Discussion Paper No.100
Threshold Measurement Model for
Perceived Service Quality
Toshikuni Sato
August, 2019

 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Threshold Measurement Model for Perceived Service Quality
Toshikuni Sato**Graduate School of Economics and Management, Tohoku University
Abstract
This study proposes a threshold measurement model based on the prospect theory
and zone of tolerance for the SERVQUAL scale to measure the latent perceived
service quality. The concept of zone of tolerance is where customers are willing toaccept a service discrepancy within a standard they recognize. The discussion
focuses on three stages of consumers’ mental state and how they relate to observable
perceived service quality. It then proposes a model that employs a thresholdspecification representing extent limit as a zone of tolerance. Because the valuefunction in prospect theory describes human perception’s dependence on theevaluation of differences, rather than absolute magnitudes, the proposed mode also
integrates asymmetric and nonlinear properties. Empirical analysis was
implemented using the data collected from several different service sectors, and the
proposal model showed better performance as against other competitive models. The
results provide an insight into the asymmetric and nonlinear latent structures ofconsumers’ perceived service quality. Clustering was conducted by applying
estimated thresholds and factor scores to obtain three different kinds of consumer
segments.
Keywords: Nonlinear measurement model, Nonlinear factor analysis, Measuring
perceived service quality, Zone of tolerance, Prospect theory11. Introduction
The method of measuring service quality is an essential topic in management because theperceived service quality influences customer satisfaction in consumer behaviors (Cronin et al.
2000). If managers or marketers can obtain quality understanding of their consumers’ perceived
service quality, then the company can compare their position with its competitors. In the field of
marketing, the SERVQUAL model (Parasuraman et al. 1985, 1988) is a primary method thatutilizes measurement scale to measure the perceived service quality. Although there are many
theoretical and statistical issues in the SERVQUAL scale, it fundamentally contributes to theexisting service quality models (e.g., Cronin & Taylor 1992, Rust & Oliver 1994, Brady &Cronin 2001, Brady et al. 2002, Kang & James 2004).
The perceived service quality in SERVQUAL is defined as a discrepancy between
expectations and perceived performances; therefore, the measurement scale of SERVQUAL iscalled the “difference score”. Utilizing the difference score to measure perceived service quality
is one of the issues in SERVQUAL, and the discussions have been ongoing for quite some time(e.g., Cronin & Taylor 1992; 1994, Parasuraman et al. 1994a; 1994b, Brady et al. 2002, Carrillat
et al. 2007). Nearly all previous discussions regarding the issue of difference score have
implicitly assumed linearity when observing the perceived service quality. Hence, theSERVQUAL model and other models are also defined within the linear measurement model
based on the simple Classical Test Theory (CTT) (Traub 1997; Novick 1966; Lewis 2007).
In contrast, the prospect theory represents human judgments and perceptions as attuned tothe evaluation of changes or differences, rather than absolute magnitudes (Kahneman & Tversky
1979, p.277), and defines value function with nonlinearity and asymmetry properties(Kahneman & Tversky 1979, p.279). Moreover, consumers may approve of the differences
because they have a “zone of tolerance,” defined as the extent to which customers recognize andare willing to accept the service discrepancies (Zeithaml et al. 1993, p.6). Because previous
studies have not sufficiently discussed the relationship between these topics and themeasurement models, it is necessary to address in the nonlinear mental process for perceived
service quality evaluations.
In this paper, we discuss the functional form of the measurement model for observable
perceived service quality, and reconsider the practical applications of the SERVQUAL modelwith difference score. Section 2 summarizes related literatures; section 3 introduces someextended SEVQUAL models. Section 4 presents the empirical results using the data from several
service industries. Finally, the results from proposed models and future scope are discussed in
section 5.

2. Related Literature
2.1. SERVQUAL Model
Service quality has different characteristics when compared with the quality of goods. The threebasic characteristics of service quality are: intangibility, heterogeneity, and inseparability(Parasuraman et al. 1985, p.42; 1988, p.13). These characteristics make it difficult to measureservice quality, thus inspiring many researchers to conceptualize a plethora of service quality
models (e.g., Wolfinbarger & Gilly 2003; Parasuraman et al. 2005; Kang 2006; Lin & Hsieh
2011; Orel & Kara 2014; Blut 2016). The SERVQUAL method, which was developed in linewith the expectation disconfirmation theory (Oliver 1980), is the first attempt to overcome these
difficulties; Martínez & Martínez (2010) summarize the other representative service quality
models (see also Grönroos 1984, Cronin & Taylor 1992, McDougall & Levesque 1995, Rust &Oliver 1994, Dabholkar et al. 1996, Brady & Cronin 2001; Kang & James 2004).
The SERVQUAL scale constitutes 22 questionnaires for each expectation and actual
perception. Difference score is then calculated by subtracting the expectation score from theperception score. The SERVQUAL model identified as a factor analysis model with fivedimensions (Figure 1). Although Parasuraman et al. (1993, 1994a, 1994b) confirm the validity2

of the SERVQUAL scale and model, the issues in this method have been widely discussed
among many researchers (e.g., Babakus & Boller 1992; Cronin & Taylor 1992, 1994; Brown etal. 1993; Peter et al. 1993; Carman 1990; Prakash 1984). This paper briefly divides these issues
into two parts, and suggests additional problems.
Figure 1: SERVQUAL model
2.2. Issues in SERVQUAL
The first issue is the measurement of service expectations. Based on the expectation
disconfirmation theory and the assumption that consumers evaluate service quality depends ontheir subjectivity, Parasuraman et al. (1986) define perceived service quality as being result of acomparison between consumer expectation and the actual service performance. However, thedifference score makes it difficult to specify the dissimilarities between service quality andsatisfaction (Cronin & Taylor 1992), and results in a reduction of the reliability coefficient(Prakash 1984, Peter et al. 1993).
The second issue is the instability of dimensions. Although a factor analysis model requires
original dimensions when the measurement scales being used, the SERVQUAL model with thedifference score often provides different dimensions from the original five (Babakus & Boller
1992; Cronin & Taylor 1992,1994). This issue implies that the construct validity, such as theconvergent and discriminant validity of SERVQUAL, is not sufficient. Therefore, Cronin andTaylor (1992; 1994) recommend a performance-only measurement, i.e., SERVQUAL scale
without the expectation score, because the SERVPERF model is specified by a one-factor modelwith this measurement and reports better results when compared with the difference score(Cronin & Taylor 1992; 1994).
2.3. Nonlinearity and Zone of Tolerance
Few researchers discuss the nonlinear and asymmetric properties of perceived service quality.
Based on the context of prospect theory, Mittal et al. (1998) examine the nonlinear effects of
attribute-level performance on the overall satisfaction for services and products. They mention
the possibility that the relationship between SERVQUAL dimensions and the overall quality is
nonlinear and asymmetric (Mittal et al. 1998, p.34). Sivakumar et al. (2014) discuss thetheoretical application of the prospect theory regarding the perceived service quality withexpectations. They define service failure and delight as, service performances that fall below
expectations and exceed expectations, respectively (Sivakumar et al. 2014, p.41). This is in linewith expectation disconfirmation theory proposed by Parasuraman et al. (1985, p.48; 1988).
According to the prospect theory, the mental process of service failure and delight
communicate the value function of the observable perceived service quality. The function isdefined as concave for service delights and convex for service failures; the impact of service
failures is more than that of service delights (Sivakumar et al. 2014; Kahaneman & Tversky1979). Moreover, Zeithaml et al. (1993) and Parasuraman et al (1993) discuss the zone oftolerance. It is defined as a mental space between the adequate and desired service, which is thestandard of services that the customer will accept and hopes to receive. The zone of tolerance
indicates that a consumer’s mental space of perceived service quality has thresholds where they
are wiling to accept the discrepancy. Although Teas (1993) proposed a modified SERVQUAL
scale, comprising the measurement of ideal points corresponding to the thresholds, it would also
be impactful to consider specifying the zone of tolerance as a model of measurement.
In spite of these two important mental properties, the original SERVQUAL model has been
misspecified by linear measurement model; hence, the nonlinear measurement model should be
investigated. The next section focuses on the nonlinearity and threshold for perceived service
quality, and discusses the marketing applications of the SERVQUAL model with difference
scores. A few nonlinear SERVQUAL models with threshold specifications based on the prospect3

theory and zone of tolerance are also proposed. Results of the empirical analysis provides aninsight on the performance of proposed models and practical applications.
3. Model Development
3.1. Basic Concepts
According to the CTT, the linear measurement model with a construct is defined as the following
equation:z ji  a j t i   ji ,(1)where 𝑖 is the number of individuals, 𝑗 is the number of items, 𝑧𝑗𝑖 is the observed score, 𝑡𝑖
is the true score, 𝜀𝑗𝑖 is the measurement error, and 𝑎𝑗 is the item discrimination that indicatesthe effectiveness of the construct to the 𝑗th item. The true score replaces the latent variable, anda linear factor analysis is adapted to estimate this model. In contrast, this paper considers thelatent nonlinear mental process between observed and latent variables as follows:z ji  f  ti    ji .

(2)

To introduce the properties of prospect theory and zone of tolerance to the SERVQAL
measurement model, the observed perceived service quality is through a nonlinear andasymmetric process when the latent discrepancy crosses the thresholds. Three types of difference
scores are subsequently observed as perceived service qualities based on the value function withthresholds. The three types of difference score are as follows:
i. A positive difference score is observed when the latent positive discrepancies (service
delights) cross over the positive threshold.
ii. A negative difference score is observed when the latent negative discrepancies (service
failures) cross over the negative threshold.
iii. A difference score of 0 is observed when a consumer does not recognize the discrepancies
or the latent discrepancies within the thresholds.
The proposed model uses the second-order SERVQUAL model (Figure 2) to express theaforementioned assumptions, and modifies this model based on a nonlinear factor analysis
model (e.g., Zhu & Lee 1999).
Figure 2: Second order factor model for proposed model
3.2. Base Model for Proposed Model
The base model for second-order SERVQUAL (for 𝑖 = 1, … , 𝑛) is defined asy i  Λω i  ε i ,ω i  G ζ i   τ i ,
𝑇

(3)
(4)

where 𝐲𝑖 = {𝑦𝑖,1 , ⋯ , 𝑦𝑖,22 } is a (22 × 1) random vector of observed variables for difference
scores to measure “Tangibles (𝑗 = 1, ⋯ ,4) ,” “Reliability (𝑗 = 5, ⋯ ,9) ,” “Responsiveness(𝑗 = 10, ⋯ ,13) ,” “Assurance (𝑗 = 14, ⋯ ,17) ,” and “Empathy (𝑗 = 18, ⋯ ,22) ,” 𝚲 is a𝑇
(22 × 5) factor loading matrix, 𝛚𝑖 = {𝜔𝑖,1 , ⋯ , 𝜔𝑖,5 } is a (5 × 1) random vector of firstorder latent variables corresponding to “Tangible (𝑘 = 1) ,” “Reliability (𝑘 = 2) ,”“Responsiveness (𝑘 = 3) ,” “Assurance (𝑘 = 4) ,” and “Empathy (𝑘 = 5),” 𝛆𝒊 is a random

4vector of error measurements assumed as 𝛆𝒊 ~𝑖. 𝑖. 𝑑. 𝑁(𝟎, 𝚿𝝐 ), 𝚿𝝐 = 𝑑𝑖𝑎𝑔{𝜓𝜖,1 , ⋯ , 𝜓𝜖,22 }. For
the second measurement equation, 𝐆( ) is a function proposed in the next sections, and 𝜻𝑖 isa (1 × 1) second-order latent variable defined as baseline quality that indicates a latentcommon discrepancy and assumed as 𝜁𝑖 ~𝑖. 𝑖. 𝑑. 𝑁(0, 𝜎𝜁2 ) , 𝛕𝑖 is a random vector of error
measurements assumed as 𝛕𝑖 ~𝑖. 𝑖. 𝑑. 𝑁(𝟎, 𝚫τ ), 𝚫𝜏 = 𝑑𝑖𝑎𝑔{𝛿𝜏,1 , ⋯ , 𝛿𝜏,5 }, and 𝛜 ⊥ 𝚭 ⊥ 𝛕.
In this model, the first corresponding elements of 𝚲 between observed variable and latent
factor is fixed by 1, and the other corresponding and remaining elements of 𝚲 are freeparameters and the reaming elements of 𝚲 are fixed by 0, respectively. The linear model forsecond-order equation is defined as 𝐆(𝜻𝑖 ) = 𝚪𝛇𝑖 , where 𝚪 is a (5 × 1) matrix of factor
loadings. The above-stated model can be expressed asy i  Λ G  ζ i   τ i   ε i  ΛG  ζ i   Λτ i  ε i .

(5)Equation (5) explains that the proposed model is modified by adding a nonlinear term instead ofassuming the factor correlations in the original linear SERVQUAL model. The next section
proposes a few assumptions for 𝐆.
3.3. Proposed Model
To express the zone of tolerance, let 𝜂 + and 𝜂 − be a positive and negative threshold parameter,
respectively. The threshold logistic model (TLGM) is defined as

1
1 

G  ζi ; Γ  , Γ  ,  ,    Γ  I  ζi     0  1  exp   ζi    2 
,


1
1


 Γ  I  ζi     0   1  exp   ζi    2 

(6)where 𝑰 is the indicator function taking 1 if a condition in ( ) is satisfied. 𝚪 + =𝑇
𝑇{𝛾1,𝑘 + , ⋯ , 𝛾1,5 + } and 𝚪 − = {𝛾2,𝑘 − , ⋯ , 𝛾2,5 − } are assumed to be service delight and failure
parameters, respectively. This model is specified by a logistic function because it uses one of
the “S”-shaped curves as a value function, where 𝚪 − is expected to be larger than 𝚪 + . The
estimates for 𝜂 + and 𝜂 − , which correspond to a lower and upper limits for zone of tolerance,
represent a level for adequate and desired services, respectively.
In addition, the other two threshold models and three asymmetric models are considered toinvestigate a better functional form. The threshold linear model (TLM) and threshold quadraticmodel (TQM) are defined asG  ζi ; Γ  , Γ  ,  ,    Γ  I  ζi     0 ζi     Γ  I  ζi     0 ζi    

,


 .
 Γ I  ζ    0    ζ    G  ζi ; Γ  , Γ  ,  ,    Γ  I  ζi     0   ζi    




i

5

2



i

(7)

2

(8)

The asymmetric linear model (ALM), asymmetric quadratic model (AQM), and asymmetric
logistic model (ALGM) are defined asG  ζi ; Γ , Γ   Γ I  ζi  0ζi   Γ I  ζi  0ζi  ,(9)G  ζi ; Γ , Γ   Γ I  ζi  0 ζi2   Γ I  ζi  0  ζi2  ,

(10)


1
1
G  ζi ; Γ  , Γ    Γ  I  ζi  0   1  exp   ζi  2 
11 Γ  I  ζi  0   1  exp   ζi  2 

.

(11)

Table 1 summarizes all of the prosed models for model comparison, and Figure 3 shows eachfunction described by the threshold and asymmetric models.
Table 1: Summary of the proposed models
Figure 3: Proposed functions

4. Empirical Applications
4.1. Data Description
The data were gathered through a research company from two types of hotels, banks, and retailstores in Japan. The questionnaires were referred to Parasuraman et al. (1988; 1991; 1994b), anda total of 300 respondents were gathered in each service industry. Hotel B is a business hotel
offering select services in low prices. Hotel A is a city hotel with some restaurants and shops
located near a large station. Bank B is a local bank focusing on local customers and companies.
Bank A is a megabank providing diverse services in domestic and overseas market. Retail B isa supermarket primarily selling commodities and food. Retail A is a department store withseveral specialty shops.
4.2. Model Estimation
The proposed models used the Bayesian estimation via Markov chain Monte Carlo (MCMC)
algorithm with Gibbs sampling for estimation. In Gibbs sampling, conditional distributions foreach random variable are considered; therefore, the algorithm of nonlinear factor analysis modelis almost the same as that of the linear factor analysis model (Zhu & Lee 1999, Lee 2007), which
is a major advantage of the Bayesian approach with the Gibbs sampler. However, simulatingfrom p(𝛇𝑖 | −), 𝑝(𝜂 + | −), and 𝑝(𝜂 − | −), which are nonstandard and complex, is not an easytask. Hence, to simulate from these distributions, the random walk Metropolis-Hastings (RWMH) method is used within the MCMC algorithm.
4.3. Model Comparison
Tables 2 and 3 report that the model fits for each model were evaluated using the widely
applicable information criterion (WAIC) (Watanabe 2010a, 2010b, Gelman et al. 2013) and thewidely applicable Bayesian information criterion (WBIC) (Watanabe 2013). These indexes
represent an information criterion for model selection in terms of prediction and the logarithmof Bayes marginal likelihood, respectively. The smaller WAIC indicates a more accurate model.
The WBIC is interpreted as a minus logarithm of Bayes marginal likelihood (Watanabe 2013),
so that the smaller WBIC also suggests a better model fitting.

6

Although the TLM is supported in Bank A and Retail B by the WAIC, the TLGM displays
better WBIC for all service industries. This result also indicates that consumers’ perceived
service qualities are driven by latent nonlinear structure along with the thresholds.
Table 2: WAIC
Table 3: WBIC
4.4. Estimation Results
The Appendix provides the estimates of parameters in the TLGM, and Tables 4 and 5 show theestimated asymmetric and threshold parameters. Figure 4 describes the estimated function anddistribution of unique factors for each SERVQUAL dimension.
Table 4: Estimated threshold parameters
Table 5: Estimated asymmetric parameters
Figure 4: Estimated function and uniqueness
Table 4 provides estimates for threshold parameters and the ranges reveal the latent zone oftolerance. This result indicates that consumers evaluate perceived service quality with anacceptable discrepancy between expectations and perceptions. The larger the positive threshold,the more difficult it is for consumers to experience service delight. In contrast, at a smaller
negative threshold, consumers find it easier to tolerate service failure. When comparing theabsolute value of each estimated threshold parameter in Table 4, the negative threshold in Hotel
A, Bank B, Retail B, and Retail A are estimated to be larger than the positive threshold.
They ,therefore, obtained better results, while both thresholds mostly displayed similar estimatesin Hotel B and Bank A, respectively (see also Figure 4). The absolute value of estimates in Bank
A is the largest; thus, indicating that customers might accept discrepancies more easily in Bank
A. Hotel B is required to pay more attention to service failures because of the smallest absolutevalue of the negative threshold.
“Delight” and “Failure” in Table 5 indicate the estimates for delight and failure parameters,
and the standardized coefficients are shown in std.D and std.F. According to the 95% highest
probability density interval (HPDI), all estimates are not 0. P{D < F} shows that of the twoparameters, failure is greater than delight. Although some failure parameters are smaller thandelight parameters in Hotel B, Bank A, and Retail B, whereas all failure parameters are larger
than delight parameters in Hotel A, Bank B, and Retail A, which is parallel to the assumptionsof value function. These results indicate that delight and failure parameters primarily follow theprospect theory, and that service failure, which is a negative discrepancy, has significantly more
influence on the observed perceived service quality than service delight. Therefore, consumers’
evaluation process of perceived service quality has an asymmetric structure.
In Table 5, std.U indicates standardized estimated variances of each uniqueness factor for
SEVQUAL dimensions that show the dependent efficacy of each SERVQUAL dimension. The
precisions of the five dimensions’ qualities are presumed to be unequal because corresponding
distributions look different. The distribution becomes flat if the factor’s uniqueness has a larger
effect, whereas it becomes shaper with a smaller effect (see Figure 4). Smaller uniquenessindicates that the sub-dimension depends on the higher dimension (common factor) rather thanon the uniqueness itself. On the contrary, larger uniqueness indicates that the sub-dimension has
some unique features in comparison to other sub-dimensions. For example, estimates for
Tangible in Hotel A (see Table 5) indicates that the Tangible factor is almost independent fromthe other factors, and has a larger effect than the baseline quality, so that it possesses larger
uniqueness than the other factors.

7

4.5. Segmentation for the Customer by Threshold Parameters
Figure 5 illustrates the proportion of segments for the customers in each service industry divided
by the threshold parameters. P.PSQ, or the top portion of the bar plots, describes the proportionof customers whose baseline quality (second-order factor) score exceeds the positive threshold,indicating that the customers perceived positive service quality. S.PSQ, in the middle of the bar
plots, indicates the class of customers whose baseline quality score is inside both positive andnegative threshold parameters, whereas N.PSQ, at the bottom of the bar plots, indicates the classof customers whose baseline quality score is less than the negative threshold parameter. Theseplots enable the comparison of potential perceived service quality for each service industry that
does not meet customer expectations.
For example, over 30 % of customers in Hotel A perceived that services exceeded
expectations. Bank A achieved better service perception than other service industries; however,almost all customers might evaluate that the service is neither good nor bad because of highly
proportion of S.PSQ. In Hotel B, the each segment is divided as almost equally, and the
proportion of customers who perceived negative service quality is the largest among theseindustries, which suggests that it may be useful to improve their services.
5. Implications and Conclusions
Three possible implications from the proposed model are investigated and future research is
discussed in this study.
First, the common nonlinear effects and independent linear effects of each SERVQUAL
dimension are estimated using the second-order factor analysis with nonlinear structure. In
addition, a nonlinear structure for customers’ perceived service quality is established bycomparing several nonlinear measurement models. Second, a comparison of different
magnitudes of effects between service delights and failures is possible by estimating theasymmetric parameters. Third, considering the threshold parameter in the measurement model,
it is possible to estimate consumers’ zone of tolerance. Moreover, the properties of the proposedmodel can be visualized by constructing a plot, as shown in Figure 4. The threshold parameters
are also helpful in classifying the customers into three categories as in Figure 5.
In this study, the nonlinear and asymmetric measurement model with threshold isestablished to measure the perceived service quality. Finally, the threshold logistic model is
specified, and demonstrates better results when compared with the original SERVQUAL modeland the other candidate models. Moreover, using the difference score enables a properinterpretation of the threshold logistic model because both the prospect theory and zone oftolerance assume the evaluation with some reference point, such as expectation. Additional workis warranted to develop a nonparametric measurement model to explore and estimate thefunctional form directly. Finally, the construct validation must be extended to confirm thevalidity for the nonlinear measurement model. Future research can focus on investigating thoseissues.

8

Figures and Tables
Figure 1: SERVQUAL model

Figure 2: Second order factor model for proposed model

Note: the observed variables and error variables are abbreviated.

9

Figure 3: Proposed functions

10

Figure 4: Estimated function and uniqueness

11

12

13

Figure 5: Proportion of segments

Model
1_factor
Original
2nd_order
ALM
AQM
ALGM
TLM
TQM
TLGM

Table 1: Summary of the proposed models
Nonlinearity
AsymmetryThreshold
No
No
No
No
No
No
No
No
No
No
Yes
No
Yes
Yes
No
Yes
Yes
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes

Function
LinearLinearLinearLinearQuadratic
Logistic
LinearQuadratic
Logistic

Note: 1_factor is the first-order factor analysis model with only one latent variable. Original is
the SERVQUAL model proposed by Parasuraman et al. (1988), and 2nd_order is the linear
second-order factor analysis model.

14

Table 2: WAIC
Hotel B

original
15545.63

1_factor
16687.35

2nd_factor
15528.51

ALM
15515.58

AQM
15512.13

ALGM
15513.99

TLM
15487.12

TQM
15528.36

TLGM
15456.88

resultTLGMHotel A

14341.73

15511.40

14334.90

14317.28

14322.45

14324.11

14288.40

14321.04

14281.28

TLGM

Bank B

16057.84

17047.89

16041.60

16007.22

15997.80

16012.55

15971.49

16003.18

15968.51

TLGM

Bank A

14261.95

15428.69

14252.64

14254.08

14227.90

14273.41

14183.03

14227.22

14197.54

TLM

Retail B

15759.58

16983.10

15751.12

15741.73

15717.25

15793.99

15694.44

15721.36

15751.09

TLM

Retail A

14854.60

15787.66

14848.88

14846.96

14825.1014850.17

14801.11

14834.09

14787.83

TLGM

Table 3: WBIC
Hotel B

original
7402.83

1_factor
8219.37

2nd_factor
7420.27

ALM
7389.84

AQM
7380.46

ALGM
7348.89

TLM
7361.30

TQM
7387.29

TLGM
7331.83

resultTLGMHotel A

6818.23

7666.57

6812.966782.97

6799.916752.826789.55

6811.246746.56

TLGM

Bank B

7649.33

8415.567656.097618.587619.35

7605.237601.537615.127568.46

TLGM

Bank A

6756.167587.34

6736.316745.136733.92

6701.566717.43

6740.276682.58

TLGM

Retail B

7490.738380.04

7482.02

7475.157485.33

7476.887463.937478.277456.48

TLGM

Retail A

7090.217796.33

7069.997078.227047.49

7016.30

7043.057049.40

7007.85

TLGM

Table 4: Estimated threshold parameters

Hotel B
Hotel ABank B
Bank A
Retail B
Retail A

[
[
[
[
[
[

negative
positive-0.191 , 0.193 ]-0.413 , 0.151 ]-0.518 , 0.193 ]-0.746 , 0.632 ]-0.427 , 0.188 ]-0.406 , 0.233 ]15

Table 5: Estimated asymmetric parameters
95%HPDI

Delight

95%HPDI

FailureP{D < F}

std.D

std.F

std.U

Hotel B
Tangilbles←BQ

1.000

1.737

[

1.019

,

2.438

]

0.984

0.200

0.342

0.838

Reliability ←BQ

1.299

[

0.790

,

1.878

]

1.496

[

0.975

,

2.055

]

0.709

0.340

0.389

0.726

Responsiveness←BQ

2.146

[

1.539

,

2.784

]

2.143

[

1.568

,

2.781

]

0.497

0.453

0.449

0.588

Assurance←BQ

2.122

[

1.538

,

2.781

]

1.905

[

1.325

,

2.469

]

0.293

0.462

0.411

0.612

Empathy ←BQ

2.003

[

1.350

,

2.648

]

1.839

[

1.260

,

2.553

]

0.352

0.405

0.369

0.693

Hotel A
Tangilbles←BQ

1.000

1.745

[

1.112

,

2.424

]

0.993

0.245

0.374

0.796

Reliability←BQ

1.692

[

1.215

,

2.206

]

2.183

[

1.631

,

2.726

]

0.932

0.431

0.493

0.567

Responsiveness←BQ

1.987

[

1.480

,

2.522

]

2.818

[

2.217

,

3.409

]

0.993

0.435

0.547

0.508

Assurance←BQ

2.399

[

1.740

,

3.023

]

3.397

[

2.704

,

4.133

]

0.986

0.412

0.517

0.558

Empathy ←BQ

1.835

[

1.269

,

2.408

]

2.648

[

2.035

,

3.355

]

0.981

0.376

0.481

0.622

Bank B
Tangilbles←BQ

1.000

1.351

[

0.467

,

2.212

]

0.779

0.184

0.207

0.918

Reliability ←BQ

1.129

[

0.542

,

1.685

]

1.615

[

0.988

,

2.280

]

0.888

0.307

0.369

0.761

Responsiveness←BQ

1.302

[

0.724

,

1.927

]

2.068

[

1.401

,

2.798

]

0.965

0.330

0.442

0.689

Assurance←BQ

1.680

[

1.006

,

2.409

]

2.277

[

1.557

,

3.072

]

0.875

0.346

0.395

0.717

Empathy ←BQ

1.366

[

0.734

,

2.052

]

2.239

[

1.470

,

2.979

]

0.966

0.298

0.412

0.734

Bank A
Tangilbles←BQ

1.000

1.882

[

1.141

,

2.685

]

0.990

0.185

0.322

0.858

Reliability←BQ

1.000

1.656

[

1.114

,

2.267

]

0.993

0.254

0.388

0.781

Responsiveness←BQ

2.608

[

1.889

,

3.328

]

2.659

[

1.924

,

3.445

]

0.544

0.434

0.413

0.636

Assurance←BQ

2.979

[

2.114

,

3.774

]

2.915

[

2.065

,

3.733

]

0.447

0.436

0.399

0.646

Empathy ←BQ

2.800

[

1.890

,

3.604

]

3.117

[

2.264

,

4.061

]

0.699

0.387

0.403

0.683

Retail B
Tangilbles←BQ

1.000

1.923

[

1.058

,

2.733

]

0.988

0.175

0.178

0.877

Reliability ←BQ

0.953

[

0.392

,

1.553

]

2.052

[

1.292

,

2.876

]

0.991

0.201

0.210

0.794

Responsiveness←BQ

1.359

[

0.854

,

1.869

]

1.694

[

1.159

,

2.273

]

0.843

0.409

0.402

0.640

Assurance←BQ

1.956

[

1.276

,

2.637

]

2.010

[

1.371

,

2.677

]

0.544

0.442

0.429

0.661

Empathy ←BQ

1.490

[

0.890

,

2.127

]

2.057

[

1.318

,

2.762

]

0.898

0.328

0.332

0.723

Retail A
Tangilbles←BQ

1.000

1.499

[

1.000

,

2.107

]

0.970

0.282

0.383

0.769

Reliability←BQ

2.173

[

1.530

,

2.779

]

2.650

[

1.951

,

3.278

]

0.858

0.425

0.476

0.588

Responsiveness←BQ

1.892

[

1.284

,

2.469

]

2.302

[

1.628

,

2.920

]

0.852

0.417

0.466

0.603

Assurance←BQ

2.428

[

1.788

,

3.123

]

3.223

[

2.483

,

3.987

]

0.952

0.414

0.504

0.570

Empathy ←BQ

1.866

[

1.225

,

2.460

]

2.138

[

1.527

,

2.803

]

0.746

0.387

0.407

0.679

Note 1: std.D, std.F, and std.U are standard coefficients for Delight, Failure, and Uniqueness,
respectively.
Note 2: Bank A fixed two parameters to avoid improper solutions, whereas the other industries
fixed only one parameter in factor loadings.

16

Appendix: MCMC Algorithm for Threshold Logistic Model
A.1. Details of Second-order Measurement Equation for Threshold Logistic Model
The second-order measurement equation in base model (4) can be expressed asω i  G(ζ i )  τ i ΓF(ζ i )  τ i

,

(12)

where,

  1,1Γ  
 1 ,2

 2,1 3,1 4,1 5,1  2,2 3,2 4,2 5,2 

,

(13)

F(ζi )   f   ζ i    , f   ζ i    ,

(14)


1
1
f  ( x)  I {x  0} ,
1  exp  x  2 

(15)


1
1
f  ( x)  I {x  0} .
1  exp  x  2 

(16)

T

T

and

A.2. Prior Distribution
Parameter

Settings
 j ,0  0 , H  . j ,0  I

 j |  . j ~ N  j , 0 ,  . j H  . j , 0   , j ~ IG   , j , 0 ,   , j ,0   , j , 0  0.01 ,   , j , 0  0.01

k |   .k ~ N k , 0 ,   .k H  ,k , 0 k , 0  0 , H  ,k , 0  I

  ,k ~ IG  ,k , 0 ,  ,k ,0  ,k , 0  0.01 ,  ,k ,0  0.01

  | Z ~ trN0 ,max Z 0 , 0 

0  0 ,  0  1

  | Z ~ trNmin Z,0 0 , 0 
A.3. Full Conditional Distribution[1]ω | Λ , Ψ , Δ , ζ , Γ , η , η , y 



i

ε

τ

i

i



1
N  Δτ 1  Λ T Ψε1 Λ   Λ T Ψε1 y i  Δ τ1 G i  ,  Δ τ 1  Λ T Ψ ε1 Λ   .
 Λ j |  , j , Ω , Y N  a  , j ,  , j A , j 
1

[2]



where, A  , j  H 1, j ,0  ΩΩT[3]

 and a
1

, j





n
IG   , j ,0  ,  , j  .
21
where,   , j    , j , 0  YTj Y j  a T , j A 1, j a  , j  Λ Tj.0 H Tj ,0 Λ j ,0 .
2



17

(18)

 A  , j H ,1j ,0 Λ j.0  ΩYj .

  , j | Ω , Y 



(17)

(19)

Set 𝑡(= 1, ⋯ , 𝑇) as a number of MCMC iterations and the RW-MH algorithm for𝑝(𝜁𝑖 |𝚪, 𝚫𝜏 , 𝜎𝜁2 , 𝛚𝑖 , 𝜂 + , 𝜂 − ) , 𝑝(𝜂 + |𝚭, 𝚪, 𝚫𝜏 , 𝜎𝜁2 , 𝛀, 𝜂 − ) , and 𝑝(𝜂 − |𝚭, 𝚪, 𝚫𝜏 , 𝜎𝜁2 , 𝛀, 𝜂 + ) arefollowing
[4]

ζi t

N  ζi t 1 ,  ζ  .

(20)

The probability of accepting is p  ζi t | Γ , Δ ,  2 , ωi ,  ,   
min 
,1 .
t 1
2


 p  ζi | Γ , Δ ,   , ωi , ,  


 


,  .

[5]  ,ttrN0 ,max  Z k , t 1 ,  .

[6]

  ,ttrNmin Z,0

 , t 1

(21)

(22)
(23)



The probability of accepting is p   , t ,  , t | Γ , Δ ,  2 , Ω , Z  p   , t 1   , t  p   , t 1   , t  
min 
,1 .
 , t 1,  , t 1 | Γ , Δ ,  2 , Ω , Z  p   , t   , t 1  p  , t  , t 1   p 

(24)𝜎𝜁 , 𝜈𝜂+ , and 𝜈𝜂− are step-size parameters which are given so that each acceptance rate
becomes approximately 0.25 (Gelman et al. 1995; Zhu & Lee 1999).
[7]N  a  ,k ,  ,k A ,k  .

Γk |  ,k , Z , Ω ,  ,Ω  .

where, A  ,k  H ,1k ,0  Fk ,k 1FTk ,k 1and a  ,k  A ,k H ,1k ,0 Γ k .0  Fk ,k 1
[8]

(25)

1

k

n
IG   ,k ,0  ,  ,k  ,21
where,  ,k   ,k , 0  ΩTk Ω k  a T,k A 1,k a  ,k  ΓTk.0 H T,k ,0 Γ k , 0 .
2 ,k | Z , Ω ,  



(26)



The above results are valid for situations where all elements of 𝚲 and 𝚪 are freeparameters. As an example, consider that 𝚲𝑗𝑇 and the 𝑗th row of 𝚲 contain fixed parameters.
Let 𝒄𝑗 be the corresponding 1 × 𝑞 row vector such that 𝑐𝑗𝑘 = 0 if 𝜆𝑗𝑘 is a fixed parameter,and 𝑐𝑗𝑘 = 1 if 𝜆𝑗𝑘 is an unknown parameter. As for 𝑗 = 1, ⋯ , 𝑝 and 𝑘 = 1, ⋯ , 𝑞 , let 𝑟𝑗 =𝑐𝑗1 + ⋯ + 𝑐𝑗𝑞 be the number of unknown parameters in 𝚲𝑗𝑇 , 𝚲𝑗∗𝑇 be a 1 × 𝑟𝑗 row vector that
contains the only unknown parameters in 𝚲𝑗𝑇 , and 𝛀𝑗∗ be a 𝑟𝑗 × 𝑛 submatrix of 𝛀 such that∗∗
all the rows corresponding to 𝑐𝑗𝑘 = 0 are deleted. Let, 𝐘𝑗∗𝑇 = (𝑦1,𝑗, ⋯ , 𝑦𝑛,𝑗) with yi , j    j ,k k ,i 1  c jk  .
q

y

*
i, j

k 1

18

(27)

Equation (27) subtracts the constant terms from 𝐘𝑗 . Hence, the conditional distributions with
𝚲𝑗 , 𝐘𝑗 , and 𝛀 in part of (𝚲, 𝚿𝜖 ) must be replaced by 𝚲𝑗∗ , 𝐘𝑗∗ , and 𝛀𝑗∗ , respectively. This++
procedure is also adapted in full conditional distribution for (𝚪, ∆𝜏 ) because 𝛾1,1and 𝛾1,2
are+fixed by 1 in Bank A, and 𝛾1,1 is fixed by 1 in the other industries to avoid improper solutions.
Moreover, 𝜎𝜁2 , the variance of 𝜁𝑖 , is fixed by 1 to identify this model. The basic and related
algorithm is explained in Xing et al. (2016), Song & Lee (2010), Lee (2007), and Zhu & Lee(1999).

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21