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CHAPTER 1 INTRODUCTION Section 1 Rationale
Structural Equation Modeling (SEM) is a general statistical modeling technique commonly used in the social and behavioral sciences. In the SEM literature, two models, namely the reflective and formative measurement models, have been
advanced to depict the relationship between a latent variable and its measures. In both of these models, the measures have conceptual unity in that they correspond to a theoretical definition of the concept represented by a latent variable (Bollen &
Bauldry, 2011). The reflective measurement model, based on factor analysis
(Spearman, 1904) and classical test theory (Lord & Novick, 1968), is characterized by the underlying latent variable determining its measures (Bollen & Lennox, 1991), each of which has its own unique factor (i.e., measurement error). Under this
representation, as the measures are considered to be effects of the latent variable, the indicators are referred to as effect indicators (Bollen, 1989; Bollen & Bauldry, 2011;
Bollen & Lennox, 1991; MacCallum & Browne, 1993). On the other hand, with the formative measurement model, it is the measures together with a disturbance term that determine the latent variable (Bollen & Lennox, 1991); the disturbance term
represents the impact of all remaining causes outside of those represented by the already included indicators, with which it is assumed to be uncorrelated
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(Diamantopoulos, 2006). In this model, the latent variable is defined as a linear function of its measures plus a disturbance term; hence, the indicators are referred to as causal indicators (Bollen, 1989; Bollen & Bauldry, 2011; Bollen & Lennox, 1991;
MacCallum & Browne, 1993). For the sake of clarity, latent variables measured with effect and causal indicators will be referred to in the current research as reflective
latent variables (e.g., Brown, 2006; Diamantopoulos, 2006) and formative latent
variables (e.g., Diamantopoulos, 2006; Diamantopoulos, Riefler & Roth, 2008;
Treiblmaier, Bentler, & Mair, 2011), respectively. Note that each of these two classes of latent variables represents a first-order latent variable, which means that there is a direct relation between latent variables and their observed indicators (Hoyle, 2011).
For a graphical representation of these two measurement models, refer to Figures 1A and 1B.
Over the past 20 years, the social and behavioral sciences have witnessed a trend toward an increasing number of empirical studies involving exogenous or endogenous
(A) first-order reflective latent variable (1stR) with multiple effect indicators
1 F 1 R
(B) first-order formative latent variable (1stF) with multiple causal indicators
Figure 1. Reflective versus formative measurement models
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interaction (also called moderation) between reflectively-measured latent variables.
As but one empirical example in which the exogenous interaction model was applied, Pollack, Vanepps, and Hayes (2012) examined the psychological experience of entrepreneurs in response to economic stress. One of the findings was that entrepreneurial withdrawal intentions are determined by the interaction between economic stress and contact with business-related social ties (i.e., the relation between stress and withdrawal intentions was stronger among entrepreneurs having less
contact with social ties and weaker among entrepreneurs having more contact with social ties). Exemplifying the employment of the endogenous interaction model, Cole, Walter, and Bruch (2008) investigated affective mechanisms linking dysfunctional behavior to performance in work teams. They specified negative team affective tone as a mediator between dysfunctional team behavior and team performance, whereas nonverbal negative expressivity was specified as a moderator of the
behavior-performance relationship. Results confirmed that levels of nonverbal negative expressivity are contingent on the relation of team affective tone on team performance. In general, examples of empirical studies modeling interaction effects between reflective latent variables are quite abundant in the literature of various disciplines, including business and management (e.g., Cole, Bedeian, & Bruch, 2011;
Cole et al., 2008; Pollack et al., 2012), psychology (e.g., Bagozzi, Moore, & Leone,
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2004; Diestel & Schmidt, 2009, 2010; Hukkelberg, Hagtvet, & Kovac, 2014; Koring et al., 2012; Luszczynska et al., 2010; Morgan-Lopez, Castro, Chassin, & MacKinnon, 2003; Parade, Leerkes, & Blankson, 2010; Popan, Kenworthy, Frame, Lyons, &
Snuggs, 2010; Takeuchi, Yun, & Wong, 2011; Wiedemann, Schüz, Sniehotta, Scholz,
& Schwarzer, 2009), education (e.g., Camminatiello, Paletta, & Speziale, 2012;
Ganzach, 1997) and marketing communications (e.g., Chang, 2010; Slater, Hayes, &
Ford, 2007; Van Rompay, De Vries, & Van Venrooij, 2010), among others.
In contrast to the multitude of empirical studies modeling interaction effects of reflective latent variables, interaction studies utilizing formatively-measured latent variables as moderator variables are relatively sparse. It can be speculated that this tendency may have arisen from the traditional lack of attention to formative
measurement models in the literature, a phenomenon which may in turn have impeded researchers from undertaking a nonlinear effect analysis involving formative latent variables (Henseler, Fassott, Dijkstra, & Wilson, 2012). However, considering the increasing prevalence of formative measurement models appearing in various disciplines such as marketing and consumer research (cf., Diamantopoulos, et al., 2008; Jarvis, MacKenzie, & Podsakoff, 2003), information systems (cf., Petter &
Straub, & Rai, 2007), and strategic management (cf., Podsakoff, Shen, & Podsakoff, 2006), as well as the growing number of methodological papers devoted to validating
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formative measurement models and advancing clarity guidelines for analyzing
formative latent variables (e.g., Bagozzi, 2011; Bollen, 2011; Bollen & Bauldry, 2011;
Bollen & Davis, 2009a, 2009b; Bollen & Ting, 2000; Diamantopoulos, 2006;
Diamantopoulos, et al., 2008; Diamantopoulos & Siguaw, 2006; Diamantopoulos &
Winklhofer, 2001; Edwards & Bagozzi, 2000; Jarvis et al., 2003; MacKenzie, Podsakoff, & Jarvis; 2005; MacKenzie, Podsakoff, & Podsakoff, 2011; Petter et al., 2007; Podsakoff et al., 2006; Williams, Edwards, & Vandenberg, 2003), the time appears to be ripe for progressing toward theoretical and empirical research concerned with modeling formatively-measured latent variables as moderator variables. In fact, in the most recent decade or so, interaction involving formative latent variables has begun to emerge in the empirical literature (e.g., Conner et al., 2013; Jeffers, Muhanna, & Nault, 2008; Jokela & Keltikangas-Järvinen, 2011; Kankanhalli, Pee, Tan, & Chhatwal, 2012; Reinartz, Krafft, & Hoyer, 2004). To illustrate the use of formative latent variables as moderators, two additional example studies will be briefly described here. In one study, Yu, Mishra, Gopal, Slaughter and
Mukhopadhyay (2015) investigated the impact of electronic procurement
(e-procurement) infusion, composed of the two dimensions intensity of e-procurement use and organizational acceptance of e-procurement, on firm procurement
performance. Results showed that intensity of e-procurement use and organizational
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acceptance of e-procurement, specified as formative latent variables, are positively related to firm procurement performance, while their interaction effect on firm procurement performance is negative. In another study, Tucker-Drob and Briley (2012) examined the effects of socioeconomic status (SES) and adolescents’
domain-specific interests, respectively specified as formative and reflective latent variables, on adolescents’ domain-specific knowledge acquisition through 11 academic, vocational/professional, and recreational domains. The results indicated that in each of the content domains (except for the farming domain), SES plays a role in moderating interest-knowledge associations, with the association being strongest among individuals living in higher SES contexts.
From the above-cited empirical interaction examples, one can surmise that the substantive variables of interest utilized to establish causal connections in the corresponding theoretical models are generally treated as latent variables, each of which is a theoretical definition of a concept and measured by effect or causal indicators. More specifically, the interaction (or moderation) effects that researchers are interested in are often embedded in the context of the structural model
representing the relationships among latent variables while the components of each cross-product interaction term are defined in the context of the measurement model representing the relationships between observed and latent variables. In general, at
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least three possible types of interaction effects have appeared in empirical research, namely, (a) interaction between first-order reflective latent variables (e.g., Pollack et al., 2012), (b) interaction between first-order formative latent variables (e.g., Yu et al., 2015), and (c) interaction between first-order reflective and formative latent variables (e.g., Tucker-Drob & Briley, 2012). The conceptual and statistical diagrams of each type of latent interaction model1 are respectively illustrated in Figures 2A, 2B and 2C.
It can be seen that in each type of model the causal relationships among latent variables are similar in that L is affected by 3 L , 1 L and the interaction term 2
1 2L
L (shown in dashed boxes), with the only differences across the three types being in whether causal or effect indicators are utilized to measure L and 1 L (shown in 2 dotted boxes). In other words, each of the three types of interaction effects has its own unique identity, an identity that is based on the respective measurement model
specification for L and 1 L . 2
1 Our discussion here focuses on model specification rather than model identification of the three types of interaction effects. However, it should be noted that the last two types of interaction models will have the identification problem associated with the presence of formative latent variable(s).
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L
L L
L
L L
L
L L
(A) Interaction between first-order reflective latent variables
(B) Interaction between first-order formative latent variables
(C) Interaction between first-order reflective and formative latent variables
Figure 2. Conceptual diagrams (left-hand side) and statistical diagrams (right-hand side) of each type of latent interaction effect
SES
Interest Knowledge
Acceptance Intensity
Performance Social ties
Economic Stress Withdrawal Intentions
(e.g., Pollack, Vanepps, & Hayes, 2012)
(e.g., Tucker-Drob & Briley, 2012) (e.g., Yu et al., 2015)
L
L L
L L
Social ties
Economic Stress
Economic Stress × Social ties
Withdrawal Intentions
L
L L
L L
Performance Intensity
Acceptance
Acceptance × Intensity
L
L L
L L
SES
Interest Knowledge
Interest × SES
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Considering the model specification of the three types of interaction effects shown above, it should be appropriate to utilize SEM rather than moderated multiple regression (e.g., Aiken & West, 1991; Edwards & Lambert, 2007; Hayes, 2013;
Preacher, Rucker, & Hayes, 2007), the latter being one of the most common methods for testing moderating relationships, as a preferred analytical technique for estimating nonlinear effects of latent variables by virtue of the fact that SEM is capable of dealing with multivariate models and multiple measures of latent variables while controlling for measurement errors in observed variables (Bollen & Noble, 2011) in addition to allowing researchers to examine structural and measurement models simultaneously (Gefen, Straub, & Boudreau, 2000).
Out of multiple recent lines of covariance-based SEM research, a variety of approaches have been developed for the estimation of latent nonlinear effects. Most approaches can be divided into several major categories: product indicator approaches (e.g., Algina & Moulder, 2001; Chen & Cheng, 2014; Coenders, Batista-Foguet, &
Saris, 2008; Jaccard & Wan, 1995; Jöreskog & Yang, 1996; Kelava & Brandt, 2009;
Kenny & Judd, 1984; Little, Bovaird, & Widaman, 2006; Marsh, Wen, & Hau, 2004;
Wall & Amemiya, 2001), maximum likelihood (ML) estimation methods (e.g., Cudeck, Harring, & du Toit, 2009; Klein & Moosbrugger, 2000; Klein & Muthén, 2007; Lee, Song, & Lee, 2003; Lee & Zhu, 2002; Wall & Amemiya, 2007), Bayesian
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estimation methods (e.g., Arminger & Muthén, 1998; Lee, Song, & Tang, 2007; Lee
& Zhu, 2000), method of moments approaches (e.g., Lyhagen, 2007; Mooijaart &
Bentler, 2010; Wall & Amemiya, 2003) and semiparametric approaches (e.g., Bauer, 2005; Pek, Sterba, Kok, & Bauer, 2009). To the best of our knowledge most of these approaches have been developed primarily to estimate interaction and/or quadratic effects of reflective latent variables, particularly focusing on exogenous reflective latent variables (e.g., Figure 2A), while leaving nonlinear effects of endogenous formative latent variables2 (e.g., Figures 2B and 2C) unaccounted for. Unfortunately, this means that there is a growing divide between the capability of available nonlinear SEM approaches, on the one hand, and the interests of social science empirical
researchers on the other. In particular, current scholars, while continuing to do
interaction research involving exogenous reflective latent variables, have increasingly attempted to model endogenous reflective and formative latent variables as
moderators in their theoretical models.
2 In this study, the distinction between exogenous and endogenous variables, whether observed or latent variables, is based on the definition described by Bollen (1989, p. 46). In other words, a formative latent variable which is influenced by its causal indicators and the disturbance term is defined as endogenous rather than exogenous.
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While each of the afore-mentioned nonlinear SEM approaches could conceivably be developed to incorporate latent nonlinear relations involving endogenous reflective or formative latent variables3, the resulting mathematical theories may be overly complex which could lead to some potential problems. For example, within the classes of ML and Bayesian estimation methods, Wall (2009) mentioned that when latent nonlinear models increase in complexity (e.g., a larger number of latent variables), the computational algorithms are likely to fail to reach convergence, and even if they do, may become less numerically precise. As another example, within the class of product indicator approaches, the elaborate and tedious nature of the model specification procedure may limit its potential usefulness. Even so, considering the ever-increasing number of complicated latent nonlinear relations (e.g., a combination of mediation and moderation) and different types of latent nonlinear effects (e.g., Figures 2B and 2C) appearing in empirical applications, it is still imperative that research efforts be made to develop and generalize current nonlinear SEM approaches.
3 Kelava et al. (2011) pointed out that the latent moderated structural equations (LMS) approach (Klein
& Moosbrugger, 2000), one of the class of ML estimation methods, can allow researchers to build nonlinear models involving endogenous latent variables in the Mplus package (Muthén & Muthén, 1998-2012). However, the general nonlinear SEM framework demonstrated in Klein and
Moosbrugger's paper appears to have been developed to treat interaction and quadratic effects of exogenous latent variables, leaving nonlinear effects involving endogenous latent variables seemingly nonaccommodable. The incorporation of nonlinear effects involving endogenous latent variables into their framework is an issue worth further exploration.
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Pursuing this research aim, Chen and Cheng (2014) generalized Jöreskog and Yang's (1996) constrained approach, one of the product indicator methods, to process interaction and quadratic effects involving endogenous reflective latent variables.
Importantly, as an advantage of Chen and Cheng's framework, in contrast to specifying constraints in equation form as in Jöreskog and Yang's framework,
specifying constraints in matrix form simplifies the constraint specification procedure and the process of model specification. While Chen and Cheng's framework extends previous research to incorporate nonlinear effects of endogenous reflective latent variables, nonlinear effects involving formative latent variables remain unconsidered, which, as discussed earlier, represents a limitation of the framework. Fortunately, while it appears that an implementation of formative latent variables into a nonlinear covariance-based SEM approach has not yet been demonstrated in the literature, it is in fact feasible to do just that, as will be shown in the current research.
Section 2 Purpose of the Study
The first study of this doctoral dissertation implements formative latent variables into the constrained approach by extending Chen and Cheng's (2014) research to create two highly generalized frameworks in which a first-order formative latent variable(s) can be modeled as the moderator(s). Prototypes of the newly created frameworks are represented by the interaction models shown in Figures 2B and 2C,
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while the prototype of Chen and Cheng's framework is shown in Figure 2A, Note that the frameworks developed by these two studies, if packaged together, can encompass the three types of latent nonlinear effects that have arisen in empirical applications, namely the interaction and/or quadratic effects between first-order reflective latent variables, between first-order formative latent variables, and between first-order reflective and formative latent variables. For the sake of brevity and better legibility, these three latent nonlinear frameworks will be abbreviated as the R-R, F-F, and R-F frameworks in the present dissertation. Importantly, the R-R, F-F, and R-F
frameworks are complementary to each other in the sense that the latter two, developed in the current study, retain the matrix partitioning technique which was employed on the former in Chen and Cheng's research, thus substantially reducing problems inherent to the traditional constrained approach, which, to reiterate, include the tedious nature of the model specification procedure and the complexity of
derivations of constraints. Taking a broader perspective, by respectfully implementing endogenous reflective and formative latent variables into traditional constrained approaches (Algina & Moulder, 2001; Jaccard & Wan, 1995; Jöreskog & Yang, 1996;
Kenny & Judd, 1984; referred to in Marsh et al., 2004; Marsh, Wen, Hau, &
Nagengast, 2013) through a matrix framework(s), Chen and Cheng and Study 1 of the current dissertation further increase the usefulness of the class of product indicator
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approaches as a whole by allowing other approaches within this class, notably, the generalized appended product indicator (GAPI) approach (Wall & Amemiya, 2001), the unconstrained approach (Marsh, et al., 2004), the extended unconstrained
approach (Kelava & Brandt, 2009) and Coenders et al. (2008), to be generalized to a wider range of situations (e.g., different types of latent nonlinear effects or complex relationships with multiple latent nonlinear terms).
While the R-R, F-F, and R-F frameworks can accommodate a broad range of nonlinear models having a single or multiple latent interaction and/or quadratic term(s) each of which may be composed of exogenous or endogenous variables, the three frameworks are limited by the fact that they are individually devised to support only one specific type of latent nonlinear effect. In order to extend the potential
applicability of these frameworks to situations in which different types of latent nonlinear effects are to be modeled simultaneously, the second study of this doctoral dissertation aggregates the results of Chen and Cheng (2014) and Study 1 to build four advanced frameworks (denoted as F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R frameworks) individually capable of dealing with the three combinations of two of the three types of latent nonlinear effects and the full set of all three types of latent
nonlinear effects. As such, the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R frameworks will henceforth be referred to as integrated frameworks, while the term
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fundamental frameworks will be used to refer to the R-R, F-F and R-F frameworks. To
help make the distinctions among the seven frameworks more clear, Figure 3 provides a graphical representation of the prototype nonlinear models4 characterizing each of the fundamental and integrated frameworks. As can be seen, Models A, B and C, encompassing the three individual types of latent nonlinear effects L21, L22 and
1 2L
L , can be embedded into the three fundamental frameworks, whereas Models D, E, F and G, composed of all unique combinations of different types of latent nonlinear effects (L21L22), (L2L1L22), (L21L2L1) and (L21L2L1L22), can be embedded into the four integrated frameworks.
4 Similar to the two types of latent interaction models illustrated in Figures 2B and 2C, Models B to G shown in Figure 3 have the identification problem associated with the presence of formative latent variable(s).
L : formative latent variable.
L : reflective latent variable.
L : quadratic term of formative latent variables.
L : quadratic term of reflective latent variables.
L L : interaction term between reflective and formative latent variables.
Figure 3. Statistical diagram for seven prototype nonlinear models L
L L
L L L L
Framework Example
F-F fundamental Model A : L R-F fundamental Model B : L L R-R fundamental Model C : L F-F/R-R integrated Model D : L +L R-F/R-R integrated Model E : L L +L F-F/R-F integrated Model F : L +L L F-F/R-F/R-R integrated Model G : L +L L +L
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Not surprisingly, the integrated frameworks that will be devised in Study 2 preserve the matrix partitioning technique that is at the heart of the fundamental frameworks from Chen and Cheng (2014) and the current Study 1. Taken together, the three fundamental frameworks and the four integrated frameworks form a
complementary and comprehensive set of seven generalized frameworks that
encapsulates all possible combinations of interaction and/or quadratic effects between first-order latent variables measured by effect or causal indicators, thereby further broadening the potential usefulness of the entire class of product indicator approaches, most notably the constrained approaches.
This doctoral dissertation is divided into five chapters. Following the present introduction, the subsequent chapter will lay the necessary groundwork for the development of the general nonlinear frameworks of the current Studies 1 and 2, including a model reformulation procedure, a model partitioning technique, and a filter matrix implementation to obtain the general forms of nonlinear variables. With these prerequisites fulfilled, the third and fourth chapters will be devoted to the actual
This doctoral dissertation is divided into five chapters. Following the present introduction, the subsequent chapter will lay the necessary groundwork for the development of the general nonlinear frameworks of the current Studies 1 and 2, including a model reformulation procedure, a model partitioning technique, and a filter matrix implementation to obtain the general forms of nonlinear variables. With these prerequisites fulfilled, the third and fourth chapters will be devoted to the actual