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fundamental frameworks is established. The precise formulation of these frameworks is left as the first major topic in Chapter 3.
Section 4 Model Partitioning Scheme Applied on the Integrated Frameworks
In this section, the partitioning scheme which has already been demonstrated on reformulated prototype nonlinear models characterizing the F-F and R-F fundamental frameworks, will now be adapted to fit into the context of four reformulated prototype nonlinear models characterizing the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks. It is important to be aware that the aim of this exercise is to provide a practical context from which to establish the partitioning pattern of the integrated frameworks.
In constructing the four prototype models, the basic model (previously defined as the submodel not including any nonlinear terms) of Figure 6A (see page 31) is utilized on an across-the-board basis as it is set up to generate any combination of the three types of latent nonlinear effects, L21, L22 and L2L1. Here, L21 and L22 respectively represent the quadratic effects of formative latent variable L and reflective latent 1 variable L while 2 L2L1 represents the interaction effect of a combination of the two; as such, the three types of nonlinear effects will also be referred to as the F-F, R-R and R-F effects. With the appropriate combination of nonlinear effects generated from this basic model, the four prototypes characterizing the F-F/R-R, R-F/R-R,
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F-F/R-F and F-F/R-F/R-R integrated frameworks can be created by incorporating (L21L22), (L2L1L22), (L21L2L1) and (L21L2L1L22) into the basic models as shown in Figures 8A, 9A, 10A and 11A.
Examining the structural parts of these nonlinear models6 shown in Figures 8A to 11A, the selected combination of latent nonlinear effects from the set (L21, L22, L2L1) are all arbitrarily presumed to influence the same outcome variable L , but it should 4 be pointed out that L3 and L5 are also permissible outcome variables. Meanwhile, an inspection of the measurement parts of these models reveals that the three types of latent nonlinear effects L21, L22 and L2L1 are respectively measured by the sets of variables (y , 1 y , 2 y12, y2y1, y22) , (y , 23 y4y3, y24) and (L , 2 L2y1, L2y2). It is important to note that the direction of the relationship between a given latent
nonlinear variable and its measures is determined by the type of latent nonlinear effect involved. Specifically, for R-R effects (meaning L22 in Figures 8A, 9A and 11A), the path is from the nonlinear variable to its associated measures while for F-F effects (meaning L21 in Figures 8A, 10A and 11A) and R-F effects (meaning L2L1 in Figures 9A, 10A and 11A), the path is from the associated measures to the nonlinear variable.
6 These nonlinear models, analogous to Models D to G in Figure 3 (see page 15), resolve the identification problem in the same way as in Examples 1 and 2, as explained in Footnote 5.
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(A) The original model with product terms
(B) The reformulated model with product terms
Figure 8. The prototype model characterizing the F-F/R-R framework
y
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(A) The original model with product terms
(B) The reformulated model with product terms
y
Figure 9. The prototype model characterizing the R-F/R-R framework
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(A) The original model with product terms
(B) The reformulated model with product terms
y
Figure 10. The prototype model characterizing the F-F/R-F framework
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(B) The reformulated model with product terms
y
(A) The original model with product terms R-F effect
Figure 11. The prototype model characterizing the F-F/R-F/R-R framework
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Table 1
Partitioning scheme applied on the reformulated models
Fundamental framework Integrated framework
Nonlinear effect(s) F-F R-F F-F/R-R R-F/R-R F-F/R-F F-F/R-F/R-R
2
L1 L2L1 (L21 L22) (L2L1L22) (L21L2L1) (L21L2L1L22)
Partitioned vector The structural part The structural part
η S ηS : F-F effect L21(42) L21(42) L21(42) L21(42)
~
ηS : R-F effect L2L1(
3
4) L2L1(
3
4) L2L1(
3
4) L2L1(
3
4)
ηF R-R effect L22(32) L22(32) L22(32)
Causes of ηS (12 ,21 ,22) (12 ,21 ,22) (12 ,21 ,22) (12 ,21 ,22)
Causes of η~S (31 ,32) (31 ,32) (31 ,32) (31 ,32)
ηF (1 ,2) (1 ,2 ,3) (1 ,2 ,3) (1 ,2 ,3) (1 ,2 ,3) (1 ,2 ,3)
ηS ( 4) ( 4) ( 4) ( 4) ( 4) ( 4)
ηT (3 ,5 ,6 ,7) (5 ,6 ,7) (5 ,6 ,7) (5 ,6 ,7) (5 ,6 ,7) (5 ,6 ,7) (Continued)
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Fundamental framework Integrated framework
Nonlinear effect(s) F-F R-F F-F/R-R R-F/R-R F-F/R-F F-F/R-F/R-R
2
L1 L2L1 (L21L22) (L2L1L22) (L21 L2L1) (L21L2L1L22)
Partitioned vector The measurement part The measurement part
yF Indic. of R-R effect (y23 ,y4y3 ,y24) (y32 ,y4y3 ,y24) (y23 ,y4y3 ,y24)
Indic. of causes of ηS (y12 ,y2y1 ,y22) (y12 ,y2y1 ,y22) (y12 ,y2y1 ,y22) (y12 ,y2y1 ,y22) Indic. of causes of η~S (y3y1 ,y4y1 ,
) y y , y y3 2 4 2
, y y , y (y3 1 4 1
) y y , y y3 2 4 2
, y y , y (y3 1 4 1
) y y , y y3 2 4 2
, y y , y (y3 1 4 1
) y y , y y3 2 4 2
yF Indic. of ηF (y1 ,y2) (y1 ,y2 ,y3 ,y4) (y1 ,y2 ,y3 ,y4) (y1 ,y2 ,y3 ,y4) (y1 ,y2 ,y3 ,y4) (y1 ,y2 ,y3 ,y4) yT Indic. of ηT (y3 , ,y10) (y5 , ,y10) (y5 , ,y10) (y5 , ,y10) (y5 , ,y10) (y5 , ,y10) Notes: The gray blocks mark each type of latent nonlinear effect (F-F, R-F and R-R effects). Indic. is an abbreviation for Indicator.
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In light of the fact that the partitioning scheme described in Section 3 represents the relationships among variables based on the model notation of Muthén's (1984) Case A, which, to reiterate, presumes all observed indicators are effects of latent variables, the four prototype models characterizing the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks, shown in Figures 8A to 11A, need to be re-specified through the reformulating technique to correspond to Muthén's notation before proceeding to the partitioning procedure. While the reformulating procedure has already been described at great length in Section 2, it is still appropriate to make a few remarks on how the procedure is applied in the current context. Stated briefly, in applying the reformulating technique to these prototype models, first-order formative latent variables (i.e., L in Figures 8A to 11A), F-F effects (i.e., 1 L21 in Figures 8A, 10A and 11A) and R-F effects (i.e., L2L1 in Figures 9A, 10A and 11A) are all redefined as second-order latent variables, whereas first-order reflective latent
variables (i.e., L to 2 L5 in Figures 8A to 11A) and R-R effects (i.e., L22 in Figures 8A, 9A and 11A) all maintain their original definition. The graphical representations of the reformulating procedure applied to the prototype models are depicted in Figures 8B to 11B. As can be seen, 4(L1), 42(L21) and/or 34(L2L1) in these reformulated models are respectively measured by the sets of first-order reflective latent variables (1, 2), (1, 2, , 12 21, ) and (22 , 3 , 31 32) associated
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with the sets of observed effect indicators (y , 1 y ), (2 y , 1 y , 2 y12, y2y1, y22) and ((y3, y ), (4 y3y1, y4y1), (y3y2, y4y2)).
Having specified the reformulated prototype models characterizing the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks, we can now look at how the partitioning scheme described in the previous section can be modified to
accommodate these models. To begin, it should be recognized that if the R-R effects (marked by dashed boxes) of the reformulated models shown in Figures 8B and 9B were disregarded, these two models could respectively be partitioned in the same manner as the reformulated prototype models representing the F-F and R-F
fundamental frameworks. Incorporating the R-R effects of the models represented by Figure 8B and 9B into the partitioning scheme is simply a matter of broadening the definition of ηF to not only aggregate all nonlinear causes of η (the original S purpose of ηF within the F-F and R-F fundamental frameworks) but also include the R-R effects that the researcher is interested in. It can similarly be recognized that if only the F-F (R-F) effect were included in the reformulated model shown in Figure 10B, this model could be partitioned in the same way as the reformulated prototype model representing the F-F (R-F) fundamental frameworks. Concurrently
incorporating both the F-F and R-F effects of the model represented by Figure 10B into the partitioning scheme simply involves expanding η such that it consists of a S
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combination of η and S η (recall that this is allowed by the definition ~S η on page S 41). By the same logic, the F-F, R-F, and R-R effects of the reformulated model shown in Figure 11B can all be assimilated into the partitioning scheme by concurrently broadening η to combine S η and S η and extending ~S ηF to encompass not only all nonlinear causes of η but also any R-R effects. The S graphical representations of the partitioning scheme applied to these reformulated prototype models are depicted in the gray blocks of Figures 8B to 11B and
summarized in Table 1.
In summary, the partitioning scheme previously established in Section 3 can be applied to the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks by merely modifying the purposes of ηF and/or η . Meanwhile, the other subvectors S from the structural part (i.e., η , F ηS and η ) as well as the full set of subvectors T from the measurement part (i.e., yF, y and F y ) are defined exactly the same as T they were in the partitioning scheme applied on the F-F and R-F fundamental frameworks. With the partitioning of the F-F/R-R, R-F/R-R, F-F/R-F and
F-F/R-F/R-R integrated frameworks having been covered in this section, the precise formulations of these frameworks remains as the central topic of Chapter 4.