## 2 Materials and Methods
### Research Methodology--Models of Analysis  
This research relies on a complex modelling endeavor comprising GGMs for the network analysis and SEM deployed to capture the patterns and overall linkages between 29 cephalometric measurement units performed on the two samples (Prader-Willi and Control).  
The network analysis is based on GGMs which are innovative exploratory research tools, very useful to infer the connections/dependencies between variables (particularly biological variables and genes). For this research, GGMs were configured and processed through the extended Bayesian information criteria (EBIC) with graphical (g) least absolute shrinkage and selection operator (lasso) (EBICglasso) and partial correlation (Pcor). A GGM is a graph in which all random variables are continuous and jointly Gaussian [17; 18] and it is based on conditional independence, respectively if \(\Omega=\{w_{v1,v2}\}\), two variables v1 and v2 are conditionally independent if \(\omega_{v1,v2}=0\), namely there are 0 entries of the precision matrix \(\Omega=\Sigma^{-1}\). \(\Sigma\) is the positive definite covariance matrix and \(\Omega\) is the precision matrix of the distribution, defined as the inverse of \(\Sigma\).  
If \(\Sigma\) is positive definite, distribution has density on \(f(x\Big{|}\xi,\Sigma)=(2\pi)^{-d/2}(\text{det}\Omega)^{1/2}e^{-(x-\xi)^{T} \Omega(x-\xi)/2}\). The sample covariance matrix is represented by \(\bar{\Sigma}=\frac{1}{n-1}\sum\limits_{i=1}^{n}{(x_{i}-\xi)(x_{i}-\xi)}^{T}\)[17; 18].  
GGMs \"entail an undirected network of partial correlation coefficients (both positive and negative). They are graphically reflected through the absolute strengths (width and saturation of the edges between nodes), thus being a network model of conditional associations and avoiding spurious correlation\".  
The partial correlation (pcor) determined in the GGMs can be calculated as in Equation (1) [19]:  
\[r_{xy\cdot z}=\frac{r_{xy}-r_{xz}r_{yz}}{\sqrt{1-r_{xz}^{2}}\sqrt{1-r_{yz}^{2}}} \tag{1}\]  
where: \(r\) represents the correlation degree.  
In our GGM networks, positive partial correlations are generally visualized with blue edges and negative partial correlations with red edges.  
GGMs are complemented in our research with another modern technique and approach to modelling longitudinal data, namely SEM. Both techniques \"imply a variance-covariance matrix, aiming to identify how variables are related to each other, namely the direct and indirect effects of one variable on another, having their origin in path analysis\" [19].  
SEM are modern research instruments that allowed us to identify and assess the latent interconnections between numerous cephalometric measurements specific for the Prader-Willi syndrome diagnostic, so as to infer the positive and negative influences in this regard. The general configuration of the SEM models designed in our research is presented in Figure 3.  
\"Going beyond the classical linear regression analyses, SEM examines the causal relationships among variables, while controlling simultaneously for measurement error as a greatest advantage in empirical researches\". SEM allowed us to determine the degree of correlation (path coefficients) that capture the importance of a certain path of influence from cause to effect [20].