## 2 Materials and Methods
### Data Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Study variables**} & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** \\ \hline
1 & Availability of social support & & & & & & & & & & & & & \\
2 & Satisfaction of social support & 0.41 \({}^{*}\) & & & & & & & & & & & & \\
3 & Family support for autonomy & 0.27 \({}^{*}\) & 0.27 \({}^{*}\) & & & & & & & & & & & \\
4 & Family support for competence & 0.23 \({}^{*}\) & 0.36 \({}^{*}\) & 0.20 \({}^{*}\) & & & & & & & & & \\
5 & Family support for relatedness & 0.16 & 0.15 & 0.47 \({}^{*}\) & 0.19 \({}^{*}\) & & & & & & & \\
6 & BPNS for autonomy & 0.15 & 0.17 & 0.34 \({}^{*}\) & 0.32 \({}^{*}\) & 0.19 \({}^{*}\) & & & & & & \\
7 & BPNS for competence & 0.17 & 0.13 & 0.04 \({}^{*}\) & 0.19 \({}^{*}\) & \(-\)0.01 & 0.59 \({}^{*}\) & & & & & \\
8 & BPNS for relatedness & 0.22 \({}^{*}\) & 0.31 \({}^{*}\) & 0.38 \({}^{*}\) & 0.34 \({}^{*}\) & 0.31 \({}^{*}\) & 0.63 \({}^{*}\) & 0.54 \({}^{*}\) & & & & \\
9 & BPNF for autonomy & \(-\)0.05 \(-\)0.29 \({}^{*}\) & \(-\)0.19 \({}^{*}\) & \(-\)0.36 \({}^{*}\) & \(-\)0.30 \({}^{*}\) & \(-\)0.30 \({}^{*}\) & \(-\)0.18 \({}^{*}\) & \(-\)0.34 \({}^{*}\) & & & \\
10 & BPNF for competence & \(-\)0.14 \(-\)0.37 \({}^{*}\) & 0.04 \({}^{*}\) & \(-\)0.24 \({}^{*}\) & 0.06 \({}^{*}\) & \(-\)0.25 \({}^{*}\) & \(-\)0.32 \({}^{*}\) & \(-\)0.20 \({}^{*}\) & 0.49 \({}^{*}\) & & & \\
11 & BPNF for relatedness & \(-\)0.15 \(-\)0.32 \({}^{*}\) & \(-\)0.08 \({}^{*}\) & \(-\)0.52 \({}^{*}\) & \(-\)0.18 \({}^{*}\) & \(-\)0.20 \({}^{*}\) & \(-\)0.33 \({}^{*}\) & \(-\)0.27 \({}^{*}\) & 0.25 \({}^{*}\) & 0.43 \({}^{*}\) & 0.41 \({}^{*}\) & \\
12 & Depressive feelings & \(-\)0.10 \(-\)0.40 \({}^{*}\) & 0.06 \(-\)0.42 \({}^{*}\) & 0.04 \({}^{*}\) & \(-\)0.22 \({}^{*}\) & \(-\)0.33 \({}^{*}\) & \(-\)0.27 \({}^{*}\) & 0.25 \({}^{*}\) & 0.43 \({}^{*}\) & 0.41 \({}^{*}\) & \\
13 & Purpose in life & 0.27 \({}^{*}\) & 0.31 \({}^{*}\) & 0.08 \({}^{*}\) & 0.30 \({}^{*}\) & 0.07 \({}^{*}\) & 0.38 \({}^{*}\) & 0.42 \({}^{*}\) & 0.34 \({}^{*}\) & \(-\)0.28 \({}^{*}\) & \(-\)0.52 \({}^{*}\) & \(-\)0.37 \({}^{*}\) & \(-\)0.47 \({}^{*}\) & \\  & _Atari_ & 3.41 & 4.78 & 3.73 & 4.33 & 3.82 & 4.39 & 4.15 & 4.09 & 1.71 & 2.30 & 1.63 & 2.38 & 3.84 \\  & _Standard deviation_ & 2.13 & 0.89 & 0.88 & 0.69 & 0.80 & 0.58 & 0.59 & 0.55 & 0.66 & 0.75 & 0.68 & 2.43 & 0.56 \\  & _Siveness_ & 1.04 & \(-\)1.24 \({}^{*}\) & \(-\)0.39 & \(-\)1.27 & \(-\)0.18 & \(-\)1.06 & \(-\)0.60 \({}^{*}\) & 0.40 & 0.80 & 0.14 & 1.18 & 1.30 \({}^{*}\) & \(-\)0.16 \\  & _Kurtosis_ & 0.53 & 3.12 & \(-\)0.73 & 1.58 & \(-\)0.78 & 1.08 & 0.34 & 0.08 & \(-\)0.20 & \(-\)0.62 & 1.10 & 1.79 \({}^{*}\) & \(-\)0.81 \\ \hline \hline \end{tabular} Note. BPNS = basic psychological need satisfaction; BPNF = basic psychological need frustration; \({}^{*}\)_p_ \(<\) 0.05.  
\end{table} 
Table 1: Descriptive statistics and correlation matrix. A two-step modelling approach was used to better identify the sources of poor overall model fit [27]. The first step (outer model) allowed us to focus on the factor structure underlying the items and/or parcels of each construct. This analysis also allowed us to examine the psychometric properties of each of the constructs. Standardised factor loadings, composite reliability values (\(\rho\)), average variance extracted (AVE) values and an eigenvalue analysis of the correlation matrix of each set of manifest variables were used to assess the quality of the measurement model [27; 28; 29]. An AVE value of 0.50 or greater and a \(\rho\) value of 0.70 or greater indicated acceptable reliability of the scores [28]. The first eigenvalue should be larger than 1, whereas the second eigenvalue should be smaller than 1 [28]. Then, the second step (inner model) consisted of simultaneously testing the structural and measurement models and allowed us to focus on conceptual connections among the latent factors [27].