形成性潛在變項於非線性效果之模型設定 : 限制式方法 - 政大學術集成

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(1)國立政治大學心理學系博士論文 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Psychology, National Chengchi University. 政治大. 形成性潛在變項於非線性效果之模型設定 : 限制式方法. 立. 學. ‧ 國. Model Specification for Nonlinear Effects of Formative Latent Variables: The Constrained Approach. ‧. Nat. sit. 陳淑萍. n. C hShu-Ping Chen engchi. er. io. al. y. by. i n U. v. 指導教授：鄭中平博士 Advisor: Chung-Ping Cheng Ph.D.. 中華民國一○五年七月 July, 2016.

(2) 立. 政治大. 學. ‧ 國. Copyright © 2016 - Shu-Ping Chen All rights reserved.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i n U. v.

(3) ABSTRACT Modeling latent nonlinear effects is a significant issue in the social and behavioral sciences. A variety of approaches have recently been developed for the estimation of nonlinear structural equation modeling. To the best of our knowledge most of these approaches have been developed primarily to estimate interaction and/or quadratic effects of reflectively-measured latent variables, while leaving nonlinear effects of. 政治大. formatively-measured latent variables unaccounted for. The current study implements. 立. formatively-measured latent variables into Jöreskog and Yang’s (1996) constrained. ‧ 國. 學. approach by extending Chen and Cheng’s (2014) research to create a unified set of six. ‧. generalized nonlinear frameworks, each capable of differentially or collectively. Nat. io. sit. y. modeling the three types of latent nonlinear effects that have arisen in empirical. er. applications (i.e., interaction and/or quadratic effects between reflective latent variables,. al. n. v i n C hand between reflective between formative latent variables, e n g c h i U and formative latent. variables). By preserving the inherent advantage of Chen and Cheng, i.e., the matrix partitioning technique, while at the same time further generalizing its applicability, it is expected that the current framework enhances the potential usefulness of the constrained approach as well as the entire class of product indicator approaches. Keywords: formatively-measured latent variables, latent nonlinear effects, the constrained approach. iii.

(4) 摘要社會行為科學領域中，潛在非線性關係常為研究者所關切，發展潛在變項間非線性效果方法有其重要性。近年來，已有許多統計方法致力於非線性結構方程模型之估計。就作者所知，大多數方法主要侷限在以反映性測量模式 (reflective measurement model) 為基礎之潛在非線性效果估計，而忽略以形成性測量模式 (formative measurement model) 為基礎之潛在非線性效果估計。本研究衍生 Chen. 政治大. 與 Cheng (2014) 於反映性測量模式基礎下所建立的非線性效果方法，拓展至以. 立. 形成性測量模式為基礎之非線性效果方法。本研究建立的六個廣義性非線性架構，. ‧ 國. 學. 可獨立或同時嵌入三種類型非線性效果，包含反映性潛在變項間交互作用與二次. ‧. 項效果、形成性潛在變項間交互作用與二次項效果和反映性與形成性潛在變項間. Nat. io. sit. y. 交互作用效果。值得注意地，每個非線性架構皆保有 Chen 與 Cheng 矩陣分割技. er. 術，可簡化模型設定的過程，並類推至更多情境的非線性模型。整體來說，本研. al. n. v i n Ch 究促進限制式方法與交乘項指標方法的發展，希冀提升方法發展與研究者在實務 engchi U 研究的應用。關鍵詞:形成性潛在變項、潛在非線性效果、限制式方法. iv.

(5) TABLE OF CONTENTS PAGE ENGLISH ABSTRACT............................................................................................... iii CHINESE ABSTRACT............................................................................................... iv LIST OF TABLES...................................................................................................... ix LIST OF FIGURES...................................................................................................... xi. 政治大. CHAPTER 1 INTRODUCTION.............................................................................. 1. 立. Section 1 Rationale............................................................................................... 1. ‧ 國. 學. Section 2 Purpose of the Study.......................................................................... 12. ‧. CHAPTER 2 PREPROCESSING OF MODEL SPECIFICATION........................ 17. Nat. io. sit. y. Section 1 The R-R Fundamental Framework..................................................... 20. er. Part 1: The Partitioning Scheme of the R-R Framework............................ 20. al. n. v i n C h Matrix Representation Part 2: A Structural Equation of engchi U the R-R Framework ............................................................................ 22 Part 3: The Constraint Specification for the R-R Framework..................... 25 Section 2 Model Reformulating Procedure......................................................... 29 Example 1: Interaction between Formative Latent Variables.................... 32 Example 2: Interaction between Reflective and Formative Latent Variables.............................................................................................. 35. v.

(6) Section 3 Model Partitioning Scheme Applied on the Fundamental Frameworks..................................................................... 38 Section 4 Model Partitioning Scheme Applied on the Integrated Frameworks.......................................................................... 44 Section 5 General Forms of Nonlinear Variables............................................... 54 Case 1: The Model Characterizing the F-F Fundamental Framework......... 56 Case 2: The Model Characterizing the R-F Fundamental Framework......... 57 Case 3: The Model Characterizing the F-F/R-F/R-R Integrated Framework......................................................................... 57. 立. 政治大. ‧ 國. 學. CHAPTER 3 MODEL SPECIFICATION FOR THE F-F AND R-F FUNDAMENTAL FRAMEWORKS................................................................. 60. ‧. Section 1 The Basic Model across Each Current Framework............................. 62. sit. y. Nat. Section 2 The F-F Fundamental Framework....................................................... 65. n. al. er. io. Step 1: Expanding the Nonlinear Vectors................................................... 66. Ch. i n U. v. Step 2: Formulating the Structural Equation Matrix Representation ......... 68. engchi. Step 3: Implementing the Constraint Specification..................................... 70 Section 3 The R-F Fundamental Framework...................................................... 73 Step 1: Expanding the Nonlinear Vectors................................................... 74 Step 2: Formulating the Structural Equation Matrix Representation ......... 76 Step 3: Implementing the Constraint Specification..................................... 77 Section 4 Simulated Examples............................................................................ 80. vi.

(7) Illustrating the F-F Effect through Yu et al.'s Model............................... 84 Illustrating R-F Effects through Tucker-Drob and Briley's Model............. 86 Validating the Model Specification of the F-F and R-F Fundamental Frameworks................................................................... 88 CHAPTER 4 MODEL SPECIFICATION FOR THE F-F/R-R, R-F/R-R, F-F/R-F AND F-F/R-F/R-R INTEGRATED FRAMEWORKS....................................... 99 Section 1 The Inter-Compatibility of the Model Specifications of the R-R, F-F and R-F Fundamental Frameworks...................................... 101. 治政 Section 2 The F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R 大立 Integrated Frameworks............................................................................. 107 ‧ 國. 學. Model Specification of the F-F/R-R Framework...................................... 112. ‧. Step 1: Expanding the nonlinear vectors........................................... 112. Nat. io. sit. y. Step 2: Formulating the structural equation matrix representation... 112. er. Step 3: Implementing the constraint specification............................ 113. al. n. v i n Ctheh R-F/R-R Framework...................................... Model Specification of 114 engchi U Step 1: Expanding the nonlinear vectors........................................... 115 Step 2: Formulating the structural equation matrix representation... 115 Step 3: Implementing the constraint specification............................ 115 Model Specification of the F-F/R-F Framework....................................... 115 Step 1: Expanding the nonlinear vectors........................................... 116 Step 2: Formulating the structural equation matrix representation... 116 vii.

(8) Step 3: Implementing the constraint specification............................ 118 Model Specification of the F-F/R-F/R-R Framework............................... 120 Step 1: Expanding the nonlinear vectors........................................... 120 Step 2: Formulating the structural equation matrix representation... 120 Step 3: Implementing the constraint specification............................ 120 Section 3 Simulated Examples.......................................................................... 122. 政治大. Illustrating the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R Effects through Yu et al.'s Reformulated Model........................................... 123. 立. ‧ 國. 學. Validating the Model Specification of the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R Integrated Frameworks......................................... 132. ‧. CHAPTER 5 DISCUSSION..................................................................................... 156. sit. y. Nat. Section 1 Discussion of Findings...................................................................... 158. n. al. er. io. Section 2 Limitations and Directions for Further Research.............................. 161. Ch. i n U. v. REFERENCES.......................................................................................................... 171 APPENDICES. engchi. A. Expansions of ηF and ηS .......................................................................... 192 B. Derivations of the Partitioned Matrices α and Ψ ...................................... 196 C. Expansions of η~S and y F ........................................................................... 199 D. Derivations of the Partitioned Matrices α , ν , Ψ and Θ ......................... 202 E. The Derivation of the Constraint Ψ S~S  ....................................................... 204. viii.

(9) LIST OF TABLES PAGE Table 1. Partitioning scheme applied on the reformulated models............................. 50 Table 2. Simulation results for Yu et al.'s model....................................................... 90 Table 3. Simulation results for Tucker-Drob and Briley's model.............................. 94 Table 4. Model partitioning pattern of the R-R, F-F and R-F fundamental frameworks.............................................. 104. 治政 Table 5. Model specification of the 大 R-R, F-F and R-F立 fundamental frameworks.............................................. 106 ‧ 國. 學. Table 6. Model partitioning pattern of the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks..... 108. ‧. io. sit. y. Nat. Table 7. Model specification of the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks..... 121. n. al. er. Table 8. Partitioning scheme applied on the four models encompassing F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R effects.............................. 131. Ch. engchi. i n U. v. Table 9. Simulation results for Yu et al.'s reformulated model with F-F/R-R effects.......................................................................................... 134 Table 10. Simulation results for Yu et al.'s reformulated model with R-F/R-R effects......................................................................................... 139 Table 11. Simulation results for Yu et al.'s reformulated model with F-F/R-F effects.......................................................................................... 144 Table 12. Simulation results for Yu et al.'s reformulated model with F-F/R-F/R-R effects................................................................................... 149. ix.

(10) Table 13. Performance of latent nonlinear effects within each integrated framework........................................................................ 154. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. x. i n U. v.

(11) LIST OF FIGURES PAGE Figure 1. Reflective versus formative measurement models........................................ 2 Figure 2. Conceptual diagrams and statistical diagrams of each type of latent interaction effect.............................................................. 8 Figure 3. Statistical diagram for seven prototype nonlinear models.......................... 15 Figure 4. The conceptual diagram of the R-R partitioning scheme............................. 20. 治政 Figure 5. Interaction between first-order formative latent 大variables.......................... 30 立 ‧ 國. 學. Figure 6. Interaction between first-order reflective and formative latent variables... 31 Figure 7. Partitioning the reformulated models from Examples 1 and 2................... 39. ‧. Figure 8. The prototype model characterizing the F-F/R-R framework.................... 46. sit. y. Nat. io. n. al. er. Figure 9. The prototype model characterizing the R-F/R-R framework.................... 47. i n U. v. Figure 10. The prototype model characterizing the F-F/R-F framework..................... 48. Ch. engchi. Figure 11. The prototype model characterizing the F-F/R-F/R-R framework........... 49 Figure 12. Yu et al.'s model with an F-F effect............................................................ 82 Figure 13. Tucker-Drob and Briley's Model with R-F Effects................................... 83 Figure 14. Yu et al.'s reformulated model with F-F/R-R effects............................... 125 Figure 15. Yu et al.'s reformulated model with R-F/R-R effects.............................. 126 Figure 16. Yu et al.'s reformulated model with F-F/R-F effects.............................. 127 Figure 17. Yu et al.'s reformulated model with F-F/R-F/R-R effects..................... 128 xi.

(12) 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. 政治大. of these models, the measures have conceptual unity in that they correspond to a. 立. theoretical definition of the concept represented by a latent variable (Bollen &. ‧ 國. 學. Bauldry, 2011). The reflective measurement model, based on factor analysis. ‧. (Spearman, 1904) and classical test theory (Lord & Novick, 1968), is characterized by. Nat. io. sit. y. the underlying latent variable determining its measures (Bollen & Lennox, 1991),. er. each of which has its own unique factor (i.e., measurement error). Under this. al. n. v i n representation, as the measures C areh considered to be effects e n g c h i U 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. 1.

(13) (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. 政治大. variables (e.g., Diamantopoulos, 2006; Diamantopoulos, Riefler & Roth, 2008;. 立. Treiblmaier, Bentler, & Mair, 2011), respectively. Note that each of these two classes. ‧ 國. 學. 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).. Nat. sit. n. al. er. io. and 1B.. y. For a graphical representation of these two measurement models, refer to Figures 1A. 1 R. Ch. engchi. iv n U 1. F. (B) first-order formative latent variable (1stF). (A) first-order reflective latent variable (1stR). with multiple causal indicators. with multiple effect indicators. Figure 1. Reflective versus formative measurement models 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. 2.

(14) 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. 政治大. stress and withdrawal intentions was stronger among entrepreneurs having less. 立. contact with social ties and weaker among entrepreneurs having more contact with. ‧ 國. 學. social ties). Exemplifying the employment of the endogenous interaction model, Cole,. ‧. Walter, and Bruch (2008) investigated affective mechanisms linking dysfunctional. Nat. io. sit. y. behavior to performance in work teams. They specified negative team affective tone. er. as a mediator between dysfunctional team behavior and team performance, whereas. al. n. v i n nonverbal negative expressivityC was U of the hspecified e n g casha imoderator. 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,. 3.

(15) 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, &. 政治大. Ford, 2007; Van Rompay, De Vries, & Van Venrooij, 2010), among others.. 立. In contrast to the multitude of empirical studies modeling interaction effects of. ‧ 國. 學. reflective latent variables, interaction studies utilizing formatively-measured latent. ‧. variables as moderator variables are relatively sparse. It can be speculated that this. Nat. io. sit. y. tendency may have arisen from the traditional lack of attention to formative. er. measurement models in the literature, a phenomenon which may in turn have impeded. al. n. v i n researchers from undertaking a C nonlinear analysis involving formative latent h e neffect gchi U 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. 4.

(16) 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.,. 政治大. 2007; Podsakoff et al., 2006; Williams, Edwards, & Vandenberg, 2003), the time. 立. appears to be ripe for progressing toward theoretical and empirical research concerned. ‧ 國. 學. with modeling formatively-measured latent variables as moderator variables. In fact,. ‧. in the most recent decade or so, interaction involving formative latent variables has. Nat. io. sit. y. begun to emerge in the empirical literature (e.g., Conner et al., 2013; Jeffers,. er. Muhanna, & Nault, 2008; Jokela & Keltikangas-Järvinen, 2011; Kankanhalli, Pee,. al. n. v i n C Krafft, Tan, & Chhatwal, 2012; Reinartz, 2004). To illustrate the use of h e n&gHoyer, chi U 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. 5.

(17) 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. 政治大. academic, vocational/professional, and recreational domains. The results indicated. 立. that in each of the content domains (except for the farming domain), SES plays a role. ‧ 國. 學. in moderating interest-knowledge associations, with the association being strongest. ‧. among individuals living in higher SES contexts.. Nat. io. sit. y. From the above-cited empirical interaction examples, one can surmise that the. er. substantive variables of interest utilized to establish causal connections in the. al. n. v i n Care corresponding theoretical models h egenerally h i Uas latent variables, each of n g ctreated 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. 6.

(18) 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.. 政治大. It can be seen that in each type of model the causal relationships among latent. 立. variables are similar in that L3 is affected by L1 , L 2 and the interaction term. ‧ 國. 學. L 2 L1 (shown in dashed boxes), with the only differences across the three types being. ‧. in whether causal or effect indicators are utilized to measure L1 and L 2 (shown in. Nat. io. sit. y. dotted boxes). In other words, each of the three types of interaction effects has its own. n. al. er. unique identity, an identity that is based on the respective measurement model specification for L1 and L 2 .. 1. Ch. engchi. i n U. v. 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).. 7.

(19) L Social ties. L. Social ties. Economic Stress. Withdrawal Intentions. Economic Stress. L. L. L. L. Withdrawal Intentions. L L Economic Stress × Social ties. (A) Interaction between first-order reflective latent variables (e.g., Pollack, Vanepps, & Hayes, 2012). Acceptance. Intensity. L. L. 學. L. 立. L. Acceptance. Performance. L L. L Performance. ‧. ‧ 國. Intensity. 政治大 L. Acceptance × Intensity. y. Nat. n. al. er. (e.g., Yu et al., 2015). io SES. sit. (B) Interaction between first-order formative latent variables. L. Ch. engchi. i Ln U. v. SES. L. L. L. Interest. Knowledge. Interest. L Knowledge. L L Interest × SES. (C) Interaction between first-order reflective and formative latent variables (e.g., Tucker-Drob & Briley, 2012). Figure 2. Conceptual diagrams (left-hand side) and statistical diagrams (right-hand side) of each type of latent interaction effect. 8.

(20) 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. 政治大. dealing with multivariate models and multiple measures of latent variables while. 立. controlling for measurement errors in observed variables (Bollen & Noble, 2011) in. ‧ 國. 學. addition to allowing researchers to examine structural and measurement models. ‧. simultaneously (Gefen, Straub, & Boudreau, 2000).. Nat. io. sit. y. Out of multiple recent lines of covariance-based SEM research, a variety of. er. approaches have been developed for the estimation of latent nonlinear effects. Most. al. n. v i n C h major categories: approaches can be divided into several e n g c h i U 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. 9.

(21) 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. 政治大. latent variables (e.g., Figure 2A), while leaving nonlinear effects of endogenous. 立. formative latent variables2 (e.g., Figures 2B and 2C) unaccounted for. Unfortunately,. ‧ 國. 學. 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. Nat. io. sit. y. researchers on the other. In particular, current scholars, while continuing to do. er. interaction research involving exogenous reflective latent variables, have increasingly. al. n. v i n attempted to model endogenousC reflective formative latent variables as h e n and gchi U 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.. 10.

(22) 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. 政治大. variables), the computational algorithms are likely to fail to reach convergence, and. 立. even if they do, may become less numerically precise. As another example, within the. ‧ 國. 學. class of product indicator approaches, the elaborate and tedious nature of the model. ‧. specification procedure may limit its potential usefulness. Even so, considering the. Nat. io. sit. y. ever-increasing number of complicated latent nonlinear relations (e.g., a combination. er. of mediation and moderation) and different types of latent nonlinear effects (e.g.,. al. n. v i n Figures 2B and 2C) appearing inCempirical it is still imperative that h e n gapplications, chi U 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.. 11.

(23) 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. 政治大. and the process of model specification. While Chen and Cheng's framework extends. 立. previous research to incorporate nonlinear effects of endogenous reflective latent. ‧ 國. 學. variables, nonlinear effects involving formative latent variables remain unconsidered,. ‧. which, as discussed earlier, represents a limitation of the framework. Fortunately,. Nat. io. sit. y. while it appears that an implementation of formative latent variables into a nonlinear. er. covariance-based SEM approach has not yet been demonstrated in the literature, it is. al. n. v i n in fact feasible to do just that, asC will be shown in the U he n g c h i 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,. 12.

(24) 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,. 政治大. these three latent nonlinear frameworks will be abbreviated as the R-R, F-F, and R-F. 立. frameworks in the present dissertation. Importantly, the R-R, F-F, and R-F. ‧ 國. 學. 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. Nat. io. sit. y. employed on the former in Chen and Cheng's research, thus substantially reducing. er. problems inherent to the traditional constrained approach, which, to reiterate, include. al. n. v i n the tedious nature of the model C specification and the complexity of h e n gprocedure chi U. 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. 13.

(25) 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).. 政治大. While the R-R, F-F, and R-F frameworks can accommodate a broad range of. 立. nonlinear models having a single or multiple latent interaction and/or quadratic term(s). ‧ 國. 學. 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. Nat. io. sit. y. one specific type of latent nonlinear effect. In order to extend the potential. er. applicability of these frameworks to situations in which different types of latent. al. n. v i n C hsimultaneously, theUsecond study of this doctoral nonlinear effects are to be modeled engchi 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. 14.

(26) 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. L 2 L1 , can be embedded into the three fundamental frameworks, whereas Models D, E,. 政治大. F and G, composed of all unique combinations of different types of latent nonlinear. 立. effects ( L21  L22 ), ( L 2 L1  L22 ), ( L21  L 2 L1 ) and ( L21  L 2 L1  L22 ), can be embedded. ‧ 國. 學. into the four integrated frameworks.. F-F/R-F integrated F-F/R-F/R-R integrated. Model D : L +L. al. L. n. R-F/R-R integrated. Model C : L. io. F-F/R-R integrated. y. L. Model B : L L. sit. R-R fundamental. Nat. R-F fundamental. Model A : L. Model E : L L +L. Ch. i U e h n c g Model G : L +L L +L Model F : L +L L. er. F-F fundamental. Example. ‧. Framework. v ni. L. L. L : formative latent variable. L : reflective latent variable.. L. L : quadratic term of formative latent variables. L : quadratic term of reflective latent variables.. L L. L L : interaction term between reflective and formative latent variables.. Figure 3. Statistical diagram for seven prototype nonlinear models 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).. 15.

(27) 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. 政治大. first-order latent variables measured by effect or causal indicators, thereby further. 立. broadening the potential usefulness of the entire class of product indicator approaches,. ‧ 國. 學. most notably the constrained approaches.. ‧. This doctoral dissertation is divided into five chapters. Following the present. Nat. io. sit. y. introduction, the subsequent chapter will lay the necessary groundwork for the. er. development of the general nonlinear frameworks of the current Studies 1 and 2,. al. n. v i n including a model reformulationCprocedure, technique, and a U h e n ga cmodel h i partitioning 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 construction of the frameworks for Studies 1 and 2 as well as the specification of constraints in matrix form. Finally, a general discussion and comments on some limitations of the current research will conclude the dissertation in the fifth chapter.. 16.

(28) CHAPTER 2 PREPROCESSING OF MODEL SPECIFICATION This chapter describes the prerequisites for the development of the F-F and R-F fundamental frameworks as well as the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks. In Section 1, we will first take a step back and briefly describe Chen and Cheng's (2014) nonlinear framework (abbreviated as the R-R fundamental framework in the. 政治大. current dissertation) as it plays a leading role in developing the model specification of. 立. the above-mentioned fundamental and integrated frameworks. In particular, the. ‧ 國. 學. techniques of adopting the model notation of Muthén (1984) Case A and. ‧. implementing the matrix partitioning technique and filter matrices, first introduced in. Nat. sit er. io. frameworks.. y. the context of the R-R framework, will be cornerstones in developing the proposed. al. n. v i n C technique In Section 2, a reformulating be implemented on two types of h e n gwill chi U. latent interaction models similar to those shown in Figures 2B and 2C (see page 8) in which a first-order formative latent variable(s) can be modeled as the moderator(s). Through the reformulating procedure, the two types of latent nonlinear models characterizing the F-F and R-F fundamental frameworks will be represented in the model notation of Muthén (1984) Case A which presumes all observed indicators are effects of latent variables. Importantly, the adoption of Muthén's notation in the F-F. 17.

(29) and R-F frameworks makes the set of all three fundamental frameworks compatible with each other. Furthermore, as will later be apparent, the use of a consistent notation across the fundamental frameworks will be instrumental in developing the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks. In Section 3, the reformulated models described in Section 2 will serve as the basis for demonstrating the partitioning technique which will, in Chapter 3, come to. 政治大. be employed on the F-F and R-F fundamental frameworks developed in Study 1 of. 立. this dissertation. Subsequently, in Section 4, the reformulating technique and. ‧ 國. 學. partitioning scheme established in Sections 2 and 3 will be applied to the four. ‧. combinations of different types of latent nonlinear models analogous to Models D, E,. Nat. io. sit. y. F and G in Figure 3 (see page 15). Similarly, this preprocessing is necessary to. er. facilitate Chapter 4 in which the partitioning scheme will be employed on the F-F/R-R,. al. n. v i n C hintegrated frameworks R-F/R-R, F-F/R-F and F-F/R-F/R-R e n g c h i U developed in Study 2 of the dissertation. Importantly, in a similar manner to what was shown for the R-R. framework, the matrix partitioning scheme applied to the proposed fundamental and integrated frameworks will come to simplify the model specification procedure proposed by Jöreskog and Yang's (1996) constrained approach in addition to further generalizing its applicability.. 18.

(30) Lastly, Section 5 will establish the general forms of nonlinear terms of each of the six proposed frameworks through filter matrices (Chen & Cheng, 2014); this will be done in order to create a more flexible partitioned framework from which the latent nonlinear variables and product indicators can be selected by the researcher.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 19. i n U. v.

(31) Section 1 The R-R Fundamental Framework The purpose of this section is to establish concepts necessary for developing the proposed fundamental and integrated frameworks through an overview of the model specification of the R-R fundamental framework (Chen & Cheng, 2014), which will be examined in three major parts, namely the partitioning scheme of the R-R framework, a matrix representation of the structural equations of the R-R framework,. 政治大. and the constraint specification for the R-R framework. It is important to point out. 立. that the techniques described in Part 1 will be utilized in Sections 2 through 5 of this. ‧ 國. 學. chapter in preprocessing the proposed frameworks, while the procedures described in. ‧. Parts 2 and 3 will come to serve as a guide for Chapters 3 and 4 in which. Nat. io. er. Part 1: The Partitioning Scheme of the R-R Framework. sit. y. corresponding procedures will be implemented on the proposed frameworks.. al. n. v i n The partitioning scheme ofC thehR-R fundamental U e n g c h i framework is demonstrated. through the conceptual diagram shown below.. η. ηI. ηO. ηI. yI. yO. y I. y Figure 4. The conceptual diagram of the R-R partitioning scheme. 20.

(32) The design of the R-R partitioning scheme was based on the model notation of Muthén (1984) Case A, which assumes a linear structure system (i.e., η  α  Βη  ζ , where η is a vector of latent endogenous variables, α and ζ are vectors of intercepts and disturbance terms associated with η , and Β is a coefficient matrix for latent endogenous variables of η ) and a linear measurement structure (i.e.,. y  ν  Λη  ε , where y is a vector of observed indicators of η , ν and ε are. 政治大. vectors of intercepts and measurement errors for y , and Λ is a coefficient matrix. 立. relating y to η ). More specifically, with regard to the structural part of the R-R. ‧ 國. 學. partitioning scheme, latent variables were partitioned into two vectors (denoted as ηI. ‧. and ηO ) to support the integration of the vector of latent nonlinear variables (denoted. io. sit. y. Nat. as ηI ). In particular, ηI was devised to include nonlinear effects of first-order. er. reflective latent variables that the researcher may be interested in, ηI was devised to. al. n. v i n C hin forming nonlinear include the latent variables involved e n g c h i U variables of. ηI , and ηO. was devised to include the dependent variables affected by latent nonlinear variables of ηI together with any remaining latent variables. Likewise, with regard to the measurement part of the R-R partitioning scheme, observed indicators were partitioned into two vectors utilized as observed indicators of ηI and ηO (denoted as y I and y O ) which in turn support the integration of product indicators of ηI (denoted as y I ). Importantly, in order to create a more flexible partitioned. 21.

(33) framework from which nonlinear effects of reflective latent variables and product indicators can be selected by the researcher, the two vectors ηI and y I were respectively defined as W1 vec(ηI ηIT ) and W2 vec(y I y IT ) , where the vec operator vectorizes a matrix by stacking its columns (Seber, 2007), and W1 and W2 serve as filter matrices (Finkbeiner, 1979) to select the considered nonlinear variables. Subsequently, the R-R partitioning scheme was established by integrating the. 政治大. partitioned vector of latent variables η  [ ηTI | ηOT | ηI T ]T and the partitioned vector of. 立. observed indicators y  [y TI | y OT | y I T ]T , where the first and second rows of η and y. ‧ 國. 學. (i.e., [ ηIT | ηOT ]T and [y IT | y OT ]T , respectively) constitute the basic model, which was. io. sit. Nat. observed product indicators (i.e., ηI and y I ).. y. ‧. defined as the submodel that does not include nonlinear effects of latent variables and. er. Part 2: A Structural Equation Matrix Representation of the R-R Framework. al. n. v i n Ch In Part 1, we described how Chen and Cheng (2014) e n g c h i U established the R-R. partitioning scheme. This scheme served as a template from which the authors formulated a matrix representation of the structural equations of the R-R fundamental framework, as we will summarize in the following paragraphs. First, the structural equations for the basic model (i.e., [ ηIT | ηOT ]T and [y IT | y OT ]T , respectively) were specified, with the structural and measurement parts shown in Equations 1 and 2, respectively.. 22.

(34)  ηI   α I   B II η   α   B  O   O   OI. B IO ( 0)  ηI   ζ I   η   ζ . BOO  O   O . (1).  y I   ν I   Λ II y    ν    Λ  O   O   OI. Λ IO ( 0)  ηI   ε I   η   ε . Λ OO  O   O . (2). Here, α i and ζi (for i  I, O ) are vectors of intercepts and disturbance terms associated with ηi while B ij (for i, j  I, O ) represents the coefficient matrix relating ηi to η j , with all diagonal elements of B ij (for i  j ) specified as zero.. 政治大. Meanwhile, ν i and εi (for i  I, O ) are vectors of intercepts and measurement. 立. errors associated with y i , while Λ ij (for i, j  I, O ) represents the factor loading. ‧ 國. 學. matrix relating y i to η j . It should also be noted that the vectors of disturbance. ‧. terms and measurement errors ζi and ε j (for i, j  I, O ) were assumed to be. Nat. n. al. Ch. sit.  0  Ψ II     0 Ψ OI  0  ,  0     0    0. Ψ OO 0.    .   Θ OO  . er. io.  ζI  ζ   O ~ N  εI    ε O . y. distributed as a multivariate normal distribution as follow:. Θ II. e n 0g c Θh i. OI. i n U. v. (3). Subsequently, Chen and Cheng (2014) formulated the expanded forms of ηI and y I . For the sake of reducing the complexity of the derivations of ηI and y I , the authors restricted the coefficient submatrices B IO of Equation 1 and Λ IO of Equation 2 to be null. With this restriction, the expanded forms of ηI and y I were obtained by plugging the equations ηI  α I  B II ηI  ζ I from Equation 1 and. y I  ν I  Λ II ηI  ε I from Equation 2 into the equations ηI  W1 vec(ηI ηTI ) and 23.

(35) y I  W2 vec(y I y IT ) described in Part 1. The resulting simplified expanded forms of ηI and y I can be expressed as below:. ηI ( W1 vec(ηI ηIT ))  C1  ζ I .. (4). y I ( W2 vec(y I y IT ))  C2  C3 ηI  C4 ηI  ε I .. (5). Note that C1 to C4 , whose complete expanded forms can be found on page 97 of Chen and Cheng (2014), were defined as constraint matrices. ζ I , a function of ζ I ,. 政治大. was defined as the disturbance term of ηI , while ε I  , a function of ε I , was defined. 立. as the measurement error of y I .. ‧ 國. 學. Building on the groundwork laid in the previous steps, a matrix representation of. ‧. the structural equations for the R-R framework was established by combining the. Nat. io. sit. y. structural and measurement parts of the basic model described in Equations 1 and 2. n. al. er. with the expanded forms of ηI and y I described in Equations 4 and 5, as shown in. Ch. Equations 6 and 7, respectively.. η.  α. .  ηI   α I    η   α     O  O   ηI   C1  . engchi B. B II. B IO ( 0). B OI B II ( 0). B OO B IO ( 0). i n U. v. η.  ζ. B II ( 0)   ηI   ζ I       (6) B OI   ηO   ζ O . B II ( 0)  ηI  ζ I . In this equation, η , the 3 1 partitioned vector of latent variables, is associated with the 3 1 partitioned vectors of intercepts and disturbance terms α and ζ as well as the 3 3 partitioned coefficient matrix B . Meanwhile, the four coefficient submatrices relating ηI to ηI , ηI to ηO , ηI to ηI , and ηI to ηI (denoted as 24.

(36) B I I , B IO , B II  and B II  , respectively) were set to null to be consistent with. Equation 4, while the coefficient submatrix relating ηO to ηI (denoted as B OI  ) was specified as non-null to capture nonlinear effects of reflective latent variables.. y.  ν. . Λ.  yI   νI   y   ν     O  O   yI  C2  . ΛII. ΛIO ( 0). ΛOI. ΛOO. ΛII ( C3 ). ΛIO ( 0). η.  ε. ΛII ( 0)  ηI   εI      ( 7 ) ΛOI ηO   εO . ΛII ( C4 ) ηI  εI . In this equation, y , the 3 1 partitioned vector of observed indicators, is associated. 治政 with the 3 1 partitioned vectors of intercepts and measurement 大 errors 立. ν and ε as. ‧ 國. 學. well as the 3 3 partitioned factor loading matrix Λ . Similar to Equation 6, the four factor loading submatrices relating y I to ηI , y I to ηO , y I to ηI , and y I. ‧. to ηI (denoted as Λ II , Λ IO , Λ II and Λ II , respectively) were set to be C 3 , 0 ,. sit. y. Nat. io. n. al. er. C4 and 0 to be consistent with Equation 5, while the factor loading submatrix. v. relating y O to ηI (denoted as Λ OI ) was specified as non-null to measure the effect of ηI on y O .. Ch. engchi. i n U. Part 3: The Constraint Specification for the R-R Framework Having briefly described the procedures behind the matrix representation of the structural equations of the R-R framework from Chen and Cheng (2014), we can at last examine the constraint specification for the R-R framework. The implied mean vector and covariance matrix of the partitioned vector of observed indicators y  [y TI | y OT | y I T ]T , conforming to the model notation of Muthén's 25.

(37) (1984) Case A, were represented as functions of the six partitioned matrices α , B ,. Ψ , ν , Λ and Θ , where Ψ and Θ are the covariance matrices of the partitioned vectors of disturbance terms and measurement errors ζ  [ζ TI | ζ OT | ζ IT ]T and ε  [ε IT | ε OT | ε IT ]T , respectively. Importantly, constraints were embedded into the six partitioned matrices and specified in matrix form. The simplified resulting forms of the six partitioned matrices are expressed as below:.   B   . B 治 B ( 0) 政 B B 大.   α  .   αO . α I ( C5 ).   Ψ  . Ψ II. Ψ OI. Ψ OO. Ψ II ( C6 ). Ψ IO ( C7 ).   ν  .     νO . Λ    ν I ( C9 ) . 立. Θ II ( C10 ). OI. OO. B II ( 0). B II ( 0)   BOI . B II ( 0). B IO ( 0).   . Ψ II ( C8 ). Λ OI. Λ OO. Λ II ( C 3 ). Λ IO ( 0). Ch. Θ OO. engchi. Θ IO ( C11 ). y. Λ IO ( 0). sit. Λ II. er. ‧ 國. al. n. Θ OI. IO. (8). ‧. io. Θ II. II. 學. νI. Nat.   Θ  . αI. i n U. v. Λ II ( 0)   Λ OI . Λ II ( C4 ).   . Θ II ( C12 ). Note that C 5 to C12 , whose complete derived forms can be found on pages 98 and 99 of Chen and Cheng (2014), were, similar to C1 to C4 , defined as constraint matrices. In particular, each C i ( i  1, , 12 ) is a function of one or more of submatrices associated with ηI and/or submatrices associated with y I .. 26.

(38) In general, each of the six partitioned matrices shown above can be thought of as being composed of a combination of two essential parts. One part is associated with the submatrices corresponding to the basic model (marked by the dotted boxes in Equation 8). The other part is associated with the matrices (except for B OI  and Λ OI ), consisting of pertinent combinations of submatrices from the basic model,. which were derived and constrained based on the normality assumption of the basic. 政治大. model described in Equation 3 as well as the structural equations of the R-R. 立. framework described in Equations 6 and 7.. ‧ 國. 學. In summary, Chen and Cheng (2014) generalized Jöreskog and Yang's (1996). ‧. constrained approach to a matrix form, in which the model specification of the R-R. Nat. io. sit. y. fundamental framework was developed through the three procedures described in. er. Parts 1 through 3 above. Initially, the R-R partitioning scheme was established by first. al. n. v i n C h (1984) Case AUand then implementing the adopting the model notation of Muthén engchi. matrix partitioning technique and filter matrices. Secondly, the structural equations of the R-R framework were formulated by combining the expansions of the vectors of latent nonlinear variables and product indicators into the structural equations of the basic model. Lastly, constraint specification was implemented by considering the normality assumption of the basic model as well as the structural equations of the R-R framework. As will be seen in the following sections of this chapter, the elementary. 27.

(39) techniques described in Part 1 will be applied in preprocessing the F-F and R-F fundamental frameworks as well as the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks of the current research. Moreover, the advanced procedures summarized in Part 2 and Part 3 will later be applied in Chapters 3 and 4 to formulate the model specifications of these six current frameworks, which will come to represent a much broader and more powerful generalization of the constrained approach.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 28. i n U. v.

(40) Section 2 Model Reformulating Procedure In this section, two examples of latent nonlinear models5, one representing the interaction between first-order formative latent variables (see Figure 5A) and the other representing the interaction between first-order reflective and formative latent variables (see Figure 6A), are introduced; these models form the basis from which reformulated models (see Figures 5B and 6B) are generated to be expressed in the. 政治大. model notation of Muthén (1984) Case A. It is important to note at the outset that. 立. Muthén's notation is used in order to be compatible with Chen and Cheng (2014),. ‧ 國. 學. from which our current framework extends. In contrast to the LISREL model notation. ‧. (Jöreskog & Sörbom, 1993), Muthén's notation does not distinguish between. Nat. io. sit. y. exogenous and endogenous variables, meaning the generalization of latent nonlinear. n. al. er. effects between exogenous and/or endogenous variables is relatively easier when this latter notation is applied.. 5. Ch. engchi. i n U. v. To solve the identification problem associated with the presence of formative latent variables, in both examples the submodels defined as not including nonlinear terms of latent and observed variables are devised to conform to the identification rules (e.g., 2+ emitted paths rule and exogenous X rule) established by Bollen and Davis (2009a; 2009b).. 29.

(41) y. b. y. D L. b. y. y. b. y. D. y. L. L. b. y. L. y. y y. y. L. y y. y. L L. y y y y. (A) The original model with product terms. y. 1. y. 1. y. 1. 立. 治 L L 政大. y y y. L. L. ‧ 國. 1. 學. y. y. y. L. ‧ y. L L. sit. y. Nat. n. al. er. io. (B) The reformulated model without product terms. y. 1. y. 1. y. 1. y. 1. y y. 1. y y. 1. y y. 1. y y. 1. Ch L. engchi. i n U. v. L. y. L. L. y. y y. L. y y. L L. (C) The reformulated model with product terms. Figure 5. Interaction between first-order formative latent variables. 30.

(42) y. b. y. b. D L. y y. L. y. L. y. y. L. y y. L. L y. y. L L. L y. (A) The original model with product terms. y. 1. y. 1. 立. y. y y. L. L. ‧ 國. y. 學. y. 治 L L政大. y y. L. ‧ y. L L. sit. y. Nat. n. al. er. io. (B) The reformulated model without product terms. y y. 1 1. Ch L. engchi. y y. i n U. v. L. y. L. L. y. y y. y y. L. y y. y y. L L y y y y. (C) The reformulated model with product terms. Figure 6. Interaction between first-order reflective and formative latent variables. 31.

(43) Example 1: Interaction between Formative Latent Variables Looking at the overall structure of this first example as shown in Figure 5A (see page 30), y1 to y 4 are the causal indicators of the formative latent variables L1 and L 2 , where L1 and L 2 are assumed to not be associated with any effect indicators but are determined by their casual effects together with their respective disturbance terms D1 and D2 . Meanwhile, y 5 to y10 are the effect indicators. 政治大. associated with the reflective latent variables L 3 to L 5 . In this case, the direct. 立. effects of L1 on L 3 and L 4 , L 2 on L 4 and L 5 , and L 2 L1 (representing the. ‧ 國. 學. interaction between first-order formative latent variables) on L 4 are presumed to be. ‧. of interest to the researcher.. Nat. sit. n. al. er. io. L 2 L1  (a 2  b 3 y 3  b 4 y 4  D 2 )(a 1  b1 y1  b 2 y 2  D1 ) 4.  (a 2 a1 )  (a1  b j y j j 3. 4. y. The expanded form of L 2 L1 can be expressed as follows:. i n C  a h b y   b b y U engchi y ) 2. 2. 4. i. i 1. i. 2. j  3 i 1. j. i. j. v. (9). i. 2.  [(a 2   b j y j )D1  (a1   b i y i )D 2  D 2 D1 ] . j 3. i 1. Here, a 1 and a 2 are the intercepts of L1 and L 2 , while b i and b j represent the direct effects of y i on L1 and y j on L 2 (for i  1, 2 and j  3, 4 ). The right side of Equation 9 can be broken down into the sum of three components: a constant term, the effects of observed indicators (including non-product and product terms) on L 2 L1 , and a disturbance term whose addends are all factors of D1 and/or. 32.

(44) D2 . As can be seen both graphically from Figure 5A and formally from Equation 9, the original terms y1 to y 4 and product terms y 3 y1 , y 4 y1 , y3 y 2 and y 4 y 2 serve as the causal indicators of L 2 L1 . In order to allow the causal indicators to be specified by the model notation of Muthén (1984) Case A which presumes all observed indicators are effects of latent variables, the above model is re-specified by introducing a phantom latent variable for. 政治大. each causal indicator (e.g., Bollen, 1989, p. 311; Bollen & Davis, 2009a; Williams, et. 立. al., 2003). Each causal indicator y1 to y 4 is set to have a factor loading of one from. ‧ 國. 學. its respective phantom latent variable; the intercepts are fixed zeros and there is no. ‧. measurement error. As illustrated in the path diagram in Figure 5B, this reformulation. Nat. sit. y. technically transforms observed causal indicators into effect indicators. In other words,. er. io. each phantom latent variable 1 to  4 is defined as a first-order latent variable. al. n. v i n whose direction path runs to its C own reflective latent U hindicator, e n g ci.e., h ia first-order. variable. In accordance with this reformulated model, the expanded form of.  65 ( L 2 L1 ) can be rewritten as follows:  6 5  ( 6   63 3   64 4   6 )( 5   511   52 2   5 ) 4. 2. 4. j 3. i 1. (10). 2.  ( 6 5 )  ( 5   6 j j   6   5ii    6 j  5i ji ) 4. 2. j 3. i 1. j  3 i 1.  [( 6    6 j j ) 5  ( 5    5ii ) 6   6 5 ] . In this equation,  5 and  6 ,  5 and  6 , respectively identical to a 1 and a 2 , D1. 33.

(45) and D2 from Equation 9, are intercepts and disturbance terms of  5 (  L1 ) and.  6 (  L 2 ) , whereas  5i and  6 j , identical to b i and b j from Equation 9, represent the effects of i on  5 and  j on  6 (for i  1, 2 and j  3, 4 ). Importantly, the path diagram of the reformulated model shown in Figure 5B is equivalent to that of the original model shown in Figure 5A but with  5 and  6 influenced by latent variables rather than observed causal indicators. In other words,. 政治大. the formative latent variables  5 (  L1 ) and  6 (  L 2 ) , while defined as first-order. 立. latent variables in the original model, are defined as second-order latent variables. ‧ 國. 學. having first-order reflective latent variables as their casual indicators (e.g.,. ‧. Diamantopoulos, et al., 2008; Jarvis, et al., 2003) in the reformulated model. It should. io. sit. y. Nat. be further noted that in both the reformulated and original models,  65 ( L 2 L1 ) is. er. not associated with any product term of existing observed effect indicators because. al. n. v i n wereC previously to not have associated effect h e n gassumed chi U.  5 (  L1 ) and  6 (  L 2 ). indicators. Moreover, it is important to be aware that, in contrast to the original model, the reformulated model in Figure 5B cannot be identified despite the fact that scaling restrictions were properly imposed on particular parameters to set the metric of the latent variables. This can be explained by evoking the finding by Jöreskog and Yang (1996) that the constrained model cannot be identified unless at least one product term of existing observed variables is used as the indicator of the latent nonlinear variable.. 34.

(46) Fortunately, the above-described problem associated with the reformulated model can be solved by taking product terms of existing observed variables as indicators of the latent interaction variables  31 ,  41 ,  3 2 and  4 2 as shown in Figure 5C. As previously established, each causal indicator y1 to y 4 perfectly measures its own phantom latent variable. Hence, the product indicator y j y i (for. i  1, 2 and j  3, 4 ) is equivalent to  j i , meaning that the intercept of y j y i , the. 政治大. coefficient relating y j y i to  ji , and the measurement error for y j y i are specified. 立. to be zero, one and zero, respectively. Note that in this case, each  ji has only one. ‧ 國. 學. measured product indicator as its direct effect.. ‧. Example 2: Interaction between Reflective and Formative Latent Variables. Nat. io. sit. y. The graphical representation of this model, shown in Figure 6A (see page 31), is. er. similar to that of the interaction model of Figure 5A, but with L 2 assumed to be the. n. al. i n C reflective latent variable associatedhwith effect indicators engchi U (y. v. 3. and y 4 ). Further, it can. be seen that the relationships among the latent variables are assumed to be identical to those of Example 1 in that L 3 and L 5 are respectively affected by L1 and L 2 while L 4 is affected not only by L1 and L 2 but also by L 2 L1 . It should also be noted that in this example model, L 2 L1 represents the interaction between first-order reflective and formative latent variables, not just between first-order formative latent variables as in Example 1. The expanded form of L 2 L1 is expressed as shown here:. 35.

(47) L 2 L1  L 2 (a1  b1 y1  b 2 y 2  D1 ). (11). 2.  (a1L 2   b i L 2 y i )  (L 2 D1 ) , i 1. where a1 is the intercept of L1 and b i represents the direct effects of y i on L1 (for i  1, 2 ). In this equation, it can be seen that L 2 L1 is determined not only by the effects of L 2 and L 2 yi on L 2 L1 but also by a disturbance term which is a factor of. D1 , as shown graphically in Figure 6A. Note that L 2 yi represents the interaction. 政治大. between the latent and observed variables.. 立. Next, applying the same technique described in Example 1, the reformulated. ‧ 國. 學. model is constituted to fit into Muthén's notation (1984) Case A by creating the. ‧. phantom latent variables 1 and  2 which solely influence the causal indicators y1. Nat. sit. y. and y 2 , respectively, as illustrated in Figure 6B. It should be noted that, analogous to. er. io. 1 to  4 from Example 1, 1 and  2 are defined here as first-order reflective. al. n. v i n C hform of   ( LUL ) can be rewritten as latent variables. Thus, the expanded engchi 3 4. 2. 1. follows:.  3 4   3( 4   411   42 2   4 ). (12). 2.  ( 43    4i 3i )  (3 4 ) , i 1. where  4 and  4 , identical to a1 and D1 from Equation 11, are the intercept and disturbance term of  4 ( L1 ) . Meanwhile,  4i (for i  1, 2 ), identical to b i from Equation 11, represents the effect of i on  4 . Importantly, the path diagram of the. 36.

(48) reformulated model from Figure 6B is equivalent to that of the original model from Figure 6A with two adjustments: first,  4 ( L1 ) is affected by 1 and  2 rather than y1 and y 2 , meaning that  4 , while defined as a first-order latent variable in the original model, is defined here as a second-order latent variable; second,. 3 4 ( L 2 L1 ) is affected by  31 and  3 2 instead of L2 y1 and L 2 y 2 . As a result of this second adjustment,  31 and  3 2 can be unambiguously defined as. 政治大. latent variables in the reformulated model.. 立. Similar to the reformulated model of Example 1, the reformulated model in. ‧ 國. 學. Figure 6B is not identified because there are no product indicators associated with. ‧.  31 and  3 2 . This is remedied as shown in Figure 6C by using all possible product. sit. y. Nat. terms as indicators of each  3 i , i.e., y 3 y1 and y 4 y1 , y3 y 2 and y 4 y 2 are used as. er. io. observed indicators of  31 and  3 2 , respectively. However, unlike in Example 1,. al. n. v i n j  3, 4C ) does perfectly measure its own latent variable. h enot ngchi U. y j y i (for i  1, 2 and. More specifically, for this example, the expanded forms of y i and y j are respectively expressed as  i and  j   j 3 3   j , where  j and  j are the intercept and measurement error of y j , and  j 3 represents the loading of y j on.  3 . Hence, the product indicator y j y i can be expanded as  j i   j 3 3 i   i j , which means that the coefficients relating y j y i to  i and  3 i are respectively constrained as  j and  j 3 , while the measurement error for y j y i is equal to  i j .. 37.

(49) To briefly summarize this section, the two types of latent interaction models, i.e., Examples 1 and 2, are reformulated to be congruous with the R-R framework developed by Chen and Cheng (2014) based on a technique in which a phantom variable is created for each causal indicator. Through this procedure, first-order formative latent variables in the original models are redefined as second-order latent variables specifying first-order reflective latent variables as their causal indicators in. 政治大. the reformulated models. Likewise, the two different types of interaction effects from. 立. the reformulated models of the above examples are both redefined as second-order. ‧ 國. 學. latent interaction variables as they employ first-order latent variables (including. ‧. non-product and product terms) as their indicators (e.g., Ping, 2007). Each of these. Nat. al. er. io. reflective latent variables and a disturbance term.. sit. y. second-order latent interaction variables is determined by its respective first-order. n. v i n Section 3 Model Partitioning C Scheme on the Fundamental Frameworks h e nApplied gchi U In this section, the partitioning technique that will come to be utilized on the F-F. and R-F fundamental frameworks (and later on the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks) is demonstrated in the context of the above-described reformulated models with product indicators shown in the path diagrams of Figures 5C and 6C. The complete graphical representations of the partitioning scheme applied to the two examples are shown in Figures 7A and 7B.. 38.

(50) ηF. yF y. 1. y. 1. y. 1. y. 1. y y. ηT. ηS. yT y. L. L. y y. L. L. y y. L. 1. y. y y. 1. y y. 1. y y. 1. L L. ηS 政 ( η S ). η F. y F. 立. 治. 大. 1. y. Nat. y. L. L. io. y. al. L. n. y y y y. y. L. sit. y. y. L. er. 1. yT. ‧. y. ηT. ηS. y. ηF. 學. yF. ‧ 國. (A) Example 1: interaction between formative latent variables. n U engchi L L. Ch. iv. y y. y. y y y y. y F. η F. ηS  ( η~S ). (B) Example 2: interaction between reflective and formative latent variables. Figure 7. Partitioning the reformulated models from Examples 1 and 2. As a convenient way to distinguish the two different latent nonlinear effects epitomized by the two examples, the vector of interaction and/or quadratic effect(s). 39.

(51) between formative latent variables is denoted as ηS while the vector of interaction effect(s) between reflective and formative latent variables is denoted as η~S , where the subscript stands for “Second-order latent nonlinear effects”. In applying the partitioning technique on the structural part of the overall framework, the block of second-order latent nonlinear effects can be composed of ηS , η~S or a combination of the two (a topic that will be revisited in Section 4); thus, when making general. 政治大. references to this block, the notation ηS will be used.. 立. As in Chen and Cheng (2014), the submodel that does not include nonlinear. ‧ 國. 學. effects of latent variables and product indicators is referred to as the basic model. In. ‧. the structural part of the basic model, latent variables are partitioned into three. io. sit. y. Nat. subvectors (denoted as ηF , ηS and ηT , where the subscripts respectively stand for. n. al. er. “First layer”, “Second layer” and “Third layer”) to support the integration of the. Ch. vector of second-order latent nonlinear variables ηS. engchi. iv n and the vector of first-order U. latent nonlinear variables (denoted as η F , where the subscript stands for “First-order latent nonlinear effects”), the latter of which is formulated to constitute all the nonlinear causes of ηS . In light of the fact that the compositions of ηS and η F effectively determine the distinction among ηF , ηS and ηT , a clear explanation of the proposed partitioning technique should begin with more detailed descriptions of ηS and η F .. 40.