Transcript Introduction to Structural Equation Modeling and Basic PLS

KF Qualitätsmanagement
Vertiefungskurs V
Messung und statistische
Analyse von
Kundenzufriedenheit
Outline





Customer satisfaction measurement
The Structural Equation Model (SEM)
Estimation of SEMs
Evaluation of SEMs
Practice of SEM-Analysis
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The ACSI Model
Ref.: http://www.theacsi.org/model.htm
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ACSI-Model: Latent Variables
 Customer Expectations: combine customers’
experiences and information about it via media,
advertising, salespersons, and word-of-mouth from other
customers
 Perceived Quality: overall quality, reliability, the extent
to which a product/service meets the customer’s needs
 Customer Satisfaction: overall satisfaction, fulfillment
of expectations, comparison with ideal
 Perceived Value: overall price given quality and overall
quality given price
 Customer Complaints: percentage of respondents who
reported a problem
 Customer Loyalty: likelihood to purchase at various
price points
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Base
line*
Q2
1995
Q2
1996
Q2
1997
Q2
1998
Q2
1999
Q2
2000
Q2
2001
Q2
2002
Q2
2003
Q2
2004
%
Chan
ges
%
Chang
es
79.2
79.8
78.8
78.4
77.9
77.3
79.4
78.7
79.0
79.2
78.3
-1.1%
-1.1%
Personal Computers
78
75
73
70
71
72
74
71
71
72
74
2.8%
-5.1%
Apple Computer, Inc.
77
75
76
70
69
72
75
73
73
77
81
5.2%
5.2%
Dell Inc.
NM
NM
NM
72
74
76
80
78
76
78
79
1.3%
9.7%
Gateway, Inc.
NM
NM
NM
NM
76
76
78
73
72
69
74
7.2%
-2.6%
All Others
NM
70
73
72
69
69
68
67
70
69
71
2.9%
1.4%
Hewlett-Packard Company
– HP
78
80
77
75
72
74
74
73
71
70
71
1.4%
-9.0%
Hewlett-Packard Company
– Compaq
78
77
74
67
72
71
71
69
68
68
69
1.5%
-11.5%
MANUFACTURING/DURA
BLES
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The European Customer
Satisfaction Index (ECSI)
Ref.: http://www.swics.ch/ecsi/index.html
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ACSIe-Model for Food Retail
0,33
Perceived
Quality
Emotional
Factor
0,35
0,37
0,36
0,73
0,53
Expectations
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Hackl et al. (2000)
(-0,01)
Customer Satisfaction
0,34
(0,06)
Latent variables and
path coefficients
0,34
Loyalty
Value
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Austrian Food Retail Market
 Pilot for an Austrian National CS Index (Zuba, 1997)
 Data collection: December 1996 by Dr Fessel & GfK
(professional market research agency)
 839 interviews, 327 complete observations
 Austria-wide active food retail chains (1996: ~50%
from the 10.5 B’EUR market)
Billa: well-assorted medium-sized outlets
Hofer: limited range at good prices
Merkur: large-sized supermarkets with
comprehensive range
Meinl: top in quality and service
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The Data
Indicators
Latent
total expected quality (EGESQ), expected compliance with
demands (EANFO), expected shortcomings (EMANG)
Expectations
(E)
total perceived quality (OGESQ), perceived compliance with
needs (OANFO), perceived shortcomings (OMANG)
Perceived
Quality (Q)
value for price (VAPRI), price for value (PRIVA)
Value (P)
total satisfaction (CSTOT), fulfilled expectations (ERWAR),
comparison with ideal (IDEAL)
Customer Satisfaction (CS)
number of oral complaints (NOBES), number of written
complaints (NOBRI)
Voice (V)
repurchase probability (WIEDE), tolerance against pricechange (PRVER)
Loyalty (L)
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The Emotional Factor
Principal component analysis of satisfaction drivers
staff (availability, politeness)
outlet (make-up, presentation of merchandise, cleanliness)
range (freshness and quality, richness)
price-value ratio (value for price, price for value)
customer orientation (access to outlet, shopping hours,
queuing time for checkout, paying modes, price information,
sales, availability of sales)
identifies (Zuba, 1997)
staff, outlet, range: “Emotional factor”
price-value ratio: “Value”
customer orientation: “Cognitive factor”
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Structural Equation Models
Combine three concepts
 Latent variables


Pearson (1904), psychometrics
Factor analysis model
 Path analysis


Wright (1934), biometrics
Technique to analyze systems of relations
 Simultaneous regression models

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Econometrics
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Customer Satisfaction
Is the result of the customer‘s
comparison of


his/her expectations with
his/her experiences
has consequences on


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loyalty
future profits of the supplier
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Expectation vs. Experience
 Expectation reflects




customers‘ needs
offer on the market
image of the supplier
etc.
 Experiences include



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perceived performance/quality
subjective assessment
etc.
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CS-Model: Path Diagram
Expectations
Customer Satisfaction
Perceived
Quality
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Loyalty
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A General CS-Model
Expectations
Voice
Customer Satisfaction
Perceived
Quality
Loyalty
Profits
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CS-Model: Structure
EX: expectation
PQ: perceived
quality
CS: customer
satisfaction
LY: loyalty
Recursive structure:
triangular form of
relations
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to
from
EX
EX
PQ
CS
LY
X
X
0
X
0
PQ
0
CS
0
0
LY
0
0
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0
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CS-Model: Equations
PQ = a1 + g11EX + z1
CS = a2 + b21PQ + g21EX + z2
LY = a3 + b32CS + z3
Simultaneous equations model
in latent variables
Exogenous: EX
Endogenous: PQ, CS, LY
Error terms (noises): z1, z2, z3
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Simple Linear Regression
Model: Y = a + gX + z
Observations: (xi, yi), i=1,…,n
Fitted Model: Ŷ = a + cX
OLS-estimates a, c:
s
c
, a  y - cx
xy s
sx2
minimize the sum of squared residuals
2
ˆ
 min
i ( yi - yi )  S (a, g) 
a, g
sxy: sample-covariance of X and Y
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Criteria of Model Fit
R2: coefficient of determination
the squared correlation between Y and Ŷ:
R2 = ryŷ2
t-Test: Test of H0: g=0 against H1:g≠0
t=c/s.e.(c)
s.e.(c): standard error of c
F-Test: Test of H0: R2=0 against H1: R2≠0
R2
2
F
1 - R2 n - 2
follows for large n the F-distribution with n-2 and
2 df
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Multiple Linear Regression
Model: Y = a + X1g1 + ... + Xkgk + z = a + x’g + z
Observations: (xi1,…, xik, yi), i=1,…,n
In Matrix-Notation: y = a + Xg + z
y, z: n-vectors, g: k-vector, X: nxk-matrix
Fitted Model: ŷ = a + Xc
OLS-estimates a, c:
c  ( X ' X )-1 X ' y, a  y - c1x1 - ... - ck xk
R2 = ryŷ2
F-Test
t-Test
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Simultaneous
Equations Models
A 2-equations model:
PQ = a1 + g11EX + z1
CS = a2 + b21PQ + g21EX + z2
In matrix-notation: Y = BY + GX + z
with
 z1 
 PQ 
Y 
 , X  ( EX ) , z   z 
 CS 
 2
 0 0
 a1 g 11 
path coefficients
B
, G  

 b 21 0 
 a 2 g 21 
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Simultaneous
Equations Models
Model: Y = BY + GX + z
Y, z: m-vectors,
B: (mxm)-matrix
G: (mxK)-matrix,
X: K-vector
Some assumptions:
z: E(z)=0, Cov(z) = S
Exogeneity: Cov(X,z) = 0
Problems:
Simultaneous equation bias: OLS-estimates of
coefficients are not consistent
Identifiability: Can coefficients be consistently
estimated?
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Path Analytic Model
d1
Var(d1) = sEX2
PQ = g11EX + z1
CS = b21PQ + g21EX + z2
EX
g21
g11
PQ
CS
b21
z1
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z2
 z 1   s 12 0 
Var    
2
z 2   0 s 2 
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Path Analysis
 Wright (1921, 1934)
 A multivariate technique
 Model: Variables may be
 structurally related
 structurally unrelated, but correlated
 Decomposition of covariances allows to write
covariances as functions of structural parameters
 Definition of direct and indirect effects
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Example
d1
sCS,EX = g21s2EX + b21sPQ,EX
= g21s2EX + g11b21s2EX
EX
g11
PQ
s YX   i(Y
X)
Yis iX
CS
b21
z1
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z2
g21
with standardized variable EX:
rCS,EX = g21 + g11b21
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Direct and Indirect Effects
rCS,EX = g21 + g11b21
 Direct effect: coefficient that links independent with
dependent variable; e.g., g21 is direct effect of EX
on CS
 Indirect effect: effect of one variable on another via
one or more intervening variable(s), e.g., g11b21
 Total indirect effect: sum of indirect effects between
two variables
 Total effect: sum of direct and total indirect effects
between two variables
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Decomposition of Covariance syx
s YX   I(Y
I  (Y
X)
YI s IX
X ): variable on path from X to Y
YI: path coefficient of variable I to Y
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First Law of Path Analysis
Decomposition of covariance sxy between Y and X:
s YX   i(Y X ) Yis iX
Assumptions:
 Exogenous (X) and endogenous variables (Y) have
mean zero
 Errors or noises (z)
 have mean zero and equal variances across
observations
 are uncorrelated across observations
 are uncorrelated with exogenous variables
 are uncorrelated across equations
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Identification
PQ = g11EX + z1
CS = b21PQ + g21EX + z2
Y1 = g11X + z1
Y2 = b21Y1 + g21X + z2
In matrix-notation: Y = BY + GX + z
2
0
0
g

s1 0 


 11 
2
B
 , G    ,   (s EX ),   
2
 b 21 0 
 g 21 
 0 s2 
Number of parameters: p=6
Model is identified, if all parameters can be expressed
as functions of variances/covariances of observed
variables
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Identification, cont’d
Y1 = g11X + z1
Y2 = b21Y1 + g21X + z2
s1X =g11 sX
s2X = b21s1X + g21sX2
s21 = b21s12 + g21s1X
sX 2 = sX 2
sy12 = g11s1X+s12
sy22 = b21s21 + g21s2X+s22
2
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p=6
first 3 equations allow
unique solution for path
coefficients, last three for
variances of d and z
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Condition for Identification
 Just-identified: all parameters can be uniquely
derived from functions of variances/covariances
 Over-identified: at least one parameter is not
uniquely determined
 Under-identified: insufficient number of
variances/covariances
Necessary, but not sufficient condition for
identification: number of variances/covariances at
least as large as number of parameters
A general and operational rule for checking
identification has not been found
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Latent variables and Indicators
Latent variables (LVs) or constructs or factors are
unobservable, but
We might find indicators or manifest variables (MVs)
for the LVs that can be used as measures of the
latent variable
Indicators are imperfect measures of the latent
variable
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Indicators for “Expectation”
d1
d2
d3
From: Swedish CSB
Questionnaire, Banks:
Private Customers
E1
E2
E3
EX
E1, E2, E3: „block“ of LVs
for Expectation
E1: When you became a customer of AB-Bank, you probably knew
something about them. How would you grade your expectations
on a scale of 1 (very low) to 10 (very high)?
E2: Now think about the different services they offer, such as bank
loans, rates, … Rate your expectations on a scale of 1 to 10?
E3: Finally rate your overall expectations on a scale of 1 to 10?
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Notation
d1
d2
d3
X1
X2
l1
l2
l3
x
X3
x: latent variable, factor
Xi: indicators, manifest
variables
li: factor loadings
di: measurement errors,
noise
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X1=l1x+d1
X2=l2x+d2
X3=l3x+d3
“reflective” indicators
Some properties:
LV: unit variance
noise di: has mean zero,
variance si2, uncorrelated with other noises
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Notation
d1
d2
d3
X1
X2
l1
l2
l3
x
X3
x: latent variable, factor
Xi: indicators, manifest
variables
li: factor loadings
di: measurement error,
noise
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X1=l1x+d1
X2=l2x+d2
X3=l3x+d3
In matrix-notation:
X = Lx + d
with vectors X, L, and d
e.g., X = (X1, X2, X3)‘
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CS-Model: Path Diagram
d1
d2
d3
e1
e2
e3
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E1
E2
E3
EX
Q2
Q3
g21
g11
Q1
PQ
z2
C1
CS
b21
e4
e5
C2
e6
C3
z1
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SEM-Model: Path Diagram
d1
d2
d3
e1
e2
e3
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X1
X2
X3
x
Y3
g21
h1
h2
e5
Y5
e6
Y6
b21
z1
e4
Y4
g11
Y1
Y2
z2
h = Bh + Gx + z
X = Lxx+d, Y = Lyh+e
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SEM-Model: Notation
h = Bh + Gx + z
Inner relations, inner model
2

s
0
 0 0
 g 11 
2
1
B
 , G    ,   (s EX ),   
2
 b 21 0 
 g 21 
 0 s2 
Outer relations, measurement model
X, d: 3-component vector X = Lxx+d, Y = Lyh+e
Y, e: 6-component vector
 l11 l12 l13 0 0 0 
Lx  ( l11 l12 l13 ) , Ly  
 , d , e
 0 0 0 l11 l12 l13 
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Statistical Assumptions
 Error terms of inner model (z) have




zero means
constant variances across observations
are uncorrelated across observations
are uncorrelated with exogenous variables
 Error terms of measurement models (d, e) have




zero means
constant variances across observations
are uncorrelated across observations
are uncorrelated with latent variables and with each
other
 Latent variables are standardized
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Covariance Matrix
of Manifest Variables
Unrestricted covariance matrix (order: K = kx+ky)
S = Var{(X’,Y’)’}
Model-implied covariance matrix
 A1 A2 
S( )  
,   (B, G, ,  , L x , L y , d , e )

 A2 A3 
A1  L x ( I - B) -1 (GG   )[( I - B) -1 ]L y  e
A2  L y ( I - B) -1 G[L x ]
A3  L x [L x ]  d
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Estimation of the Parameters
 Covariance fitting methods
 search for values of parameters  so that the modelimplied covariance matrix fits the observed
unrestricted covariance matrix of the MVs
 LISREL (LInear Structural RELations): Jöreskog
(1973), Keesling (1972), Wiley (1973)
 Software LISREL by Jöreskog & Sörbom
 PLS techniques
 partition of  in estimable subsets of parameters
 iterative optimizations provide successive
approximations for LV scores and parameters
 Wold (1973, 1980)
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Discrepancy Function
The discrepancy or fitting function
F(S;S) = F(S; S())
is a measure of the “distance” between the modelimplied covariance-matrix S() and the estimated
unrestricted covariance-matrix S
Properties of the discrepancy function:
 F(S;S) ≥ 0;
 F(S;S) = 0 if S=S
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Covariance Fitting (LISREL)
 Estimates of the parameters are derived by
F(S;S())  min
 Minimization of (K: number of indicators)
F(S;S) = log|S| – log|S| + trace (SS-1) – K
gives ML-estimates, if the manifest variables are
independently, multivariate normally distributed
 Iterative Algorithm (Newton-Raphson type)
 Identification
 Choice of starting values is crucial
 Other choices of F result in estimation methods like OLS
and GLS; ADF (asymptotically distribution free)
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PLS Techniques
 Estimates factor scores for latent variables
 Estimates structural parameters (path coefficients,
loading coefficients), based on estimated factor
scores, using the principle of least squares
 Maximizes the predictive accuracy
 “Predictor specification”, viz. that E(h|x) equals the
systematic part of the model, implies E(z|x)=0: the
error term has (conditional) mean zero
 No distributional assumptions beyond those on 1st
and 2nd order moments
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The PLS-Algorithm
Step 1: Estimation of factor scores
1.
2.
3.
4.
Outer approximation
Calculation of inner weights
Inner approximation
Calculation of outer weights
Step 2: Estimation of path and loading coefficients by
minimizing Var(z) and Var(d)
Step 3: Estimation of location parameters (intercepts)
 Bo from h = Bo + Bh + Gx + z
 Lo from X = Lo + Lxx + d
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Estimation of Factor Scores
Factor hi: realizations Yin, n=1,…,N
Yin(o): outer approximation of Yin
Yin(i): inner approximation of Yin
Indicator Yij: observations yijn; j=1,…,Ji; n=1,…,N
1. Outer approximation: Yin(o)=Sjwijyijn s.t. Var(Yi(o))=1
2. Inner weights: vih=sign(rih), if hi and hh adjacent;
otherwise vih=0; rih=corr(hi,hh) (“centroid weighting”)
3. Inner approximation: Yin(i)=ShvihYhn(o) s.t. Var(Yi(i))=1
4. Outer weights: wij=corr(Yij,Yi(i))
Start: choose arbitrary values for wij
Repeat 1. through 4. until outer weights converge
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Example
d1
d2
d3
e1
e2
e3
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E1
E2
E3
EX
Q2
Q3
g21()
g11()
Q1
PQ
z2
C1
CS
b21()
e4
e5
C2
e6
C3
z1
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Example, cont’d
Starting values wEX,1,…,wEX,3,wPQ,1,…,wPQ,3,wCS,1,…,wCS,3
Outer approximation:
EXn(o) = SjwEX,jEjn; similar PQn(o), CSn(o);
standardized
Inner approximation:
EXn(i) = + PQn(o) + CSn(o)
PQn(i) = + EXn(o) + CSn(o)
CSn(i) = + EXn(o) + PQn(o)
standardized
Outer weights:
wEX,j = corr(Ej,EX(i)), j=1,…,3; similar wPQ,j, wCS,j
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Choice of Inner Weights
Centroid weighting scheme: Yin(i)=ShvihYhn(o)
vij=sign(rih), if hi and hh adjacent, vij=0 otherwise
with rih=corr(hi,hh); these weights are obtained if vih are
chosen to be +1 or -1 and Var(Yi(i)) is maximized
Weighting schemes:
hh predecessor
centroid
hh successor
sign(rih)
sign(rih)
factor, PC
rih
rih
path
bih
rih
bih: coefficient in regression of hi on hh
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Measurement Model: Examples
Latent variables from Swedish CSB Model
1. Expectation
2.
E1: new customer feelings
E2: special products/services expectations
E3: overall expectation
Perceived Quality
Q1:
Q2:
Q3:
Q4:
Q5:
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range of products/services
quality of service
clarity of information on products/services
opening hours and appearance of location
etc.
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Measurement Models
Reflective model: each indicator is reflecting the latent
variable (example 1)
Yij = lijhi + eij
Yij is called a reflective or effect indicator (of hi)
Formative model: (example 2)
hi = py'Yi + di
py is a vector of ki weights; Yij are called formative or
cause indicators
Hybrid or MIMIC model (for “multiple indicators and multiple
causes”)


Choice between formative and reflective depends on the
substantive theory
Formative models often used for exogenous, reflective
and MIMIC models for endogenous variables
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Estimation of Outer Weights
 “Mode A” estimation of Yi(o): reflective
measurement model
weight wij is coefficient from simple regression of
Yi(i) on Yij: wij = corr(Yij,Yi(i))
 “Mode B” estimation of Yi(o): formative
measurement model
weight wij is coefficient of Yij from multiple
regression of Yi(i) on Yij, j=1,…,Ji
multicollinearity?!
 MIMIC model
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Properties of Estimators
A general proof for convergence of the PLS-algorithm
does not exists; practitioners experience no
problems
 Factor scores are inconsistent but “consistent at
large”: consistency is achieved with increasing
sample size and block size
 Loading coefficients are inconsistent and seem to
be overestimated
 Path coefficients are inconsistent and seem to be
underestimated
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ACSI Model: Results
Perceived
Quality
0,90
0,73
0,95
0,53
Expectations
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-0,38
-0,29
0,78
0,47
(-0,15)
0,12
-0,24
(0,06)
Value
Customer Satisfaction
0,57
0,35
0,40
0,35
Voice
0,17
(0,06)
Loyalty
EQS-estimates
PLS-estimates
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Evaluation of SEM-Models
 Depends on estimation method
 Covariance-fitting methods: distributional
assumptions, optimal parameter estimates, factor
indeterminacy
 PLS path modeling: non-parametric, optimal
prediction accuracy, LV scores
 Step 1: Inspection of estimation results (R2,
parameter estimates, standard errors, LV scores,
residuals, etc.)
 Step 2: Assessment of fit
 Covariance-fitting methods: global measures
 PLS path modeling: partial fitting measures
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Inspection of Results
 Covariance-fitting methods: global optimization
 Model parameters and their standard errors; do they
confirm theory?
 Correlation residuals: sij-sij()
 Graphical methods
 PLS techniques: iterative optimization of outer
models and inner model
 Model parameters
 Resampling procedures like blindfolding or jackknifing
give standard errors of model parameters
 LV scores
 Graphical methods
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Fit Indices
 Covariance-fitting methods: covariance fit
measures such as





Chi-square statistics
Goodness of Fit Index (GFI), AGFI
Normed Fit Index (NFI), NNFI, CFI
Etc.
Basis is the discrepancy function
 PLS path modeling: prediction-based measures
 Communality
 Redundancy
 Stone-Geisser’s Q2
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Chi-square Statistic







Test of H0: S = S() against non-specified alternative
Test-statistic X2=(N-1)F(S;S(ˆ))
If model is just identified (c=p): X2=0 [c=K(K+1)/2, p:
number of parameters in ]
Under usual regularity conditions (normal distribution,
ML-estimation), X2 is asymptotically 2(c-p)-distributed
Non-significant X2 indicate: the over-identified model
does not differ from a just-identified version
Problem: X2 increases with increasing N
Some prefer X2/(c-p) to X2 (has reduced sensitivity to
sample size); rule of thumb: X2/(c-p) < 3 is acceptable
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Goodness of Fit Indices
Goodness of Fit Index (Jöreskog & Sörbom):
F[ S , S(ˆ)]
GFI  1 -
F[ S , S(O)]
 Portion of observed covariances explained by the
model-implied covariances
 “How much better fits the model as compared to no
model at all”
 Ranges from 0 (poor fit) to 1 (perfect fit)
 Rule of thumb: GFI > 0.9
 AGFI penalizes model complexity:
 c  F[S , S(ˆ)]
AGFI  1 - 

c
p

 F[S , S(O)]
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Other Fit Indices

Normed Fit Index, NFI (Bentler & Bonett)




Comparative Fit Index, CFI (Bentler)


Less depending of sample size than NFI
Non-Normed Fit Index, NNFI (Bentler & Bonett)



Similar to GFI, but compares with a baseline model,
typically the independence model (indicators are
uncorrelated)
Ranges from 0 (poor fit) to 1 (perfect fit)
Rule of thumb: NFI > 0.9
Also known as Tucker-Lewis Index
Adjusted for model complexity
Root mean squared error of approximation, RMSEA
(Steiger):
RMSEA  F [ S , S(ˆ)]/(c - p)
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Assessment of PLS Results
 Not a single but many optimization steps; not a
global measure but many measures of various
aspects of results
 Indices for assessing the predictive relevance
 Portions of explained variance (R2)
 Communality, redundancy, etc.
 Stone-Geisser’s Q2
 Reliability indices
 NFI, assuming normality of indicators
 Allows comparisons with covariance-fitting
results
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Some Indices
Assessment of diagonal fit (proportion of explained
variances)
 SMC (squared multiple correlation coefficient) R2:
(average) proportion of the variance of LVs that is
explained by other LVs; concerns the inner model
 Communality H2: (average) proportion of the variance of
indicators that is explained by the LVs directly connected
to it; concerns the outer model
 Redundancy F2: (average) proportion of the variance of
indicators that is explained by predictor LVs of its own LV
 r2: proportion of explained variance of indicators
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Some Indices, cont’d
Assessment of non-diagonal fit
 Explained indicator covariances
rs = 1- c/s
with c = rms(C), s = rms(S); C: estimate of Cov(e)
 Explained latent variable correlation
rr = 1- q/r
with q = rms(Q), r = rms(Cov(Y)); Q: estimate of
Cov(z)
 reY = rms (Cov(e,Y)), e: outer residuals
 reu = rms (Cov(e,u)), u: inner residuals
rms(A) = (SiSj aij2)1/2: root mean squared covariances (diagonal
elements of symmetric A excluded from summation)
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Stone-Geisser’s Q2
 Similar to R2
E
Q  1O
E: sum of squared prediction errors; O: sum of
squared deviations from mean
 Prediction errors from resampling (blindfolding,
jackknifing)
 E.g., communality of Yij, an indicator of hi
2
Qijc2  1 -
ˆ Y )]2
[
y
(
l
 n ijn ij in
 [y
n
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ijn
- yij ]2
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Lohmöller’s Advice
 Check fit of outer model
 Low unexplained portion of indicator variances and
covariances
 High communalities in reflective blocks, low
residual covariances
 Residual covariances between blocks close to zero
 Covariances between outer residuals and latent
variables close to zero
 Check fit of inner model
 Low unexplained portion of latent variable indicator
variances and covariances
 Check fit of total model
 High redundancy coefficient
 Low covariances of inner and outer residuals
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ACSI Model: Results
Perceived
Quality
0,90
0,73
0,95
0,53
Expectations
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-0,38
-0,29
0,78
0,47
(-0,15)
0,12
-0,24
(0,06)
Value
Customer Satisfaction
0,57
0,35
0,40
0,35
Voice
0,17
(0,06)
Loyalty
EQS-estimates
PLS-estimates
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Diagnostics: EQS
3.12.2004
ACSI
ACSIe
2
247.5
378.7
df
81
NNFI
0.898
0.930
RMSEA
0.079
0.060
173
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Diagnostics: PLS (centroid weighting)
ACSI
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ACSI
e
Hui
Schenk
R2
0.29
0.35
0.43
0.40
Q2
0.36
0.41
0.58
0.49
rr
0.47
0.55
0.58
0.59
H2
0.71
0.64
0.64
0.64
F2
0.22
0.24
0.30
0.26
r2
0.63
0.63
0.57
0.60
reY
0.26
0.24
0.19
0.09
reu
0.19
0.17
0.16
0.08
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Practice of SEM Analysis
 Theoretical basis
 Data
 Scaling: metric or nominal (in LISREL not standard)
 Sample-size: a good choice is 10p (p: number of
parameters); <5p cases might result in unstable
estimates; large number of cases will result in large
values of X2
 Reflective indicators are assumed to be uni-dimensional;
it is recommended to use principal axis extraction,
Cronbach’s alpha and similar to confirm the suitability of
data
 Model
 Identification must be checked for covariance fitting
methods
 Indicators for LV can be formative or reflective; formative
indicators not supported in LISREL
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Practice of SEM Analalysis
cont’d
 Model
 LISREL allows for more general covariance structures
e.g., correlation of measurement errors
 Estimation
 Repeat estimation with varying starting values
 Diagnostic checks






Use graphical tools like plots of residuals etc.
Check each measurement model
Check each structural equation
Lohmöller’s advice
Model trimming
Stepwise model building (Hui, 1982; Schenk, 2001)
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LISREL vs PLS
 Models
 PLS assumes recursive inner structure
 PLS allows for higher complexity w.r.t. B, G, and L;
LISREL w.r.t.  and 
 Estimation method






Distributional assumptions in PLS not needed
Formative measurement model in PLS
Factor scores in PLS
PLS: biased estimates, consistency at large
LISREL: ML-theory
In PLS: diagnostics much richer
 Empirical facts
 LISREL needs in general larger samples
 LISREL needs more computation
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The Extended Model
Perceived
Quality
0,87
0,73
0,85
0,53
Expectations
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(0,20)
0,33
0,58
0,37
(-0,14)
(0,06)
Emotional
Factor
0,55
0,36
(-0,14)
(-0,01)
Value
0,31
0,35
Customer Satisfaction
0,48
0,34
0,41
0,34
Loyalty
EQS-estimates
PLS-estimates
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Diagnostics: EQS
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ACSI
ACSI
2
247.5
378.7
df
81
NNFI
0.898
0.930
RMSEA
0.079
0.060
e
173
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Diagnostics: PLS (centroid weighting)
ACSI
3.12.2004
ACSI
e
Hui
Schenk
R2
0.29
0.35
0.43
0.40
Q2
0.36
0.41
0.58
0.49
rr
0.47
0.55
0.58
0.59
H2
0.71
0.64
0.64
0.64
F2
0.22
0.24
0.30
0.26
r2
0.63
0.63
0.57
0.60
reY
0.26
0.24
0.19
0.09
reu
0.19
0.17
0.16
0.08
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Model Building: Hui’s Approach
Perceived
Quality
0,43
Emotional
Factor
0,31
-0,18
0,10
0,33
-0,18
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0,35
0,36
0,42
Expectations
0,61
0,17
0,63
Value
0,12
Customer Satisfaction
0,21
0,23
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Loyalty
75
Model Building: Schenk’s Approach
0,32
Perceived
Quality
Emotional
Factor
0,35
0,31
0,32
0,73
0,34
0,60
Expectations
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Customer Satisfaction
Value
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The end
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Data-driven Specification
 No solid a priori knowledge about
relations among variables
 Stepwise regression
Search of the “best” model
Forward selection
Backward elimination
Problem: omitted variable bias
 General to specific modeling
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Stepwise SE Model Building
 Hui (1982): models with interdependent
inner relations
 Schenk (2001): guaranties causal
structure, i.e., triangular matrix B of path
coefficients in the inner model
η=Bη+ζ
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Stepwise SE Model Building
Hui’s algorithm
Stage 1
1. Calculate case values Yij for LVs ηi as principal
component of corresponding block, calculate R =
Corr(Y)
2. Choose for each endogenous LV the one with highest
correlation to form a simple regression
3. Repeat until a stable model is reached
a. PLS-estimate the model, calculate case values, and
recalculate R
b. Drop from each equation LVs with t-value |t|<1,65
c. Add in each equation the LV with highest partial
correlation with dependent LV
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Stepwise SE Model Building
Hui’s algorithm, cont’d
Stage 2
1. Use rank condition for checking identifiability of
each equation
2. Use 2SLS for estimating the path coefficients in
each equation
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Hui’s vs. Schenk’s Algorithm
Hui’s algorithm is not restricted to a causal
structure; allows cycles and an arbitrary
structure of matrix B
Schenk’s algorithm



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uses an iterative procedure similar to that used
by Hui
makes use of a priori information about the
structure of the causal chain connecting the
latent variables
latent variables are to be sorted
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Stepwise SE Model Building
Schenk’s algorithm
1. Calculate case values Yij for LVs ηi as principal
component of corresponding block, calculate R =
Corr(Y)
2. Choose pair of LVs with highest correlation
3. Repeat until a stable model is reached
a. PLS-estimate the model, calculate case values, and
recalculate R
b. Drop LVs with non-significant t-value
c. Add LV with highest correlation with already included
LVs
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Data, special CS dimensions
Staff
2 availability1 (PERS), politeness1 (FREU)
Outlet
3 make-up1 (GEST), presentation of merchandise1 (PRAE), cleanliness1 (SAUB)
Range
2 freshness and quality (QUAL), richness
(VIEL)
Customerorientation
7 access to outlet (ERRE), shopping hours
(OEFF), queuing time for checkout1
(WART), paying modes1 (ZAHL), price
information1 (PRAU), sales (SOND),
availability of sales (VERF)
1
Dimension of “Emotional Factor”
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References
C. Fornell (1992), “A National Customer Satisfaction Barometer:
The Swedish Experience”. Journal of Marketing, (56), 6-21.
C. Fornell and Jaesung Cha (1994), “Partial Least Squares”, pp.
52-78 in R.P. Bagozzi (ed.), Advanced Methods of Marketing
Research. Blackwell.
J.B. Lohmöller (1989), Latent variable path modeling with partial
least squares. Physica-Verlag.
H. Wold (1982), “Soft modeling. The basic design and some
extensions”, in: Vol.2 of Jöreskog-Wold (eds.), Systems under
Indirect Observation. North-Holland.
H. Wold (1985), “Partial Least Squares”, pp. 581-591 in S. Kotz,
N.L. Johnson (eds.), Encyclopedia of Statistical Sciences, Vol.
6. Wiley.
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