Discussion Paper No. 56

Xing Aijing, Nobuhiko Terui and P.K.Kannan

April 2016

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANHow customer satisfaction affects loyalty:
Insights from nonlinear hierarchical Bayes modeling of customer satisfaction index

Xing Aijing, Nobuhiko Terui1 and P. K. Kannan
Tohoku University and University of Maryland

April, 2016

Abstract
The paper investigates the nonlinear relationship between customer satisfaction and loyalty.
We extend the relationship proposed by the customer satisfaction index (CSI) model to include anonlinear functional form between satisfaction and loyalty. We examine different functional
forms on how satisfaction affects loyalty and propose a model that reflects intrinsic
characteristics of nonlinear effects, such as saturation-attainable limit of effectiveness,
non-constant marginal return, and asymmetric response between satisfied and dissatisfied
customers, in a parsimonious way. The model is estimated via a hierarchical Bayes model toaccommodate structural heterogeneity of companies surveyed in the analysis.
The key
contributions of the paper include a nonlinear structural equation model that includes nonlinear
term of endogenous latent variable and an efficient algorithm of MCMC in terms of multi-movesampler by using Gibbs sampling.
The empirical analysis by using survey data shows that (1) hierarchical Bayes models
estimated by borrowing other companys’ data are better than the independent model using theirown data in terms of not only goodness of fit measures but also in the number of significant
model estimates, (2) nonlinear models perform better than linear models, (3) nonlinear modelwith asymmetric marginal returns and attainable limits is found to be the best model. The
managerial implications for loyalty management include: (i) there are limits to attainablelevels of loyalty through satisfaction; (ii) the phenomenon of loss aversion is observed incustomers’ responses; (iii) marginal return of satisfaction is asymmetric across satisfied anddissatisfied customers, i.e., increasing for dissatisfied customers and decreasing for satisfied
customers, (iv) in general, direct effect of satisfaction is more significant than indirect effectthrough recommendation intention.
Finally, based on the estimated response curve of loyalty as a function of satisfaction andthe empirical distribution of customers on the dimensions of CSI scores, we evaluate theefficiency of loyalty programs under assumptions of full and limited access to customers.

1

Terui acknowledges the grant by JSPS KAKENHI Grant Number (A)25245054.
1

1. Introduction
The relationship between customer satisfaction and loyalty has been one of the widely
studied relationships in marketing and services literature (Dong et al 2011; Kumar et al 2013).
The premise that customer satisfaction, a construct that underlies customers’ perceptions
regarding their overall consumption experiences (Anderson and Salisbury 2003), significantly
impacts customer loyalty, a construct that drives customer retention and repurchase behavior, is
key to firms’ customer orientation. It is this this relationship that forms the basis for measuringmarketing effectiveness (Fornell 1992, Bolton and Lemon 1999, Anderson et al. 2004), and forfirms’ market and financial performance and for firm value (Anderson and Mittal 2000; Gupta,
Lehmann and Stuart 2004, Gupta and Zeithaml 2006). The fact that this relationship has been
extensively studied in marketing and services over several decades highlights its important andcritical role in determining the effectiveness of marketing programs and ensuring the creating offirm value through marketing action.
In this paper, we examine the relationship between customer satisfaction and customer
loyalty by using the survey data used for developing customer satisfaction index.

The

framework of analysis uses the customer satisfaction index (CSI) model as the starting point topropose a nonlinear structural equation model which includes a nonlinear function form
between satisfaction and loyalty as one equation in the set of equations.

We contribute to theexisting literature that examines satisfaction and loyalty variables which are measured bycomprehensive system of equations (Dong et al 2011; Kumar et al 2014), in contrast to previous
studies using just the metrics of satisfaction in isolation of their context.
As for functional forms of the nonlinear relationship, we consider piecewise linear andS-shaped functions.

The former is motivated by the ease of estimation, being close to linear

model and the latter specification is justified by prospect theory of Kahneman and Tversky(1979) and empirically supported by Ngobo (1999), whose research objective and dependent
variable of loyalty are common with ours.
From the methodological point of view, there are quite a few extant papers on nonlinear
structural equation models, for example, Lee (2007) discusses a model with nonlinearity onlywith respect to exogenous latent variables.

This article contributes to the modeling literatureby the modeling nonlinear structural equations that include nonlinear terms of endogenous
latent variable.

By using the recursive property of system for CSI model, implying that latentvariables are determined sequentially, we provide an efficient algorithm of MCMC in terms of2

multi-move sampler for latent variables by using Gibbs sampling.
In addition, we employ a hierarchical Bayes model to deal with structural heterogeneity
across individual companies in the survey data.

The model connects the structural models forrespective company, and it leads to higher reliability of model estimates than the original
customer satisfaction index model.

This is accomplished by using the insights from Terui et al.

(2011).
Section 2 discusses extant work on nonlinear relationship between satisfaction and outcomes
including loyalty in the literature.

We also discuss the perspective for building the frameworkof our analysis.

In section 3, we propose the model, including possible alternative

specifications.

Section 4 reports the empirical results of model comparison, parameter

estimates, the interpretation of estimates and derived managerial implications.

Section 5

presents the conclusions.

2. Nonlinear Relationship between Customer Satisfaction and Loyalty
2.1. Nonlinear Relations of Satisfaction
There are many extant works on nonlinear relationships between satisfaction and outcome
variables.

They investigate customer satisfaction relationship with one of firm’s outcomes byusing different metrics.

Most of the studies show that satisfaction has a positive and nonlinear

asymmetric impact on firm’s outcomes, and different functional forms are supported by thesestudies (Dong et al 2011, Kumar et al 2014).
Fornell (1992) empirically showed that the relationship between customer satisfaction andtheir intention to repurchase goods or services is nonlinear, and dissatisfaction has greater
influence than satisfaction on customers’ repurchase intentions. Mittal and Kamakura (2001)
examined levels of customer satisfaction and dissatisfaction impacting purchase intention andactual purchase behavior. They found that linear methods may underestimate the influence ofsatisfaction and suggest nonlinear relations whose patterns are moderated by consumer
heterogeneity across the attributes.

Keiningham et al (2003), Bowman and Narayandas (2004),

and Cooil et al. (2007) showed that the satisfaction affects the share of wallet nonlinearly.
Homburg, Koschate and Hoyer (2005) suggest inverse-S shaped for willingness to pay by using
experimental study.

On the other hand, Ngobo (1999) examined the relation between

satisfaction and loyalty (purchase and word-of mouth intentions) and suggested S-shaped3

function for the relationship.
2.2 Nonlinearity and Moderating Effects
Nonlinearity has been investigated in the context of moderating effects on the relationshipbetween satisfaction and outcome variables.

In particular, Jones and Sasser (1995) clarified

that the competitive environment of the market affects the nonlinear relationship between
satisfaction and loyalty.

Their empirical research involved more than 30 companies in fiveindustries (local telephone companies, airlines, hospitals, personal computers, andautomobiles) and showed that the competitive environment greatly influences the relationshipbetween customer satisfaction and loyalty.
Bloemer and Ruyter (1998) demonstrated nonlinearity by incorporating involvement as the
key parameter between customer satisfaction and loyalty.

Based on customers expressing

equal levels of satisfaction, their comparative study showed that highly involved customers
exhibit greater loyalty than customers with low involvement.

Mittal and Kamakura (2001)

and Cooil et al. (2007) discuss nonlinearity in the context of the moderating effect of consumer
characteristics.

Recently, Eisenbeiss, Corneliben, Backhaus and Hoyer (2014) investigate

nonlinear and asymmetric return on satisfaction to willingness-to-pay by considering two kindsof moderating effects – firm reputation and consumer’s involvement.
2.3 Framework of Our Study
In most prior research, satisfaction and loyalty variables are directly measured by using a survey
directed to respondents. The impact of other antecedents on satisfaction is also not taken into
consideration.

On the other hand, Fornell (1992) discusses the need to use a comprehensive

system of post-purchase outcomes in the way that satisfaction is part of the overall outcome thatis measured.

This is motivated by the fundamental principles that a variable should take onmeaning depending on the context (Fornell, 1982, 1988; Fornell and Yi, 1992), survey variables
contain some degrees of errors (Andrews, 1984), and satisfaction is not directly observable(Howard and Sheth, 1969, Oliver, 1981, Westbrook and Riley, 1983).

In addition, he insists

that, if satisfaction variable is measured in isolation of the context and it is used retrospectively
to estimate the relationship, we tend to have the results with low reliability and strongly biased
parameter estimates.
Given these reasons, we use the system of structural equations that contain satisfaction and4loyalty as latent variables, i.e., customer satisfaction index (CSI) model by Fornell, et al. (1996).
However, CSI model premises a linear relationship between latent variables, and we extend themodel to propose a nonlinear structural equation model which includes nonlinear function fromsatisfaction to loyalty.

3. Nonlinear CSI Model
3.1 Customer Satisfaction Index Model
The customer satisfaction index uses the only uniform measure of customer satisfaction thatallows comparison between companies and bench-marking across industries.

It also illustrates

how customer satisfaction is embedded in a system of cause–effect relationships.

Furthermore,this index is significant as a leading indicator of the financial results of the company (Anderson
et al., 2004; Fornell et al., 1996, 2010).

They employ the adopted expectancy disconfirmation

as a basic theory which was proposed by Oliver (1980).

It is a model in which the level ofcustomer satisfaction is decided by the degree of disconfirmation between perceived quality
after a purchase and customer expectation before a purchase.
The CSI model describes that customer expectations drive perceived quality and perceived
value, and these three latent variables generate customer satisfaction.
turn directly affects customers’ voice and loyalty.

Customer satisfaction in

The model estimation employs 17 manifest

variables, which are ordered categorical variables based on survey questions rated on a scale of
1–10 (low–high).

The scores of customer satisfaction are factor scores for n sampled

customers’ satisfaction, and they are reported as standardized metrics between 0 and 100 points.
Full description of model is provided in Appendix A.

Using identical structure model forcompanies allows us to compare satisfaction level between companies, and the changes in it
over years.
3.2. Nonlinear Model on the Satisfaction to Loyalty
Customer loyalty (LOY) is a function of customer satisfaction (CS) and recommendation toothers (RE) (“Voice” in original CSI) and the model is described as one equation in the set of
equations of CSI model by

LOYi   CSi   REi   iwhere i means the index for respondent (customer),  i is the normally distributed error term,
5(1)and  ,  are path coefficients.
The linear specification is reasonable as local approximation to possibly more complicatedrelations, but it has limitations in failing to accommodate some characteristics discussed in theliterature, i.e., (i) not constant return to scale, (ii) saturation effects and (iii) asymmetricresponse.

These are well captured by nonlinear models.

We model the accommodation of these characteristics in a parsimonious way as follows:



1
1 
1
1 
    (  ) 1  I      REi   i
LOYi   (  ) I 
1  exp  CSi  2 
1  exp  CSi  2 
where I is the indicator function taking 1 if CSi  0 and zero otherwise.
function is depicted in Figure 1.

(2)

The shape ofThis function is in line with Prospect Theory (Kahamenan

and Tversky, 1979) and empirically supported by Ngobo (1999) whose research objective anddependent variable of loyalty closely resemble ours in the literature.

Figure 1: Nonlinear Model for Satisfaction to Loyalty

This function captures nonlinear effects in terms of logistic function.

That is, it hasasymmetric response around the inflection point, i.e., CS  0 , which is fixed for
identification of model, and also the marginal return to LOY is changing at every level of CS.
The upper limit represents the satiation level and the lower limit indicates the baseline
independent of CS.

They are respectively provided by1 ()1 and   (  ) . Thus the22estimates of coefficients  (  ) and  (  ) determine the maximum and minimum levels of
loyalty caused by satisfaction. They also define the speed to reach these limits in respective
regimes.

They provide several interesting implications for service management.

The

company with larger  (  ) value will attain the attainable limit of loyalty quickly byadditional effort on satisfaction as long as it stays in the positive regime.

On the other hand,the smaller value of  (  ) shows that the increasing rate of return is slower although thelowest limit might not be all that detrimental when it is in negative regime.
6

That is, themodel suggests that, given a level of CS score, companies obtain useful information on how
additional effort on the customer satisfaction dimension impacts the relative to the speed of
return to CS increases and the attainable limit to loyalty.

We call model (2) as “asymmetric

logit model.”When we replace the logistic function by simply CSi as a piece-wise linear term, we
obtain the “asymmetric linear model” which we use for comparison purposes in the empiricalapplication.

LOYi   (  ) I CSi    (  ) 1  I CSi    REi   iThis model defines different slopes of linear response in respective regimes.

(3)
It approximatesnonlinear relation by piecewise linear functions, and it does not have attainable limits since it
models the relationship locally, even though it has more useful information than a simple linear
model.

In addition, we consider the inverse S-shaped models, as is discussed in Homburg etal. (2005), by introducing thresholds, r1 and r2 , in the satisfaction domain, which define thezone of tolerance.

LOYi   (  , r1 ) I CSi    ( r2 , ) 1  I CSi    REi   i(4)where  (  , r1 ) takes some value when CSi  r1  0 and zero otherwise,  ( r2 , ) takes somevalue when 0  r2  CSi and zero otherwise.

We call this “threshold linear model”.

Similarly, “threshold logit model” is specified as

1
1 
1
1 


    ( r2 , ) 1  I      REi   i (5)LOYi   (  ,r1 ) I 
1  exp  CSi  2 
1  exp  CSi  2 
We compare these alternative models in the empirical application.

The functions of thesemodels are depicted in Figure 1 for comparison.
We employ these nonlinear models just for the customer satisfaction to loyalty relationshipnd keep linear structural equations for other relations in CSI model.
model is given at Appendix A.

7

Full description of our3.2. Hierarchical Bayes Modeling for Stable Estimation
The CSI model assumes that every company has the identical structure on customer
satisfaction for the purpose of comparing the services across different companies, and thus isaggregated to industry groups and national levels.

However, each company should havestructural heterogeneity on customer satisfaction measures.

The structural heterogeneity

appears to produce the result that some path coefficient estimates are not significant for somecompanies, which leads to reduce the credibility of scores.

To overcome this problem, Terui etal. (2011) proposed the hierarchical Bayes modeling of customer satisfaction index to increase
reliability of model estimates not only on the goodness of fit, but also by the number ofsignificant estimates of path coefficient.

That is, HB model produces larger number ofsignificant estimates in the model and better goodness of fit than independently estimated model.
This result comes from the property that HB modeling borrows information of neighbors bypooling data to get the stable estimate of parameters on the assumption that they sharehomogeneity in some aspects regardless of independent information.
In this study, we employ HB modeling which relates the model of each companyh  1,..., H such thatηh  N k  0, V 

β h  Θ ' z h  ηh ;,(6)

where β h is the vector of path coefficients between latent variables and z h is attribute data
for the company h, and ηh is error term.

This is prior distribution on the path coefficients

β h , and this means that the path coefficients are not independent and restricted by the common
parameter ( Θ,V ).

The prior specification (6), together with appropriate prior specifications

for other parameters including latent variables of ω hi , are combined with the Gaussian8likelihood based to constitute joint posterior density,p β h  , ω hi  , Θ, V | Data  .

(7)

The numerical evaluation of this density is conducted by Markov chain Monte Carlo (MCMC),
and its algorithm is described together with prior specifications in Appendix B.

4. Empirical Results
4.1. Data
The dataset is available from the Japanese CSI development working group managed by theJapanese Agency of Service Productivity and Innovation Growth. We use the data for survey
conducted in 2008 year, and it includes 21 companies in three industries—mobile
telecommunications (4), convenience stores (5), hotels (12).

The sample sizes used in analysis

are: mobile telecommunications (company1 = 456, company2 = 456, company3 = 360),
convenience stores (company1 =456, company2 = 456, company3 = 360), hotels (company1 =300, company2 = 300, company3 = 300).
4.2. Data Transformation and Full Conditional Posterior Density for Estimation
We employ Bayesian inference on the model estimation as is explained in Lee (2007) and Teruiet al.(2011) on the grounds of distributional property of observations as well as derived
distribution of estimated satisfaction scores.

The data are measured by 10 point Likert scale.

Thus the ordered categorical data are not consistent with normality assumption.

On the other

hand, the structural equation model is developed on the assumption of normality on variables.
The American CSI model employs PLS method for model estimation since PLS does notassume any distribution on the error terms to estimate the model parameters.
no free lunch.

However, there is

In fact, Terui et al. (2011) compares the estimates by Bayesian MCMC methodwith those by PLS, and it demonstrated that the distribution of estimated satisfaction score by
PLS method is mostly skewed, on the other hand, the score distribution evaluated by Bayesian
MCMC algorithm is stable and symmetric.

The satisfaction score is calculated by taking

sample mean of estimated respondent’s scores, and being standardized to be inside 0 and 100
points, and the satisfaction score must be reasonable only when the distribution is symmetrical.
In order to be consistent with our inference below, we first transform ordered categorical9

data into continuous variable, which follows the specified normal distribution by way of data
augmentation (Lee, 2007; Terui et al., 2011).

We introduce a set of cut points across thenormal distribution to decompose it into 10 segments that may be categorized on a scale of 1 to
10.

Thus, the probability of each region corresponds to the probability mass of each orderedcategory.

When we have a categorical sample, we generate the samples from the truncatednormal distribution whose cut points are defined by the corresponding segment.
The algorithm for Bayesian inference of linear structural equation model is given by Lee(2007).

By using the special properties of CSI model that the latent variables   ,1,2 ,3 ,4 ,5  ' are determined sequentially by the initial driving force of “expectation

 ”, in the way that 1  2  3  4  5 and also the nonlinear equation of 5 (LOY)by 3 (CS) is positioned in the last. The efficient algorithm is available for generating
posterior distribution of latent variables. We first decompose the set of latent variables
into linear and nonlinear parts 1   ,1 , 2 , 3 , 4  ' and 5 .

Then we express thejoint prior density is defined by p    p 1  p 5 | 1  to derive marginal posterior
density of linear latent variables 1 and conditional density of nonlinear latent variable of 5
on 1(i) p 1 | x,    p 1  p[1 ]  x |  ,  (ii) p  5 |  1 , x ,    p  5 |  1  p [ 5 ]  x |  ,  where  means the set of model parameters including factor loadings, path coefficients, andvariances, and x is data.

The algorithm for linear part (i) is given by Lee (2007).

The

nonlinear part (ii) is the product of normal prior and normal likelihood, and thus the posterior
density is analytically derived by using conjugate property.
available for latent variables of our model.

That is, multi-move sampler is

The path coefficient parameters are defined aslinear in our model, and the algorithm for linear structural equation model is available, together
10with other parameters of factor loadings and variance, in Lee (2007).

The details are explainedin the algorithm section in Appendix B.
Finally, the conditional posterior density of model parameter, p  |  , x  , is available in
Lee (2007) since this is the same structure conditional on latent variables  with linear structural
equation model.

The details of full conditional posterior density are provided in Appendix B.

4.3. Model Comparison
We estimated the parameter using Markov chain Monte Carlo (MCMC) by the use of Gibbs
sampling.

This section reports results of the comparison between models by comparing thevalues of Deviance Information Criterion (DIC), an information criterion of Bayesian analysis
as well as log of marginal likelihood (LML).
We compare (i) linear model, (ii) logit model, (iii) asymmetric linear model, and (iv)asymmetric logit model in their HB estimations.

As a benchmark model, we also set theoriginal CSI model, denoted by (0) independent linear model.

Table 1：Model Comparison: DIC and log of Marginal Likelihood

Table 1 shows the calculated values of DIC and LML for the different models.

First of all,both measures support the HB models than independent linear model, and the advantage of HBmodeling is more evident for the measure of LML.

The comparison between linear andnonlinear models supports nonlinear models by both criteria, and within groups, asymmetricresponse models are supported more than symmetric models: HB asymmetric linear model isbetter than HB linear model, and HB asymmetric logit model performs better than HB logit
model in case of DIC.

We note that LML of HB logit model slightly shows better fit than HB

asymmetric logit model.

However, the latter model contains double number of response

parameter as the former, and we employ DIC which discounts the number of parameter more
appropriately than LML.
Table 2 tabulates the number of path coefficients that were not significant in the sense of
95% highest probability density (HPD) region for respective models.

The total number of

insignificant estimates in each model is shown at the bottom of table.

The effect of HBmodeling that borrows other company’s data on the estimates is evident.
11

The number ofinsignificant estimates in (0) independent model is drastically reduced from 27 to 16 for (i) HB
linear; 17 for (ii) HB asymmetric linear; 13 for (iii), (iv) HB (asymmetric) logit models.

The

heterogeneity in industry is evident to see that hotels have relatively more insignificant
estimates.

Table 2: The Effect of HB modeling on Estimate of Path Coefficients

The examination of results on model fit criteria in Table 1 and the number of significant
parameters estimate in Table 2 shows the order of better model is HB asymmetric logit, HB logit,
HB asymmetric linear, and HB linear models.

The asymmetric response is better supported

and furthermore the model gets more advantageous if the saturation effects are incorporated in
the model.

4.4. Parameter Estimates of Nonlinear Term from CS to LOY
Table 3.1 shows the estimates (posterior mean) of coefficient of  (  ) and  (  ) of nonlinear
term from satisfaction to loyalty for individual companies.
to the estimate.

95% HPD region is also given next

The industry level estimates given in Table 3.2 are derived from posteriormeans of industrial dummy in the hierarchical model.

Table 3.1: Parameter Estimates (Company Level)
Table 3.2: Parameter Estimates (Industry Level)First of all, path coefficients are significant for all companies as HDP region does notinclude zero with the level of 95% probability.
greater than that of  (  ) for all cases.

Pr  (  )   (  )Second, we observe the estimate of  (  ) isMore precisely, the posterior probability

 is given in the fifth column of the Table 3.1 to show that it holds with highprobability for most companies, and Table 3.2 shows that probability of at industry level is thehighest 92.02% for mobile telecommunication and the lowest 71.22% for convenient store

12

industry.
limit (

These coefficients respectively determine the lower limit ( 

1 () ) and the upper2

1 () ) of loyalty over satisfaction dimension, and it turns out that the speed of reaching2

the limit is slower in positive regime than negative regime.
is observed for every company across industries.

This means that the loss aversion

The customers recognizing dissatisfaction

induce great depreciation of loyalty compared with the same amount of increase of satisfactionin positive regime.
Finally, we observe that these estimates of convenience store industry are relatively larger
than those in other industry.

The mean value of the estimated difference  (  ) -  (  ) is 0.72 formobile telecommunication, 0.24 for convenience store; and 0.40 for hotels.

This implies thatthe loss aversion is most pronounced in telecommunication industry, and next is hotels although
the situation is rather heterogeneous within hotels.

Next we consider the band width

1 () 1 ()  between the upper and lower limits. This is a measure of importance of2
2satisfaction on the variation of loyalty.

The mean value of band width is estimated as 1.24 formobile telecommunication, 1.69 for convenience store and 1.21 for hotels.

These suggest thatthe satisfaction in convenience stores industry is most likely to produce significant impact onloyalty.

On the other hand, mobile telecommunication companies are not relatively wellplaced to gain the loyalty by means of satisfaction.

4.5. Estimated Functional Form
Figure 2 depicts the figure of estimated functional form of CS  LOY.

We observe

asymmetric responses in respective regimes and upper limits are smaller than negative of lower
limits, implying loss aversion in relationship between satisfaction and loyalty.

This is the mostevident for M2 of mobile telecommunications as the difference is - 0.40 (= 0.41 - 0.81).
opposite situation happens for C5 in convenience stores, i.e., -0.07 (=0.92-0.85).

Figure 2: Estimated Functional Form and Upper and Lower Limits

13

The

4.6. Distribution of Satisfaction Score
We express the levels of satisfaction and loyalty on 100-point scale as is usually reported incustomer satisfaction index (CSI).

The CSI score is calculated as the standardized factor

scores for customer satisfaction by

CSi  min[CSi ] 100 .
max[CSi ]  min[CSi ]distribution of respondent’s scores for each company.

Figure 3 depicts the empiricalIn the figure, the statistics of mean,

median, and standard deviations as well as number of respondents are shown as legends.
score distributions are heterogeneous among companies.

The

However, the distributions are

relatively stable and symmetric since the difference of mean and median is small for every
company.

This is consistent with the study by Terui et al. (2011) on the ground of estimation

after transformation of original categorical data to normal distributed data by data augmentationfor Bayes modeling.

Thus the sample mean would be reasonable estimate of CSI score evenfor nonlinear model.

Figure 3: Distribution of Satisfaction Score

4.7. Marginal Effects and Indirect Effect of Satisfaction
Loyalty is determined not only by satisfaction, but also by the intention to recommend to others.
According to the model (2), the marginal effects of satisfaction and recommendation intention
are respectively measured byLOYi
exp(CS i )exp(CS i ) ˆ (  ) I
 ˆ (  ) I  12
CS i
1  exp(CS i ) 2
1  exp(CS i ) 
LOYi
 ˆ
REi

(8)

The marginal effect of satisfaction is not constant, changing with the level of satisfaction.
In contrast, the marginal effect of recommendation intention is constant ̂ .
these effects.

Figure 4 depicts

The marginal effect of satisfaction among dissatisfied customers (negative

regime) is increasing up to1 ˆ ( )1 ˆ ()from the left, and then it is decreasing from4
4

toward zero.

Satisfaction and recommendation have positive impact to loyalty since their marginal
14

effects are positive over the domain of satisfaction for every company because of positive
parameter estimates reported in Table 4.
level of satisfaction.

However, which is more influential depends on the

According to the relation depicted in Figure 4, the satisfaction is more

influential on loyalty than recommendation for the customers with the central level of satiation.
The recommendation intention, in turn, is more effective for extreme customers.

The intervalof satisfaction where satisfaction is more important is reported as [d1, d2], and the transformed
score by 100 points scale as [b1, b2] in Table 4.

The percentage of customers who belong tothis interval is also given as “Customer Ratio” and it shows that the satisfaction is more
effective for 89% customers for convenient stores; 54% customers for telecommunicationindustry and 37% customers for hotel industry which has strong heterogeneity inside.

Figure 4: Marginal and Indirect Effects

Table 4: Marginal and Indirect Effects

Following the customer satisfaction model, RE is determined by CS as a structural equation

REi   CSi  ei .

(9)

 i  ˆCS ,The indirect effect of CS to LOY is defined as marginal effect of estimated RE by CS, REi

ˆˆ.
and it is given by 

It means the effect of CS by way of RE to LOY.

In (9), CS is assumed to have a positive effect on RE, and in fact ˆ is estimated positive for every
company.

Then the indirect effect of CS by way of RE to LOY is interpreted as marginal effect of

RE discounted by ˆ .

We also define the direct effect of CS as the marginal effect in (8), and wecompare these effects over the domain of satisfaction. It is evident that direct effect is much more
influential for loyalty by the interpretation of estimated indirect effects above.

We define, similarly

before, the interval where direct effect of CS is greater than indirect effect, and their “Customer

Ratio”.

These are reported in Table 4. It shows that the direct effect is most pervasive with over

99% customers in wider range of interval in convenience stores; in particular, it is dominant (100%)for the company C4.

The hotels have rather significant impacts of indirect effect, 88% in average.
15

The indirect effect is more important for the hotel H10 (41.33% customer ratio) and comparable forH8 (64%).

The CS campaign leading to recommendation would be necessary for thesecompanies.

4.8. Managerial Implications
We consider two kinds of measure for managerial implications derived from our models.

The

first measure is the expected incremental loyalty that is defined by the expected values ofincremental loyalty on unit change of satisfaction with respective to customer distribution on
CSI scores.

This measure is useful for overall evaluation of loyalty program when we assume

that the firm approaches every customer.

Under the limited budget for loyalty program, thesecond measure finds a customer segment optimizing the incremental loyalty.

(i) Expected Incremental Loyalty
According to satisfaction and loyalty scores obtained from the empirical study of our model,
company managers can review their situations and formulate their strategies.

First, theexpected incremental loyalty (EIL) can be used to forecast future profitability of loyalty
program by combining estimated response function and customers distribution over the same
domain.

Based on the empirical distribution of CSI scores in 4.6, we first set the cut-off point

vector (0, 5, 10, 15, 20,..., 100) for CSI score dimension, and calculate the frequency of eachcell to get the empirical distribution of CSI scores p cs i | data  , i=1,2,..., 20.

Then we define

a middle point vector (cs1 ,cs2 ,cs3 ...cs19 ,cs20) by (2.5, 7.5, 12.5,..., 97.5) for calculating
estimate of marginal incremental loyalty, f ' (csi ) defined in (6).

Then the EIL is formallydefined by
20

EIL   f ' (csi ) p(csi | data) .

(10)

i 1

EIL shows the future profitability when loyalty program has a full access to their customers.
Table 5 at the first column shows the measure of EIL for individual companies.

The

industry of convenience store has the highest EIL, and it will get the largest loyalty increment
when the customers’ satisfaction level is improved.

EIL and the band width given in Table 3.1.
than H11, having 1.27 and 1.28 respectively.
(0.2693).

Another point is the difference between

The band width of company M1 is a little lower
However, M1 has higher EIL (0.2741) than H11

It means the extensible space of loyalty in M1 is not as wide as those in H11,

however customers in M1 have more concentration around neutral point where it has the most16

sensitive change of loyalty.

(ii) Targeted Customer Interval for Efficient Loyalty Program
Next, we consider a situation that company might just offer loyalty program to a limited
proportion of customers due to their budget constraint.

Our model provides the framework to

consider how they should target some customers effectively subject to their budget for loyalty
program.

Assume that a company manager prospects that she/he is allowed to provide loyalty

program for only 30% customers.

Then the problem is to specify the set of customers underconstrained optimization,CS b

max  f ' (CS i )

(11)

CS a

s.t. PCS a  CS i  CS b   0.3

Figure 5: Frequency and Increment Loyalty

We call the interval [ CS a , CSb ] 30% targeted customer interval (TCI), implying that thecustomer segment maximizes incremental loyalty induced by loyalty program.

Figure 5 shows

the smoothed frequency distribution of customer’s CSI scores on the left, and the marginal
loyalty curve over CSI score dimension on the right.

The customers whose CSI scores arelocated in this interval of [ CS a , CSb ] are most attractive to be targeted for increasing theirloyalty. TCI is constructed in the same way as the highest probability density (HPD) region forBayesian confidence interval.

That is, we incorporate customers into TCI in order with higher

incremental loyalty until the interval contains 30% customers.

Table 5: EIL and TCI

The second column of Table 5 shows the TCI for individual companies.

Under theassumptions of limited access and identification of CSI scores of their customers, every
company can find the customers to be targeted by their loyalty program.

17

5. Concluding Remarks
This study investigated the effects of customer satisfaction on loyalty by focusing on
nonlinear characteristics represented as attainable limit of loyalty induced by satisfaction,
asymmetric response between satisfied and dissatisfied customers, and not-constant marginal
returns over the domain of satisfactions.

There are a few extant works on investigating therelation between satisfaction and loyalty, in particular, and this is the first model to measurenonlinear relation based on a uniform measure of customer satisfaction index in terms of systemequation by using structure that the loyalty is determined by customer satisfaction in theconnections of related other constructs.

As is discussed in Fornell (1992), the investigation byusing the system approach leads to higher reliability than the results obtained under theperspective being limited to two variables.
We introduced hierarchical Bayes modeling for estimation to improve the measurement byconsidering company heterogeneity, to which identical model structure must be applied.

In all,our study’s contributions to the modeling literatures are that (i) nonlinear term is embedded in
the structural model of customer satisfaction index, and (ii) hierarchical Bayes modeling of
nonlinear structural equation model for measuring customer satisfaction index to accommodate
heterogeneity of surveyed companies.

To our knowledge, this is the first study on nonlinear

structural equation model which includes nonlinear term of endogenous latent variable.

We

propose an efficient algorithm of MCMC, i.e., multi-move sampler for latent variables by using
Gibbs sampling.
In the empirical application, we compared comprehensive sets of specifications and theasymmetric nonlinear function with attainable limits is best supported by two kinds of criteria,
goodness of fit measures and the number of significant parameter estimates.

We obtained

managerial implications for loyalty management such as attainable limits; customer’s loss
aversion response; asymmetric marginal returns between satisfied and dissatisfied customers,i.e., increasing for dissatisfied customers and decreasing for satisfied customers, direct effect ofcustomer satisfaction is more significant than recommendation in general.

As managerial

implications, we derived the measures for efficient loyalty program by combining informationof estimated response curve of satisfaction to loyalty and empirical distribution of customers on
the dimension of CSI scores under assumptions of fully and limited access to customers.
Other studies by using not loyalty but other outcome, for example, willingness to pay(Homburg et al., 2005), suggested the inverse S-shaped function which means having negligible18change for customers with medium level satisfaction in consistent with the concept of zone oftolerance.

The inverse S-shaped function represents unrealistic situation since unlimited effectcan be expected for highly satisfied (delighted) customers.

Then the nonlinear function withneutral zone as well as attainable limits can be devised by modifying S-shaped function so thatit has three regimes by two additional parameters which split the domain of satisfaction to plugzone of tolerance at the mid regime, and loss and gain regimes with attainable limits at theextremes.

There is another nonlinear relationship between other constructs, for example,

Mittal, Ross and Baldasare (1998) showed nonlinear relations from attribute performance(perceived quality in CSI), to satisfaction, suggesting inverse S-shaped function for it.
nonlinear CSI model including additional nonlinear terms is also possible.
extensions for future research.

19

The

We leave theseReferences
Andrews, F.M. (1984), “Construct Validity and Error Component of Survey Measures,” Public

Opinion Quarterly, 48, 409-442.
Anderson, E. W., C. Fornell, and S. Mazvancheryl (2004), “Customer Satisfaction andShareholder Value,” Journal of Marketing, 68, 172–185.
Anderson, Eugene W. and Vikas Mittal (2000), “Strengthening the Satisfaction-Profit
Chain,” Journal of Service Research, 3 (2), 107-120.
Anderson, Eugene W. and Linda C. Salisbury (2003), “The Formation of Market-Level
Expectations and Its Covariates,” Journal of Consumer Research, 30 (June), 115-124.
Bloemer, J. and K. de Ruyter (1998), “On the Relationship between Store Image, Store
Satisfaction and Store Loyalty,” European Journal of Marketing, 32, 499–513.
Bolton, Ruth N. and Katherine N. Lemon (1999), “A Dynamic Model of Customers’
Usage of Services: Usage as an Antecedent and Consequence of Satisfaction,”

Journal of Marketing Research, 36 (2), 17-186.
Bowman, D. and D. Narayandas (2004), “Linking customer management effort to customer
profitability in business markets,” Journal of Marketing Research, 41, 433–447
Cooil, B., T.L. Keiningham, L. Aksoy and M. Hsu (2007), “A longitudinal analysis of customer
satisfaction and share of wallet: investigating the moderating effect of customer
characteristics,” Journal of Marketing, 71, 67–83.
Dong, Songting, Min Ding, Rajdeep Grewal, and Ping Zhao (2011), "Functional Forms of theSatisfaction-Loyalty Relationship," International Journal of Research in Marketing, 28 3850.
Eisenbeiss, M.,

M. Cornelißen. K. Backhaus and W.D. Hoyer (2014), “Nonlinear andasymmetric returns on customer satisfaction: do they vary across situations and consumers?,”

Journal of the Academy of Marketing Science, 42, 242–263.
Fornell, C. and B. Wernerfelt (1988), “A model for customer complaint management,”

Marketing Science, 7, 287-298.
Fornell, C. (1992), “A National Customer Satisfaction Barometer: The Swedish Experience,”

Journal of Marketing, 56, 1–21.
Fornell, C. and Y. Yi (1992), “Assumptions of the Two Step Approach to Latent Variable
20Modeling,” Sociological Methods and Research 20, 291-320.
Fornell, C., M.D. Johnson, E.W. Anderson, J. Cha, and B.E. Bryant (1996), “The American
Customer Satisfaction Index: Nature, Purpose, and Findings,” Journal of Marketing, 60, 7–
18.
Fornell, C., R.T. Rust, and M.G. Dekimpe (2010), “The Effect of Customer Satisfaction on
Consumer Spending Growth,” Journal of Marketing Research, 47, 28–35.
Homburg, C., N. Koschate and W.D. Hoyer (2005), “Do satisfied customers really pay more? A
study of the relationship between customer satisfaction and willingness to pay,” Journal ofMarketing, 69, 84–96.
Howard, J.A. and J.N. Sheth (1969), The Theory of Buyer Behavior, New York: John Wiley &Sons. Inc.
Gupta, Sunil and Valarie Zeithaml (2006), “Customer Metrics and Their Impact on
Financial Performance,” Marketing Science, 25 (6), 718-739.
Gupta, Sunil, Donald R. Lehmann, and Jennifer Ames Stuart (2004), “Valuing
Customers,” Journal of Marketing Research, 41 (1), 7-18.
Jones, T. O. and W. Earl Sasser (1995), “Why Satisfied Customers Defect,” Harvard Business

Review, 73, 88–99.
Khaneman, Daniel and Amos Tversky (1979) "Prospect Theory: An Analysis of Decision Under
Risk," Econometrica, 47, 2, 263-291
Keiningham, T. L., Perkins-Munn, T. and Evans, H. (2003), “The impact of customer
satisfaction on share-of-wallet in a business-to-business environment,” Journal of Service

Research, 6, 37–50.
Kumar, V., Ilaria Dalla Pozza and Jaishankar Ganesh (2013), “Revisiting the Satisfaction–
Loyalty Relationship: Empirical Generalizations and Directions for Future Research”,

Journal of Retailing, 89 246-262.
Lee, S. Y. (2007), Structural Equation Modeling: A Bayesian Approach, Wiley, New York.
Mittal, V. and W. A. Kamakura (2001), “Satisfaction, Repurchase Intent, and Repurchase
Behavior: Investigating the Moderating Effect of Customer Characteristics,” Journal ofMarketing Research, 38, 131–142.
Mittal, V., W.T. Ross, and P.M. Baldasare (1998), “The Asymmetric Impact of Negative andPositive Attribute-Level Performance on Overall Satisfaction and Repurchase Intentions,”

21

Journal of Marketing, 62 (January), 33–47.
National Quality Research Center (NQRC; 2005), “American Customer Satisfaction Index(ACSI): Methodology Report,” Stephen M. Ross School of Business at the University ofMichigan.
Ngobo, Paul-Valentin (1999), “Decreasing Returns in Customer Loyalty: Does It Really Matter
to Delight the Customers?”, Advances in Consumer Research, 26 (1), 469–76.
Oliver, R.L. (1980), “A Cognitive Model of the Antecedent and Consequences of Satisfaction
Decisions,” Journal of Marketing, 17, 460–469.
Oliver, R.L. (1981), “Measurement and Evaluation of Satisfaction Process in Retail Setting,”

Journal of Retailing, 17, 460–469.
Terui, N., S. Hasegawa, T. Chun, and K. Ogawa (2011), “Hierarchical Bayes Modeling of the
Customer Satisfaction Index,” Service Science, 3(2), 127–140.
Westbrook, R.A. and M.D. Reilly (1983), “Value-Percept Disparity: An Alternative to the
Disconfirmation of Expectation Theory of Consumer Satisfaction,” Advances in Consumer

Research, 10, 256-261.

22

Figure 1: Linear and Nonlinear CSI Model(a) Asymmetric Linear(b) Asymmetric Logit(c) Threshold Linear(d) Threshold Logit

23

Figure 2: Estimated Functional Form and Upper and Lower Limits

Figure 3: Distribution of Satisfaction Score

24

Figure 4: Marginal and Indirect Effects

Figure 5: Frequency and Loyalty Increment

25

Table 1：Model Comparison: DIC and log of Marginal Likelihood

Independent
DIC
LML

238539.1-98516.1

HB
LinearSymmetric Asymmetric
234960.5 234912.3-96844.0-96577.7

Threshold234963.2-96578.3

Logit
Symmetric Asymmetric Threshold
234908.7 234908.5 234937.7-96568.3-96575.5 -96570.8

Table 2: The Effect of HB modeling on Estimate of Path Coefficients

HB

Independent
Symmetric
Total

LinearAsymmetricThreshold

Symmetric

Logit
AsymmetricThreshold14.29% (27/189) 8.47%(16/189) 8.10%(17/210) 7.79%(20/231) 6.88%(13/189) 6.19%(13/210) 6.06%(14/231)

MobileTelecommunication

19.44%(7/36)

5.56%(2/36)

5%(2/40)

11.36%(5/44)

5.56%(2/36)

5%(2/40)

4.55%(2/44)

Convenience stores

4.44%(2/45)

2.22%(1/45)

2%(1/50)

1.82%(1/55)

2.22%(1/45)

2%(1/50)

1.82%(1/55)

Hotels

16.67%(18/108) 12.04%(13/108) 11.67%(14/120) 10.61%(14/132) 9.26%(10/108) 8.33%(10/120) 8.33%(11/132)

The number means the percentage of significant estimates in the model.

The ratio is given in parenthesis.

26

Table 3.1: Parameter Estimates (Company Level)
Company

β(-)

HPD

β(+)

HPD

Pr{β(-) >β(+)}

Band Width

α

HPD

M1
M2
M3
M4
Industry mean
C1
C2
C3
C4
C5
Industry mean
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
Industry mean

1.63
1.62
1.46
1.68
1.60
1.79
1.75
1.70
2.00
1.84
1.81
1.42
1.31
1.44
1.34
1.28
1.48
1.54
1.26
1.36
1.54
1.47
1.44
1.41

[0.98, 2.28][1.06, 2.18][0.82, 2.04][1.09, 2.28]

0.91
0.82
0.82
0.99
0.88
1.43
1.57
1.40
1.77
1.71
1.58
1.19
0.93
1.06
0.95
1.10
0.87
1.06
0.79
1.11
0.92
1.10
0.99
1.01

[0.29, 1.56][0.25, 1.41][0.19, 1.44][0.42, 1.56]

91.12%
95.54%
89.28%
92.34%
92.07%
77.56%
66.70%
70.92%
69.46%
61.88%
69.30%
75.38%
85.44%
89.36%
89.58%
68.60%
97.66%
91.78%
89.30%
77.10%
95.36%
87.08%
91.80%
86.54%

1.27
1.22
1.14
1.33
1.24
1.61
1.66
1.55
1.89
1.78
1.69
1.31
1.12
1.25
1.15
1.19
1.18
1.30
1.03
1.24
1.23
1.28
1.22
1.21

0.22
0.24
0.24
0.23
0.23
0.30
0.15
0.22
0.26
0.25
0.24
0.26
0.31
0.28
0.29
0.35
0.21
0.33
0.31
0.22
0.36
0.27
0.24
0.29

[0.12, 0.33][0.15, 0.34][0.15, 0.34][0.13, 0.33]

Total mean

1.54

83.49%

1.33

0.26[1.18, 2.37][1.25, 2.26][0.99, 2.34][1.46, 2.54][1.30, 2.37][0.97, 1.88][0.83, 1.80][1.06, 1.83][0.92, 1.75][0.77, 1.76][1.08, 1.91][1.09, 2.02][0.75, 1.76][0.91, 1.83][1.06, 2.05][1.04, 1.92][1.03, 1.89][0.85, 2.01][1.06, 2.12][0.72, 2.08][1.23, 2.37][1.19, 2.25][0.74, 1.66][0.42, 1.44][0.63, 1.50][0.54, 1.38][0.57, 1.63][0.45, 1.28][0.58, 1.54][0.26, 1.30][0.63, 1.63][0.41, 1.42][0.62, 1.57][0.54, 1.46]

1.12[0.21, 0.39][0.07, 0.23][0.11, 0.35][0.17, 0.35][0.16, 0.35][0.16, 0.37][0.21, 0.44][0.20, 0.44][0.20, 0.39][0.23, 0.46][0.12, 0.31][0.23, 0.45][0.20, 0.42][0.13, 0.34][0.25, 0.48][0.17, 0.38][0.15, 0.34]

Table 3.2: Parameter Estimates (Industry Level)

The posterior mean of parameter estimate, and 95% HPD region are given for respective
parameters.

The table contains the column for band width

limits.

27

1 () 1 ()   for attainable22

Table 4: Marginal and Indirect Effects
Company
M1
M2
M3
M4
Industry mean
C1
C2
C3
C4
C5
Industry mean
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
Industry mean
Total mean

¼β(-)
0.41
0.41
0.37
0.42
0.40
0.45
0.44
0.42
0.50
0.46
0.45
0.36
0.33
0.36
0.34
0.32
0.37
0.38
0.32
0.34
0.38
0.37
0.36
0.35
0.39

¼β(+)
0.23
0.21
0.20
0.25
0.22
0.36
0.39
0.35
0.44
0.43
0.39
0.30
0.23
0.26
0.24
0.27
0.22
0.27
0.20
0.28
0.23
0.27
0.25
0.25
0.28

α
0.22
0.24
0.24
0.23
0.23
0.30
0.15
0.22
0.26
0.25
0.24
0.26
0.31
0.28
0.29
0.35
0.21
0.33
0.31
0.22
0.36
0.27
0.24
0.29
0.26

[d1,d2](CS) [d1,d2](CSI score) Customer Ratio[-1.64, 0.32][25.3, 54.4]
61.07%[-1.51, 0.00][22.7, 52.3]
45.41%[-1.32, 0.00][27.0, 45.9]
44.07%[-1.66, 0.59][21.5, 52.5]
65.00%
53.89%[-1.33, 0.87][22.4, 66.7]
77.85%[-2.27, 2.14][27.4, 80.8]
98.46%[-1.69, 1.39][ 8.8, 60.2]
90.28%[-1.70, 1.52][18.5, 81.3]
92.50%[-1.62, 1.51][31.2, 77.8]
90.00%
89.82%[-1.16, 0.75][42.9, 69.7]
65.33%[-0.43, 0.00][35.9, 45.5]
24.00%[-1.00, 0.00][21.6, 40.2]
34.33%[-0.76, 0.00][30.4, 43.2]
27.67%[ 0.00, 0.00][48.6, 48.6]
0.00%[-1.58, 0.40][24.9, 57.6]
64.00%[-0.78, 0.00][39.4, 51.3]
25.67%[-0.35, 0.00][46.6, 53.6]
18.00%[-1.35, 0.96][24.3, 65.0]
79.00%[-0.49, 0.00][43.7, 52.0]
17.67%[-1.11, 0.00][29.8, 49.5]
32.00%[-1.34, 0.44][32.4, 59.3]
56.33%
37.00%
52.79%

28

α×γ
0.07
0.12
0.15
0.13
0.12
0.13
0.05
0.06
0.11
0.08
0.08
0.13
0.22
0.21
0.17
0.20
0.14
0.21
0.19
0.16
0.26
0.20
0.17
0.19
0.15

[b1,b2](CS) [b1,b2](CSI score) Customer Ratio[-3.04, 2.37][ 4.6, 84.8]
98.21%[-2.40, 1.48][ 5.3, 81.2]
96.20%[-2.00, 1.09][17.2, 61.6]
87.25%[-2.38, 1.68][11.5, 67.6]
96.00%
94.41%[-2.50, 2.23][0.00, 94.0]
99.12%[-3.46, 3.35][13.0, 95.5]
99.34%[-3.36, 3.15][0.00, 89.6]
99.17%[-2.79, 2.65][0.00, 100]
100.00%[-3.05, 2.97][ 9.9, 99.6]
99.33%
99.39%[-2.22, 2.00][27.8, 87.2]
97.00%[-1.34, 0.56][15.8, 57.8]
74.00%[-1.51, 0.93][12.1, 57.5]
79.67%[-1.76, 1.22][13.6, 63.6]
87.67%[-1.45, 1.19][22.2, 70.1]
86.33%[-2.11, 1.35][16.1, 73,3]
91.67%[-1.64, 1.00][26,4, 66.4]
80.00%[-1.49, 0.42][23.8, 61.9]
64.00%[-1.83, 1.54][15.9, 75.1]
91.67%[-1.32, 0.00][29.6, 52.0]
41.33%[-1.67, 1.19][19.7, 70.7]
87.33%[-1.86, 1.29][24.6, 72.1]
88.00%
80.72%
87.78%

Table 5: EIL and TCI(30%)

29Appendix A: Full Description of Model and Inference Procedure
The CSI model assumes six latent variables i  (1i ,2i ,...,5i , i ) and these are extracted'

by 17 questions of survey.

For the vector of question items yi  ( y1i ,…, y17i ) , which are ordered'categorical variables, we first transform them into continuous data xi  ( x1i ,…, x17i ) following'

normal distribution. This transformation is conducted by data-augmentation whenyi  ( y1i ,…, y17i )' is given at the conditional posterior density in Appendix B.
The structural equation model has measurement model to extract the latent variables from
data, and structural model which describes the relation between latent variables.

Then we set

the measurement model for xi by factor model and we define the structure on the factors as
structural model.
(i) Measurement model
The observable vector of xi  ( x1i ,…, x17i ) has a factor analytic representation with six'

factorsx i    i   i , i  1, ...n(A1)where  represents the factor loading matrix, i  (i ,1i ,2i ,...,5i ) is factor score vector'

for i represents. The error term vector  i  (1i ,…,17i ) is assumed to follow'

 i ~ N17  0,   where   diag 1...,  17  .
(ii) Structural model
The structural equation model assumes that the factor scores i  (1i ,2i ,...,5i , i )'have the relation each other in terms of set of equations:
Perceived Quality: 1   16  1

(A2)

Perceived Value: 2   211   26   2

(A3)
30

Customer Satisfaction: 3   311   322   36   3(A4)

Recommendation Intention: 4   433   4(A5)

Customer Loyalty:

1
1 
1
1 


    53(  ) 1  I      544   5 (A6)1  exp  3  2 
1  exp  3  2 

5   53(  ) I 

where   1 ,...,  5  ' ~ N5  0,   ,   diag  1 ,..., 5  .
More specifically, the model describes that customer expectation  drives perceived
quality 1 and perceived value  2 .

These three latent variables next generate customer

satisfaction 3 , which directly affects recommendation intention 4 and customer loyalty 5 .
Equation (A6) indicates the extension of this study.

Bayesian Inference of Nonlinear SEM
The structural models (A2)-(A6) play a role of prior for the likelihood defined by themeasurement model (A1) for Bayesian inference.

The joint prior density of  1 , 2 ,..., 5 ,   is decomposed by using their recursive relation between endogenous'

latent variable 1 , 2 ,...,5  byp    p   p 1 |   p  2 | 1 ,   p 3 | 1 , 2 ,   p  4 | 3  p 5 | 3 , 4 (A7)

On the other hand, we denote the likelihood function for  conditional on parametersn   ,    and data x as p  x | ,    p  xi | i ,  , then full conditional posteriori 1density is as follows:(i) p  | x,    p   p[ ]  x |  ,  (ii) p 1 |  , x,    p 1 |   p[1 ]  x |  ,  (iii) p  2 | 1 ,  , x,    p  2 | 1 ,   p[2 ]  x |  ,  
31

(iv) p 3 | 1 , 2 ,  , x,    p 3 | 1 , 2 ,   p[3 ]  x |  ,  (v) p  4 | 3 , x,    p  4 | 3  p[4 ]  x |  ,  (vi) p 5 | 3 , 4 x,    p 5 | 3 , 4  p[5 ]  x |  ,  where p[ z ]  x |  ,   means the part of joint likelihood regarding the latent variable z.
These conditional posteriors are analytically evaluated to be normal distribution since both priorand likelihood functions are normal density.
Then, staring from initial value (0) (1(0) ,2 (0) ,...,5(0) ,  (0) )' , we iterate the Gibbs

sampling from the conditional posterior density to obtain the joint posterior density p  | x,   .
This is a single move-sampler for MCMC.
The multi-move sampler is available for our model by the use of recursive system of CSI
model to derive more efficient algorithm by using linearity of subsystem on1  1 , 2 ,3 , 4 ,   ' . That is, we set the joint prior density byp    p 1  p(5 | 3 ,4 )(A8)

and the conditional posterior density is obtained by multi-move sampler for 1 following by
Lee (2007) since full conditional posterior density is as follows:(i) p 1 | x,    p 1  p[1 ]  x |  ,  (ii) p 5 | 3 , 4 x,    p 5 | 3 , 4  p[5 ]  x |  ,  
The details of algorithm are described in Appendix B.
Appendix B: MCMC Algorithms
The prior setting and conditional posterior density are described in this appendix for our model.
The measurement model connecting observed data and latent variables in the form of factor
model (A1), and structural model relating latent variables (A2)-(A6) are compactly written by32 xhi   hhi   hi ; hi ~ N (0,  h )hi   hhi   h hi   hi  hhi   hi ;  hi ~ N (0,  h ),  hi ~ N (0,  h )



1
1 
1
1 
    h( 53) 1  I      h 54 h 4   h 5
1  exp  h 3  2 
1  exp   h 3  2 hi 5   h( 53) I 

(B1)

(B2)

(1) Prior Density
The diffuse priors are set on the model parameters, and these are shown in next table:
Parameter

Settinghk ~ N (0 ,V0 )0  0 K 1 , V0  I K K 100

hk ~ N (0 ,  hk H 0 )0  100, H 0  100  hk ~ IG( 0 ,  0 ) 0  2,  0  2

  hj ~ N (  ' j z h ,  hj H  0 )H  0  I J  J  100  hj ~ IG (  0 ,   0 )  0  2,  0  2

 h ~ IW ( R01 ,  0 )R01  3, 0  1

 | V   vec | V ~ N vec ,V  D   0 H Z , D  I H H  100

Vh ~ IG (v 0 , V 0 )v 0  2, V 0  2





1



where  k is the k th row of  ,   k is k th element of   ( k  1,  ,17) ,   j is the jth row of  , and   j is j th element of ( j  1,  , 5) . H means the number ofcompanies, and V h is covariance matrix of path coefficient in structure model of company h.
is the number of path coefficients.

L

Z is the number of variables of attributes for company, and weuse industrial dummy variables for this.
33

(2) Conditional Posterior Density
In the below, the subscripts of heterogeneous parameters are not used for readability where
not confused.
(i) Measurement model(a)xi | yi ,  , , i ,  (i  1,  , n) (Data Augmentation)

We transform categorical data yi to continuous data xi by

xi ~ N [ yi 1 ,  yi ](   i ,  )(B3)

where the cut-off points vector  k  ( k1 ,  k 2 ,...,  k 8 ,  k 9 ) ' for the rating distribution ofquestion k are determined by n
 p    I ( yi  p ) / n  , p  1, , 9
 i 11where 1

(B4)

is the inverse of cumulative distribution function of standard normal distribution,and I ( yi  p ) is the indicator function, if yi  p , I ( yi  p ) = 1. However, we set0  , 10   .
(b) i | xi ,  , ,   ,  i ,   , ,(i  1,  , n)i  (1i ,2i ,3i ,4i ,5i ,  h ) '(b.1) i



[ 5]

 (1i , 2i , 3i , 4i ,  h )' :[ 5 ]i| xi[ 5] , [ 5] , ,  [ 5] ,  [ 5] , [ 5] , [ 5]i[ 5] ~ N   1  [ 5] [ 5] [ 5]
T

1


1

[ 5] T

1





T

1

[ 5] xi[ 5] ,  1  [ 5] [ 5] [ 5]( I   [ 5] ) 1 ( [ 5]  [ 5] T   [ 5] )( I   [ 5] ) T
Where   [ 5 ] T[ 5 ] T(I   ) [5] means the parameter and data matrix respect to  i[ 5 ] .
34

 
1

(B5)

( I   [ 5] ) 1  [ 5]  
 , and

(b.2)  i[ 5]   5i : | x ,  , ,  ,  ,  , , , 
5i

[5]
i

[5]

[5]







53

5i ~  1/  5  [5] ' [5] [5]
1


53

54

3i

5

4i

  /   '
1

5i

[ 5]

5

[5] 1 [5]ix, 1/   '
5

[ 5]

[5] 1 [5]





 
1

(B6)



1
1  ( )11
1  I     53
    54 4i
1  exp(3i ) 2 
1  exp(3i ) 2 

where 5i   53 I ()xi[ 5]  x12



[5]

x13

x14 ' , [ 5]  1 13,5

14 ,5 '00   12  0   13
0 
 0
0   14 (c) k | xi , , i ,  ( k  1,  ,17)



n





i 1



k ~ N (V01  n k1 )1 (V010  k1 ( xi   ki ), (V01  n k1 )1 (B7)(d)  k | xi , i ,  ( k  1,  ,17)k ~ N ( H 01   k1kk ) 1 ( H 01 0  k xk ), ( H 01   k1k k ) 1



(B8)


where k  (1k ,  , nk) , xk  ( x1k ,, xnk ) .

(e)   k | x i , , i ~ IG ( n / 2    0 ,   k ) ( k  1,  ,17)
where  k   0 

n

(B9)

( x      ) / 2.
2

ik

k

k

i

i 1

(ii) Structural model(e)  j | i ,  ( j  1,  , 5)

[1]j , where [1]j is the linear part, j=1,2,3,4, and [2] is the nonlinear part,[ 2]   

Let  j  35then we estimate [1]j and separately   .
[ 2]





(e.1) [1]k |  [1] , [1k] ,  [1] , z h , H [10] ~ N (a* k , [1k] A*k )

(B10)





*T
*T
where A* k  ( H [1]0 k 1  *Tk *k ) 1 , a* k  A* k H [1]0 k 1[1]k ' z   k  k , and  k is the k throw of  (1) , which corresponding to [1]k .k=1,2,3,4.
()(e.2) [2 ]  (0, 0,  53
|  53(  ) ,  54 , 0)

The prior distribution of  54 is N (' 54 z h , H [ 254] ) and then we have

 54 |  3i , 4i , 5i ,  53(  ) ,  53(  ) ,  5 ,  54 , z h , H ( 254)



 54 ~ N  H ( 254)   15 4i ' 4i
1

 H
1

( 2 ) 1

 54







1

'54 z h   15 4i '  5i  G  53(  ) ,  53(  ) , 3i ，H ( 254)   15 4i ' 4i

 
1

(B11)




1
1
1
1
( )
   (1  I )   53
 
 
 1  exp( 3i ) 2 
 1  exp(3i ) 2 

( )
( )
where G( 53,  53,3i )  I   53(  )  

(e.3)

For

 53(  ) and  53(  ) , the prior distribution is

N (' 53(  ) z h , H [ 253] ) andN (' 53(  ) z h , H [ 253] ) , they share a common prior variance. Then we have

 53(  ) |  3i (  ) , 4i , 5i ,  54 ,  5 ,   53(  ) , z h , H ( 253)

 53(  ) ~ N  H [ 253]   15 3(i ) ' 3(i )
1

 H
1

[ 2 ] 1 53

' 53(  ) z h   15 3(i ) '  5i   54 4i  ，H [ 253]   15 3(i ) ' 3(i )
1

 
1

(B12)

 53(  ) |  3i (  ) , 4i , 5i ,  54 ,  5 ,   53(  ) , z h , H [ 253]

 53(  ) ~ N  H [ 253]   15 3(i ) ' 3(i )
1

 H
1

[ 2 ] 1 53

' 53(  ) z h   15 3(i ) '  5i   54 4i  ，H [ 253]   15 3(i ) ' 3(i )
1

(B13)
(f)   j |  , i ( j  1,  , 5)

36

 
1



n





i 1



  j ~ IG  0  n / 2,  0  ( j   ji )2 / 2 for j  1, , 4 , (B14)


nr





i 1



  5 ~ IG  0  n / 2,  0  (5  G ( 53(  ) ,  53(  ) ,3i )   54 4i ) 2 / 2 (B15)

where nr is the number of rows of i .
(g)  | i

 ~ IW (   R01 , n   0 )(B16)

where   (1 ,, n ) .
(iii) Hierarchical Bayes Regression(h)  | V ,  , zh   vec   | V ,  , zh  ~ N d,V W
where W   zh ' zh  D  ,1 h  1,2...21

(B17)

 

d  vec( D ) , D  W 1 zh ' B  D vec    , B    1 ,  2 ,... 21  ',







 h    16, 21 ,... 53(  ) ,  53(  ) ,  54 



1
h

(i) V


T 



 z h '   h  z h '  Lh| , z h ,  h , v 0 ,V 0 ~ Gamma v0  ,V 0 

2
2






37

(B18)