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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 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.
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