## 5 Discussion
### Statistical Analysis  
At certain BER value, the collected packet loss rate, delay, and jitter data are continuous, while the Ethernet QoS scores can be regarded as ordinal data, measured and classified as best, optimal or tolerable [21] (Table 1). Furthermore, let us define the ordinary Ethernet score as a union of the three events, defined by either packet loss rate, or packet delay, or packet jitter exceeding their respective RFC2544 thresholds of tolerance, specifically for the LTE (S1 interface), as presented in (1):  
\[D=\left\{ \begin{array}{cc}0:&Delay\ \leq\ 20\ \mathrm{msec}\\ 1:&Delay>\ 20\ \mathrm{msec}\end{array} \right. ;J=\left\{ \begin{array}{cc}0:&Jitter \ \leq\ 8\ \mathrm{msec}\\ 1:&Jitter>\ 8\ \mathrm{msec}\end{array} ;PLR=\left\{ \begin{array}{cc}0:&Packet\ LossRate\ \leq\ 0.5\ \%\\ 1:&Packet\ LossRate\ >\ 0.5\ \%\end{array} \right. \tag{1}\]  
\[QoS=D+J+PLR=\left\{ \begin{array}{cc}0&\Rightarrow\ \mathrm{tolerable\ QoS}\\ 1,2,3\Rightarrow\ \mathrm{intolerable\ QoS}\end{array} \right. \tag{2}\]  
Therefore, we need to investigate whether there is a relationship between RR unavailability and the Ethernet QoS score (2). However, common parametric tests are not a good choice for statistical analysis of ordinal data of the sort that we were to conduct here, so we used the nonparametric Spearman rank-order correlation test to find the correlation coefficient (\(\rho\)), which is a good measure of strength and direction of the relationship between the continuous and the ordinal random variables. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Packet Length (Bytes)** & **Average Burst (frms)** & **Average Burst (s)** \\ \hline
64 & 26,470 & 0.022 \\
512 & 18,990 & 0.084 \\
1500 & 19,420 & 0.240 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximal lossless packet transmission rate (back-to-back packets). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Packet Length** & **Avg Pkt** & **Max Avg Pkt** & **Pass Rate** & **Pass Rate** & **Pass Rate** & **Pause** & **Pass/Fail** \\ **(Bytes)** & **Jitter (us)** & **Jitter (us)** & **(Mbps)** & **(\%)** & **(\%)** & **(frm/s)** & **Detected** & **Pass/Fail** \\ \hline
64 & 0 & 0 & 200.1 & 20.01 & 235,862 & No & PASS \\
512 & 0 & 33 & 200.1 & 20.01 & 45,129 & No & PASS \\
1500 & 0 & 59 & 200.1 & 20.01 & 16,213 & No & PASS \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Jitter. Thus, considering the QoE score and RR unavailability as being represented by paired observations, our preliminary investigation revealed monotonic relationship between them. The null and the alternative hypotheses, respectively, are as follows:  
**Hypotheses 0 (H0)**. _There is no association between the QoE score and RR unavailability._  
**Hypotheses 1 (H1)**. _There is an association between the QoE score and RR unavailability._  
The significance of a test result is articulated with the \(p\)-value; the smaller it is, the more significant is the result. In this respect, the comparison reference that is commonly referred to as significance (\(\alpha\)) is in fact the probability of a \"false positive\" decision about null hypothesis rejection when it should be accepted [22]. So, for example, if the null hypothesis is rejected (\(p<\alpha\) found), then a smaller \(\alpha\) value implies stronger evidence that the finding is statistically significant [22]. In our analysis, we adopted a moderate value of \(\alpha=1\%\) to find whether and to what extent (excessive) RR unavailability incurs notable QoS degradation. Accordingly, our tests revealed that a direct relationship exists between the end-to-end QoS score, and the RR unavailability prediction at any given bit error rate (BER). Specifically, In Table 5, the values of the Spearman correlation coefficient between the end-to-end Ethernet QoS score and the average RR unavailability are presented for particular BER and \(p\)-values.