Integrated Generalized Structured Component Analysis: On the Use of Model Fit Criteria in International Management Research

Abstract

Structural equation modeling (SEM) has remained two mutually exclusive domains, factor-based vs. component-based, depending on whether a construct is modeled by either a factor or a component (i.e., weighted composite of indicators). Research in international management (IM) and international business (IB), however, needs to accommodate a more general model that considers a wide range of constructs from different disciplines at the same time, representing some constructs as factors (e.g., cultural distance and institutional distance) and others as components (e.g., international experience and export intensity). Integrated generalized structured component analysis (IGSCA) is a recently developed statistical method for estimating such models with both factors and components. IGSCA can provide overall fit indexes for model evaluation, including the goodness-of-fit index (GFI) and the standardized root mean squared residual (SRMR). However, the performance of these indexes in IGSCA is not yet investigated. Addressing this limitation, we (a) highlight the limitations of the dominantly used SEM approaches, (b) review the use of different SEM approaches in IM/IB research in the last decade, (c) conduct a simulation study, confirming that both GFI and SRMR distinguish well between correct and misspecified models with both factors and components, and (d) we illustrate the indexes’ efficacy using a model concerning the role of personality traits and international experience in shaping cultural intelligence. Based on the review and the results of the simulation study and the illustrative example, we also propose rules-of-thumb cutoff criteria for each index in IGSCA.

Appendix

1.1 Appendix 1: Model Specification and Estimation in IGSCA

Here we provide a brief description of IGSCA’s model specification and estimation based on Hwang et al. (2021). IGSCA involves specification of three sub-models: measurement, structural, and weighted relation models. The measurement model specifies the relationships between indicators and constructs that are represented by either factors or components, whereas the structural model expresses the relationships between constructs. The weighted relation model is used to specify a construct as a component with or without measurement errors explicitly incorporated. Let z₁ and z₂ denote vectors of J₁ composite indicators and J₂ effect indicators, respectively. Let γ₁ and γ₂ denote vectors of P₁ and P₂ constructs associated with z₁ and z₂, respectively. Assume that all indicators and constructs are normalized. Let C₁ and C₂ denote J₁ by P₁ and J₂ by P₂ matrices of loadings relating γ₁ and γ₂ to z₁ and z₂, respectively. Let u denote a vector of J₂ unique variables for z₂. Let D denote a diagonal matrix of J₂ unique loadings for z₂.

The measurement model for IGSCA is a combination of those used in GSCA and GSCA_M. The measurement model for GSCA is given as

$$ {\mathbf{z}}_{{1}} = {\mathbf{C}}_{{1}} {{\varvec{\upgamma}}}_{{1}} + {{\varvec{\upvarepsilon}}}_{{1}} $$

(A-1)

where ε₁ is the error term that represents the portion of z₁ left unexplained by γ₁ (Hwang & Takane, 2014, Chapter 2). The measurement model for GSCA_M is given as

$$ {\mathbf{z}}_{{2}} = {\mathbf{C}}_{{2}} {{\varvec{\upgamma}}}_{{2}} + {\mathbf{Du}} + {{\varvec{\upvarepsilon}}}_{{2}} , $$

(A-2)

where C₂γ₂ and Du indicate common and unique parts of z₂, respectively, and ε₂ is the portion of z₂ left unexplained by their common and unique parts (Hwang et al., 2017). We assume that γ₂ is uncorrelated with u (i.e., γ₂u' = uγ₂' = 0) and u is column-wise orthonormalized (i.e., uu' = ${\mathbf{I}}_{{J_{2} }}$, where ${\mathbf{I}}_{{J_{2} }}$ is the identity matrix of order J₂).

The measurement model for IGSCA is then given as

$$\begin{array}{l}\left[\begin{array}{c}{\text{z}}_{1}\\ {\text{z}}_{2}\end{array}\right]=\left[\begin{array}{cc}{\text{C}}_{1}; {0}\\ {0}; {\text{C}}_{2}\end{array}\right]\left[\begin{array}{c}{\gamma}_{1}\\ {\gamma}_{2}\end{array}\right]+\left[\begin{array}{c}{0}\\ {\text{Du}}\end{array}\right]+\left[\begin{array}{c}{\varepsilon}_{1}\\ {\varepsilon}_{2}\end{array}\right]\\ {\text{ z }}= {C\gamma + s + \varepsilon},\end{array}$$

(A-3)

where z = [z₁; z₂], C = $\left[\begin{array}{cc}{\mathbf{C}}_{1}&\mathbf{0}\\ \mathbf{0}& {\mathbf{C}}_{2}\end{array}\right]$, γ = [γ₁; γ₂], s = [0; Du], and ε = [ε₁; ε₂].

The weighted relation model for IGSCA is also a combination of those for GSCA and GSCA_M. The weighted relation model for GSCA is given as

$$ {{\varvec{\upgamma}}}_{{1}} = {\mathbf{W}}_{{1}} {\mathbf{z}}_{{1}} , $$

(A-4)

where W₁ is a P₁ by J₁ matrix of weights assigned to z₁ (Hwang & Takane, 2014, Chapter 2). The weighted relation model for GSCA_M is given as

$$ {{\varvec{\upgamma}}}_{{2}} = {\mathbf{W}}_{{2}} \left( {{\mathbf{z}}_{{2}} {-} \, {\mathbf{Du}}} \right), $$

(A-5)

where W₂ is a P₂ by J₂ matrix of weights assigned to z₂, of which the unique parts are removed (Hwang et al., 2017). Then, the weighted relation model for IGSCA is given as

$$\begin{array}{l}\left[\begin{array}{l}{\varvec{\upgamma}}_{1}\\ {\varvec{\upgamma}}_{2}\end{array}\right]=\left[\begin{array}{ll}{\mathbf{W}}_{1}& \mathbf{0}\\ \mathbf{0}& {\mathbf{W}}_{2}\end{array}\right]\left[\begin{array}{l}{\mathbf{z}}_{1}\\ {\mathbf{z}}_{2}-{\mathbf{Du}}\end{array}\right]\\ {{\varvec{\upgamma}}}= {\mathbf{W(z - s}}),\end{array}$$

(A-6)

where W = $\left[\begin{array}{cc}{\mathbf{W}}_{1}& \mathbf{0}\\ \mathbf{0}& {\mathbf{W}}_{2}\end{array}\right]$. This sub-model shows that both γ₁ and γ₂ are defined as components rather than factors. Nonetheless, as (A-5) shows, γ₂ represents components of effect indicators, of which the unique parts are removed. In this way, measurement errors in the indicators are taken into account, meaning the parameters involving γ₂ are similar to those in factor-based SEM (Hwang et al., 2017).

Let B denote a P by P matrix of path coefficients relating γ among themselves, where P = P₁ + P₂. The structural model for IGSCA is expressed as

$${\varvec{\upgamma}} = {\mathbf{B}\varvec{\upgamma}} + {\varvec{\upzeta}},$$

(A-7)

where ζ is the error term for γ. The B matrix contains path coefficients relating components to factors, as well as those among either factors or components only.

Our approach integrates the sub-models into a unified formulation, referred to as the IGSCA model, as follows:

$$\begin{array}{*{20}l} {\left[ {\begin{array}{*{20}c} \mathbf{z} \\ {\varvec{\upgamma}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c}\mathbf{C} \\ \mathbf{B} \\ \end{array} } \right]{\varvec{\upgamma}} + \left[ {\begin{array}{*{20}c} \mathbf{s} \\ \mathbf{0} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\varvec{\upvarepsilon}} \\ {\varvec{\upzeta}} \\ \end{array} } \right]} \hfill \\ {\varvec{\uppsi} = \mathbf{A}{\varvec{\upgamma}} + \mathbf{v} + \mathbf{e},} \hfill \\ \end{array}$$

(A-8)

where ψ = [z; γ], A = [C; B], v = [s; 0], and e = [ε; ζ]. As in GSCA, IGSCA does not make a distributional assumption of e, i.e., multivariate normality.

Let e_i denote the error term in (A-8) for a single observation of a sample of N observations (i = 1, …, N). To estimate all parameters in W, A and v, IGSCA aims to minimize the following objective function

$$ \varphi = \sum\limits_{i = 1}^{N} {{\mathbf{e}}_{i} ^{\prime}{\mathbf{e}}_{i} } $$

(A-9)

subject to diag(γγ') = I_P, γ₂u' = uγ₂' = 0, and uu' = ${\mathbf{I}}_{{J_{2} }}$, where diag() denotes a diagonal matrix. An iterative algorithm is used to minimize the objective function by a simple adoption of the existing algorithms for GSCA and GSCA_M (Hwang et al., 2021). This algorithm repeats several steps, each of which updates one set of parameters while the other sets are fixed, until no substantial differences in parameter estimates occur between iterations. Specifically, in step 1, the unique variables u and unique loadings D are updated. In step 2, the component weights W are updated using the GSCA algorithm for W₁ and the GSCA_M algorithm for W₂. In the next steps, the loadings C and path coefficients B are estimated.

1.2 Appendix 2: Model Fit Indexes

IGSCA offers two overall fit indexes, GFI and SRMR, which summarize the size of the differences between the sample and model-implied covariances. The implied covariances obtained from the parameter estimates on convergence are also called reproduced covariances. Although in principle, IGSCA does not assume indicators have a specific covariance structure to derive the reproduced covariances, we assume that composite indicators per component are correlated to each other, leading the covariance matrix of ε₁ to be block-diagonal, as is typically assumed in component-based SEM (e.g., Bollen & Bauldry, 2011; Cho & Choi, 2020; Dijkstra, 2017; Grace & Bollen, 2008). Also, we assume that the common factor analytic model holds for a sample, meaning ε₂ will be zero (Velicer & Jackson, 1990). Further, we assume that cov(v, e) = 0 and cov(ε, ζ) = 0. Then, we can re-express (A-8) as

$$\begin{array}{l}\left[\begin{array}{c}{\mathbf{I}}_{J}\\ {\mathbf{W}}\end{array}\right]{(}{\mathbf{z}}{ - }{\mathbf{s}}) =\left[\begin{array}{cc}\mathbf{0}& {\mathbf{C}}\\ \mathbf{0}& {\mathbf{B}}\end{array}\right]\left[\begin{array}{c}{\mathbf{I}}_{J}\\ {\mathbf{W}}\end{array}\right]({\mathbf{z}}-{\mathbf{s}}{) +}\left[\begin{array}{c}{\varvec{\upepsilon}}\\ {\varvec{\upzeta}}\end{array}\right]\\ \mathbf{V}{(}{\mathbf{z}}{ - }{\mathbf{s}}) = {\mathbf{TV}}\left[\begin{array}{c}{\mathbf{I}}_{J}\\ {\mathbf{W}}\end{array}\right]{(}{\mathbf{z}}{ - }{\mathbf{s}}) + {\mathbf{e}}\\ \mathbf{V}({\mathbf{z}}-{\mathbf{s}}) = {{(}{\mathbf{I}}_{K} \, { - }{\mathbf{T}}{)}}^{{-}{1}} + {\mathbf{e}},\end{array}$$

(A-10)

where J = J₁ + J₂, K = J + P, and T = $\left[\begin{array}{cc}\bf{0}& {\bf{C}}\\ \bf{0}& {\bf{B}}\end{array}\right]$. The implied covariance matrix of z, denoted by Σ, is then given as

$${{\varvec{\Sigma}}} = {\mathbf{G}}({\mathbf{I}}_{K} {-}{\mathbf{T}})^{{{-}{1}}} {\text{E}}\left( {{\bf{ee}}^{\prime}} \right)(({\mathbf{I}}_{K} {-}{\mathbf{T}})^{{{-}{1}}} )^{\prime}{\mathbf{G}}^{\prime} + {\text{E}}\left( {{\mathbf{ss}}^{\prime}} \right),$$

(A-11)

where G = [I_J, 0], E(ee') is a block-diagonal covariance matrix of e, and E(ss') is a diagonal covariance matrix of s, of which the non-zero entries are equal to D².

Let S and $\widehat{\mathbf{\Sigma }}$ denote the sample and the reproduced covariance matrices. Let s_ij and ${\widehat{\sigma}}_{\text{ij}}$ respectively denote an observed covariance in S and the corresponding reproduced covariance in $\widehat{\mathbf{\Sigma }}$. Then, the GFI and SRMR are calculated as

$${\text{GFI }} = { 1 }{-}\frac{{{\text{trace}}([{\mathbf{S}} - {\hat{\mathbf{\Sigma }}}]^{2} )}}{{{\text{trace}}({\mathbf{S}}^{2} )}},{\text{ and}}$$

(A-12)

$${\text{SRMR }} = \sqrt {\left\{ {2\sum\limits_{j = 1}^{J} {\sum\limits_{l = 1}^{j} {\left[ {(s_{jl} - \hat{\sigma }_{jl} )/(s_{jj} s_{ll} )} \right]^{2} } } } \right\}/J(J + 1)}$$

(A-13)

As the above formulas show, GFI values close to 1 and SRMR values close to 0 indicate a small degree of the covariance discrepancies.