MEM and SEM in the GME framework: Modelling Perception and Satisfaction - Maurizio Carpita, Enrico Ciavolino. December, 10 2013 Ies2013

1. SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and
Policy Interventions
MEM and SEM in the GME
framework: Modelling
Perception and
Satisfaction
Maurizio Carpita, University of Brescia
Enrico Ciavolino, University of Salento
Ies2013. Milan – December, 10 2013

2. Innovation and Society
Metodi statistici per la valutazione
Milano
-­‐
December
10,
2013
MEM
and
SEM
in
the
GME
framework:
Modelling
Percep9on
and
Sa9sfac9on
Maurizio
Carpita
DEM
–
University
of
Brescia
Enrico
Ciavolino
DSS
–
University
of
Salento
This
research
is
supported
by
Project
SYRTO
(SYstemic
Risk
TOmography:
Signals,
Measurements,
Transmission
Channels
and
Policy
Interven9ons;
syrtoproject.eu),
funded
by
the
European
Union
under
the
7th
Framework
Programme
(FP7-­‐SSH/2007-­‐2013),
Grant
Agreement
n.
320270

3. Objective and contents
• To
review
the
Measurement
Errors
Model
(MEM)
and
the
Structural
Equa;ons
Model
(SEM)
used
to
represent
rela9ons
between
subjec9ve
percep9ons
(as
job
sa9sfac9on)
in
the
framework
of
the
Generalized
Maximum
Entropy
(GME)
es;mator

4. Objective and contents
• To
review
the
Measurement
Errors
Model
(MEM)
and
the
Structural
Equa;ons
Model
(SEM)
used
to
represent
rela9ons
between
subjec9ve
percep9ons
(as
job
sa9sfac9on)
in
the
framework
of
the
Generalized
Maximum
Entropy
(GME)
es;mator
• The
talk
is
in
three
parts:
1.
Introducing
the
GME
es9mator
2.
The
MEM
with
one
composite
indicator
3.
The
SEM
with
many
Rasch
measures

5. 1.
Introducing
the
GME
es9mator

6. Introducing the GME estimator
• Consider
the
simple
linear
regression
model:
y = β·x + ε

7. Introducing the GME estimator
• Consider
the
simple
linear
regression
model:
y = β·x + ε
• Idea:
re-­‐parameterize
it
in
the
classical
Shannon’s
Maximum
Entropy
Framework
β = Σk zk
β pk
β
(expecta9on
of
the
r.v.
Zβ)
ε = Σh zh
ε ph
ε
(expecta9on
of
the
r.v.
Zε)

8. Introducing the GME estimator
• Consider
the
simple
linear
regression
model:
y = β·x + ε
• Idea:
re-­‐parameterize
it
in
the
classical
Shannon’s
Maximum
Entropy
Framework
β = Σk zk
β pk
β
(expecta9on
of
the
r.v.
Zβ)
ε = Σh zh
ε ph
ε
(expecta9on
of
the
r.v.
Zε)
• Problem:
es9mate
probabili9es
pβ and
pε in
presence
of
data
and
model
constraints

9. Introducing the GME estimator
• Solu;on:
using
a
sample
( yi , xi)
of
n
data,
maximize
the
Entropy
Func;on
H( pβ, pε) = - Σk pk
β log( pk
β) - Σhi phi
ε log( phi
ε)

10. Introducing the GME estimator
• Solu;on:
using
a
sample
( yi , xi)
of
n
data,
maximize
the
Entropy
Func;on
H( pβ, pε) = - Σk pk
β log( pk
β) - Σhi phi
ε log( phi
ε)
subject
to
the
system
of
restric;ons
1.
yi = (Σk zk
βpk
β)·xi + (Σh zh
εphi
ε) ∀i
2. pk
β ≥ 0 and phi
ε ≥ 0 ∀k, h, i
3.
Σk pk
β = 1 and Σh phi
ε = 1 ∀i

11. Introducing the GME estimator
• Advantages:
-­‐
No
distribu9onal
errors
assump9ons
are
required
-­‐
Robustness
for
a
general
class
of
error
distribu9ons
-­‐
Good
with
small
samples
and
ill-­‐posed
design
matrices
-­‐
Allows
to
use
inequality
constraints
on
parameters

12. Introducing the GME estimator
• Advantages:
-­‐
No
distribu9onal
errors
assump9ons
are
required
-­‐
Robustness
for
a
general
class
of
error
distribu9ons
-­‐
Good
with
small
samples
and
ill-­‐posed
design
matrices
-­‐
Allows
to
use
inequality
constraints
on
parameters
• Drawbacks:
-­‐
Cumbersome
for
models
with
many
pars/errs
-­‐
Not
very
suitable
for
“big
data”
problems

13. 2.
The
MEM
with
one
composite
indicator

14. the MEM with one composite indicator
• Consider
the
MEM
with
mul;ple
indicators:
y = η + ε = β·ξ + ε
xj = ξ + δj j = 1, 2,…, J
with
(η,ξ)
latent
vars.
and
structural
parameter
β

15. the MEM with one composite indicator
• Classical
solu;on:
use
the
(equal
weight)
composite
indicator
ξ ^ = Σj xj /J
to
compute
β ^
OLS = Cov(Y, ξ^ )/Var(ξ ^
)

16. the MEM with one composite indicator
• Classical
solu;on:
use
the
(equal
weight)
composite
indicator
ξ ^ = Σj xj /J
to
compute
β ^
OLS = Cov(Y, ξ^ )/Var(ξ ^
)
and
obtain
the
OLS
Adjusted
for
a`enua9on
β ^
OLSA = β ^
OLS /κ^
ξ
with
the
es9mate
of
the
reliability
index
X
J ⋅
r
X
+ − ⋅
=
1 ( 1)
J r
ˆξ κ

17. the MEM with one composite indicator
• GME
solu;on:
using
a
sample
( yi , xij)
of
n
data,
maximize
the
Entropy
Func;on
H( pβ, pδ, pε)
for
the
data-­‐model
yi = β·(ξ ^ i – δ i) + εi
= ∀i
= (Σk zk
β)·(ξ ^
βpk
i – Σh zh
δph i
δ) + (Σh zh
εphi
ε)
subject
to
the
related
system
of
restric;ons

18. the MEM with one composite indicator
• Choice
of
the
support
points:
-­‐
As
usual,
for
zk
β
we
use
(-100, -50, 0, 50, 100)

19. the MEM with one composite indicator
• Choice
of
the
support
points:
-­‐
As
usual,
for
zk
β
we
use
(-100, -50, 0, 50, 100)
-­‐
For
zh
δ
and
zh
ε
we
use
the
3σ
rule
with
Var(δ) = Var(ξ ^
)·(1 – κ^
ξ )
Var(ε) = Var( y)·(1 – ρ^
ξ y )
and
the
es9mated
adjusted
correla;on
ξ y = Cor(ξ ^
ρ^
, y)/(κ^
ξ )1/2

20. the MEM with one composite indicator
• Advantages:
-­‐
Consider
the
apriori
informa9on
on
δ
and
ε

21. the MEM with one composite indicator
• Advantages:
-­‐
Consider
the
apriori
informa9on
on
δ
and
ε
-­‐
Obtain
an
es9mate
of
the
error
terms
δ ^
GME = Σh zh
i
δ p^
δ
hi
i = 1, 2,..., n

22. the MEM with one composite indicator
• Advantages:
-­‐
Consider
the
apriori
informa9on
on
δ
and
ε
-­‐
Obtain
an
es9mate
of
the
error
terms
δ ^
GME = Σh zh
i
δ p^
δ
hi
i = 1, 2,..., n
and
therefore
an
es9mate
of
the
latent
variable
ξ^
i
GME = ξ ^
i – δ ^
i
GME
i = 1, 2,..., n

23. the MEM with one composite indicator
• Simula;on
scenario:
-­‐
Normal
distribu9ons
for
ξ, δj and ε
-­‐
Four
con9nuous
mul9ple
indicators
xj
-­‐
One
structural
parameter
β = 0.5
-­‐
Six
reliability
levels
κ ξ = 0.70 (0.05) 0.95
-­‐
Two
sample
sizes
n = 30, 60
-­‐
Average
results
with
2,000
random
replica9ons

24. the MEM with one composite indicator
• Results
for
the
case
n = 30:
0.65$
0.60$
0.55$
0.50$
0.45$
0.40$
0.35$
0.30$
0.25$
0.20$
Averages ± Standard Errors
0.65$ 0.70$ 0.75$ 0.80$ 0.85$ 0.90$ 0.95$
Reliability
OLS$ OLSA$ GME$
0.14#
0.12#
0.10#
0.08#
0.06#
0.04#
0.02#
0.00#
Root Mean Square Errors
0.65# 0.70# 0.75# 0.80# 0.85# 0.90# 0.95#
Reliability
OLSA# GME#
1.00$
0.95$
0.90$
0.85$
0.80$
0.75$
0.70$
Correlation with latent variable
0.65$ 0.70$ 0.75$ 0.80$ 0.85$ 0.90$ 0.95$
Reliability
Simple$Mean$ SM$with$GME$correc;on$

25. the MEM with one composite indicator
• Innova;on
example:
concerns
27
Countries
of
the
EU
from
the
Global
Innova9on
Index
2012
Report,
to
the
study
their
innova9on
level
Correlation matrix X1 X2 X3
Know. workers - X1 1
Innovat. linkages - X2 0.713 1
Know. absorption - X3 0.486 0.426 1
Output Index - Y 0.826 0.753 0.556
Mean corr. of Xs ( rX ) 0.542 Reliability (κˆξ ) 0.780
Regression results
R2 = 0.720 Estimate Std.Err. t Stat.
ˆβ
OLS 0.826 0.13 6.354
ˆβ
OLSA 1.073 0.193 5.560
ˆβ
GME 1.023 0.154 6.643
!
75
65
55
45
35
25
15
20 30 40 50 60 70 80
ˆξ
Y
0.130

26. the MEM with one composite indicator
• We
have
also
studied
the
case
of
the
MEM
with
discrete
mul;ple
indicators
• We
consider
the
case
of
the
Likert-­‐type
scale
in
the
case
of
parallel
measures
j = 1, 2,..., J

27. the MEM with one composite indicator
• Likert-­‐type
scale
with
parallel
measures
−4 −2 0 2 4
0.0 0.2 0.4
Standard Normal Variable
Probability density
1 2 3 4 5
Discrete Variable Optimal (O)
Probability mass
0.0 0.2 0.4
Probability density 1 2 3 4 5
−4 −2 0 2 4
0.0 0.2 0.4
Standard Normal Variable
Discrete Variable Right−Skewed (R)
Probability mass
0.0 0.2 0.4
−4 −2 0 2 4
0.0 0.2 0.4
Standard Normal Variable
Probability density
1 2 3 4 5
Discrete Variable Left−Skewed (L)
Probability mass
0.0 0.2 0.4

28. the MEM with one composite indicator
• Simula;on
results
1:

29. the MEM with one composite indicator
• Simula;on
results
2:

30. the MEM with one composite indicator
• McDonald
example:
Y
is
the
overall satisfaction
measured
on
a
10
points
scale,
the
composite
indicator
is
obtained
using
a
5
points
Likert-­‐type
scale
(1:
very
bad,
2:
bad,
3:
equal,
4:
good,
5:
very
good)
with
4
aspects:
X1 = Product variety
X2 = Food taste
X3 = Quality ingredients
X4 = Nutritional quality

31. the MEM with one composite indicator
• McDonald
example
(n = 100)

32. 3.
The
SEM
with
many
Rasch
measures

33. the SEM with many Rasch measures
• Consider
the
standard
linear
SEM:
η = Bη + Γξ + τ
y = ΛYη + ε
x = ΛXξ + δ

34. the SEM with many Rasch measures
• Consider
the
standard
linear
SEM:
η = Bη + Γξ + τ
y = ΛYη + ε
x = ΛXξ + δ
• The
GME
es9mator
use
the
re-­‐parameteriza9on
in
term
of
expecta9ons
of
the
matrices
B,
Γ,
Λ
and
the
errors
τ,
ε,
δ for
the
data-­‐model
yi = ΛY(I – B)– s[ΓΛX
– 1(xi – δi
) + τi
] + εi
∀i

35. the SEM with many Rasch measures
• The
ICSI-­‐SEM
example:
a
representa9on
of
the
subjec9ve
quality
of
work
in
the
Italian
social
coopera9ves
(ICSI2007
survey)
➸
9
composite
indicators
and
5
latent
variables

36. the SEM with many Rasch measures
• Two-­‐step
es;ma;on
approach:
1st
Step
-­‐
from
the
discrete
mul9ple
indicators
(Likert-­‐type
data)
construct
the
composite
indicators
with
the
Rasch
-­‐
Ra,ng
Scale
Model

37. the SEM with many Rasch measures
• Two-­‐step
es;ma;on
approach:
1st
Step
-­‐
from
the
discrete
mul9ple
indicators
(Likert-­‐type
data)
construct
the
composite
indicators
with
the
Rasch
-­‐
Ra,ng
Scale
Model
• 2nd
Step
-­‐
use
the
GME
es9mator
of
the
parameters
considering
for
the
errors
the
reliability
levels
of
the
composite
indicators

38. the SEM with many Rasch measures
• GME
measurement
parameters
and
errors:

39. the SEM with many Rasch measures
• GME
structural
parameters
and
errors:
• Correla;on
matrix
of
the
GME
es;mated
LVs:

40. Epilogue
• Simula9on
suggest
that
the
GME
es9mator
performs
as
well
as
the
OLSA
es9mator
with
rela9vely
small
samples
• The
two
step
approach
have
same
advantages
(reliability
versus
substan9ve
research)
• The
GME
allows
the
reconstruc9on
of
the
LVs
• Some
computa9onal
problems
with
big
datasets

41. Epilogue
• Simula9on
suggest
that
the
GME
es9mator
performs
as
well
as
the
OLSA
es9mator
with
rela9vely
small
samples
• The
two
step
approach
have
same
advantages
(reliability
versus
substan9ve
research)
• The
GME
allows
the
reconstruc9on
of
the
LVs
• Some
computa9onal
problems
with
big
datasets
Thank
you

42. This project has received funding from the European Union’s
Seventh Framework Programme for research, technological
development and demonstration under grant agreement n° 320270
www.syrtoproject.eu