# Construction Quality of Prefabricated Buildings Using Structural Equation Modeling
## 3 Materials and Methods
### Introduction of SEM  
SEM is a multivariate statistical method that includes path and factor analyses. This method can analyze the relationship between measurement variables, latent variables, and errors. Moreover, this method can be used to determine the effect on dependent variables from the analysis of independent variables. This method is helpful to analyze the relationship between the factors affecting the construction quality of prefabricated buildings. It can also analyze the effect of this latent variable on its corresponding explicit variable in the construction process, construction organization management, and precast component. SEM uses AMOS (analysis of moment structures) software to draw and browse the model through the visualization module. This method can also fit the data collected through the questionnaire with various indexes of the model and output the best model. The model is considered scientifically valid when the fit indexes meet the established standards. The indexes include relative-fit and absolute-fit indexes. The main indexes and standards are shown in Table 4.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Basic Information** & **Type** & **Quantity** & **Proportion** \\ \hline \multirow{8}{*}{Work unit type} & Government-related units & 15 & 2.83\% \\  & Colleges and universities & 49 & 9.24\% \\  & Real estate development enterprises & 103 & 19.43\% \\  & Component supplier enterprises & 24 & 4.52\% \\  & Building construction enterprises & 196 & 37\% \\  & Supervisory units & 132 & 24.90\% \\  & Other (graduate students) & 11 & 2.08\% \\ \hline \multirow{4}{*}{Years of work/research in a prefabricated building} & Less than 1 & 96 & 18.11\% \\  & 1–3 & 297 & 56.04\% \\ \cline{1-1}  & 4–5 & 128 & 24.15\% \\ \cline{1-1}  & Over 5 & 9 & 1.70\% \\ \hline \multirow{4}{*}{Degree of understanding the construction quality of a prefabricated building} & Very well known & 102 & 19.25\% \\ \cline{1-1}  & Fairly well known & 135 & 25.47\% \\ \cline{1-1}  & Generally know & 164 & 30.94\% \\ \cline{1-1}  & Very little & 73 & 13.77\% \\ \cline{1-1}  & Do not know & 56 & 10.57\% \\ \hline \end{tabular}
\end{table}
Table 3: Questionnaire sample data statistics.  
SEM can be divided into exploratory factor analysis and confirmatory factor analysis (CFA). CFA is a method for measuring whether the relationship between the measured factors and the measured items is consistent with the researcher's predictions. SEM consists of a measurement model (Equations (1) and (2)) and a structural model (Equation (3)).  
\[X=\Lambda_{x}\xi+\delta \tag{1}\]  
\[Y=\Lambda_{y}\eta+\varepsilon \tag{2}\]  
Equation (1) defines the relationship between the explicit observation variable \(X\) and the implicit observation variable \(\xi\). Equation (2) defines the relationship between the explicit latent variable \(Y\) and the implicit latent variable \(\eta\). \(\delta\) is the residual of the dominant observed variable, \(X\). \(\eta\) is the residual of the dominant latent variable, \(Y\). \(\Lambda_{x}\) is the regression coefficient of \(\xi\). \(\Lambda_{y}\) is the regression coefficient of \(\eta\).  
\[\eta=\beta\eta+\Gamma\xi+\zeta \tag{3}\]  
Equation (3) represents the relationship between the implicit observation variable \(\xi\) and the implicit latent variable \(\eta\). \(\beta\) is the coefficient matrix between the hidden and the latent variables. \(\Gamma\) is the coefficient matrix composed of hidden observed variables and hidden latent variables. \(\zeta\) is the residual of the equation.