1.1. Model and Parameter Estimation

GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} involves three sub-models—weighted relation, component measurement, and structural models (Hwang and Takane, Reference Hwang and Takane2004, Reference Hwang and Takane2014). Let z ${}_{\mathrm{std}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std} =$$\end{document} [z ${}_{std,1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std,1}$$\end{document} , z ${}_{std,2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std,2}$$\end{document} , $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , z ${}_{std,J}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std,J}$$\end{document} ] ${}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\prime $$\end{document} denote a J by 1 random vector of standardized indicators, where z ${}_{std,j}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std,j}$$\end{document} is the jth standardized indicator, i.e., E(z ${}_{std,j})=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std,j}) =$$\end{document} 0 and var(z ${}_{std,j})=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std,j}) =$$\end{document} 1 ( $j=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j =$$\end{document} 1, 2, $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , J). The mean of z ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} is a zero vector and the correlation matrix of z ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} is denoted by ${\Sigma }_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma } _{std}$$\end{document} . Let ${\gamma }_{\mathrm{std}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\upgamma }_{std} =$$\end{document} [ ${\gamma }_{std,1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{std,1}$$\end{document} , ${\gamma }_{std,2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{std,2}$$\end{document} , $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , ${\gamma }_{std,P}{]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{std,P}]^{\prime }$$\end{document} denote a P by 1 random vector of standardized components, where ${\gamma }_{std,p}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{std,p}$$\end{document} is the pth standardized component, i.e., $E\left({\gamma }_{std,p}\right)=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(\upgamma _{std,p}) =$$\end{document} 0, var( ${\gamma }_{std,p})=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{std,p}) =$$\end{document} 1 ( $p=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p =$$\end{document} 1, 2, $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , P). Let W ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} denote a J by P matrix consisting of component weights assigned to indicators. Let C ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} denote a P by J matrix of loadings relating components to indicators. Let B ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} denote a P by P matrix of path coefficients relating components to each other. Let $\xi =$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upxi }} =$$\end{document} [ ${\xi }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upxi _{{1}}$$\end{document} , ${\xi }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upxi _{{2}}$$\end{document} , $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , ${\xi }_J{]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upxi _{J}]^{\prime }$$\end{document} denote a J by 1 random vector of errors in the component measurement model, where ${\xi }_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upxi _{j}$$\end{document} is an error for the jth indicator. Let $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upzeta }}$$\end{document} $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} [ ${\zeta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upzeta _{{1}}$$\end{document} , ${\zeta }_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upzeta _{{2}}$$\end{document} , $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , ${\zeta }_P{]}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upzeta _{P}]^{\prime }$$\end{document} denote a P by 1 random vector of errors in the structural model, where ${\zeta }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upzeta _{p}$$\end{document} is an error for the pth component. The three sub-models of GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} are expressed as follows.

(1) $\begin{array}{cc}{\gamma }_{\mathrm{std}}\equiv & W_{\mathrm{std}}^{\prime }{z}_{\mathrm{std}}\left(\text{weighted relation model}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{\upgamma }}_{std}\equiv & {} {{{\textbf {W}}}}_{std}'{{{\textbf {z}}}}_{std} \,(\text {weighted relation model}) \end{aligned}$$\end{document}

(2) $\begin{array}{cc}{z}_{\mathrm{std}}=& C_{\mathrm{std}}^{\prime }{\gamma }_{\mathrm{std}}+\xi \left(\text{component measurement model}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{{\textbf {z}}}}_{std}= & {} {{{\textbf {C}}}}_{std}'{\varvec{\upgamma }}_{std} + {\varvec{\upxi }} \,(\text {component measurement model}) \end{aligned}$$\end{document}

(3) $\begin{array}{cc}{\gamma }_{\mathrm{std}}=& B_{\mathrm{std}}^{\prime }{\gamma }_{\mathrm{std}}+\zeta \left(\text{structural model}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{\upgamma }}_{std}= & {} {{{\textbf {B}}}}_{std}'{\varvec{\upgamma }}_{std} + {\varvec{\upzeta }} \,(\text {structural model}). \end{aligned}$$\end{document}

The weighted relation model (1) shows that (standardized) components are defined as a linear combination of standardized indicators. The component measurement and structural models (2) and (3) express the directional relationships between the indicators and components and those among the components, respectively. As (2) and (3) can be seen as systems of linear regression equations, their model parameters, including loadings and path coefficients, can be interpreted in the same manner as standardized regression coefficients. The three sub-models are combined into the following equation,

(4) $\begin{array}{cc}& \text{[}{z}_{\mathrm{std}};{\gamma }_{\mathrm{std}}]=[C_{\mathrm{std}},B_{\mathrm{std}}{\text{]}}^{\prime }{\gamma }_{\mathrm{std}}+\left(\xi ;\zeta \right)\\ & \leftrightarrow \text{[}{I}_J,W_{\mathrm{std}}{\text{]}}^{\prime }{z}_{\mathrm{std}}={[C_{\mathrm{std}},B_{\mathrm{std}}]}^{\prime }{W_{\mathrm{std}}^{\prime }z}_{\mathrm{std}}+\left(\xi ;\zeta \right)\\ & \leftrightarrow {V_{\mathrm{std}}^{\prime }z}_{\mathrm{std}}=A_{\mathrm{std}}^{\prime }{W}_{\mathrm{std}}^{\prime }{z}_{\mathrm{std}}+e,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} \text {[}{{{\textbf {z}}}}_{std};{\varvec{\upgamma }}_{std}]=[{{{\textbf {C}}}}_{std},{{{\textbf {B}}}}_{std}\text {]}^\prime {\varvec{\upgamma }}_{std}+\left[ {\varvec{\upxi };\varvec{\upzeta }} \right] \nonumber \\{} & {} \quad {\leftrightarrow }\text {[}{{{\textbf {I}}}}_{J},{{{\textbf {W}}}}_{std}\text {]}^\prime {{{\textbf {z}}}}_{std}=[ {{{\textbf {C}}}}_{std},{{{\textbf {B}}}}_{std}]'{{{{\textbf {W}}}}_{std}'\textbf{z}}_{std}+\left[ {\varvec{\upxi };\varvec{\upzeta }} \right] \nonumber \\{} & {} \quad {\leftrightarrow }{{{{\textbf {V}}}}_{std}'\textbf{z}}_{std}={{{\textbf {A}}}}_{std}'{{{\textbf {W}}}}_{std}'{{{\textbf {z}}}}_{std}+\textbf{e}, \end{aligned}$$\end{document}

where I ${}_J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{J}$$\end{document} is the identity matrix of order J, $V_{\mathrm{std}}\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {V}}}}_{std} \equiv $$\end{document} [I ${}_J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{J}$$\end{document} , $W_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {W}}}}_{std}$$\end{document} ], $A_{\mathrm{std}}\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {A}}}}_{std} \equiv $$\end{document} $\text{[}{C}_{\mathrm{std}},B_{\mathrm{std}}\text{]}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {[}{{{\textbf {C}}}}_{std},{{{\textbf {B}}}}_{std}\text {]}$$\end{document} , e $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} [ $\xi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upxi }}$$\end{document} ; $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upzeta }}$$\end{document} ] and a semicolon within brackets is an operator to vertically concatenate two vectors in the array. The equation (4) is called the GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} model.

Let 1 ${}_Q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{Q}$$\end{document} denote a column vector of Q ones. Let SS(X) $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} tr(X ${}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\prime }$$\end{document} X) for any matrix X. Let vecdiag() denote an operator that returns a column vector stacking the diagonal elements of a square matrix one below another. GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} estimates model parameters ( $W_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {W}}}}_{std}$$\end{document} and $A_{\mathrm{std}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {A}}}}_{std})$$\end{document} by minimizing the following objective function

(5) $\begin{array}{cc}& f_{\mathrm{std}}(W_{\mathrm{std}},A_{\mathrm{std}})\\ & =tr\left(E\left(e_{\mathrm{std}}{e}_{\mathrm{std}}^{\prime }\right)\right)\\ & =E(SS({[z_{\mathrm{std}};{\gamma }_{\mathrm{std}}]}^{\prime }-z_{\mathrm{std}}^{\prime }{W}_{\mathrm{std}}{A}_{\mathrm{std}}),\\ & =E(SS\left(z_{\mathrm{std}}^{\prime }([I_J,W_{\mathrm{std}}]-W_{\mathrm{std}}{A}_{\mathrm{std}})\right)\\ & =tr\left({(V_{\mathrm{std}}-W_{\mathrm{std}}{A}_{\mathrm{std}})}^{\prime }{\Sigma }_{\mathrm{std}}(V_{\mathrm{std}}-W_{\mathrm{std}}{A}_{\mathrm{std}})\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} f_{std}({{{\textbf {W}}}}_{std},{{{\textbf {A}}}}_{std})\nonumber \\{} & {} \quad =tr(E({{{\textbf {e}}}}_{std}{{{\textbf {e}}}}_{std}'))\nonumber \\{} & {} \quad =E(SS([{{{\textbf {z}}}}_{std};{\varvec{\upgamma }}_{std}]'-{{{\textbf {z}}}}_{std}'{{{\textbf {W}}}}_{std}{{{\textbf {A}}}}_{std}),\nonumber \\{} & {} \quad =E(SS({{{\textbf {z}}}}_{std}'([{{{\textbf {I}}}}_{J},{{{\textbf {W}}}}_{std}]-{{{\textbf {W}}}}_{std}{{{\textbf {A}}}}_{std}))\nonumber \\{} & {} \quad =tr (({{{\textbf {V}}}}_{std}-{{{\textbf {W}}}}_{std}{{{\textbf {A}}}}_{std})'{\varvec{\Sigma }}_{std}({{{\textbf {V}}}}_{std}-{{{\textbf {W}}}}_{std}{{{\textbf {A}}}}_{std})) \end{aligned}$$\end{document}

subject to vecdiag( $W_{\mathrm{std}}^{\prime }{\Sigma }_{\mathrm{std}}{W}_{\mathrm{std}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {W}}}}_{std}'{\varvec{\Sigma }}_{std}{{{\textbf {W}}}}_{std})$$\end{document} $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=\, $$\end{document} 1 ${}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} . Thus, GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} estimates the model parameters by minimizing the sum of error variances for all variables in the model given ${\Sigma }_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_{std}$$\end{document} . In general, ${\Sigma }_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_{std}$$\end{document} is replaced with the sample correlation matrix of indicators, denoted by S ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} . The objective function (5) also shows that GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} aims to create components that explain the total variances of variables in the model rather than their covariances, as with PCA or other component-based methods. The error terms in the GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} model are not considered independent entities that cause the variation of indicators but simply treated as residuals that are unexplained by independent components. Thus, GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} typically makes no assumptions about the correlation structure of the error terms of indicators, leaving them freely correlated. This is distinct from the common factor model, where the error terms are typically assumed to be uncorrelated. Nonetheless, no error covariances between different blocks of indicators may be assumed in some special cases of GSCA (Cho et al., Reference Cho, Hwang, Sarstedt and Ringle2020, Reference Cho, Sarstedt and Hwang2022).

Note that (1) defines a component as a weighted sum of indicators, which is also the case in PCA. However, this equation itself is not identified because there would exist infinitely different ways of deciding the component weights. Thus, we need a certain rule or criterion to determine the component weights. PCA’s criterion is one of the most widely used ones in statistics that the weights are to be determined in such a way that their corresponding components explain the maximum total variance of the indicators. The regression coefficients of indicators on their component are (component) loadings. These relationships between components and their indicators are expressed in the component measurement model (2). Thus, GSCA can have confirmatory PCA (Takane, Kiers, & de Leeuw, 1995) as a special case when it considers (1) and (2) only.

As the minimization problem (5) cannot be solved in closed form, an alternating least squares (ALS) algorithm was developed for iteratively finding the minimum point of (5). In the ALS algorithm, W ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} and A ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} are updated alternately with the other fixed until the difference in (5) between consecutive iterations decreases beyond a pre-specified tolerance level (e.g., 10 ${}^{-\text{5}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-\text {5}})$$\end{document} (see Hwang & Takane, Reference Hwang and Takane2014, Chapter 2, for a full description of the ALS algorithm). Let ${\hat{\Gamma }}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Gamma }}}}_{std}$$\end{document} denote an N by P matrix of the standardized score estimates of components, $D_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {D}}}}_{std}$$\end{document} denote an N by J matrix of the standardized scores of indicators, and N is the number of cases in the sample. Let us suppose that we obtain the estimates of W ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} and A ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} that minimize (5), denoted by ${\hat{W}}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\textbf {W}}}}_{std}$$\end{document} and ${\hat{A}}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{{\textbf {A}}}}}_{std}$$\end{document} . Then, a matrix of standardized component scores is obtained by

(6) $\begin{array}{c}{\hat{\Gamma }}_{\mathrm{std}}\equiv D_{\mathrm{std}}{\hat{W}}_{\mathrm{std}}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{{\widehat{\Gamma }}}}_{std}\,\,{\equiv }\,\,{{{\textbf {D}}}}_{std}{\widehat{{\textbf {W}}}}_{std}. \end{aligned}$$\end{document}

1.2. Unstandardized Weight Estimates in GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document}

Let D $=1_N{\hat{\mu }}^{\prime }+D_{\mathrm{std}}{\hat{\Delta }}_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{=}\,{{{\textbf {1}}}}_{N}{\varvec{{\hat{\upmu }}}}'\, +\, {{{\textbf {D}}}}_{std}{\varvec{{\widehat{\Delta }}}}_{\text {z}}$$\end{document} denote an N by J matrix of the unstandardized scores of indicators, where $\hat{\mu }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\hat{\upmu }}}}$$\end{document} is a J by 1 sample mean vector and ${\hat{\Delta }}_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Delta }}}}_{\text {z}}$$\end{document} is a diagonal matrix whose entries are sample standard deviations of unstandardized indicators. Conventionally, unstandardized component means are subsequently computed by transforming ${\hat{W}}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\textbf {W}}}}_{std}$$\end{document} as follows. As it follows from (6) that

(7) $\begin{array}{cc}& {\hat{\Gamma }}_{\mathrm{std}}=(D-1_N{\hat{\mu }}^{\prime }){\hat{\Delta }}_{\text{z}}^{-1}{\hat{W}}_{\mathrm{std}}\\ & {\leftrightarrow 1}_N{\hat{\mu }}^{\prime }{\hat{\Delta }}_{\text{z}}^{-1}{\hat{W}}_{\mathrm{std}}+{\hat{\Gamma }}_{\mathrm{std}}=D{\hat{\Delta }}_{\text{z}}^{-1}{\hat{W}}_{\mathrm{std}}\\ & {\leftrightarrow 1}_N{\hat{\mu }}^{\prime }{\hat{W}}_{\mathrm{uni}}+{\hat{\Gamma }}_{\mathrm{std}}=D{\hat{W}}_{\mathrm{uni}},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} {\varvec{{\widehat{\Gamma }}}}_{std}=({{{\textbf {D}}}}-\textbf{1}_{N}{\varvec{{\hat{\upmu }}}}'){\varvec{\widehat{\Delta }}}_{\text {z}}^{{-1}}{{{\widehat{{\textbf {W}}}}}}_{std} \nonumber \\{} & {} {{{\leftrightarrow 1}}}_{N}{\varvec{\hat{\upmu }}}'{\varvec{\widehat{\Delta }}}_{\text {z}}^{{-1}}{{\widehat{{\textbf {W}}}}}_{std}+{\varvec{{\widehat{\Gamma }}}}_{std}{=\textbf{D}}{\varvec{\widehat{\Delta }}}_{\text {z}}^{{-1}}{\widehat{{\textbf {W}}}}_{std} \nonumber \\{} & {} {{{\leftrightarrow 1}}}_{N}{\varvec{\hat{\upmu }}}'{\widehat{{\textbf {W}}}}_{uni}+{\varvec{\widehat{\Gamma }}}_{std}{=\textbf{D}}{\widehat{{\textbf {W}}}}_{uni}, \end{aligned}$$\end{document}

where ${\hat{W}}_{\mathrm{uni}}\equiv {\hat{\Delta }}_{\text{z}}^{-1}{\hat{W}}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\textbf {W}}}}_{uni} \equiv {\varvec{\widehat{\Delta }}}_{\text {z}}^{{-1}}{\widehat{{\textbf {W}}}}_{std}$$\end{document} , GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} computes unstandardized component scores, denoted here by ${\hat{\Gamma }}_{\mathrm{uni}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Gamma }}}}_{uni}$$\end{document} , as ${\hat{\Gamma }}_{\mathrm{uni}}\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Gamma }}}}_{uni}\,\equiv \,\, $$\end{document} D ${\hat{W}}_{\mathrm{uni}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\textbf {W}}}}_{uni}$$\end{document} (Hwang & Takane, Reference Hwang and Takane2014, p. 26).

As shown in the last line of (7), however, ${\hat{\Gamma }}_{\mathrm{uni}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Gamma }}}}_{uni}$$\end{document} can be simply seen as a variant of standardized component scores whose means are only relocated a posteriori by $1_N\hat{\mu }{{}^{\prime }\hat{W}}_{\mathrm{uni}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {1}}}}_{N}{\varvec{\hat{\upmu }}}{'{\widehat{{\textbf {W}}}}}_{uni}$$\end{document} in that ${\hat{\Gamma }}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\widehat{\Gamma }}}_{std}$$\end{document} remains standardized irrespective of the sample variances of the original indicators. Consequently, it is not guaranteed that the scores of ${\hat{\Gamma }}_{\mathrm{uni}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Gamma }}}}_{uni}$$\end{document} are within the same range of the unstandardized scores of their indicators, which will be empirically shown in Sect. 4. Also, as illustrated in Sect. 1, GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} tends to assign smaller unstandardized weights to original indicators with relatively large variances in forming ${\hat{\Gamma }}_{\mathrm{uni}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\widehat{\Gamma }}}}_{uni}$$\end{document} . That is because minimizing (5) involves imposing a relatively large penalty on an original indicator with a relatively large variance, which is shown in Appendix 1. This disproportionate penalization for original indicators can inadvertently amplify the influence of an original indicator with a small variance on GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} ’s parameter estimation. Such an approach could be deemed unsuitable when one aims to obtain an unstandardized component of original indicators on a single scale.

1.3. Convex Component and Its Six Properties

Let ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p}$$\end{document} denote the pth component ( $p=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p =$$\end{document} 1, 2, $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , P) that is assumed to have the mean ${\tau }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uptau _{p}$$\end{document} and variance ${\varphi }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upphi _{p}$$\end{document} . Let z ${}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} denote a $J_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{p}$$\end{document} by 1 vector of indicators for ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p}$$\end{document} , where $J_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{p}$$\end{document} is the number of indicators for ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p}$$\end{document} . We call the vector z ${}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} a block of indicators for ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p}$$\end{document} , which is assumed to have the mean vector ${\mu }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upmu }}_{p}$$\end{document} and covariance matrix ${\Sigma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }_{p}$$\end{document} . Let w ${}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} denote a $J_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{p}$$\end{document} by 1 vector of weights for z ${}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} . Let 0 ${}_{k\times l}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{k\times l}$$\end{document} denote a k by l matrix of zeros, where k and l are any scalars. If ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p}$$\end{document} is defined as a convex component, it can be expressed as

(8) $\begin{array}{c}{\gamma }_p\equiv w_p^{\prime }{z}_p\text{subject to}{w}_p^{\prime }{1}_{\mathrm{Jp}}=1\text{and}{w}_p\ge 0_{J\times 1}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \upgamma _{p} \equiv {{\textbf {w}}}_{p}'{{\textbf {z}}}_{p} \text { subject to } {{\textbf {w}}}_{p}'{{\textbf {1}}}_{Jp} = 1 \text { and }{{\textbf {w}}}_{p} \ge {{\textbf {0}}}_{J \times 1}. \end{aligned}$$\end{document}

A convex component has six useful properties as follows.

2. Convex GSCA

2.1. Model Specification

Convex GSCA (GSCA ${}_{\mathrm{cvx}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx})$$\end{document} introduces a convex component with original indicators into the GSCA model. The GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model also consists of three sub-models: weighted relation, component measurement, and structural models(Hwang and Takane, Reference Hwang and Takane2004, Reference Hwang and Takane2014). Let $\gamma =$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upgamma }} =$$\end{document} [ ${\gamma }_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upgamma }} _{\text {1}}$$\end{document} , ${\gamma }_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upgamma }}_{\text {2}}$$\end{document} , $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , ${\gamma }_P$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\upgamma }}_{P}$$\end{document} ] ${}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\prime $$\end{document} denote a P by 1 random vector of components. Each component ( ${\gamma }_p)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p})$$\end{document} can be either a convex or standardized component. If a block of indicators (z ${}_p)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p})$$\end{document} has the same measurement unit within the block, ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{p}$$\end{document} is defined as a convex component as expressed in (8). Otherwise, ${\gamma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upgamma }}_{p}$$\end{document} is defined as a standardized component, whose indicators (z ${}_p)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p})$$\end{document} are also assumed to be standardized such that ${\tau }_p=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\uptau }_{p} =$$\end{document} 0, ${\varphi }_p=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upphi _{p} =$$\end{document} 1, ${\mu }_p=0_{Jp\times 1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upmu }}_{p}={{\textbf {0}}}_{Jp \times 1}$$\end{document} , and vecdiag ( ${\Sigma }_p)=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }_{p}) =$$\end{document} 1 ${}_{\mathrm{Jp}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{Jp}$$\end{document} . Let W denote a J by P matrix consisting of component weights assigned to z. Let C denote a P by J matrix of loadings relating $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upgamma }}$$\end{document} to z. Let B denote a P by P matrix of path coefficients relating $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma $$\end{document} to each other. Let $c_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {c}}_0$$\end{document} and $b_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {b}}_0$$\end{document} denote the column vectors of intercepts for the component measurement and structural models, respectively. The three sub-models of GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} are expressed as follows.

(9) $\begin{array}{cc}\gamma \equiv & W^{\prime }z\left(\text{weighted relation model}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{\upgamma }}\equiv & {} {{\textbf {W}}}'{{\textbf {z}}} \,(\text {weighted relation model}) \end{aligned}$$\end{document}

(10) $\begin{array}{cc}z=& c_{\text{0}}+C^{\prime }\gamma +\xi \left(\text{component measurement model}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\textbf {z}}}= & {} {{\textbf {c}}}_{\text {0}} + {{\textbf {C}}}' {\varvec{\upgamma }} + {\varvec{\upxi }} \,(\text {component measurement model}) \end{aligned}$$\end{document}

(11) $\begin{array}{cc}\gamma =& b_{\text{0}}+B^{\prime }\gamma +\zeta \left(\text{structural model}\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{\upgamma }}= & {} {{\textbf {b}}}_{\text {0}} + {{\textbf {B}}}' {\varvec{\upgamma }} + {\varvec{\upzeta }} \,(\text {structural model}). \end{aligned}$$\end{document}

In GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} , the weighted relation model (9) shows that each component is defined as a weighted sum of standardized or unstandardized indicators. As GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} may involve unstandardized variables, intercept terms ( $c_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {c}}_0}$$\end{document} and $b_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {b}}}_0$$\end{document} ) are newly included into the component measurement and structural model (10) and (11). Each model parameter in (10) and (11)—intercepts, loadings, and path coefficients—can be interpreted in the same manner as the intercepts and regression coefficients in linear regression model with unstandardized variables. The three sub-models are combined into the following equation,

(12) $\begin{array}{cc}& [z;\gamma ]=[c_{\text{0}};b_{\text{0}}]+{[C,B]}^{\prime }\gamma +\left(\xi ;\zeta \right)\\ & \leftrightarrow {[I_J,W]}^{\prime }z=[c_{\text{0}};b_{\text{0}}]+{[C,B]}^{\prime }{W}^{\prime }z+\left(\xi ;\zeta \right)\\ & \leftrightarrow V^{\prime }z=a_{\text{0}}+A^{\prime }{W}^{\prime }z+e,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} {}[\textbf{z};\varvec{\upgamma }]=[{{{\textbf {c}}}}_{\text {0}};{{{\textbf {b}}}}_{\text {0}}]+[\textbf{C},\textbf{B}]'\varvec{\upgamma }+\left[ {\varvec{\upxi };\varvec{\upzeta }} \right] \nonumber \\{} & {} {\leftrightarrow }\, [{{{\textbf {I}}}}_{J},\textbf{W}]'\textbf{z}=[{{{\textbf {c}}}}_{\text {0}}; {{{\textbf {b}}}}_{\text {0}}]+[\textbf{C},\textbf{B}]'\textbf{W}'\textbf{z}+\left[ {\varvec{\upxi };\varvec{\upzeta }} \right] \nonumber \\{} & {} {\leftrightarrow }\, {{{\textbf {V}}}'\textbf{z}=}\, {{{\textbf {a}}}}_{\text {0}} +\textbf{A}'\textbf{W}'\textbf{z}+\textbf{e}, \end{aligned}$$\end{document}

where $a_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {a}}}_0$$\end{document} $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} [ $c_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {c}}}_0$$\end{document} ; $b_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {b}}}_0$$\end{document} ], V $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} [I ${}_J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{J}$$\end{document} , W], A $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} [C, B], and e $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} [ $\xi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upxi }}$$\end{document} ; $\zeta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upzeta }}$$\end{document} ]. The equation (12) is called the GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model. If every indicator and component is standardized, the GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model (12) becomes identical to the GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} model (4)

2.2. Parameter Estimation

Let ${\sigma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upsigma }}_{p}$$\end{document} denote a $J_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{p}$$\end{document} by 1 vector of standard deviations (SD) of z ${}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} . If the pth component is defined as standardized ones, ${\sigma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upsigma }}_{p}$$\end{document} is equivalent to 1 ${}_{\mathrm{Jp}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{Jp}$$\end{document} . Let O ${}_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {z}}$$\end{document} denote a J by J diagonal matrix whose jth element is $J_p^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{p}^{-1}$$\end{document} 1 ${}_{\mathrm{Jp}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{Jp}$$\end{document} $\prime$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\prime }$$\end{document} ${\sigma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upsigma }}_{p}$$\end{document} if the jth indicator in the pth block is a dependent variable and zero otherwise. Let O ${}_{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\upgamma }}$$\end{document} denote a P by P diagonal matrix whose pth element is $J_p^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{p}^{-1}$$\end{document} 1 ${}_{\mathrm{Jp}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{Jp}$$\end{document} $\prime$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\prime }$$\end{document} ${\sigma }_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upsigma }}_{p}$$\end{document} if the pth component is a dependent variable and zero otherwise. Let O $\equiv$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\equiv $$\end{document} blkdiag(O ${}_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {z}}$$\end{document} , O ${}_{\gamma })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\upgamma }})$$\end{document} . GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} estimates parameters by minimizing the following objective function

(13) $\begin{array}{cc}& f_{\mathrm{cvx}}(W,A,a_{\text{0}})\\ & =tr\left(OE\left({\mathrm{ee}}^{\prime }\right)O\right)\\ & =E\left(SS\left(({[z;\gamma ]}^{\prime }-(a_{\text{0}}^{\prime }+z^{\prime }\mathrm{WA}))O\right)\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} f_{cvx}(\textbf{W,A},{{{\textbf {a}}}}_{\text {0}})\nonumber \\{} & {} \quad =tr(\textbf{O}E (\textbf{ee}')\textbf{O})\nonumber \\{} & {} \quad =E(SS(([\textbf{z};\varvec{\upgamma }]^\prime -({{{\textbf {a}}}}_{\text {0}}'+\textbf{z}'\textbf{WA}))\textbf{O})). \end{aligned}$$\end{document}

subject to $w_p^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {w}}}}_{p}^\prime $$\end{document} ${\Sigma }_p^{\prime }{w}_p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\Sigma }}_{p}^{\prime }{{{\textbf {w}}}}_{p}$$\end{document} $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 1 or 1 ${}_{\mathrm{Jp}}{}^{\prime }{w}_p=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{Jp}\,^{\prime }{{{\textbf {w}}}}_{p} =$$\end{document} 1 ( $p=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p =$$\end{document} 1, 2, $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , P). The objective function (13) shows that components in GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} are constructed such that they can minimize the “weighted” sum of error variances for all dependent variables under the constraints. Specifically, the objective function (13) penalizes each prediction error for dependent variables differentially by dividing it by the average SD of the corresponding block of indicators. This prevents prediction errors for a block of indicators with large variances from dominating the estimation of parameters.

To help understand the role of O in (13), we illustrate how O is determined based on the standard deviations of indicators. This will also explain the characteristic of the objective function described above. Figure 1 presents an illustrative GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model involving two convex components ( ${\gamma }_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {1}}$$\end{document} and ${\gamma }_{\text{2}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {2}})$$\end{document} , each measured by three indicators that share the same scale, while the scales of two indicator blocks differ. Let us assume that ${\sigma }_{\text{1}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upsigma }}_{\text {1}} =$$\end{document} [1; 2; 3] and ${\sigma }_{\text{2}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upsigma }}_{\text {2}} =$$\end{document} [100; 200; 300], indicating that the differences in the overall magnitude of indicators’ variances between the two blocks arises from the difference in scale. In this case, without O in (13) (i.e., O $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} I), the value of (13) would predominantly rely on the error variances for z ${}_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}}$$\end{document} and ${\gamma }_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {2}}$$\end{document} , implying that the error variances for z ${}_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {1}}$$\end{document} would be rarely considered in parameter estimation due to their scale. However, GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} determines O $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\, =\, $$\end{document} blkdiag(O ${}_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {z}}$$\end{document} , O ${}_{\gamma })$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\upgamma }})$$\end{document} , where O ${}_{\text{z}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {z}} =$$\end{document} blkdiag(2, 2, 2, 200, 200, 200) ${}^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} and O ${}_{\gamma }=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\upgamma }}=$$\end{document} blkdiag(0, 200 ${}^{-1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1})$$\end{document} , and then uses it to penalize the error variances for z ${}_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}}$$\end{document} and ${\gamma }_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {2}}$$\end{document} to adjust their effects on (13). For instance, given A $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} 0 and a ${}_{\text{0}}=E$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {0}} = E$$\end{document} ([z; $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\upgamma }}$$\end{document} ]), there are substantial differences in error variances between z ${}_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {1}}$$\end{document} and z ${}_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}}$$\end{document} (i.e., [1 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ; 2 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ; 3 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ] for z ${}_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {1}}$$\end{document} and [100 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ; 200 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ; 300 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ] for z ${}_{\text{2}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}})$$\end{document} , but their error variances contribute equally to the value of (13) (i.e., (1 ${}^{\text{2}}+$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}+$$\end{document} 2 ${}^{\text{2}}+$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}} +$$\end{document} 3 ${}^{\text{2}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}})$$\end{document} /2 ${}^{\text{2}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}=$$\end{document} [100 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ; 200 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ; 300 ${}^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}}$$\end{document} ]/200 ${}^{\text{2}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {2}})$$\end{document} . This suggests that introducing O into (13) enables GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} to consider prediction errors for both z ${}_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {1}}$$\end{document} and z ${}_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}}$$\end{document} during the parameter estimation process.

Figure. 1 An illustrative GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model. Hexagons represent components, squares denote indicators, straight lines indicate weights, single-headed arrows denote loadings and path coefficients. All intercepts and error terms are omitted to make the figure concise.

Conversely, as illustrated above, the objective function (13) does not impose different penalties on indicators within the same block to take into account potential differences in their variances. Furthermore, the objective function (13) is partially scale-invariant, which means that the minimum value of (13) does not vary with a linear change of measurement scales of each block of indicators that share the same scale (e.g., a scale range from 1–10 to 0–100), leading to the same weight estimates. This property is distinct from a property of (full) scale invariant (Swaminathan and Algina, Reference Swaminathan and Algina1978) in that changing the measurement scales of each indicator differentially (e.g., standardization) is not considered. The proof for the property is provided in Appendix 3.

As the minimum point of (13) cannot be found in closed form, we developed an ALS algorithm for iteratively finding its minimum point. A detailed description of the ALS algorithm is provided in Appendix 4. Note that we do not constrain the weights for convex components to be non-negative in (13) to make the method more flexible. In some cases, researchers may wish to examine which indicators contribute to forming a component in the opposite direction to the other indicators and may be excluded during model re-specification. The negative weight of an indicator for a convex component may signify that the indicator is not suitable to form the component along with other indicators. As discussed in Appendix 4, the ALS algorithm allows for the imposition of the additional non-negativity constraints on weights, forcing the weights to be always positive.

2.3. Model Evaluation Indexes

GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} provides four overall goodness-of-fit measures, including FIT, AFIT, GFI, and SRMR, and one overall cross-validation index, out-of-bag prediction error (OPE). The FIT indicates the average explained variance of all variables in the model, whereas the AFIT is an adjusted version of FIT that takes into account the number of model parameters and sample size (Hwang & Takane, Reference Hwang and Takane2014, pp. 26–29). The GFI and SRMR evaluate the discrepancy between the sample and implied covariance matrices(Cho et al., Reference Cho, Hwang, Sarstedt and Ringle2020). The OPE aims to measure the average out-of-sample prediction error of the model for all variables via a bootstrapping-based cross validation and can be used for comparing models in terms of predictive generalizability(Cho et al., Reference Cho, Jung and Hwang2019). Whereas the GFI and SRMR can be used for GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} without modification, the FIT, AFIT, and OPE need to be modified for GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} because these measures were developed only for the condition where all variables are standardized. We revised FIT and OPE such that they can be applied for the GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model with both standardized and unstandardized variables, taking into account the variances of dependent variables only.

We propose a modified version of FIT, termed FIT for unstandardized dependent variables (FIT ${}^{\text{UD}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}})$$\end{document} , as follows.

(14) $\begin{array}{c}{\text{FIT}}^{\text{UD}}=1-\frac{SS\left(([D,D\hat{W}]-(1_N{\hat{a}}_0^{\prime }+D\hat{W}\hat{A}))\hat{O}\right)}{SS\left(([D,D\hat{W}]-1_N{\hat{\mu }}^{\prime }[I_J,\hat{W}])\hat{O}\right)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{FIT}}^{\textrm{UD}}=1-\frac{SS(([{{\textbf {D}}},{{\textbf {D}}\hat{{{{\textbf {W}}}}}}]-({{\textbf {1}}}_{N} {\hat{{{\textbf {a}}}}}_{0} '+{{\textbf {D}}\hat{{\textbf {W}}}\hat{{\textbf {A}}}})){\hat{{{\textbf {O}}}}})}{SS(([{{\textbf {D}}},{{\textbf {D}}\hat{{\textbf {W}}}}]-{{\textbf {1}}}_{N} {\varvec{\hat{{\upmu }}}}'[{{\textbf {I}}}_{J},{\hat{{{\textbf {W}}}}}]){\hat{{{\textbf {O}}}}})}. \end{aligned}$$\end{document}

The FIT ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} indicates the proportion of the explained variance of all dependent variables (including dependent convex components) to their weighted total variance. If every component and indicator is standardized, FIT ${}^{\text{UD}}=\frac{T}{T_Y}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}} = \frac{T}{T_{Y} }$$\end{document} FIT, where $T\equiv P+J$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T \equiv P + J$$\end{document} and $T_{\text{Y}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\text {Y}}$$\end{document} is the total number of dependent variables in the model. Also, we provide the following two local fit measures of FIT ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document}

(15) $\begin{array}{cc}{\text{FIT}}_{\text{M}}^{\text{UD}}=& 1-\frac{SS\left((D-(1_N{\hat{c}}_0^{\prime }+D\hat{W}\hat{C})){\hat{O}}_{\text{z}}\right)}{SS\left((D-1_N{\hat{\mu }}^{\prime }){\hat{O}}_{\text{z}}\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{FIT}}_{\textrm{M}}^{\textrm{UD}}= & {} 1-\frac{SS(({{\textbf {D}}}-({{\textbf {1}}}_{N} {\hat{{{\textbf {c}}}}}_{0} '+{{\textbf {D}}\hat{{\textbf {W}}}\hat{{\textbf {C}}}})){\hat{{{\textbf {O}}}}}_{\textrm{z}} )}{SS(({{\textbf {D}}}-{{\textbf {1}}}_{N} {\varvec{\hat{{\upmu }}}}'){\hat{{{\textbf {O}}}}}_{\textrm{z}} )}, \end{aligned}$$\end{document}

(16) $\begin{array}{cc}{\text{FIT}}_{\text{S}}^{\text{UD}}=& 1-\frac{SS\left((D\hat{W}-(1_N{\hat{b}}_0^{\prime }+D\hat{W}\hat{B})){\hat{O}}_{\gamma }\right)}{SS\left((D\hat{W}-1_N{\hat{\mu }}^{\prime }\hat{W}){\hat{O}}_{\gamma }\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{FIT}}_{\textrm{S}}^{\textrm{UD}}= & {} 1-\frac{SS(({{\textbf {D}}\hat{{\textbf {W}}}}-({{\textbf {1}}}_{N} {\hat{{{\textbf {b}}}}}_{0} '+{{\textbf {D}}\hat{{\textbf {W}}}\hat{{\textbf {B}}}})){\hat{{\textbf {{O}}}}}_{\upgamma } )}{SS(({{\textbf {D}}\hat{{\textbf {W}}}}-{{\textbf {1}}}_{N} {\varvec{\hat{{\upmu }}}}'{\hat{{{\textbf {W}}}}}){\hat{{{\textbf {O}}}}}_{\upgamma } )}, \end{aligned}$$\end{document}

where ${\hat{O}}_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{\textbf {O}}}}_{\text {z}}$$\end{document} and ${\hat{O}}_{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{\textbf {O}}}}_{{\upgamma }}$$\end{document} are sample analogies of $O_{\text{z}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {O}}}}_{\text {z}}$$\end{document} and $O_{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {O}}}}_{{\upgamma }}$$\end{document} . We refer to "local fit" as the goodness-of-fit of GSCA’s sub-models. The FIT ${}_{\text{M}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {M}}^{\text {UD}}$$\end{document} and FIT ${}_{\text{S}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {S}}^{\text {UD}}$$\end{document} can be used for evaluating the component measurement and structural models, respectively. The FIT ${}_{\text{M}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {M}}^{\text {UD}}$$\end{document} indicates the proportion of the explained variance of all dependent indicators to their weighted total variance, whereas the FIT ${}_{\text{S}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {S}}^{\text {UD}}$$\end{document} indicates the proportion of the explained variance of all dependent (convex) components to their weighted total variance.

Moreover, we propose a revised version of OPE, termed OPE for dependent variables (OPE ${}^{\text{UD}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}})$$\end{document} , to evaluate the predictive generalizability of models involving convex components, as follows.

(17) $\begin{array}{c}{\text{OPE}}^{\text{UD}}=\frac{1}{K}\sum _{k=1}^K\frac{SS\left(([D_k^{\ast },D_k^{\ast }{\hat{W}}_k]-(1_{N_k}{\hat{a}}_0^{\prime }+D_k^{\ast }{\hat{W}}_k{\hat{A}}_k)){\hat{O}}_k\right)}{SS\left(([D_k^{\ast },D_k^{\ast }{\hat{W}}_k]-1_{N_k}{\hat{\mu }}_k^{\prime }[I_J,{\hat{W}}_k]){\hat{O}}_k\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{OPE}}^{\textrm{UD}}=\frac{1}{K}\sum \limits _{k=1}^K {\frac{SS(([{{\textbf {D}}}_{k}^{*},{{\textbf {D}}}_{k}^{*} {\hat{{\textbf {{W}}}}}_{k} ]-({{\textbf {1}}}_{N_{k} } {\hat{{\textbf {{a}}}}}_{0} '+{{\textbf {D}}}_{k}^{*} {\hat{{\textbf {{W}}}}}_{k} {\hat{{\textbf {{A}}}}}_{k} )){\hat{{\textbf {{O}}}}}_{k} )}{SS(([{{\textbf {D}}}_{k}^{*},{{\textbf {D}}}_{k}^{*} {\hat{{{\textbf {W}}}}}_{k} ]-{{\textbf {1}}}_{N_{k} } {\hat{\varvec{{\upmu }}}}_{k} '[{{\textbf {I}}}_{J},{\hat{{{\textbf {W}}}}}_{k} ]){\hat{{\textbf {{O}}}}}_{k} )}}, \end{aligned}$$\end{document}

where ${\hat{W}}_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\textbf {W}}}}_{k}$$\end{document} , ${\hat{A}}_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{{\textbf {A}}}}_{k}$$\end{document} , ${\hat{a}}_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{{\textbf {a}}}}}_{k}$$\end{document} , and ${\hat{\mu }}_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{{\hat{\upmu }}}}_{k}$$\end{document} are the parameter estimates obtained from the kth bootstrap sample ( $k=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k =$$\end{document} 1,2, $\cdots$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cdots $$\end{document} , K), ${\hat{O}}_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{\textbf {O}}}}_{k}$$\end{document} is the penalty term that rescales prediction errors for all dependent variables in the kth bootstrap sample, $D_k^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\textbf {D}}}}_{k}^{*}$$\end{document} is the kth test sample consisting of observations that are not included in the kth bootstrap sample, and $N_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{k}$$\end{document} is the number of observations in the kth test sample. As shown in (17), the bootstrap sampling procedure generates pairs of mutually exclusive samples (bootstrap and test samples), over which a specified GSCA model is cross-validated (for a detailed description of OPE’s computation, refer to Cho et al., Reference Cho, Jung and Hwang2019). The OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} represents the weighted average out-of-sample prediction error of the model for dependent variables. The value of the OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} ranges from 0 to infinity, where 0 means that a specified model perfectly predicts every dependent variable, and a value over 1 indicates that the prediction accuracy of a specified model is worse than that of the null model, where all dependent variables are predicted by their sample means. Again, when every variable is standardized, OPE ${}^{\text{UD}}=\frac{T}{T_{\text{Y}}}\text{OPE}-\frac{(T-T_{\text{Y}})}{T_{\text{Y}}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}} = \frac{T}{T_{\textrm{Y}} }\textrm{OPE}-\frac{(T-T_{\textrm{Y}} )}{T_{\textrm{Y}} }$$\end{document} . In addition, we provide the following two local cross-validation indexes of OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document}

(18) $\begin{array}{cc}{\text{OPE}}_{\text{M}}^{\text{UD}}=& \frac{1}{K}\sum _{k=1}^K\frac{SS\left((D_k^{\ast }-(1_{N_k}{\hat{c}}_{0,k}^{\prime }+D_k^{\ast }{\hat{W}}_k{\hat{C}}_k)){\hat{O}}_{\text{z},k}\right)}{SS\left((D_k^{\ast }-1_{N_k}{\hat{\mu }}_k^{\prime }){\hat{O}}_{\text{z},k}\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{OPE}}_{\textrm{M}}^{\textrm{UD}}= & {} \frac{1}{K}\sum \limits _{k=1}^K {\frac{SS(({{\textbf {D}}}_{k}^{*} -({{\textbf {1}}}_{N_{k} } {\hat{{{\textbf {c}}}}}_{0,k} '+{{\textbf {D}}}_{k}^{*} {\hat{{{\textbf {W}}}}}_{k} {\hat{{\textbf {{C}}}}}_{k} )){\hat{{\textbf {{O}}}}}_{{\textrm{z}},k} )}{SS(({{\textbf {D}}}_{k}^{*} -{{\textbf {1}}}_{N_{k} } {\hat{\varvec{{\upmu }}}}_{k} '){\hat{{{\textbf {O}}}}}_{\mathrm{\textrm{z}},k} )}}, \end{aligned}$$\end{document}

(19) $\begin{array}{cc}{\text{OPE}}_{\text{S}}^{\text{UD}}=& \frac{1}{K}\sum _{k=1}^K\frac{SS\left((D_k^{\ast }{\hat{W}}_k-(1_{N_k}{\hat{b}}_{0,k}^{\prime }+D_k^{\ast }{\hat{W}}_k{\hat{B}}_k)){\hat{O}}_{\gamma ,k}\right)}{SS\left((D_k^{\ast }{\hat{W}}_k-1_{N_k}{\hat{\mu }}_k^{\prime }{\hat{W}}_k){\hat{O}}_{\gamma ,k}\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{OPE}}_{\textrm{S}}^{\textrm{UD}}= & {} \frac{1}{K}\sum \limits _{k=1}^K {\frac{SS(({{\textbf {D}}}_{k}^{*} {\hat{{\textbf {{W}}}}}_{k} -({{\textbf {1}}}_{N_{k} } {\hat{{\textbf {{b}}}}}_{0,k} '+{{\textbf {D}}}_{k}^{*} {\hat{{{\textbf {W}}}}}_{k} {\hat{{{\textbf {B}}}}}_{k} )){\hat{{\textbf {{O}}}}}_{\upgamma ,k} )}{SS(({{\textbf {D}}}_{k}^{*} {\hat{{\textbf {{W}}}}}_{k} -{{\textbf {1}}}_{N_{k} } {\hat{\varvec{{\upmu }}}}_{k} '{\hat{{\textbf {{W}}}}}_{k} ){\hat{{\textbf {{O}}}}}_{\upgamma ,k} )}}, \end{aligned}$$\end{document}

where ${\hat{O}}_{\text{z},k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{\textbf {O}}}}_{{\textrm{z}},k}$$\end{document} and ${\hat{O}}_{\gamma ,k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{\textbf {O}}}}_{\upgamma ,k}$$\end{document} are the penalty terms that rescale prediction errors for dependent indicators and components, respectively, in the kth bootstrap sample. The OPE ${}_{\text{M}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {M}}^{\text {UD}}$$\end{document} and OPE ${}_{\text{S}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {S}}^{\text {UD}}$$\end{document} can be used for evaluating the predictive generalizability of the component measurement and structural models, respectively.

3. Simulated Data Analysis

We conduct a simulation study to examine the parameter recovery of the proposed method. Figure 2 depicts the population GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model used in our simulation study. The population model involves four convex components, each of which is measured by four composite indicators. Indicators per block had different mean vectors: the mean vectors of indicators are [6, 5, 4, 3] for ${\gamma }_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {1}}$$\end{document} , [5.5, 4.5, 3.5, 2.5] for ${\gamma }_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {2}}$$\end{document} , [5, 4, 3, 2] for ${\gamma }_{\text{3}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {3}}$$\end{document} , and [4.5, 3.5, 2.5, 1.5] for ${\gamma }_{\text{4}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {4}}$$\end{document} , respectively.

Figure. 2 The population GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model used in the simulation study. Double-headed arrows represent correlations. All intercepts and error terms are omitted to make the figure concise.

We manipulate four experimental factors: the variances of indicators, correlations between indicators per component, distribution of indicators, and correlations among components. We consider the variances of indicators because this is a unique piece of information for the proposed method to use for creating components as compared to GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} . The other three factors have been frequently considered in testing the performance of GSCA (e.g., Cho et al., Reference Cho, Sarstedt and Hwang2022; Cho & Choi, Reference Cho and Choi2020; Hwang, Malhotra, et al., Reference Hwang, Malhotra, Kim, Tomiuk and Hong2010b). Specifically, we consider three levels of the variances of indicators per component: [1, 1, 1, 1], [1 2, 3, 4], and [1, 4, 9, 16]. We take into account three correlation matrices of indicators per component, which are provided in Table 1 (Cho and Choi, Reference Cho and Choi2020). We consider two distributions of indicators: normal and non-normal. The normal distribution has a skewness of 0 and a kurtosis of 3, whereas the non-normal distribution has a skewness of 1.25 and a kurtosis of 3.75 as in Hwang et al. (Reference Hwang, Ho and Lee2010a). Lastly, we consider three levels of correlations among components (0, .2, and .4) as in Cho et al. (Reference Cho, Sarstedt and Hwang2022). In total, we consider 54 population GSCA models with convex components (3 levels of indicators’ variances $\times$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 2 types of indicators’ distribution $\times$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 3 levels of indicators’ correlations $\times$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 3 levels of components’ correlations).

Table 1 Three conditions of the correlation patterns of four indicators per component in the simulation study.

Per population model, we consider five sample sizes ( $N=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N =$$\end{document} 100, 200, 400, 800, and 1500), for each of which 1000 samples are randomly generated from the multivariate distribution with the population mean vector and covariance matrix of indicators. The procedure of deriving the population covariance matrix of indicators from the prescribed parameter values of a population GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} model is explained in Appendix 5. We apply GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} Footnote 1 to each sample and obtain parameter estimates.

As parameter recovery measures, we empirically compute the absolute bias and root mean squared error (RMSE) of each parameter estimator. These measures are defined as

(20) $\begin{array}{cc}\text{Absolute}\text{bias}=& \left(\text{E}\left(\hat{\theta }\right)-\theta \right)\approx \left(\frac{1}{\text{1000}}(\sum _{i=1}^{1000}{\hat{\theta }}_i)-\theta \right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hbox {Absolute\,bias}= & {} \left| {{\textrm{E}}(\hat{{\uptheta }})-\uptheta } \right| \approx \left| {\frac{1}{\text{1000 }}(\sum \limits _{i=1}^{1000} {\hat{{\uptheta }}_{i} )-\uptheta } } \right| , \end{aligned}$$\end{document}

(21) $\begin{array}{cc}\text{RMSE}=& \sqrt{\text{E}{(\hat{\theta }-\theta )}^2}\approx \sqrt{\frac{1}{1000}\sum _{i=1}^{1000}{({\hat{\theta }}_i-\theta )}^2},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{RMSE}= & {} \sqrt{{\textrm{E}}(\hat{{\uptheta }}-\uptheta )^{2}} \approx \sqrt{\frac{1}{1000}\sum \limits _{i=1}^{1000} {(\hat{{\uptheta }}_{i} -\uptheta )^{2}} }, \end{aligned}$$\end{document}

where $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uptheta $$\end{document} is the value of each parameter, $\hat{\theta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{{\uptheta }}$$\end{document} is the estimator of $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uptheta $$\end{document} , and ${\hat{\theta }}_i$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{{\uptheta }}_{i} $$\end{document} is the estimate of $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\uptheta $$\end{document} obtained from the ith sample. We focus here on reporting the average absolute bias and RMSE values of the estimators of weights, loadings, intercepts, component means, and component variances over the population models per sample size, as the sample size is the only factor that substantially influences the absolute bias and RMSE values of the estimators. The results for each population model are provided in Supplementary Material.

Table 2 shows the average absolute bias and RMSE values of the estimators per sample size. In all sample sizes, the absolute biases of the weight, loading, and component mean estimators are small and close to zero on average. For example, when $N=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N =$$\end{document} 100, the average absolute biases of the weight, loading, and component mean estimators are .002, .022, and .008, respectively. They continue to decrease and approach zero when the sample size increases. The average RMSE values of the same estimators show a similar pattern. When $N=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N =$$\end{document} 100, the average RMSE values are around .047, .134, and .216, respectively, and becomes close to zero as the sample size increases. The average absolute bias and RMSE values of the intercept and component variance estimators are relatively large, compared to those of the other parameter estimators in the same condition. For instance, when $N=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N =$$\end{document} 100, the average absolute biases of the intercept and component variance estimators are .107 and .178, respectively, and their average RMSE values are .668 and .859, respectively. However, both of them also decrease with the sample size and become close to zero. Taken together, GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} estimators are empirically unbiased on average, improving their parameter recovery as the sample size increases.

Table 2 The average absolute bias and RMSE values of the estimators of weights, loadings, intercepts, component means, and component variances per sample size.

4. Illustration with Empirical Data

To illustrate its empirical utility, we apply GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} to American customer satisfaction index (ACSI) data. The ACSI model (Fornell et al., Reference Fornell, Johnson, Anderson, Cha and Bryant1996) is built on the established theories and has been used to produce index scores for customer satisfaction in the United States since 1994. The present ACSI data are comprised of 774 customers’ responses for fourteen items: z ${}_{\text{1}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {1}} =$$\end{document} expectation for overall quality, z ${}_{\text{2}}{=}_{}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}} =_{}$$\end{document} expectation for reliability, z ${}_{\text{3}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {3}} =$$\end{document} expectation for customization, z ${}_{\text{4}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {4}} =$$\end{document} overall quality, z ${}_{\text{5}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {5}} =$$\end{document} reliability, z ${}_{\text{6}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {6}} =$$\end{document} customization, z ${}_{\text{7}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {7}} =$$\end{document} price given quality, z ${}_{\text{8}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {8}}$$\end{document} $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} quality given price, z ${}_{\text{9}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}} =$$\end{document} perceived overall satisfaction, z ${}_{\text{10}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}} =$$\end{document} fulfilment of expectations, z ${}_{\text{11}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {11}} =$$\end{document} distance to the ideal, z ${}_{\text{12}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {12}} =$$\end{document} complaint behavior, z ${}_{\text{13}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {13}} =$$\end{document} repurchase intention, z ${}_{\text{14}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {14}} =$$\end{document} price tolerance. Twelve of the items (z ${}_{\text{1}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {1}}$$\end{document} ,z ${}_{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {2}}$$\end{document} , z ${}_{\text{3}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {3}}$$\end{document} , z ${}_{\text{4}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {4}}$$\end{document} , z ${}_{\text{5}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {5}}$$\end{document} , z ${}_{\text{6}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {6}}$$\end{document} , z ${}_{\text{7}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {7}}$$\end{document} , z ${}_{\text{8}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {8}}$$\end{document} , z ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} , z ${}_{\text{10}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}}$$\end{document} , z ${}_{\text{11}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {11}}$$\end{document} , and z ${}_{\text{13}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {13}})$$\end{document} are measured on a 10-point Likert scale (e.g., 1 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} “very negative” and 10 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} “very positive”). Within the interval [1, 5], a smaller point reflects a stronger negative response, whereas within the interval [6, 10], a larger point indicates a stronger positive response. On the other hand, z ${}_{\text{12}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {12}}$$\end{document} is a binary variable (1 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} formally complained and 0 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} otherwise) and z ${}_{\text{14}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {14}}$$\end{document} is a composite of two price tolerance measures in different metrics, which is expressed as a percentage ranging from 0 to 50 (the higher, the more tolerant). The means, covariances, minimums, and maximums of the items are provided in Table 3. Refer to Fornell et al. (Reference Fornell, Johnson, Anderson, Cha and Bryant1996) for more detailed information on the items.

Table 3 Sample covariances (in upper triangular), correlations (in lower triangular), variances (in diagonal), means, minimums, and maximums of the fourteen indicators in the ACSI example.

Figure 3 depicts the relationships among the six components and their indicators. The 14 items are used as composite indicators of the following six components: ${\gamma }_{\text{1}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {1}} =$$\end{document} customer expectations (CE), ${\gamma }_{\text{2}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {2}} =$$\end{document} perceived quality (PQ), ${\gamma }_{\text{3}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {3}} =$$\end{document} perceived value (PV), ${\gamma }_{\text{4}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {4}} =$$\end{document} customer satisfaction (CS), ${\gamma }_{\text{5}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {5}} =$$\end{document} customer complaints (CC), and ${\gamma }_{\text{6}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upgamma _{\text {6}}=$$\end{document} customer loyalty (CL). We represent all the constructs by convex components with unstandardized indicators except for the customer loyalty. As two indicators (z ${}_{\text{13}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {13}}$$\end{document} and z ${}_{\text{14}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {14}})$$\end{document} for customer loyalty are not measured on the same scale, we set this component as a standardized one with the indicators standardized.

Figure. 3 The ACSI model. The dashed line labeled b ${}_{\text{10}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}}$$\end{document} signifies an incorrectly specified path coefficient. All weights and error terms are omitted to make the figure concise. CE $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} customer expectations, PQ $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} perceived quality, PV $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} perceived value, CS $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} customer satisfaction, CC $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} customer complaints, CL $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} customer loyalty.

We use 4000 bootstrap samples for computing the standard error and 95% confidence interval of each parameter estimate. For comparison, we also apply GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} to the same data and compute unstandardized weight estimates and unstandardized component scores based on the procedure discussed in Sect. 1. As customer satisfaction is the focal component in the ACSI model, we concentrate on interpreting the scores of customer satisfaction, its statistics, and the relevant model parameters.

The model fitted by GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} shows FIT ${}^{\text{UD}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}} =$$\end{document} .714, indicating that the ACSI model accounts for 71.4% of the weighted total variance of all dependent variables in the model. It also provides GFI $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .987 and SRMR $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .022, pointing to an acceptable level of model fit(Cho et al., Reference Cho, Hwang, Sarstedt and Ringle2020). In addition, it provides that FIT ${}_{\text{M}}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {M}}^{\text {UD}}$$\end{document} $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .802 and FIT ${}_{\text{S}}^{\text{UD}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {S}}^{\text {UD}} =$$\end{document} .438. This indicates that the component measurement model explains 80.2% of the weighted total variance of all dependent indicators, whereas the structural model explains 43.8% of the weighted total variance of all dependent components.

Table 4 provides the weight and loading estimates, and their standard errors and 95% confidence intervals obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} , along with the intercept estimates in the measurement model. The unstandardized weight estimates obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} are also provided for comparison. Overall, all the weight and loading estimates obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} are large and statistically significant, indicating that all the indicators contribute to forming their components, which in turn, explain the variances of their indicators well. Among the three indicators (z ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} , z ${}_{\text{10}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}}$$\end{document} , and z ${}_{\text{11}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {11}})$$\end{document} for customer satisfaction, z ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} (perceived overall satisfaction) are the largest contributor (w ${}_{\text{9}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}=$$\end{document} .422, SE $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .015, 95% CI $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} [.393, .454]). This indicates that when each of the three indicators equally increases, leading to an increase in customer satisfaction, the contribution rate of z ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} for the increase in customer satisfaction was 42.2%, which is greater than those of the two others (z ${}_{\text{10}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}} =$$\end{document} 25.4% and z ${}_{\text{11}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {11}} =$$\end{document} 32.4%). Similarly, the unstandardized weight estimate of z ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} is the largest among the three (w ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .188, w ${}_{\text{10}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}} = $$\end{document} .107, and w ${}_{\text{11}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {11}}= $$\end{document} .131). In contrast, it is uncertain how to interpret the unstandardized weight estimates obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} .

Table 4 The weights, loading, and intercept estimates of the fourteen indicators in the ACSI model and their standard errors (SE) and 95% confidence intervals (CI) obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} , along with the unstandardized weight estimates obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} ( ${\hat{\text{W}}}_{uni})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {\hat{\textrm{W}}}_{\!uni})$$\end{document} .

Table 5 presents the path coefficient estimates and their standard errors and 95% confidence intervals obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} . Overall, the patterns of all the path coefficient estimates are consistent with those from previous studies (e.g., Hwang & Takane, Reference Hwang and Takane2014, Chapter 2). For instance, perceived quality and perceived value have statistically significant influences on customer satisfaction (b ${}_{\text{5}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {5}} =$$\end{document} .723, SE $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .033, 95% CI $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} [.659, .786]; b ${}_{\text{6}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {6}} =$$\end{document} .275, SE $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .035, 95% CI $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} [.204, .344]). Customer satisfaction have statistically significant effects on customer complaints (b ${}_{\text{7}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {7}} =$$\end{document} –.059, SE $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .006, 95% CI $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} [–.072, –.047]) and customer loyalty (b ${}_{\text{8}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {8}} =$$\end{document} .252, SE $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .015, 95% CI $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} [.222, .279]). Each individual path coefficient estimate is indicative of the expected change of the dependent component for a one-unit change in indicators of a predictor component. For instance, the estimate of the path coefficient, b ${}_{\text{8}}=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {8}} =$$\end{document} .252, implies that a one-unit increase in z ${}_{\text{9}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {9}}$$\end{document} (perceived overall satisfaction), z ${}_{\text{10}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}}$$\end{document} (expectation fulfillment), and z ${}_{\text{11}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {11}}$$\end{document} (distance to the ideal) would be associated with an increase of .252 unit in customer loyalty. The $R^{\text{2}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{\text {2}}$$\end{document} value is .331 for perceived quality, .511 for perceived value, .812 for customer satisfaction, .164 for customer complaints, and .404 for customer loyalty. Also, the intercept estimates for the dependent components in the same order as above are 3.014, .793, –.501, .558, and $-$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document} 1.756.

Table 5 The path efficient estimates and their standard errors (SE) and 95% confidence intervals (CI) obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} .

Table 6 presents the estimated means, standard deviations, and ranges of unstandardized component scores obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} and GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} . As expected, the individual scores of each convex component obtained from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} are within the range of their indicators’ scores. The individual scores of customer expectation, perceived quality, perceived value, and customer satisfaction all range from 1 to 10 and those of customer complaint were between 0 and 1, which are equivalent to the ranges of their indicators’ measurement scales. The mean of customer satisfaction from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} is 7.125, indicating that the average satisfaction level in the sample is moderately positive or equivalent to the satisfaction level of a customer whose indicator scores are all 7.125. This mean of customer satisfaction appears to be congruent with the means of its original indicators (7.585, 6.824, and 6.760). The standard deviation of customer satisfaction is 2.353, suggesting that the scores of customer satisfaction are somewhat widely spread out from the mean. This standard deviation value also seems to conform to those of its original indicators (2.489, 2.504, and 2.632).

Table 6 The means, standard deviations (SD), and ranges of the unstandardized component scores estimated from GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} and GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} . The last component (CL) is defined as a standardized component in GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} .

On the contrary, unstandardized components’ scores obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} are not always within the range of their indicators’ scores. Some scores of customer expectation, perceived quality, perceived value, and customer satisfaction are smaller than 1, which is the minimum value of their indicators on the scale. Moreover, the means of unstandardized components are also far from those of their original indicators. For instance, the mean of customer satisfaction obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} is just 3.037, even though its indicators’ means are around 7 as stated above. Thus, it is questionable whether the mean of customer satisfaction obtained from GSCA ${}_{\mathrm{std}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{std}$$\end{document} can be a good representation of the average level of customer satisfaction in the sample. Furthermore, all the standard deviations of unstandardized components are fixed to one, even though none of their indicators have standard deviations being around 1.

To illustrate the usage of OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} as a model comparison criterion, we additionally contemplate two misspecified models of the ACSI model, while assuming the original ACSI model as the true model (denoted by Model 1). One misspecified model (Model 2) is an under-specified one, where a path coefficient (b ${}_{\text{6}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {6}})$$\end{document} is omitted from Model 1. The other misspecified model (Model 3) is an over-specified one that includes an additional path coefficient from customer expectation to customer loyalty in Model 1, as displayed in Fig. 3. We apply GSCA ${}_{\mathrm{cvx}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{cvx}$$\end{document} to fit the three models to the data and compute their OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} values based on 4000 bootstrap samples. Model 1 provides the smallest OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} value (Model 1 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .2883, Model 2 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .2901, and Model 3 $=$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} .2887), indicating that the original ACSI model has the highest predictive generalizability among the three models. The OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} value of Model 2 is larger than that of Model 1 (.2901 >.2883), suggesting that excluding a path coefficient (b ${}_{\text{6}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {6}})$$\end{document} from Model 1 rather decreases the prediction accuracy of the model. On the other hand, the OPE ${}^{\text{UD}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\text {UD}}$$\end{document} value of Model 3 is larger than that of Model 1 (.2887 >.2883), indicating that specifying an additional path coefficient (b ${}_{\text{10}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\text {10}})$$\end{document} to Model 1 is not helpful to improve the predictive generalizability of the model.

5. Concluding Remarks

We proposed convex GSCA that can accommodate a new type of unstandardized components, named convex components. A convex component is defined as a convex combination of original indicators whose weights are all non-negative and summed up to one. Every individual score of a convex component is always within the range of its indicators’ scores and can be interpreted as a construct’s specific level of a person who has the same score for all its indicators as his/her component score. Moreover, the means and standard deviations of convex components are estimated along with other parameters through a single optimization procedure, which can also be interpreted in terms of indicators’ scales. Thus, introducing convex components to the GSCA model will enhance the practical utility of component scores and their summary statistics, for instance, in investigating individuals’ levels of a construct or comparing the average levels of a construct between groups.

We developed an alternating least squares (ALS) algorithm for estimating parameters of the convex GSCA model, which does not require standardizing blocks of indicators that have the same measurement scales within the blocks. The algorithm not only enables information on the variances of each block of indicators to be additionally utilized in parameter estimation, but also prevents indicators with small variances from influencing more heavily the construction of an unstandardized component than those with large variances. Furthermore, its objective function is partially scale-invariant, indicating that the minimum value of the objective function remains unchanged with a linear change in the measurement scale of each block of indicators, giving rise to the same weight estimates.

We evaluated the parameter recovery of the proposed method in a simulation study and further illustrated the merits of the proposed method via a real data analysis. In the simulation study, the proposed method empirically produced unbiased parameter estimates on average under nine GSCA models with convex components and its accuracy was further improved with large sample size. In the real data analysis, the patterns of the parameter estimates were consistent with those from previous studies, and the benefits of convex components were pronounced, compared to the unstandardized components obtained from the conventional ad-hoc procedure of rescaling weight estimates. Unlike these unstandardized components, convex components’ weight estimates were interpretable, all their individual scores fell within the range of indicators’ measurement scales or their scores, and their estimated means and standard deviations were congruous with those of their indicators. Therefore, we are confident to recommend that researchers employ the method when they are interested in the GSCA model with unstandardized components of original indicators.

Note that as an anonymous reviewer pointed out, researchers may still want to consider standardizing observed variables that are measured on the same scale. We recommend considering this option only if researchers are not interested in unstandardized component scores. If researchers apply GSCA to estimate the scores of unstandardized components after standardizing indicators of the same scale, an indicator with a small variance can be assigned a relatively large unstandardized weight, leading to a potentially inflated influence of the indicator on the estimation of the component scores, as shown in Sect. 1. This issue does not occur when researchers keep the original scales of indicators and apply convex GSCA with convex components.

In future research, we may consider incorporating convex components into various extensions of GSCA, which deal with more complex analyses, for instance, those of involving higher-order components (Hwang & Takane, Reference Hwang and Takane2014, Chapter 3), missing observations (Hwang & Takane, Reference Hwang and Takane2014, Chapter 3), multilevel components (Hwang et al., Reference Hwang, Takane and Malhotra2007), components with categorical indicators (Hwang and Takane, Reference Hwang and Takane2010), component interaction terms(Hwang et al., Reference Hwang, Cho, Jin, Ryoo, Choi and Lee2021a, Reference Hwang, Ho and Lee2010a), or factors (Hwang et al., Reference Hwang, Cho, Jung, Falk, Flake and Jin2021b). Such additional extensions will improve the usefulness of GSCA, placing components on their indicators’ scales while having their means and variances free parameters to be estimated along with others.