# Analysis of Fluid Velocity inside an Agricultural Sprayer Using Generalized Linear Mixed Models
## 2 Material and Methods
### Useful Sequencing of the Best Model
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Structure** & \(\mathbf{R^{2}_{ad}}\) & **Intercept \(\mathbf{{}^{\wedge}\beta_{0}}\)** & **Slope \(\mathbf{{}^{\wedge}\beta_{1}}\)** & **SEE** & \(\hat{\mathbf{S}}^{2}_{\text{pr}}\) & \(\overset{\wedge}{\mu_{\text{pr}}}\) \\ \hline I & 0.34044 & 7.459 & 0.336 & 2.8493 & 0.99753 & \(-6.68\times 10^{-10}\) \\ AR(I) & 0.34045 & 7.459 & 0.336 & 2.8494 & 0.99753 & \(-6.10\times 10^{-10}\) \\ ARH(I) & 0.34038 & 7.452 & 0.337 & 2.8558 & 0.99755 & \(-2.34\times 10^{-11}\) \\ ARMA(I,1) & 0.34057 & 7.458 & 0.337 & 2.8491 & 0.99755 & \(-1.00\times 10^{-11}\) \\ CS & 0.34042 & 7.459 & 0.336 & 2.8493 & 0.99753 & \(-9.31\times 10^{-10}\) \\ CSH & 0.34035 & 7.452 & 0.337 & 2.8554 & 0.99755 & \(-3.14\times 10^{-11}\) \\ D & 0.34038 & 7.453 & 0.337 & 2.8548 & 0.99755 & \(-2.39\times 10^{-11}\) \\ Toep & 0.34060 & 7.455 & 0.337 & 2.8513 & 0.99757 & \(-9.60\times 10^{-10}\) \\ UN & 0.34075 & 7.450 & 0.337 & 2.8549 & 0.99770 & \(-2.77\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Adjustment regression between the values observed and those predicted by the model; intercept and slope of the equation. \(\mathbf{R^{2}_{ad}}\): adjusted determination coefficient; SEE: Standard Error of our Estimate; \(\hat{\mathbf{S}}^{2}_{\text{pr}}\): estimated Pearson residual variance; \(\hat{\mu}_{pr}\): estimated Pearson residual mean. _P_-value was \(<0.001\) in all items, according to the structure of the covariance matrix. Figure 4: _Cont._ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Software** & **AICc** & **BIC** & \(\mathbf{\hat{S}^{2}_{r}}\) & **Chi-Square/df** & \(\mathbf{\hat{S}^{2}_{pr}}\) \\ \hline SPSS & 8368.94 & 8403.34 & 0.1828 & - & 0.995 \\ SAS & 7703.14 & 7758.17 & 0.1637 & 0.16 & 0.991 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Best model information criterion AICc and BIC, \(\mathbf{\hat{S}^{2}_{r}}\): estimation of the residual variance, Chi-square value/degrees of freedom, \(\mathbf{\hat{S}^{2}_{pr}}\): estimation of the Pearson residual variance. Figure 4: Model predicted averages for fluid velocity (cm/s) with 95% confidence intervals according to liquid level in the tank (**a**), internal pressure (**b**), and the number of active nozzles (**c**), in SPSS and SAS. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Fixed-Effect**} & \multicolumn{6}{c}{**Least Square Means of Fluid Velocity (cm/s)**} \\ \cline{3-10} \multicolumn{3}{c}{**Adjustments**} & \multicolumn{6}{c}{**Fixed Effects Levels**} \\ \hline \multicolumn{3}{c}{**SPSS (df: Satterthwaite)**} & \multicolumn{6}{c}{} \\ \hline FE & F & \(p\) & 1000 L & 8 bars & 2 & 2000 L & 10 bars & 4 & 3000 L & 12 bars \\ LT & 38.64 & 0.0015 & 15.17c & & 12.80b & & & 9.37a & \\ P & 27.94 & \(<\)0.0001 & & 11.60a & & 12.17a & & 12.88b \\ NN & 6.43 & 0.0501 & & & 13.05 & & & 11.42 & \\  & & & & **SAS (df: Between-Within)** & & & & \\ LT & 17.01 & \(<\)0.0001 & 17.79c & & 12.23b & & & 9.31a & \\ P & 9.99 & \(<\)0.0001 & & 12.03a & & & 12.61a & & 13.35b \\ NN & 8.42 & 0.0052 & & 13.48b & & & 11.88a & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Contrast statistic (F) and p-value for each fixed effect (FE): Liquid level in the Tank-L, Pressure-bar, and Number of active Nozzles. Estimated averages of fluid velocity for the different levels of the FE. df: degrees of freedom. Different letters show significant differences after [PERSON]’s test (\(p<0.05\)). The relationship between the observed and predicted values estimated using the best model is represented in Figure 5, through the linear regression that relates them. Starting from this, 95% predictor intervals are shown, whose calculation responds to what is explained in (Equation (5)) [31]. The points excluded from the prediction interval, the section they belonged to, and the dispersion statistic that characterized them were identified. \[y_{i}\leq\hat{y}_{i}\pm t_{\alpha/2,n-2}\sqrt{\hat{\sigma}^{2}\bigg{[}1+\frac{ 1}{n}+\frac{\left(x_{i}-\overline{x}\right)^{2}}{S_{xx}}\bigg{]}} \tag{5}\]