# Analysis of Fluid Velocity inside an Agricultural Sprayer Using Generalized Linear Mixed Models
## 3 Results and Discussion
### Model Comparison  
Models without interactions between FE and with an increasing level of complexity were developed, whose fit was compared with (a) the AICc corrected [33; 34], despite a high number of observations in relation to the number of parameters in the model (K), such that n/K > 40, and (b) the [PERSON]'s Bayesian criterion (BIC) [34; 35], which modifies the model parameter scores with respect to the AICc (Table 2). This obviates the need to establish a level of significance in accepting or rejecting effects, similar to the stepwise method [36]. Figure 5: Linear regression between the fluid velocity (cm/s) predicted by the SAS model and that observed during the trial (regressor), with a 95% prediction limits. As can be seen in Table 2, the best model showed a lower value of the AICc and BIC information criteria coefficients, and also the highest relative Akaike weight (w\({}_{i}\); Table 3). The model that best represented the reality measured by the data, was number 11, which included the operational parameters of liquid level in the tank, number of active nozzles, and pressure, with FE and the section of the geometric shape of the tank as a RE. Models 12 and 13, respectively, followed this with a decreasing plausibility in terms of adjustment to the observed data. Model 11, together with the Models 12 and 13, were those that estimated the lowest residual variance among the other models (Table 4). Models which excluded SC, in the fixed and the random effects (Models 1 to 10 and 14), showed the lowest relative weights (w\({}_{i}\)) in the aggregate, and the highest estimates of residual variance. The smallest values of residual variance, corresponding to the models that best fit the description of the observations [37] highlight the importance of SC, as a descriptor of the sprayer tank geometry, when assessing the velocity of the fluid inside, and the need for their inclusion as RE and not as FE. The practical use of such sprayers causes them to gain importance in the interactions between the factors [38]. Nevertheless, the introduction of double interactions: LT\(\times\)P, LT\(\times\)NN, and P\(\times\)NN, and the three-way interaction: LT\(\times\)P\(\times\)NN, among the FE, barely improves the fit by 0.41% (\(\Delta\)AICc = -179.75). It must be highlighted here that, the interaction double, LT\(\times\)P, displayed greatest influence (absorbed more variability) on the fluid velocity inside the tank. Since the formulation of an efficient model was tested here, the inclusion of the double and triple interactions were dismissed, even as RE [29] (the G matrix did not converge), to simplify the interpretation of the results. In this type of GLMM models, the normal parametric distribution for RE has been assumed [39]. This is mainly for conventional reasons [40], understanding that, despite having chosen four cross-sections for the location of the control points within the sprayer's tank, the cut-off point for the choice of the section could take any value from one end to the other. It should be remembered that it is not highly recommended that random effects have many levels so that the Z matrix can converge [27]. No further information was provided on the form of the joint distribution of the RE (SC, H, S), and, thus, the normal distribution assumption, for these RE as a whole, was not evaluated directly in the Models 12, 13, and 14 [41]. The choice of a RE only partly outweighs the natural concern about the incorrect specification of RE in GLMM models [42], and its possible impact on the FE inferences [43]. No evidence was found of the use of GLMM in the analysis of the operation of the agitation system of agricultural sprayers. Until now, studies carried out in this regard have focused on the use of linear models such as ANOVA [4] or numerical modeling using CFD [1,6]. Therefore, the GLMM application methodology proposed in this research can be extended to different tank geometries (cylindrical, asymmetric, cubic, etc.)