6. Covariance structure of Λ<sup>i</sup>

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 Λ<sup>i</sup> until now has been considered independent, identically distributed with mean zero and constant variance. Depending on the chosen model, the growth responses can be explained just by including specific coefficients for the experimental units. However, this may not be enough, and, in this case, modeling the residual dependence of the data becomes important.

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 Λ<sup>i</sup> is the one that admits heterogeneity of the ni residual variances. In this structure, which has ni parameters, we assume that the residuals associated with the observed values at the ni occasions for the i-th experimental unit are independent:

$$\mathbf{A}\_{\mathbf{i}} = \begin{bmatrix} \sigma\_1^2 & 0 & \cdots & 0 \\ 0 & \sigma\_2^2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \sigma\_{n\_i}^2 \end{bmatrix}. \tag{7}$$

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 Λ<sup>i</sup> which is widely used for longitudinal data is the first-order auto-regressive, also called ARð Þ1 :

$$\mathbf{A}\_{\mathbf{i}} = \sigma^2 \begin{bmatrix} \mathbf{1} & \rho & \cdots & \rho^{n\_i - 1} \\ \rho & \mathbf{1} & \cdots & \rho^{n\_i - 2} \\ \vdots & \vdots & \ddots & \vdots \\ \rho^{n\_i - 1} & \rho^{n\_i - 2} & \cdots & \mathbf{1} \end{bmatrix} . \tag{8}$$

the same for all plants, i.e., the individual and treatment effects do not seem to be

Response profile for the accumulated ET of the lettuce plants over 23 consecutive days for the soil water levels

Parameters ^β (W1) γ^ (W2) ^δ (W3) ϕ<sup>1</sup> �0.133363 — ϕ<sup>2</sup> 2.439746 0.935210 �0.272847 ϕ<sup>3</sup> 16.648272 0:405882NS �2.302124 ϕ<sup>4</sup> 5.765065 — — The only nonsignificant parameter NS was ϕ3<sup>i</sup> for the treatment W<sup>2</sup> with p-value >0:34. The other parameters

parameter estimative of the model is presented in Table 1.

We model this data using Eq. (2) and considering treatment W<sup>1</sup> as baseline. The random effects were added to the parameters ϕ2<sup>i</sup>, ϕ3<sup>i</sup>, and ϕ4<sup>i</sup>, and the treatments were important to explain the parameters ϕ2<sup>i</sup> e ϕ3<sup>i</sup>. Besides that, we consider the heterogeneity in the treatments and an AR(1) correlation structure for Λi. The

The parameter ϕ2<sup>i</sup> seems to be influenced by the other treatments, and its value

� 2:44 þ �ð Þ¼ 0:27 2:17 kg for W3. The inflection point, i.e., the day the accumulated ET rate was maximum, also seems to be influenced by the treatments. W<sup>1</sup> and W<sup>2</sup> were not statistically different for the parameter ϕ3<sup>i</sup>, but W3, with estimative of

The first graph presented in Figure 5 brings the accumulated ET mean in each day of the four plants in each treatment. The solid lines are fitting for the treatments

was estimated in � 2:44 kg for W1, � 2:44 þ 0:94 ¼ 3:38 kg for W2, and

� 14 days, appears to be statistically different from W<sup>1</sup> (� 17 days).

important for this parameter.

Estimative for the models fixed effects parameters.

Modeling Accumulated Evapotranspiration Over Time DOI: http://dx.doi.org/10.5772/intechopen.86913

presented p-value < 0:001.

Table 1.

77

Figure 4.

W1, W2, and W3.

This structure has only two parameters, the variance parameter σ2, always positive, and the covariance parameter ρ, which may vary between �1 and 1. This kind of structure allows the residues associated with the observations in neighboring occasions to be more correlated than those whose observations are further apart. The ARð Þ1 is preferred for datasets in which the longitudinal observations are equally spaced.

## 7. Real data example

To exemplify what we have done so far, let's work with some real data of the ET from lettuce plants grown in pots. A total of N ¼ 12 were completely randomized into three levels of water in the soil. At the first treatment, W1, the water level for the plant were kept between 50.0 and 75.0% of the substrate's retention capacity. In the other two treatments, W<sup>2</sup> and W3, the water level in the substrate was kept between 50.0 and 87.5% and between 50.0 and 100.0%, respectively. When the retention capacity of the substrate reached 50.0%, the pots were irrigated until their maximum level regarding each treatment.

The profile graphs from the accumulated ET for all pots in each treatment are shown in Figure 4. Note that the inferior asymptote, when t ! �∞, is apparently Modeling Accumulated Evapotranspiration Over Time DOI: http://dx.doi.org/10.5772/intechopen.86913

Figure 4.

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

The first covariance structure we will consider for Λ<sup>i</sup> is the one that admits heterogeneity of the ni residual variances. In this structure, which has ni parameters, we assume that the residuals associated with the observed values at the ni

<sup>1</sup> 0 ⋯ 0

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

Another covariance structure for Λ<sup>i</sup> which is widely used for longitudinal data is

This structure has only two parameters, the variance parameter σ2, always positive, and the covariance parameter ρ, which may vary between �1 and 1. This kind of structure allows the residues associated with the observations in neighboring occasions to be more correlated than those whose observations are further apart. The ARð Þ1 is preferred for datasets in which the longitudinal observations are

To exemplify what we have done so far, let's work with some real data of the ET from lettuce plants grown in pots. A total of N ¼ 12 were completely randomized into three levels of water in the soil. At the first treatment, W1, the water level for the plant were kept between 50.0 and 75.0% of the substrate's retention capacity. In the other two treatments, W<sup>2</sup> and W3, the water level in the substrate was kept between 50.0 and 87.5% and between 50.0 and 100.0%, respectively. When the retention capacity of the substrate reached 50.0%, the pots were irrigated until their

The profile graphs from the accumulated ET for all pots in each treatment are shown in Figure 4. Note that the inferior asymptote, when t ! �∞, is apparently

1 ρ ⋯ ρni�<sup>1</sup> ρ 1 ⋯ ρni�<sup>2</sup> ⋮ ⋮⋱⋮ ρni�<sup>1</sup> ρni�<sup>2</sup> ⋯ 1

<sup>2</sup> ⋯ 0 ⋮ ⋮⋱ ⋮ 0 0 ⋯ σ<sup>2</sup>

ni

: (7)

5: (8)

σ2

heterogeneity of the treatments, and we can do it by using mixed models.

0 σ<sup>2</sup>

heterogeneous variance and the first-order auto-regressive.

occasions for the i-th experimental unit are independent:

Λ<sup>i</sup> ¼

6.1 Heterogeneous variance

Water Chemistry

6.2 First-order auto-regressive

equally spaced.

76

7. Real data example

maximum level regarding each treatment.

the first-order auto-regressive, also called ARð Þ1 :

<sup>Λ</sup><sup>i</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

Response profile for the accumulated ET of the lettuce plants over 23 consecutive days for the soil water levels W1, W2, and W3.


The only nonsignificant parameter NS was ϕ3<sup>i</sup> for the treatment W<sup>2</sup> with p-value >0:34. The other parameters presented p-value < 0:001.

#### Table 1.

Estimative for the models fixed effects parameters.

the same for all plants, i.e., the individual and treatment effects do not seem to be important for this parameter.

We model this data using Eq. (2) and considering treatment W<sup>1</sup> as baseline. The random effects were added to the parameters ϕ2<sup>i</sup>, ϕ3<sup>i</sup>, and ϕ4<sup>i</sup>, and the treatments were important to explain the parameters ϕ2<sup>i</sup> e ϕ3<sup>i</sup>. Besides that, we consider the heterogeneity in the treatments and an AR(1) correlation structure for Λi. The parameter estimative of the model is presented in Table 1.

The parameter ϕ2<sup>i</sup> seems to be influenced by the other treatments, and its value was estimated in � 2:44 kg for W1, � 2:44 þ 0:94 ¼ 3:38 kg for W2, and � 2:44 þ �ð Þ¼ 0:27 2:17 kg for W3. The inflection point, i.e., the day the accumulated ET rate was maximum, also seems to be influenced by the treatments. W<sup>1</sup> and W<sup>2</sup> were not statistically different for the parameter ϕ3<sup>i</sup>, but W3, with estimative of � 14 days, appears to be statistically different from W<sup>1</sup> (� 17 days).

The first graph presented in Figure 5 brings the accumulated ET mean in each day of the four plants in each treatment. The solid lines are fitting for the treatments

References

Media; 2012

2018;36(4):791-801

Media; 2000

[1] Novk V. Evapotranspiration in the Soil-Plant-Atmosphere System. New York: Springer Science & Business

Modeling Accumulated Evapotranspiration Over Time DOI: http://dx.doi.org/10.5772/intechopen.86913

[9] Brown H, Prescott R. Applied Mixed

Models in Medicine. New York: John Wiley & Sons; 2014

[10] Demidenko E. Mixed Models: Theory and Applications with R. New York: John Wiley & Sons; 2013

[11] Fitzmaurice G, Davidian M, Verbeke G, Molenberghs G.

CRC Press; 2008

Longitudinal Data Analysis. London:

[12] West BT, Welch KB, Galecki AT. Linear Mixed Models: A Practical Guide Using Statistical Software. New York: Chapman and Hall/CRC; 2014

[13] R Core Team. R: A language and environment for statistical computing.

[14] Pinheiro J, Bates D, DebRoy S, Sarkar D, and R Core Team. Nlme: Linear and nonlinear mixed effects models. R Package Version 3. CRAN

In: R Foundation for Statistical Computing. Vienna, Austria; 2018

Repository. 2017:1-131

[2] Pereira OCN, Pereira PVC, Bertonha A, Previdelli ITS. Longitudinal data analysis of stevia rebaudiana

evapotranspiration according to water levels. Revista Brasileira de Biometria.

[3] Pereira OCN, Suguiura TPDS, Pereira AP, Bertonha A, Previdelli I. Analysis of lettuce evapotranspiration across soil water. Natural Resource Modeling. 2019;32(2):e12197

[4] Pinheiro JC, Bates DM. Mixed-Effects Models in S and S-PLUS. New York: Springer Science & Business

[5] Brouwer R, De Wit C. A Simulation Model of Plant Growth with Special Attention to Root Growth and its Consequences. In: Proceedings of the 15th Easter School in Agricultural Science. University of Nottingham. Technical Report. IBS; 1968. pp. 224-242

[6] Hansen LD, Hopkin MS, Rank DR, Anekonda TS, Breidenbach RW, Criddle RS. The relation between plant growth and respiration: A thermodynamic model. Planta. 1994;194(1):77-85

[7] Paine CT, Marthews TR, Vogt DR, Purves D, Rees M, Hector A, et al. How to nonlinear plant growth models and calculate growth rates: An update for ecologists. Methods in Ecology and Evolution. 2012;3(2):245-256

[8] Yan H-P, Kang MZ, De Reffye P, Dingkuhn M. A dynamic, architectural plant model simulating resourcedependent growth. Annals of Botany.

2004;93(5):591-602

79

Figure 5. Response profile of the accumulated ET by the lettuce plants over 23 consecutive days for the water soil levels W1, W2, and W3.

made by Eq. (2). The other graphs present all values observed for the four plants in each treatment, and the solid lines indicate the individual model for each plant.
