1) The document proposes using Bayesian mixture modeling to account for heterogeneity in structural equation models (SEMs) by allowing for different data-generating mechanisms across observations. 2) It presents an example using a customer satisfaction survey where two competing theories imply different SEM structures. 3) The results show the mixture model fits the data better and estimates different path coefficients compared to a single-structure model, indicating some observations may come from an alternative generative process.

1. Finding the Right Path –
Bayesian Mixture Modeling of
Structural Equation Models
Nino Hardt
Joachim Büschken
Catholic University Eichstätt-Ingolstadt
Ingolstadt, Germany

2. Heterogeneity in SEMs
 Assumption of homogeneous population
may be violated (Muthen 1989)
 Hierarchical models (Ansari, Jedidi, Jagpal 1999)
 Finite Mixture Model (Jedidi, Harsharanjeet, DeSarbo
1997/ Zhu, Lee 2001)
• Prior definition of components‟ structures
• Can use covariates to assign corresponding
group
• Incorporate a priori information as prior
distribution

3. Heterogeneity in Parameters vs
heterogeneity in Structure
 Most applications focus on difference in
parameters (e.g. importance of image for
product evaluation varies among customers)
 Alternative theories about causal effects give
rise to different structural equations
 Observations may results from different data
generating mechanisms
 Examples:
• Cognitive processes performed by consumers
• Production functions in firms

4. „Structural Mixture“ Idea
 Explicitly model a variety of alternative data
generating mechanisms
 “Alternative” means that
• Competing theories give rise to alternative
sets of structural equations
• Statements resulting from the structural
model concerning conditional partial
correlations differ
• Otherwise model structures are equivalent
and cannot be differentiated with any data set
(Stelzl 1986, Lee and Hershberger 1990)

5. Model equivalance
C C
A B = A B
D D
 Seminal paper by Stelzl (1986)
 When building a finite mixture of SEM
structures, checking for equivalence is even
more important
 Mixing equivalent models corresponds to
heterogeneity in parameters rather than in
structure

6. The model components
Augmentation of latent
measurement measurement Variables
model (Cowles)
  c

  Z Y
structural
model
 
 Latent constructs
Latent measurement variables

7. Estimation
 Bayesian MCMC approach
 Efficient with small sample sizes
 Prior information on component allocation
can be used
 Implemented in R

8. Empirical example
ECSI customer satisfaction data

9. Data
 Sample of 250 mobile phone customers
 Survey based on the European Customer
Satisfaction Index model

10. Data (cont„d)

11. ECSI Model (e.g. Bayol et al. 2000)
Image
Loyalty
Expect
Satisfa
Value
ction
Quality
15 possible paths, 11 defined, 4 restricted
Image Expect Quality Value Sat Loyalty

12. Alternative Model
Image
Loyalty
Expect
Satisfa
Value
ction
Quality
15 possible paths, 5 defined, 10 restricted
Image Expect Quality Value Sat Loyalty

13. Competing Theories
 Customers evaluate Image, perceived value,
perceived quality and provide information
regarding loyalty and expectations
 Constructs are interrelated and finally explain
customer‟s satisfaction
vs
 Customers do not differentiate in the proposed
manner
 Constructs are mainly driven by satisfaction

14. Mixture results
Image
Sat
1.304
CuExp
0.957
0.515 0.225 PerQu
-0.048 0.523 0.099 0.701 1.675
PerV 0.331
5.673 1.245 1.824
0.414
Sat
PerV PerQu CuExp Loyalty Image
1.17
Loyalty
87.2% 12.8%

15. Probability of being in alternative model Share of Respondents in Alternative Model
0.0 0.2 0.4 0.6 0.8 1.0
0
50
100
Customers
150
200
250
Share of „halo-type“ customers
12.8%

16. ECSI only vs mixture
Marginal Density: Marginal Density:
-9586,39 -9414,27
Image Image
1.141 1.304
CuExp CuExp
1.021 0.957
0.433 0.194 PerQu 0.515 0.225 PerQu
-0.056 0.591 0.257 -0.048 0.523 0.099
PerV 0.391 PerV 0.331
0.357 0.414
Sat Sat
1.085 1.17
Loyalty Loyalty
87.2%

17. Density Density
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5
-1.0
0.5
-0.5
1.0
0.0
1.5
0.5
2.0
Density Density
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
-0.5
0.0
Black densities: mixture model
0.5
1.0
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
Density Density
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5
0.6
0.8
0.0
1.0
0.5
1.2
1.4
1.0
1.6
Density
0.0 0.5 1.0 1.5 2.0
-0.5
Path coefficients endogenous constructs
0.0
0.5
1.0

18. Path coefficients for exogenous constructs
Black densities: mixture model
2.0
2.5
1.2
1.0
2.0
1.5
0.8
1.5
Density
Density
Density
1.0
0.6
1.0
0.4
0.5
0.5
0.2
0.0
0.0
0.0
-0.5 0.0 0.5 1.0 1.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -1.0 -0.5 0.0 0.5 1.0 1.5
 Failing to account for heterogeneity of structures
may lead to biased estimates

19. Impact of the mixture model
Wo/mixture mixture
Sat -> Loyalty
Perv -> Sat Paths among
PerQ -> Sat
endogenous
Perq -> Pev
Exp -> Sat constructs
Exp -> PerV
Exp -> PerQ Paths from
Image -> Sat
Image -> Exp exogenous
Image -> Loyalty construct

20. Findings & Implications
 Accounting for heterogeneity in structure
changes estimates
 Alternative structures should be derived
based on theory (e.g. different cognitive
processes)
 And checked for equivalence prior to
estimation
 Restrictions (pi/gamma=0) are key for non-
equivalence

21. Further steps
 Bayesian imputation of missing values
 Control for heterogeneity of scale usage
 Check against models of direct relationship
between the indicators: hypothesized latent
structures may not hold for all respondents