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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 model notation of Muthén (1984) Case A. It is important to note at the outset that Muthén's notation is used in order to be compatible with Chen and Cheng (2014), from which our current framework extends. In contrast to the LISREL model notation (Jöreskog & Sörbom, 1993), Muthén's notation does not distinguish between
exogenous and endogenous variables, meaning the generalization of latent nonlinear effects between exogenous and/or endogenous variables is relatively easier when this latter notation is applied.
5 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).
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(A) The original model with product terms D
(B) The reformulated model without product terms L
(C) The reformulated model with product terms L
Figure 5. Interaction between first-order formative latent variables
L
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(A) The original model with product terms b
(B) The reformulated model without product terms L
(C) The reformulated model with product terms L
Figure 6. Interaction between first-order reflective and formative latent variables
y y y y y y y y
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Example 1: Interaction between Formative Latent Variables
Looking at the overall structure of this first example as shown in Figure 5A (see page 30), y to 1 y are the causal indicators of the formative latent variables 4 L 1 and L , where 2 L and 1 L are assumed to not be associated with any effect 2 indicators but are determined by their casual effects together with their respective disturbance terms D and 1 D . Meanwhile, 2 y5 to y10 are the effect indicators associated with the reflective latent variables L3 to L5. In this case, the direct effects of L on 1 L3 and L , 4 L on 2 L and 4 L5, and L2L1 (representing the interaction between first-order formative latent variables) on L are presumed to be 4 of interest to the researcher.
The expanded form of L2L1 can be expressed as follows:
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 L2L1, and a disturbance term whose addends are all factors of D and/or 1
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D . As can be seen both graphically from Figure 5A and formally from Equation 9, 2
the original terms y to 1 y and product terms 4 y3y1, y4y1, y3y2 and y4y2 serve as the causal indicators of L2L1.
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 each causal indicator (e.g., Bollen, 1989, p. 311; Bollen & Davis, 2009a; Williams, et al., 2003). Each causal indicator y to 1 y is set to have a factor loading of one from 4 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 technically transforms observed causal indicators into effect indicators. In other words, each phantom latent variable 1 to 4 is defined as a first-order latent variable whose direction path runs to its own indicator, i.e., a first-order reflective latent variable. In accordance with this reformulated model, the expanded form of
)
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and D from Equation 9, are intercepts and disturbance terms of 2 5(L1) and )
L
( 2
6
, whereas and 5i , identical to 6j bi and bj from Equation 9, represent the effects of on i and 5 on j (for 6 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 and 5 6
influenced by latent variables rather than observed causal indicators. In other words, the formative latent variables 5(L1) and 6 (L2), while defined as first-order latent variables in the original model, are defined as second-order latent variables 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 be further noted that in both the reformulated and original models, 65( L2L1) is
not associated with any product term of existing observed effect indicators because )
L
( 1
5
and 6(L2) were previously assumed to not have associated effect
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.
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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 , 31 41, 32 and 42 as shown in Figure 5C. As previously established, each causal indicator y to 1 y perfectly 4 measures its own phantom latent variable. Hence, the product indicator yjyi (for
2 1,
i and j 3,4) is equivalent to ji, meaning that the intercept of yjyi, the coefficient relating yjyi to ji, and the measurement error for yjyi are specified to be zero, one and zero, respectively. Note that in this case, each ji has only one
measured product indicator as its direct effect.
Example 2: Interaction between Reflective and Formative Latent Variables
The graphical representation of this model, shown in Figure 6A (see page 31), is similar to that of the interaction model of Figure 5A, but with L assumed to be the 2 reflective latent variable associated with effect indicators (y3 and y ). Further, it can 4
be seen that the relationships among the latent variables are assumed to be identical to those of Example 1 in that L3 and L5 are respectively affected by L and 1 L 2 while L is affected not only by 4 L and 1 L but also by 2 L2L1. It should also be noted that in this example model, L2L1 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 L2L1 is expressed as shown here:
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D , as shown graphically in Figure 6A. Note that 1 L2yi represents the interaction between the latent and observed variables.
Next, applying the same technique described in Example 1, the reformulated 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 y 1 and y , respectively, as illustrated in Figure 6B. It should be noted that, analogous to 2
1 to 4 from Example 1, 1 and 2 are defined here as first-order reflective
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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 y and 1 y , meaning that 2 4, while defined as a first-order latent variable in the original model, is defined here as a second-order latent variable; second,
) L L ( 2 1
4
3
is affected by and 31 32 instead of L2y1 and L2y2. As a
result of this second adjustment, and 31 32 can be unambiguously defined as latent variables in the reformulated model.
Similar to the reformulated model of Example 1, the reformulated model in Figure 6B is not identified because there are no product indicators associated with
1 3
and 32. This is remedied as shown in Figure 6C by using all possible product
terms as indicators of each 3i, i.e., y3y1 and y4y1, y3y2 and y4y2 are used as observed indicators of and 31 32, respectively. However, unlike in Example 1,
i jy
y (for i 1,2 and j 3,4) does not perfectly measure its own latent variable.
More specifically, for this example, the expanded forms of yi and yj are respectively expressed as and i j j33 j, where and j are the j intercept and measurement error of yj, and represents the loading of j3 yj on
. Hence, the product indicator 3 yjyi can be expanded as ji j33i ij, which means that the coefficients relating yjyi to and i 3i are respectively constrained as and j , while the measurement error for j3 yjyi is equal to . ij
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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 the reformulated models. Likewise, the two different types of interaction effects from the reformulated models of the above examples are both redefined as second-order 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 second-order latent interaction variables is determined by its respective first-order reflective latent variables and a disturbance term.
Section 3 Model Partitioning Scheme Applied on the Fundamental Frameworks
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.
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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)
Figure 7. Partitioning the reformulated models from Examples 1 and 2
L
(B) Example 2: interaction between reflective and formative latent variables (A) Example 1: interaction between formative latent variables
1
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between formative latent variables is denoted as η while the vector of interaction S effect(s) between reflective and formative latent variables is denoted as η , where ~S 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 η , S η or a combination ~S of the two (a topic that will be revisited in Section 4); thus, when making general references to this block, the notation η will be used. S
As in Chen and Cheng (2014), the submodel that does not include nonlinear 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
subvectors (denoted as η , F ηS and η , where the subscripts respectively stand for T
“First layer”, “Second layer” and “Third layer”) to support the integration of the vector of second-order latent nonlinear variables η and the vector of first-order S latent nonlinear variables (denoted as ηF, 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 and S ηF effectively determine the distinction among η , F ηS and η , a clear explanation of the proposed T
partitioning technique should begin with more detailed descriptions of η and S ηF.
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By definition, each element of η has its own set of first-order latent nonlinear S causes, with the aggregate of the sets making up ηF. Each set of causes associated with a given η element consists of all the cross-product terms of the direct latent S causes of the first and second components of that η element. To put it concretely, S with regards to Example 1, being that the direct latent causes of and 6 are 5 respectively ( , 3 4) and (1, 2), the associated set of elements in ηF is ( ,31
1 4
, 32, 42). On the other hand, in the case that a given term of η belongs S
to η , the associated set of latent nonlinear variables in ~S ηF consists of all the cross-product terms of the reflective component of the η term and the direct latent ~S causes of the formative component of the η term. This can be illustrated by looking ~S at Example 2, where the reflective component is and the latent causes of the 3 formative component 4 are (1, 2), yielding ( , 31 32) as the corresponding set of ηF elements. It is important to be aware that the overall composition of ηF
can also be deduced from the expansions of each of the latent nonlinear variables in
η , which, in the case of Examples 1 and 2 means examining the expansions of S 65 and 34 shown in Equations 10 and 12 (see pages 33 and 36), respectively. Finally, it should be noted that the above description of the composition of ηF is contingent on two stipulations. First, the two components of any given element of η have no S direct effects between them. Second, the direct latent causes of formative component
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element(s) from η and/or S η as well as the reflective component element(s) from ~S
~S
η are all first-order reflective latent variables, meaning that all component elements
from ηF are defined as first-order reflective latent variables.
Having more clearly defined η and S ηF, the partitioning of the three vectors comprising the structural part of the basic model (η , F ηS and η ) becomes a T relatively straightforward task. First of all, ηS is devised to consist of all the formative component elements from η . Therefore, in the case of Example 1, being S that η consists only of a S η block, both S and 5 are included in 6 ηS. However, for Example 2, as η is made up of a S η vector, only the formative ~S component 4 is assigned to ηS. Next, η is formulated to be composed of all the F (reflective) component elements of ηF as well as their indirect and direct latent causes. Looking back at Example 1 where it was already established that ηF
consists of , 31 41, 32, 42, one can readily ascertain that η is composed F of 1 to 4. Likewise, in Example 2, where ηF consists of and 31 32, it is clear that η consists of F 1 to . Finally, 3 η is composed of any remaining T
latent variables which may or may not be affected by latent nonlinear variables from
η and/or S ηF. Therefore, η consists of T to 7 in Example 1 and 9 to 5 7 in Example 2.
With regards to the measurement part of the basic model, observed indicators are
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partitioned into two subvectors utilized as observed indicators of η and F η T (denoted to y and F y , respectively) to support the integration of product T
indicators of ηF (denoted as yF). Note that the omission of yS is not an oversight, as, for the purposes of the current study, each formative latent variable in ηS is assumed to only be measured by its own phantom latent causal variables (identical to the observed causal indicators of the original models from Examples 1 and 2), not by its own observed effect indicators. In other words, multiple indicators and multiple causes, or MIMIC (Jöreskog & Goldberger, 1975), of each formative latent variable in
ηS is not considered in the current frameworks. The fact that the elements of ηS have no associated indicators means that η , each of whose elements has at least one S
ηS component, has no associated vector of observed product indicators. Thus y is S also not defined in our frameworks. As can be seen in both Examples 1 and 2, y to 1
y constitute 4 y , F y5 to y10 make up y , while T y3y1, y4y1, y3y2 and y4y2 are created to comprise yF.
In summary, the basic model is composed of latent variables partitioned into η , F
ηS and η and observed indicators partitioned into T y and F y . By integrating T
ηF and η (composed of S η or S η ) into the structural part and ~S yF into the measurement part of the basic model, the partitioning scheme within the F-F and R-F
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fundamental frameworks is established. The precise formulation of these frameworks is left as the first major topic in Chapter 3.