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

MEM and SEM in the GME framework: Modelling Perception and Satisfaction

SYstemic Risk TOmography:

Signals, Measurements, Transmission Channels, and Policy Interventions

Maurizio Carpita, University of Brescia Enrico Ciavolino, University of Salento Ies2013. Milan – December, 10 2013

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

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

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

1. Introducing the GME es9mator

Introducing the GME estimator •  Consider the simple linear regression model:

y = β·x + ε


y = β·x + ε •  Idea: re-‐parameterize it in the classical Shannon’s Maximum Entropy Framework

β = Σk zkβ pk

β (expecta9on of the r.v. Zβ)

  ε = Σh zhε ph

ε (expecta9on of the r.v. Zε)


y = β·x + ε •  Idea: re-‐parameterize it in the classical Shannon’s Maximum Entropy Framework

β = Σk zkβ pk

β (expecta9on of the r.v. Zβ)

  ε = Σh zhε ph

ε (expecta9on of the r.v. Zε)

•  Problem: es9mate probabili9es pβ and pε in presence of data and model constraints

Introducing the GME estimator •  Solu;on: using a sample ( yi , xi) of n data,

maximize the Entropy Func;on

H( pβ, pε) = - Σk pkβ log( pk

β) - Σhi phiε log( phi

ε)

Introducing the GME estimator •  Solu;on: using a sample ( yi , xi) of n data,


H( pβ, pε) = - Σk pkβ log( pk

β) - Σhi phiε log( phi

ε)

subject to the system of restric;ons

1. yi = (Σk zkβpk

β)·xi + (Σh zhεphi

ε) ∀i

2. pkβ ≥ 0 and phi

ε ≥ 0 ∀k, h, i

3. Σk pkβ = 1 and Σh phi

ε = 1 ∀i

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

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

2. The MEM with one composite indicator

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 β structural parameter

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^ )

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 /κ̂ ξ   with the es9mate of the reliability index

X

X

rJrJ⋅−+

⋅=

)1(1ˆξκ

the MEM with one composite indicator •  GME solu;on: using a sample ( yi , xij) of n data,


H( pβ, pδ, pε) for the data-‐model

yi = β·(ξ^ i – δ i) + εi = ∀i

= (Σk zkβpk

β)·(ξ^ i – Σh zhδph i

δ) + (Σh zhεphi

ε)

subject to the related system of restric;ons

the MEM with one composite indicator •  Choice of the support points: -‐ As usual, for zk

β we use (-100, -50, 0, 50, 100)

the MEM with one composite indicator •  Choice of the support points: -‐ As usual, for zk

β we use (-100, -50, 0, 50, 100)

-‐ For zhδ and zh

ε we use the 3σ rule with

Var(δ) = Var(ξ^ )·(1 – κ̂ ξ )

Var(ε) = Var( y)·(1 – ρ̂ ξ y ) and the es9mated adjusted correla;on

ρ̂ ξ y = Cor(ξ^ , y)/(κ̂ ξ )1/2

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


-‐ Obtain an es9mate of the error terms

δ^ iGME = Σh zh

δ p̂ hi

δ i = 1, 2,..., n


-‐ Obtain an es9mate of the error terms

δ^ iGME = Σh zh

δ p̂ hi

δ i = 1, 2,..., n

and therefore an es9mate of the latent variable

ξ^ iGME = ξ^ i – δ^ i

GME i = 1, 2,..., n

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

the MEM with one composite indicator •  Results for the case n = 30:

0.20$0.25$0.30$0.35$0.40$0.45$0.50$0.55$0.60$0.65$

0.65$ 0.70$ 0.75$ 0.80$ 0.85$ 0.90$ 0.95$Reliability

Averages ± Standard Errors

OLS$ OLSA$ GME$

0.00#

0.02#

0.04#

0.06#

0.08#

0.10#

0.12#

0.14#

0.65# 0.70# 0.75# 0.80# 0.85# 0.90# 0.95#Reliability

Root Mean Square Errors

OLSA# GME#

0.70$

0.75$

0.80$

0.85$

0.90$

0.95$

1.00$

0.65$ 0.70$ 0.75$ 0.80$ 0.85$ 0.90$ 0.95$Reliability

Correlation with latent variable

Simple$Mean$ SM$with$GME$correc;on$

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 ( κ̂ξ ) 0.780

Regression results R2 = 0.720 Estimate Std.Err. t Stat.

β̂OLS 0.826 0.13 6.354 β̂OLSA 1.073 0.193 5.560 β̂GME 1.023 0.154 6.643 !

15

25

35

45

55

65

75

20 30 40 50 60 70 80

ξ̂

Y

0.130

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

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

Prob

abilit

y de

nsity

1 2 3 4 5

Discrete Variable Optimal (O)

Prob

abilit

y m

ass

0.0

0.2

0.4

−4 −2 0 2 4

0.0

0.2

0.4


Prob

abilit

y de

nsity

1 2 3 4 5

Discrete Variable Right−Skewed (R)

Prob

abilit

y m

ass

0.0

0.2

0.4

−4 −2 0 2 4

0.0

0.2

0.4


Prob

abilit

y de

nsity

1 2 3 4 5

Discrete Variable Left−Skewed (L)

Prob

abilit

y m

ass

0.0

0.2

0.4

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

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

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

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

3. The SEM with many Rasch measures

the SEM with many Rasch measures •  Consider the standard linear SEM:

η = Bη + Γξ + τ y = ΛYη + ε x = ΛXξ + δ

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

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

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

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

the SEM with many Rasch measures

•  GME measurement parameters and errors:

the SEM with many Rasch measures •  GME structural parameters and errors:

•  Correla;on matrix of the GME es;mated LVs:

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

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

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

Documents

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