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## Multidimensionality and HigherOrder Factor Models, Part 2

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### Nicole Ponder. MKT 8543. Quantitative Marketing Seminar. April 19, 2005 ... More distinctions between formative and reflective...Figure 1, page 201 ... – PowerPoint PPT presentation

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Title: Multidimensionality and HigherOrder Factor Models, Part 2

1
Multidimensionality and Higher-Order Factor
Models, Part 2
MKT 8543 Quantitative Marketing Seminar
• April 19, 2005

Mississippi State University
Nicole Ponder
2
Measurement Scales versus Indices
• Measurement scale consists of effect
indicators whose values are caused by an
underlying construct (Bollen 1989)
• The reflective model (Bollen and Lennox 1991)

var(x1) ?2 var(?1) var(?1) The only thing
that the 4 indicators should have in common is
the latent construct!
?1
x1
x2
x3
x4
?1
?2
?3
?4
3
Measurement Scales versus Indices
• Index consists of indicators that, taken
together, cause the underlying construct
(Diamantopoulos and Winklhofer 2001)
• The formative model (Bollen and Lennox 1991)

?1 ?1 x1 ?2 x2 ?3 x3 ?4 x4
?1
Often, researchers mistakenly use this model to
run with SEMand problems occur!
x1
x2
x3
x4
?1
?2
?3
?4
4
Jarvis, Mackenzie, and Podsakoff (2004)
• Question of interest How do you choose between
using reflective indicators and using formative
• Simple way by using the definitional essay!
Can you craft an item that captures the entirety
of the overall definition? Or do your items
capture the different components/dimensions of
the overall concept?
• Other guidelines, p. 203
• More distinctions between formative and
reflectiveFigure 1, page 201
• Many constructs in marketing have been treated as
reflective when they really should be treated as
formative (p. 208-209)
• Many seminal articles! This could change
structural relationships and conclusions!

5
Jarvis, Mackenzie, and Podsakoff (2004)
• Now, what happens when we move from first-order
constructs to higher-order constructs?
• If the distinction between formative and
reflective indicators is not clear, it only gets
worse when moving to multidimensional constructs!
• Figure 2, page 205Four different ways of looking
at second-order factor structures

6
Bagozzi and Edwards (1998)
• In any empirical study, it is essential to be
specific as to the depth and dimensionality of
constructs and their measures if meaningful
results are to be obtained
• Interested in studying the different levels of
abstraction of a multidimensional construct
• Used SEM to test four different levels of
aggregation of the Work Aspect Preference Scale
(WAPS)
• Concluded that work values can only be studied at
its lowest levels of abstraction i.e., overall
work values cannot be properly analyzed

7
Levels of abstraction in the Work Aspect
Preference Scale (WAPS)
Work Values
Non-work Orientation
Human/ personal concern
Freedom
Money
Life style
Detachment
Creativity
Independence
Self-develop
Co-workers
Security
8
BEs Aggregation Method
• Total disaggregation model tests the
subcomponents of work values at their lowest
level of abstraction
• n437
• the ?x parameter estimates are all statistically
significant, the fit indices show strong model
fit, and the modification indices for ?x and ??
are all nonsignificant
• Thus, at this lowest level of abstraction, each
indicator can be seen as truly reflective of the
construct it is supposed to represent.

Det
Mon
Lif
Cre
Ind
Se Dev
Cow
Sec
1
2
3
4
5
6
7
8
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
9
BEs Aggregation Method
• Partial disaggregation model
• Use of parcels reduces number of parameters to
be estimated
• Overall fit does improve over the total
disaggregation model

Det
Mon
Lif
Cre
Ind
Se Dev
Cow
Sec
12
34
56
78
ab
cd
ef
gh
qr
st
ij
kl
mn
op
uv
wx
10
Quick notes on the use of parcelsfrom Little et
al. (2002)
• Pros
• Individual items are statistically less reliable
than aggregate scores
• Overall levels of specific and random error are
reduced
• Overall fit statistics provide evidence of a
better fit of the model to the data
• Parsimony!
• Cons
• What happens if the construct is
multidimensional? And esp. if the dimensions are
not related?
• Parcels can mask true problems that exist with
the measurement model
• Researchers could play with parcels to get the
best model fit

11
BEs Aggregation Method
• Partial aggregation model

Human/ personal concern
Nonwork orientation
Freedom
det
sde
mon
lif
cow
sec
cre
ind
?2 (17, n437) 150.52, p 0.00 CFI .75 BE
concluded that one must reject the partial
aggregation model based on the goodness of fit
indices.
12
Our Proposed Aggregation Method
• Creation of reflective combinations to use in the
partial aggregation model

Det
Mon
Lif
Cre
Ind
Se Dev
Cow
Sec
1
2
3
4
5
6
7
8
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
x
Reflective measures may be created for nonwork
orientation, freedom, and human/personal concern
as followsNWO1 1 5 aNWO2 2 6
bNWO3 3 7 cNWO4 4 8 d
FR1 e i FR2 f j FR3 g k FR4 h l
HPC1 m q u HPC2 n r v HPC3 o s
w HPC4 p t x
13
Our Method
• Full variance-covariance matrix provided by BE
• Used a FORTRAN-based multivariate normal data
generator to generate 1000 individual
observations that will reproduce the given matrix
• Able to replicate results that BE got for their
models
• Used SEM to re-analyze partial and total
aggregation models using our method

14
Re-analyzed Partial Aggregation Model
Freedom
Non-work Orientation
Human/personal Concern
NWO1
NWO2
NWO3
NWO4
FR1
FR2
FR3
FR4
HPC1
HPC2
HPC3
HPC4
?2 (51, n1000) 320.08, p 0.00 CFI 0.96
GFI 0.95 AGFI 0.93 Here, SMCs are high MIs
15
BEs Aggregation Method
• Total aggregation model

Work Values
Non-work Orientation
Human/personal Concern
Freedom
?2 (2, n437) 66.78, p 0.00 CFI .62 The
equal. BE concluded that the fit of the model
was poor therefore, one can only study work
values at its facet levels.
16
Re-analyzed Total Aggregation Model
• Can use reflective combinations in the total
aggregation model as well

Freedom
Non-work Orientation
Human/personal Concern
NWO1
NWO2
NWO3
NWO4
FR1
FR2
FR3
FR4
HPC1
HPC2
HPC3
HPC4
Reflective measures may be created for work
values as follows WV1 NWO1 FR1 HPC1WV2
NWO2 FR2 HPC2WV3 NWO3 FR3 HPC3WV4
NWO4 FR4 HPC4
17
Re-analyzed Total Aggregation Model
Work Values
WV1
WV2
WV3
WV4
?2 (5, n1000) 8.60, p 0.13 CFI .99 GFI
constrained to be equal. If measured properly,
work values can be studied at a global level.
18
• Much better fit, now measures are properly
reflective
• In order for reflective combinations to be proper
indicators, it is mandatory that the total
disaggregation model displays properties of
excellent model fit
• parameter estimates for ?x must be large and
statistically significant (0.70 or higher if phi
is standardized)
• SMCs for each indicator must be large (well above
0.50, preferably 0.70 or higher)
• modification indices for ?x must be statistically
non-significant (values lt 3.84)
• modification indices for ?? must be statistically
non-significant (values lt 3.84)

19
• Need clean results at the total disaggregation
level
• Take time to develop proper conceptual
definitions of constructs
• Pay attention to the assumption of reflective
measures
• How do you know which indicators to combine?
• To create reflective indicators of NWO, the
combination of 1, 5, and a is arbitrary
• Just ensure each dimension is represented!

NWO1 1 5 aNWO2 2 6 bNWO3 3 7
cNWO4 4 8 d
Money
Detach
Lifestyle
1
2
3
4
5
6
7
8
a
b
c
d
20
Conclusions
• Better alternative than HOF models
• Forces the researcher to place importance on the
development of conceptual definitions, and to
get it right at the total disaggregation level!
• Reflective combinations approach may be applied
to other multidimensional constructs
• Trust reliability, integrity, and confidence