional obse a ions and Sma PLS so wa e
ex ensions ha can be expec ed in he nea u u e.
Case s udy andmodel es ima ion
Ou illus a ions d aw on he employee e en ion model
(Fig.1) and he da a (N = 400) used in Hai e al. (2019).
The model has wo mainelemen s (Ande son and Ge bing
1988): S uc u al and measu emen models.
The s uc u al model de ines he dependence (single-
headed a ows) and co ela ion (double-headed a ows)
among cons uc s o in e es (la ge ci cles in Fig.1).
Resea che s ypicallydis inguish be ween endogenous and
exogenous cons uc s. Endogenous cons uc s a e depend-
en in ha hey a e being explained by o he cons uc s in
he model; unexplained a iance is cap u ed by e o e ms,
ep esen ed by he small ci cles in Fig.1. Exogenous con-
s uc s only explain o he cons uc s in he model and a e
hus independen .
The employee e en ion model’s objec i e is o unde -
s and and explain he e ec s ha o ganiza ional commi men
(OC) and job sa is ac ion (JS) ha e on employees’ s aying
in en ions (SI). In addi ion, he model conside s he wo k
en i onmen pe cep ions (EP) o employees and hei a i-
udes owa d hei co-wo ke s (AC) as an eceden s o OC
and JS (Fig.1). The di ec ed pa hs demons a e he hypo h-
esized ela ionships in he model and he double-headed
a ows depic he co ela ions be ween exogenous (i.e.,
independen ) cons uc s. Hai e al. (2019) allow he co a i-
ance be ween AC and EP o be eely es ima ed (i.e., uncon-
s ained), as indica ed by he double-headed a ow be ween
hese cons uc s, since he wo cons uc s a e bo h ela ed o
he en i onmen in which hey wo k. Lea ing his a ow ou
would cons ain he co a iance be ween he wo cons uc s
o ze o, which may esul in subs an ial di e ences in he
model i and could also in luence (change) o he pa ame e
es ima es o he ela ionships be ween he cons uc s.
The measu emen models speci y how indica o s o i ems
( ec angles in Fig.1) ep esen he cons uc s o in e es .
Mo e speci ically, he indica o s a e seen as mani es a-
ions o e lec ions whose a iance he unde lying cons uc
explains. Analogous o he s uc u al model, he small ci cles
in Fig.1 ep esen he indica o s’ e o e ms, cap u ing hei
unexplained a iance.1 A cons uc is usually ope a ional-
ized by se e al indica o s aimed a ensu ing he eliabili y
and alidi y o he cons uc s mee s es ablished guidelines.
Measu emen heo y speci ies which i ems a e associa ed
wi h a pa icula cons uc and he es ima es ei he con i m
o ejec he measu emen heo y. Resea ch a icles, espe-
cially hose ocusing on scale de elopmen (e.g., Relling
e al. 2016; Becke e al. 2024) and scale handbooks (e.g.,
o ma ke ing; Bea den e al. 2011; B une 2021) p o ide
esea che s wi h in o ma ion on how o ope a ionalize
1 No e ha he measu emen models can also conside po en ial co -
ela ions be ween e o e ms.
711Co a iance-based s uc u al equa ion modeling (CB-SEM): aSma PLS 4 so wa e u o ial
cons uc s. Each o he cons uc s in he employee e en ion
model is ope a ionalized wi hbe ween ou and i e indica-
o s. Table10.2 in, Chap e 10 Hai e al. (2019) shows he
scales used o he indica o s and he su ey ques ions.
To es ima e a model such as in Fig.1, CB-SEM com-
bines aspec s o bo h con i ma o y ac o analysis (CFA)
and mul iple eg ession analysis. By cons uc ing a se ies
o equa ions depic ing bo h he di ec ela ionships be ween
cons uc s and hei indi ec e ec s (i.e., h ough ano he
cons uc in he s uc u al model), SEM can simul aneously
analyze mul iple dependen ela ionships, enabling esea ch-
e s o es ablish complex heo e ical models ha assume a
ne wo k o in e dependencies among cons uc s. Mo e spe-
ci ically, CB-SEM simul aneously es ima es pa ame e s o
all equa ions in ol ed in a model by ying o ep oduce
he obse ed co a iance ma ix and, a he same ime, mee
he equi emen s o he heo e ically imposed cons ain s.
In a model wi h no heo e ically imposed cons ain s, he
p ocedu e would pe ec ly ep oduce he obse ed co a i-
ance ma ix. Such a model would be e e ed o as sa u-
a ed because i has no deg ees o eedom and is gene ally
no o heo e ical in e es . MLE is he mos widely applied
me hod o calcula ing CB-SEM esul s. While o he op ions
a e a ailable, including gene alized leas squa es, weigh ed
leas squa es, and a a ie y o dis ibu ion- ee es ima o s
(e.g., Boomsma and Hoogland 2001), MLE p o ides a el-
a i ely obus es ima ion app oach (Iacobucci 2009; Hai
e al. 2017).
The goal o MLE is o de e mine he pa ame e s o he
speci ied model (gi en some cons ain s) o ob ain he bes
model i . The model i ep esen s how well he da a (mo e
speci ically, he obse ed co a iance ma ix S) ma ches
he a-p io i heo e ical s uc u e (mo e speci ically, he
model-implied co a iance ma ix Σ). The esul s can guide
esea che s in es ing whe he he hypo hesized heo e ical
model wi h i s es ima ed ela ionships e lec s he obse ed
da a s uc u e. In addi ion, esea che s apply se e al addi-
ional c i e ia o e alua e and ensu e he quali y o he esul s
ob ained o he es ima ed model.
Case s udy illus a ion using heSma PLS 4
so wa e
Hai e al. (2019, Chap e 10) sugges ha a comp ehensi e
CB-SEM analysis comp ises o six s ages:
• S age 1—De ining indi idual cons uc s.
• S age 2—De eloping he o e all measu emen model.
• S age 3—Designing a s udy o p oduce empi ical esul s.
• S age 4—Assessing he measu emen model (CFA).
• S age 5—Speci ying he s uc u al model.
• S age 6—Assessing he s uc u al model (CB-SEM).
S ages 1 h ough 4 ocus on he cons uc ope a ion-
aliza ion on he g ounds o measu emen heo y and hei
Fig. 1 Employee e en ion model (Hai e al. 2019, Chap. 11)
712 J.F.Hai e al.
alida ion using CFAs as highligh ed by Ande son and Ge -
bing (1988); see also Baumga ne and Weij e s (2020) and
Hai e al. (2019, Chap e 10).
S ages 5 and 6 ocus on he es o la en cons uc s uc-
u e; ha is, he di ec ional ela ionships be ween he con-
s uc s as implied by s uc u al heo y. In p ac ically all
cases, his analysis is mo e cons ained han he i s because
he ypical assump ion when con igu ing a CFA is ha all
cons uc s a e ela ed o all o he cons uc s. Thus, he s uc-
u al heo y assessmen in S ages 5 and 6 is o med by speci-
ying di ec ional ela ionships be ween ele an heo e ical
cons uc s, and by adding cons ain s o he CFA o indica e
whe e ela ionships be ween ac o s should no exis .2
A e c ea ing Hai e al.’s (2019, Chap. 11) model using
Sma PLS, and es ima ing i by means o CB-SEM, we spe-
ci ically ocus on S ages 4 and 6, which deal wi h he assess-
men p ocedu es o he measu emen and s uc u al models.
The measu emen model assessmen essen ially applies he
c i e ia used in a CFA, which Hai e al. (2019), o exam-
ple, explain in hei Chap e 10. In S age 6, assessing he
s uc u al model, esea che s a e p ima ily in e es ed in he
model i . I he measu emen model demons a es adequa e
i and o he indica o s o cons uc eliabili y and alid-
i y, hen esea che s p oceed wi h he in e p e a ion o he
s uc u al model.
Model andda a
The Sma PLS 4 so wa e has he employee e en ion model
included as a sample p ojec ile. The p ojec includes he
model, as desc ibed abo e, and he o iginal da a om Hai
e al. (2019). The da a a e syn he ic, ep esen ing employee
esponses as collec ed by an es ablished ma ke ing esea ch
company. Speci ically, he da a comp ise N = 400 esponses
om HBAT Indus ies (HBAT), an in e na ional pape
p oduc manu ac u e , and mee all he assump ions o CB-
SEM, including no mally dis ibu ed da a and o some ex en
homoskedas ici y.
To impo he p ojec , open he Wo kspace iew and go
o Sample p ojec s. Na iga e o he CB-SEM/CFA sam-
ple p ojec s and ick he boxes nex o Co a iance-based
SEM (CB-SEM) and Con i ma o y ac o analysis (CFA)
(Fig.2).
Nex , he Example—Con i ma o y ac o analysis
(CFA) and Example—Co a iance-based SEM (CB-SEM)
appea in he Sma PLS Wo kspace. The sample p ojec s
include se e al models and da ase s. Fo illus a i e pu -
poses, we dele e all he models excep he Hai e al. MDA
ex book model in he Example—Con i ma o y ac o
analysis (CFA) and Example—Co a iance-based SEM
(CB-SEM) p ojec s. When we igh -click on his selec-
ion, a dialog wi h se e al op ions opens. Selec he Dele e
esou ce op ion and con i m he dele ion in he subsequen
dialog box. Simila ly, we dele e all da ase s excep he Hai
e al. (HBAT) [400]. Each Example—Con i ma o y ac o
analysis (CFA) and Example—Co a iance-based SEM
(CB-SEM) p ojec in he Wo kspace now only con ain one
model and one da ase , as shown in Fig.3.
Double-clicking on Hai e al. (HBAT) [400] opens he
Da a View, which shows he indica o s and hei desc ip-
i e s a is ics, such as he minimum and maximum alues,
he mean alues, skewness, ku osis, e c. In addi ion, he e
is an op ion o examine he co ela ion ma ix. Nex , o s a
wi h a CFA, double-click on he Hai e al. MDA ex book
model in he Example—Con i ma o y ac o analysis
(CFA) p ojec in o de o open he Modeling View, which
displays he CFA model as shown in Fig.4. The model
shows he cons uc s and hei indica o s. One loading es i-
ma e pe measu emen model is cons ained o 1 (i.e., he
i s indica o pe cons uc in his example). Such a p o-
cedu e allows us o iden i y he scale o he cons uc . You
can se and change cons ain s by double-clicking on a ela-
ionship, which opens a dialog o his pu pose. Cons ain s
can also be se on cons uc s (o e o e ms) o cons ain
he pa ame e , which would be an al e na i e o iden i y he
scale o a cons uc . The model also displays he indica-
o s’ e o e ms. I easonable, you can also add co a i-
ances/co ela ions be ween he e o e ms by using he
Co ela ion op ion in he menu ba . They a e ep esen ed
Fig. 2 Impo ing sample p ojec s in o he Sma PLS so wa e
Fig. 3 P ojec s in he Sma PLS so wa e
2 This would be he case o ecu si e models. In he a e ins ance o
a non ecu si e model, he CFA may be mo e cons ained.
713Co a iance-based s uc u al equa ion modeling (CB-SEM): aSma PLS 4 so wa e u o ial
by double-headed a ows be ween he cons uc s.3 I is also
possible o cons ain hese co ela ions.4 Nex , we ollow
he s eps in he Sma PLS so wa e o ob ain his model’s
CFA esul s.
CFA model es ima ion
Be o e he model es ima ion, you need o sa egua d unidi-
mensional y (Hai e al. 2019, Chap. 10). Unidimensional
measu es mean ha a se o measu ed a iables (indica o s)
can be explained by only one unde lying cons uc . Unidi-
mensionali y becomes c i ically impo an when mo e han
wo cons uc s a e in ol ed. In such a si ua ion, each indica-
o is hypo hesized o ela e o only a single cons uc . All
c oss-loadings and e o a iance and co a iance a e hypo h-
esized o be ze o when unidimensional cons uc s exis .
Mo eo e , you need o ensu e he model is iden i ied as
explained by Hai e al. (2019, Chap. 10). Model iden i ica-
ion ensu es ha enough in o ma ion exis s o compu e a
solu ion o a se o s uc u al equa ions using CB-SEM. In
con as , an iden i ica ion p oblem (an uniden i ied model)
leads o an inabili y o he p oposed model o gene a e es i-
ma es as inding a solu ion is ma hema ically impossible.
The h ee possible condi ions o iden i ica ion a e o e iden-
i ied, jus -iden i ied, and unde iden i ied. The model used
in his example is iden i ied as demons a ed in de ail by
Hai e al. (2019, Chap. 10). The Sma PLS so wa e uses
a simple es o e alua e i he model is unde iden i ied and
issues a wa ning.
To es ima e he model by using he maximum likelihood
CB-SEM algo i hm, click on he Calcula e bu on in he
menu ba and selec Basic CB-SEM algo i hm. The Sma -
PLS so wa e opens a dialog box (Fig.5), which enables he
use o speci y se e al algo i hm se ings, such as he maxi-
mum numbe o i e a ions and he s op c i e ion (Table1).
We ecommend esea che s should keep he de aul se -
ings. Nex , in he Basic CB-SEM algo i hm s a dialog
box, ensu e ha he box nex o Open epo has been icked
and click on he S a calcula ion bu on. The Sma PLS
so wa e will hen es ima e he CFA model, and he Resul s
View will open.
The Resul s View ini ially shows he model and selec ed
pa ame e es ima es on he igh -hand side(Fig.6), while
di e en esul epo elemen s appea on he le -hand side.
Fig. 4 CFA o he employee e en ion model cons uc s (Hai e al. 2019, Chap. 10)
3 I you wan , you can also display he co ela ion be ween he con-
s uc s in a cu ed manne . To do his, le -click on a connec ion and
selec i . Then a poin appea s in he middle o he connec ion, which
you can click on wi h he le mouse bu on and mo e o change he
s aigh lines in o cu ed ones.
4 By de aul , i no co ela ion is d awn be ween e o e ms o exoge-
nous cons uc s he co ela ion is cons ained o ze o. D awing a co -
ela ion allows eely es ima ing he co ela ion un il i is cons ained
o a speci ic alue by he use by double-clicking on he co ela ion
and se ing a ixed alue. Co ela ions (as well as all o he pa am-
e e s) can also be cons ained o be equal by using he same s ing
(e.g., “a”) o all co ela ions ha should be equal.
714 J.F.Hai e al.
The box labeled G aphical ou pu on he lowe le side
enables you o choose di e en ypes o pa ame e es ima es
o be shown in he displayed model. Fo example, when
selec ing Pa h coe icien s (s anda dized) unde S uc-
u al model, and Weigh s/loadings (s anda dized) unde
Measu emen model, he s anda dized coe icien s’ esul s
a e shown on he model’s g aphical ou pu .
Mo ing o he esul s epo , we can, o example, selec
Fac o loadings—Ma ix (s anda dized) unde Final
esul s. When clicking on his menu i em, he Resul s View
appea s on he igh -hand side, showing he Fac o load-
ings—Ma ix (s anda dized) esul s. In his new ou pu ,
ac o loadings abo e 0.70 appea in g een, while loadings
below 0.70 a e ed—as we will discuss in he measu emen
model assessmen s age. Finally, we can sa e he esul s
epo in he menu ba o he Sma PLS so wa e (i.e., he
esul s epo appea s unde he p ojec in he Wo kspace
View), o in Excel o ma , and in HTML iles o use ou -
side o he so wa e.
Measu emen model assessmen
Following he sys ema ic p ocedu e o measu ing model
assessmen as ou lined by Hai e al. (2019, Chaps. 9
and 10), we ca y ou he ollowing analyses o e alua e
he model i as well as he eliabili y and alidi y o he
cons uc s:
• O e all i .
• Reliabili y and ac o loadings.
Fig. 5 Basic CB-SEM algo i hm s a dialog box
Table 1 CB-SEM algo i hms se ings in Sma PLS
Sou ce: Sma PLS webpage, h ps:// www. sma pls. com/ docum en a ion/ algo i hms- and- echn iques/ cbsem/, Accessed May 2025
Se ing Explana ion
Maximum i e a ions The maximum numbe o i e a ions he op imize will pe o m. This pa ame e should be high enough o ensu e
a good model solu ion. The de aul alue is 1,000, bu could be highe in mo e complex models
S a ing alue s a egy Apply con igu ed s a ing alues. By checking his op ion, he use can speci y i s own s a ing alues o he
ee model pa ame e s. I his op ion is no selec ed, he so wa e will use he de aul s a ing alues
De aul s a egy. This s a egy mimics La aan’s de aul s a ing alues. I uses Fabin-s yle es ima es o i s
loadings, 0.0 o pa h coe icien s and co a iances, a 0.5*indica o a iance o i s e o a iances, and 0.05 o
ac o e o a iances
One ze o s a egy. This s a egy applies mo e simple s a ing alues, 1.0 o loadings and a iances, and 0.0 o
pa h coe icien s and co a iances
S op c i e ion (g adien ) The op imize s ops when one o he wo s op c i e ia is ul illed, and con e gence o he op imum is assumed.
In his case, he op imize e mina es when ||g||< s op c i e ion * max(1, ||x||), whe e ||.|| deno es he Euclidean
(L2) no m. The de aul alue is 10^-6
S op c i e ion ( unc ion alue) The op imize s ops when one o he wo s op c i e ia is ul illed and con e gence o he op imum is assumed.
In his case, he op imize e mina es when he dec ease in he objec i e unc ion (maximum likelihood alue)
is smalle han he ecommended minimum. The condi ion is me i ( ′– )/ < s op c i e ion, whe e ′ is he
objec i e alue o he p e ious i e a ion and is he objec i e alue o he cu en i e a ion. The de aul alue
is 10^−9
Special assump ions Imply la en a iable co ela ions. Selec his op ion i you wan o eely es ima e he co ela ions be ween all
he exogenous cons uc s. Usually, i no co ela ion a ow is d awn in he model, he co ela ion be ween he
exogenous cons uc s is cons ained o ze o. Wi h his op ion, he co ela ions a e also es ima ed eely when
no a ow is d awn
Imply causal indica o co ela ions pe cons uc . Selec his op ion i you wan o es ima e he co ela ions
be ween all he causal indica o s o a cons uc . Usually, i no co ela ion a ow is d awn in he model, he
co ela ion be ween he causal indica o s is cons ained o ze o. Wi h his op ion, he co ela ions a e also
es ima ed eely when no a ow is d awn
Imply a a iance o 1.0 o causal indica o s. I we choose his op ion, all he a iances o causal indica o s a e
cons ained o 1.0. This also o e w i es use-speci ied alues. This op ion should help mimic he de aul La aan
esul s
715Co a iance-based s uc u al equa ion modeling (CB-SEM): aSma PLS 4 so wa e u o ial
• Validi y (i.e., con e gen alidi y, nomological alidi y,
disc iminan alidi y).5
The abo e me ics a e in e p e ed o con i m he measu e-
men models. When hese me ics mee es ablished guide-
lines, as ecommended by Hai e al. (2019), he measu e-
men models can be con i med.
The assessmen o he model’s i builds on a compa -
ison o he obse ed indica o co a iance ma ix (S) and
he model-implied co a iance ma ix (Σ) by means o a χ2
es . The smalle he di e ence be ween he wo co a iance
ma ices, he be e he model i . In “close- i ing” mod-
els, he χ2 would no di e signi ican ly om 0. Howe e ,
s a is ical powe and model complexi y se e o make ha
possibili y a e in la ge models wi h la ge samples. Bo h
concep ually and p ac ically, he model i is he mos c i ical
esul o es ing a heo e ical model in CB-SEM. Resea ch
has p oposed a se ies o di e en me ics ha quan i y he
deg ee o model i . Hai e al. (2019, Chap. 9) desc ibe hese
me ics in g ea e de ail and o e sugges ions o cu o al-
ues (i.e., hei Table9.4), depending on (1) he sample size
(N), and (2) he numbe o obse ed a iables in he model
(m). These me ics include:
• The χ2 alue and he associa ed deg ees o eedom (d ).
• Absolu e i indices, such as he goodness-o - i index
(GFI), he oo mean squa e e o o app oxima ion
(RMSEA), o he s anda dized oo mean esidual
(SRMR).
• Inc emen al i indices, such as he compa a i e i index
(CFI) o he Tucke -Lewis index (TLI).
• Badness-o - i indices (e.g., RMSEA, SRMR).
• The adjus ed heo e ical i index (ATFI) o e s a use ul
sc u iny o he heo e ical s uc u al and measu emen
models’ ela i e i .
No single “magic” i index alue sepa a es poo - i ing
and well- i ing models. Fu he , applying a single se o
cu -o ules o all measu emen models, and o ha ma -
e , o all models o any ype, is no easonable. The qual-
i y o he i depends s ongly on he model cha ac e is ics,
including he sample size and he model complexi y. The e-
o e, mul iple i indices should be used o assess a model’s
goodness-o - i . Resea che s may also apply lexible cu o
alues which conside he speci ic da a (sample size) and
model pa ame e s (Niemand and Mai 2018; McNeish and
Wol 2023).
To ob ain he model i esul s, e u n o he Sma PLS
Wo kspace and double-click on he Hai e al. MDA
Fig. 6 G aphical ou pu (displaying ac o co ela ions and s anda dized loadings)
5 No e: In addi ion, you need o ensu e ace alidi y, which is he
ex en o which he con en o he i ems is consis en wi h he con-
s uc de ini ion. Face alidi y is based solely on he esea che ’s
judgmen and was es ablished based on he con en o he co espond-
ing i ems (Hai e al. 2019, Chap. 10).
716 J.F.Hai e al.
ex book model in he Example—Con i ma o y ac o
analysis (CFA) p ojec . The Modeling View opens, which
enables you o click on he Calcula e bu on. Selec he
Basic CB-SEM algo i hm op ion o ob aining esul s o
he CFA model, S a calcula ion, and Open esul s. In he
esul s epo , unde Quali y c i e ia, he Sma PLS so -
wa e displays he Model i ou comes as shown in Fig.7.
No e ha he null model is hypo hesized o be he simples
model ha can be heo e ically jus i ied. I se es as he
baseline o compa ison s anda d used in inc emen al i
indices (i.e., ha ’s why o he indices show n/a as esul o
he null model in Fig.7). We ocus on he es ima ed model’s
esul s, which ep esen he ou comes o he heo e ically
es ablished and speci ied model shown in Fig.6.
Fo he o e all i assessmen (i.e., o es he disc ep-
ancy be ween he sample and he model-implied co a iance
ma ices), esea che s o en e e o he χ2 alue, which
ep esen s a badness o i measu e in he sense ha gene -
ally, a highe alue is associa ed wi h ela i ely wo se i
(depending on d ). Fo a close i , he s a is ical null hypo h-
esis ha he obse ed and model-implied co a iance ma i-
ces do no di e would be suppo ed. In ha case, he χ2
alue would no be s a is ically signi ican and, hus, no
indica e ha he di e ence be ween obse ed and model-
implied co a iance ma ix is di e en om ze o (e.g., Kline
2023; Chap. 10). Howe e , he χ2 es ypically displays s a-
is ical signi icance (i.e., indica ing a poo - i ing model)—
as in ou example (Fig.7).
Because o his inhe en limi a ion o he χ2- es ,
esea che s ypically epo o he i s a is ics, mos o
which a e ma hema ical a ia ions o he χ2 alue, null
model χ2 alue, d and sample size. The no med χ2, which
is he χ2 alue ela i e o he deg ee o eedom (d ), gi e
an al e na i e pic u e: Resea che s conside ha ChiSq /d
alue o 3 (in some cases, e en up o 5) o less ep esen s
a good model i (Dash and Paul 2021). In ou example,
he ChiSq /d alue is 1.327 (Fig.7), which suppo he
model i . Howe e , epo ing he no med χ2 alue wi hou
he ac ual χ2 alue and d is inapp op ia e because while
i is easy o know he no med alue om ac ual (and d ),
he e e se is no ue. Fo addi ional model i assessmen
c i e ia, Hai e al. (2019, Chap. 9) p o ide c i ical cu -o
alues. Fo he esul s shown in Fig.7, we ind ha he oo
mean squa e e o o app oxima ion (RMSEA) has a alue
o 0.029, wi h a 90 pe cen con idence in e al o 0.018
o 0.038. Al hough 0 is no in he con idence in e al, he
RMSEA alue is ela i ely low. The compa a i e i index
(CFI) and Tucke -Lewis index (TLI) ou comes o 0.986 and
0.984 a e abo e 0.94. The s anda dized oo mean squa e
esidual (SRMR) has a alue o 0.035, which is below 0.08.
Based on hese ou comes, we conclude ha he model has
ela i ely good i .
Nex , we ocus on he indica o eliabili y me ic. This
me ic is e alua ed by examining he s anda dized indica o
loadings. Hai e al. (2019, Chaps. 9 and 10) p o ide de ailed
explana ions and ules o humb. To iew he s anda dized
loadings esul s in he Sma PLS so wa e, click on Final
esul s → Ou e loadings → Ma ix (s anda dized). The
esul s in Fig.8 show mos o he indica o s’ s anda dized
loadings a e abo e he assumed minimum alue o 0.70.
Fou indica o s ha e loadings be ween 0.50 and 0.70, which
is below he ecommended guideline. While hese s anda d-
ized loadings a e lowe han desi ed, hey do con ibu e in
a meaning ul way and a e he e o e accep able in p inciple.
Thus, in line wi h Hai e al. (2019, Chap. 10), we e ain
hese indica o s in he model o suppo con en alidi y o
hese cons uc s.6
High indica o eliabili y usually esul s in high in e nal
consis ency eliabili y o he cons uc , which is assessed
based on he coe icien alpha (i.e., C onbach’s α). This
c i e ion emains widely used, e en hough esea che s
acknowledge ha his me ic p obably unde es ima es eli-
abili y. High alues in hese me ics indica e ha he i ems
consis en ly e lec he same unde lying cons uc . Resea ch-
e s usually expec ha C onbach’s α o be abo e 0.7 (Hai
Fig. 7 Model i esul s o he con i ma o y ac o analysis (CFA)
6 No e ha loadings wi h a s anda dized alue o 0.50 o highe a e
usually s a is ically signi ican . Consequen ly, we do no examine he
indi idual i ems’ s a is ical signi icance a his s age o he measu e-
men model assessmen . Howe e , in e es ed esea che s could check
whe he he loadings a e signi ican as a componen o he signi i-
cance es ing o he o e all s uc u al model ela ionships. This analy-
sis is a ailable in Lis (uns anda dized) and shows all indica o load-
ings a e signi ican .
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ju isdic ional claims in published maps and ins i u ional a ilia ions.
Joseph F. Hai is he Di ec o o he PhD p og am and he Cle e don
Chai o Business a he Uni e si y o Sou h Alabama. He has au ho ed
o e 100 books, including MKTG (Ma ke ing), Cengage Lea ning,
14 h edi ion 2024; Mul i a ia e Da a Analysis, Cengage Limi ed, U.K.,
8 h edi ion 2019 (ci ed 156,000+ imes and is in he op i e all ime
social sciences esea ch me hods ex books); Essen ials o Business
Resea ch Me hods, Rou ledge, 5 h edi ion 2024; Essen ials o Ma -
ke ing Resea ch, McG aw-Hill, 6 h edi ion 2024; A P ime on Pa -
ial Leas Squa es S uc u al Equa ion Modeling, Sage, 3 d edi ion
2022, and Ad anced Issues in Pa ial Leas Squa es S uc u al Equa-
ion Modeling, Sage, 2nd edi ion 2024. He also has published nume -
ous a icles in schola ly jou nals such as he Jou nal o Ma ke ing
Resea ch, Jou nal o Academy o Ma ke ing Science, O ganiza ional
Resea ch Me hods, Eu opean Jou nal o Ma ke ing, Jou nal o Ad e -
ising Resea ch, Jou nal o Business Resea ch, Jou nal o Long-Range
Planning, Indus ial Ma ke ing Managemen , Jou nal o Re ailing, and
o he s. His wo k has been ci ed mo e han 436,000 imes in academic
li e a u e and since 2018 he has been included in he Cla i a e Analy -
ics' Highly Resea che s lis . Mo e in o ma ion: h p:// www. sou h alaba
ma. edu/ colle ges/ mcob/ ma ke ing/ hai . h ml.
Ba y J. Babin Ph.D. (Louisiana S a e Uni e si y, 1991) is Phil B. Ha -
din P o esso o Ma ke ing and Chai o he Depa men o Ma ke ing,
Analy ics, and P o essional Sales (MAPS) a he Ole Miss Business
School. Ba y also se es as he Execu i e Di ec o o he Academy o
Ma ke ing Science® (www. ams- web. o g). He has au ho ed o e 200
p o essional publica ions wi h esea ch appea ing in he In e na ional
Jou nal o Wine Business Resea ch, Jou nal o he Academy o Ma -
ke ing Science, Jou nal o Ma ke ing, Jou nal o Re ailing, Jou nal
o Business Resea ch, Jou nal o Consume Resea ch, In e na ional
Jou nal o Resea ch in Ma ke ing, he Jou nal o Wine Resea ch, and
o he s. His esea ch emphasis a eas include wine ma ke ing, ma ke ing
analy ics, me a-analysis, sales managemen , and designing e ec i e
alue-deli e ing e ail cus ome expe iences. Ba y is pas p esiden
o he Academy o Ma ke ing Science® (AMS) and a p e ious ecipi-
en o he AMS Ha old W. Be kman Dis inguished Se ice Awa d. He
co-au ho ed se e al leading books including CB: A Consume Value
F amewo k, Mul i a ia e Da a Analysis, and Essen ials o Ma ke -
ing Resea ch. He is in e na ionally known as an expe in ma ke ing
esea ch and is a gues speake a uni e si ies and con e ences ac oss
he wo ld. He has di ec ed o e 20 Disse a ions/Theses and has se ed
in o al on mo e han 75 doc o al disse a ion o HDR commi ees.
Ch is ian M. Ringle is a Chai ed P o esso and he Di ec o o he Ins i-
u e o Managemen and Decision Sciences a he Hambu g Uni e si y
o Technology (Ge many), andan Adjunc P o esso a he James Cook
Uni e si y (Aus alia). His esea ch, which has been ci ed mo e han
300,000 imes (Google Schola ), ocuses on managemen and ma -
ke ing opics, me hod de elopmen , business analy ics, a i icial in el-
ligence,machine lea ning, and he applica ion o business esea ch
me hods o decision making. His con ibu ions ha e been published
in jou nals such as Indus ial Ma ke ing Managemen , In e na ional
Jou nal o Resea ch in Ma ke ing, In o ma ion Sys ems Resea ch,
Jou nal o Business Resea ch, Jou nal o Se ice Resea ch, Jou nal
o he Academy o Ma ke ing Science, Long Range Planning, and MIS
Qua e ly. Since 2018, Ch is ian has been included in he Cla i a e
Analy ics’ Highly Resea che s lis . He egula ly eaches doc o al
semina s on business analy ics and mul i a ia e s a is ics. Ch is ian
is a co- ounde and co-de elope o he s a is ical so wa e Sma PLS
(h ps:// www. sma pls. com). Mo e in o ma ion: h ps:// www. uhh. de/
mds/ eam/ p o - d -c- m- ingle.
Ma ko Sa s ed is a ull p o esso o ma ke ing a he Ludwig-Max-
imilians-Uni e si y Munich (Ge many) and an adjunc esea ch p o-
esso a Babes-Bolyai-Uni e si y Cluj-Napoca (Romania). His main
esea ch in e es is he ad ancemen o esea ch me hods o u he
he unde s anding o consume beha io . His esea ch has been pub-
lished in Na u e Human Beha iou , Jou nal o Ma ke ing Resea ch,
Jou nal o he Academy o Ma ke ing Science, Mul i a ia e Beha io-
al Resea ch, O ganiza ional Resea ch Me hods, MIS Qua e ly, and
Psychome ika, among o he s. His esea ch anks among he mos e-
quen ly ci ed in he social sciences wi h mo e han 200,000 ci a ions
acco ding o Google Schola . Ma ko has won nume ous bes pape and
ci a ion awa ds, including i e Eme ald Ci a ions o Excellence awa ds
and wo AMS William R. Da den Awa ds. Ma ko has been epea edly
named membe o Cla i a e Analy ics’ Highly Ci ed Resea che s Lis ,
which includes he “wo ld’s mos impac ul scien i ic esea che s.”
In Ma ch 2022, he was awa ded an hono a y doc o a e om Babes-
Bolyai-Uni e si y Cluj-Napoca o his esea ch achie emen s and con-
ibu ions o in e na ional exchange. Mo e in o ma ion: h ps:// link .
ee/ ma ko sa s ed
Jan‑Michael Becke is an associa e p o esso in he Depa men o Ma -
ke ing a he BI No wegian Business School (No way). His esea ch
in e es s and expe ise ocus on he digi al ans o ma ion o ma ke -
ing s a egy and consume beha io as well as ma ke ing analy ics,
beha io al esea ch me hods, causal in e ence, machine lea ning, and
compu a ional s a is ics. His esea ch has been published in se e al
p emie academic jou nals, including he Jou nal o he Academy o
Ma ke ing Science, In e na ional Jou nal o Resea ch in Ma ke ing,
In o ma ion Sys ems Resea ch, MIS Qua e ly, Psychome ika, Na u e
Human Beha io , Mul i a ia e Beha io al Resea ch, Jou nal o Busi-
ness Resea ch, and Ma ke ing Le e s. He is a co-de elope and co-
ounde o he s a is ical so wa e Sma PLS (h ps://www.sma pls.
com). Mo e in o ma ion: h ps:// www. bi. edu/ abou - bi/ emplo yees/ depa
men - o - ma ke ing/ jan- micha el- becke /.