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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.

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Notes

  1. The E-CQS scale is copyright-protected, but is available on request at cquery@culturalq.com.

  2. We nevertheless report the Belgium sample results in Table 9 for the sake of illustration.

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Funding

This work was partially supported by the Ministry of Education and the National Research Foundation of Korea (NRF-2019S1A5A2A03052192) to Heungsun Hwang and Younyoung Choi.

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Correspondence to Heungsun Hwang or Younyoung Choi.

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Gyeongcheol Cho and Christopher Schlaegel are co-first authors.

Appendix

Appendix

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 GSCAM. 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 GSCAM 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' = 2' = 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 GSCAM. 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 GSCAM 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 ei 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') = IP, γ2u' = 2' = 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 GSCAM (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 GSCAM algorithm for W2. In the next steps, the loadings C and path coefficients B are estimated.

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 = [IJ, 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 sij 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.

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Cho, G., Schlaegel, C., Hwang, H. et al. Integrated Generalized Structured Component Analysis: On the Use of Model Fit Criteria in International Management Research. Manag Int Rev (2022). https://doi.org/10.1007/s11575-022-00479-w

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Keywords

  • Integrated generalized structured component analysis
  • Structural equation modeling
  • Model fit
  • GFI
  • SRMR
  • Cultural intelligence