## 2 Data and Methods
### Structural Equation Model  
Structural equation modeling (SEM) is a method to establish, estimate and test causality model. The model contains both observable and latent variables. SEM can replace multiple regression, path analysis, factor analysis, covariance analysis and other methods to clearly indicate the effect of observable variables and latent variables [43]. This method assumes that there is a causal relationship between variables, and latent variables are usually formed by a linear combination of some observable variables. SEM is composed of a measurement model and structural model. The measurement model focus on how well the indicators load on the latent factors. The structural model represents the causal relationship between observable variables and latent variables, which focus on how well observable variables predict and explain latent variables and is generally expressed in the form of a path diagram. The general expression of the SEM is described as follows:  
\[\mathrm{X}=\Lambda_{\mathrm{X}}\xi+\delta, \tag{2}\]  
\[\mathrm{Y}=\Lambda_{\mathrm{Y}}\eta+\varepsilon, \tag{3}\]  
\[\eta=\beta\eta+\Gamma\xi+\zeta \tag{4}\]  
Formulas (2) and (3) are measurement models, where \(\mathrm{X}\) is the measurement variable matrix of potential independent variable \(\xi\), \(\Lambda_{\mathrm{X}}\) is the measurement coefficient matrix, and \(\delta\) is the residual matrix of measurement Equation (2). \(\mathrm{Y}\) is the measurement variable matrix of potential dependent variable \(\eta\), \(\Lambda_{\mathrm{Y}}\) is the measurement coefficient matrix, and \(\varepsilon\) is the residual matrix of measurement Equation (3). Formula (4) is the structural model in the SEM, where \(\xi\), is the potential independent variable matrix, \(\eta\) is the potential dependent variable matrix, \(\Gamma\) is the path coefficient matrix, which is affected by the potential independent variable matrix \(\xi\), that affects the potential dependent variable matrix \(\eta\), \(\beta\) is the path coefficient matrix that represents the mutual influence between the constituent factors of the potential dependent variable matrix \(\eta\), and \(\zeta\) is the residual matrix of Equation (4). Based on the analysis of the correlation between the CUE and GPP, NPP, MAP, MAT, and CO\({}_{2}\) concentrations in the Qinghai-Tibet Plateau ecosystem, the SEM is used to explain the relationship between the CUE and the other variables under three future climate scenarios.  
To evaluate the complex relationship between variables, we used the SEM to test the direct and indirect effects of each variable on CUE. We design a hypothetical model in which the observed variables MAT, MAP and CO\({}_{2}\) can directly affect GPP, NPP, and CUE. According to the fitting index and Akaike information criterion (AIC), the SEM with the lowest AIC value was selected [44]. The standardized coefficients were used to explain the direct effects of different paths [45]. We added standardized direct effects of all given exogenous variables to calculate their total impact on CUE [46]. The SEMs were fitted by using the lavaan package (Version 0.6-6, Ghent, Belgium) [47]. All analyses were performed in R (Version 3.5.1, Vienna, Austria.) [48].