## 4 Results and Discussion
### Path Model Equations  
By replacing the trimmed path model's unstandardized direct and mediating coefficients \(\gamma\) and \(\beta\) in Equation (1), considering the system's mean structure [42], and using the variances of residues \(V(\zeta_{i})\), the following equations relate the estimated model (m) outputs \(\overline{\text{acr}}^{*,m}\) and \(\overline{T}_{\text{in}}^{m}\) with the input variables \(x_{j}\):  
\[\overline{\text{acr}}^{*,m}=\overline{\text{acr}}^{*,m0}+\sum \limits_{j=1}^{6}\gamma_{1j}x_{j}+\beta_{12}\overline{T}_{\text{in}}^{m}+W_{1 }(V(\zeta_{1})) \tag{2}\] \[\overline{T}_{\text{in}}^{m}=\overline{T}_{\text{in}}^{m0}+\sum \limits_{j=1}^{6}\gamma_{2j}x_{j}+\beta_{21}\overline{\text{acr}}^{*,m}+W_{2 }(V(\zeta_{2}))\]  
where \(\overline{\text{acr}}^{*,m0}\) (=7.494) and \(\overline{T}_{\text{in}}^{m0}\) (=\(-2.781\)) stand for the path model intercepts [81] and \(W_{i}(V(\zeta_{i}))\) defines the white noise for the output variables, with zero mean and constant variance equal to \(V(\zeta_{i})\). By solving the system of equations in (2), the following path model equations for \(\overline{\text{acr}}^{*,m}\) and \(\overline{T}_{\text{in}}^{m}\) are obtained:  
\[\overline{\text{acr}}^{*,m}= -1.4754+0.1049\overline{T}_{out}+0.0193\Sigma E_{\text{sol}}-0.12 15\Sigma\text{R}^{*}+0.0252\overline{S}_{c}^{*} \tag{3a}\] \[+0.9068W_{1}(0.194)+0.1396W_{2}(1.970)\] \[\overline{T}_{in}^{m}= 8.4781+0.6810\overline{T}_{out}+0.0411\Sigma E_{\text{sol}}+0.081 0\Sigma\text{R}^{*}+0.0142\overline{S}_{c}^{*}\] \[-0.6049W_{1}(0.194)+0.9068W_{2}(1.970).\]Figure 7 presents scatter plots comparing one year of sample monitoring data for the variables \(\overline{\text{acr}}^{*,m}\) (bottom graph, \(\circ\)) and \(\overline{T}_{\text{in}}\) (top graph, \(\circ\)) with the corresponding estimates \(\overline{\text{acr}}^{*,m}\) and \(\overline{T}_{\text{in}}^{m}\) (denoted with \(+\)) determined using the path model Equations (3a) and (3b). Moving averages are also represented, with four points each for the sample data (\(-\)) and estimates (\(*\)). Figure 7 confirms that the path model is unsuited for deterministic evaluation of classroom air change rate or classroom indoor air temperature. Nonetheless, the sample and estimate moving averages are fairly similar. Moreover, despite its simplicity, the constant-variance white noise used in Equations (3a) and (3b) estimates daily ranges of output variables that are not much different from those found in the sample, a sign that the system's mean structure is correctly captured. Table 5 compares the sampled and estimated mean values of the output variables \(\overline{\text{acr}}\) and \(\overline{T}_{\text{in}}\) for different class intervals of the outdoor temperature. Table 5 confirms that the path model correctly captures the output variables' mean struc  
Figure 7: Scatter plot comparing one year of \(\overline{\text{acr}}^{*}\) and \(\overline{T}_{\text{in}}\) sample data with the corresponding estimates \(\overline{\text{acr}}^{*,m}\) and \(\overline{T}_{\text{in}}^{m}\) determined with Equation (3). Moving averages (four points each for both sample data and estimates) are also shown. outdoor air temperature, with errors not exceeding 6% and 1% for air change rate and indoor air temperature, respectively. For behavioral and decision-making studies, knowledge of the mean air change rate per class interval with respect to the outdoor air temperature may be sufficient to characterize students' and teachers' behavior and support the adoption of an improved ventilation strategy. Moreover, the capability of path diagrams and of structural equation models more generally to represent and validate causal relationships between observable (physical) variables and unobservable (personal, context-related) factors may very well offset these tools' lack of capacity for deterministic estimation.