Estimative for the models fixed effects parameters.

## Abstract

The knowledge of accumulated evapotranspiration by seasonal vegetation crops throughout their life cycle can be an important tool in decision-making when considering the economic viability of the crop. This knowledge can help understand how much the plants, subject to specific management, can evapotranspirate at the end of their cycle. This information assists in estimating the quantity of a production variable, for example, the mass of shoot fresh matter, besides indicating a more interesting period for its harvest. The objective of this chapter is, from the daily evapotranspiration estimative throughout the cycle, to model the accumulated evapotranspiration over the entire growth period of the crop. In order to do so, we must understand that the behavior of the response variable, i.e., the accumulated evapotranspiration, over time is not linear and keep in mind that the several observations performed in the same experimental unit have correlations and these correlations are more intense the closer temporally the measurements are. This understanding leads us to the analysis of longitudinal data from the nonlinear mixed effect models perspective.

### Keywords

- longitudinal data
- nonlinear mixed effect model
- growth curve
- correlation structure
- irrigation

## 1. Introduction

With available water in the soil, the water flow through the plants depends only on atmospheric demand. Therefore, physical variables as temperature, relative air humidity, and wind and solar radiation affect directly the evapotranspiration (

Seasonal vegetation crops present a small demand for water, while their root system is small, reaching maximum rates in full growths and decreasing in the final stages of development. For these species, the accumulated

The objective of this chapter is to illustrate the use of nonlinear mixed effect models to fit accumulated

## 2. Response profile

Figure 1 presents the accumulated

We wish to describe the behavior of our variable of interest or *response variable*, the accumulated

Observations from the response variable in more than one moment in the same experimental unity constitute what we call as *response profile*. Therefore, Figure 1 presents the response profile of a lettuce plant over time. This profile, apparently, presents an

At the first days of observations (Figure 1), the accumulated

## 3. A model for growth data

In order to describe the behavior of the accumulated

Regarding the accumulated

There are several functions capable of characterizing the accumulated

with

To better understand what we have said, let’s observe Figure 3. It brings the observed data shown in Figure 1 represented by the black dots and two other curves representing two different fits made from this data. The first is a fifth-degree polynomial model, and the second is given by Eq. (1). The graph in this figure was extended until the

Both the fifth-degree polynomial and the nonlinear

We also see that, from approximately the

Another important aspect, besides the parameter interpretability, is that the nonlinear

## 4. Longitudinal data

Clearly, in order to make sense, an experiment must provide data from more than one experimental unit. In our case, more than one plant should be observed over time, i.e., we must obtain more than one response profile. Studies in which the response variable is observed repeatedly throughout time in the experimental units are called *longitudinal studies*. This kind of work is common in agriculture when analyzing the increase or decrease of the response variable over time [2, 3, 5, 6, 7, 8].

Measurements performed in the same experimental unit are most likely to be correlated. Suppose two plants which its

Besides the correlation between the observations within the same experimental unit, we must consider that, most likely, these correlations are greater for observations performed between neighboring times than those performed between more distant times.

## 5. Mixed effect model

In a longitudinal study, the monitoring of the experimental units over time generates correlated dataset. As mentioned, these correlations within the same experimental unit are stronger among neighboring observations. The greater the time distance between two measures, the weaker the correlation between them. Besides that, when we observe experimental units which received the same conditions regarding growth over time and are part of the same treatment, we have a variability among them that we attribute to chance. The treatment effects, the correlations, and the variabilities in a longitudinal study indicate that we need a tool that, in addition to being flexible in specifying a mathematical model, also emphasizes each experimental unit.

In mixed effect models, we select an ordinary function to describe the response variable regarding the studied covariables, that is, the responses of the experimental units in a population. Besides that, specific coefficients of this function can be unique for each experimental unit. In a mixed effect model, we assume that the experimental units of a population have the same functional form, but the function parameters may vary among the units.

The name *mixed model* comes from the fact that this model combines *fixed effects* and *random effects*. A mixed effect model is a parametric model which describes the relations between the response variable and the covariables (fixed effects) and takes into account the individual responses of each experimental unit (random effects). In other words, the fixed effects parameters describe the relations of the response variable and the covariables in an entire population, and the random effects specify the contribution of each individual within the population [4, 9, 10, 11, 12].

To illustrate how to write a

being

with the parameters

with

where each vector

Most of the times, we consider

By using nonlinear mixed effect models, we must consider the technical difficulty in the parameter estimation. In a mixed effects linear model, the derivative of the logarithmic of the likelihood function allows, in a simple way, the algorithm implementation, like Newton-Raphson, to obtain the estimative of the models parameters. Nonlinear models can, however, present nonlinear random coefficients which make it impossible to directly explain the parameters from the likelihood function. Methods that depend on linear approximations such as the first-order Taylor approximation can be used to estimate the model.

Nonlinear mixed effect model analysis can be, preferably, made by the R software [13] with the package name [14], also, at the SAS software using PROC NLMIXED. An excellent text to learn how to use these skills is given in [4].

## 6. Covariance structure of Λ i

Mixed effect model allows the dependence between the observations to be specified in the model parameters through random effects. In other words, the experimental unit responses from a population tend to follow a nonlinear growth path; however, each experimental unit has its own growth path, and the mixed effect model allows the inclusion of specific coefficients to obtain fitted growth curves that align better with the individual responses of these experimental units.

Thus, mixed models allow relevant flexibility for the specification of the random effects correlation structure. However, the dependence structure of the observations within the experimental units

There are cases where dependence on observations not accommodated by the growth function is not well understood or, sometimes, additional covariables that could explain this dependences are absent from the model. Thus, an important resource to model this dependence is to identify the covariance structure that allows correlation between the residuals in different occasions. Then, let us relax on the assumption that the errors are independent and allow them to have heteroscedasticity and/or are correlated within the experimental units.

There are several covariance structures for the residues available in the software to help model longitudinal data. However, in our text, we will highlight only two that we consider more important for these studies, the covariance structure with heterogeneous variance and the first-order auto-regressive.

### 6.1 Heterogeneous variance

The first covariance structure we will consider for

Other variables, besides time, can also be considered with heterogeneous variance in the model. For example, there are cases in which it is important to model the heterogeneity of the treatments, and we can do it by using mixed models.

### 6.2 First-order auto-regressive

Another covariance structure for

This structure has only two parameters, the variance parameter

## 7. Real data example

To exemplify what we have done so far, let’s work with some real data of the

The profile graphs from the accumulated

We model this data using Eq. (2) and considering treatment

Parameters | |||
---|---|---|---|

−0.133363 | — | — | |

2.439746 | 0.935210 | −0.272847 | |

16.648272 | −2.302124 | ||

5.765065 | — | — |

The parameter

The first graph presented in Figure 5 brings the accumulated