## 3 Results
In this table, we can see that there were positions where the error was far from zero. The positions that stand out in this case were, for example, the halves, who on average overestimated the number of impacts by 2.38 impacts. Then, we can see that the Fullbacks and the Second Row underestimated the impacts received by 1.25 and 1.38 impacts, respectively. To study this pattern, we estimated the means and the errors independently. The results of these estimates can be seen in Table 6. Thanks to this analysis, we can see that the Centres, Fullback, Half, and the Second Row had biased estimates of the number of impacts they received. By looking at Table 5 (error segmented by position), we can see that the positions that received the most impacts of more than 8G were the ones that overestimated them the most, while the positions that received fewer impacts tended to underestimate the impacts. To see how the error relates to the magnitude of the impacts, we used a methodological framework proposed by the Bland-Altman method, where the error (the difference between measurement methods, in our case questionnaire versus MEMS) is plotted against the mean of the two methods. The classic plot of [PERSON]'s agreement analysis can be seen in Figure 4. One feature we can see is that the error correlated with the magnitude of the measurements. To study this relationship, we proposed two Bayesian models; the first one is a linear model that takes the error as the dependent variable and the magnitude of the measurements as the independent variable (this variable understood as the mean between both measurements), and the variance of the model is also estimated based on the independent variable to not depend on the assumption of homoscedasticity of the linear model. The second model is an exponential model, which assumes constant variance. The results of the model fittings can be seen in Figure 5 (linear and exponential model). We can graphically check that the exponential model fit our data more naturally; this situation can be verified in Table 7, where the results of both models are compared. To compare the goodness of fit of the models to the data, the metrics of expected log pointwise predictive density (ELPD) of Watanabe-Akaike Information Criterion (WAIC) were used, where higher values indicate models with a better balance between accuracy and complexity. A penalized version of WAIC for the number of parameters (P WAIC) was also computed, which generally allows evaluating the complexity of the model; higher values indicate that the model is more complex. In our case, we can see that the exponential model is superior to the linear model, as it fit the data better (higher ELPD WAIC) and is simpler since it has fewer parameters (lower P WAIC). The results of the estimates for their parameters can be seen in Table 8. This situation gives us indications that the relationship between the players' perception and the number of impacts they receive is not linear. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Position** & **Mean** & **HDI 2.5\%** & **HDI 97.5\%** & **MCSE Mean** & **ESS** & **R Hat** \\ \hline Centre & \(-0.698\) & \(-1.473\) & \(-0.008\) & \(0.004\) & \(8000.0\) & \(1\) \\ \hline Flanker & \(0.897\) & \(-0.239\) & \(2.007\) & \(0.006\) & \(8705.0\) & \(1\) \\ \hline Fly Half & \(-0.285\) & \(-1.278\) & \(0.643\) & \(0.007\) & \(6289.0\) & \(1\) \\ \hline Fullback & \(-1.217\) & \(-3.231\) & \(-0.965\) & \(0.013\) & \(7372.0\) & \(1\) \\ \hline Half & \(2.344\) & \(0.224\) & \(4.755\) & \(0.015\) & \(7715.0\) & \(1\) \\ \hline Hooker & \(0.130\) & \(-1.364\) & \(1.554\) & \(0.009\) & \(6758.0\) & \(1\) \\ \hline Prop & \(0.983\) & \(-0.619\) & \(2.643\) & \(0.009\) & \(8408.0\) & \(1\) \\ \hline Second Row & \(-1.380\) & \(-2.800\) & \(-0.012\) & \(0.009\) & \(7536.0\) & \(1\) \\ \hline Wing & \(-0.074\) & \(-1.216\) & \(1.091\) & \(0.007\) & \(7931.0\) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results of the estimated error means for the positions.