Models for Scatter and Intensity of the Roots’ Colonisation

III.2.1 A Motivational Reference Example

We consider below a real example arising from a study on the effects of different liming and phosphorous fertilisation techniques in a field experiment (see Christensen 2017 and Christensen et al. 2021). This example will be used to expose the modelling approach studied in this article. In this study, an experimental field cultivated with spring barley was split into three blocks containing four plots; in each block, four fertilisation treatments were randomly allocated to the plots. In each plot, two minirhizotron tubes were installed. Three soil depth zones were considered in the analyses below: the superficial layer (termed horizon A), the intermediate layer (called horizon B), and the subsoil (termed horizon C). The minirhizotron tubes had six observation windows in the superficial layer and twelve windows in the other two layers. The observation windows of all the 24 tubes were examined in three time points corresponding to different development stages of the culture of spring barley (see details in Christensen 2017). For simplicity of the exposition, we ignore the block

and plot structure of the experiment.

The primary interest in the study referred above was to characterise the develop- ment of the root system in each soil depth zone when different fertilisation treatments are used. Here we approach a different question of characterising how the depen- dence between the intensity andscatter vary over the three observed development stages of the culture in the field. This problem involves studying the dependence of quantities of different stochastic nature. Indeed, while we will characterise the intensity using the counts of number of times the roots cross the reference lines in the observation windows, the scatter will be characterised examining the incidence of roots in observation windows.

The strategy we will adopt to analyse this example is to construct suitable

multivariate generalised linear mixed models describing the rootingintensitiesand the scatters at the three developmental stages (so the model will be six-dimensional). The one-dimensional generalised linear mixed models describing these two characteristics of the rooting system were first described in Labouriau (2019) and are presented in detail in Section III.2. The idea we will explore is that the fixed effects of the models will adjust for the expected differences due to the treatments and the soil depth zones.

Each of these models will contain a Gaussian random component taking the same value for each observation arising from the same minirhizotron tube (here we interpret the tubes as the experimental units). Those random components represent the local variation of theintensityor thescatter present at each experimental unit after having corrected for the effects of the depth zones and the fertilisation treatments. The multivariate model we will consider will allow us to represent different covariance structure of the six Gaussian random components corresponding to the six observed responses. The covariance structure of the random components will determine the covariance structure of the responses, as we discuss below.

III.2.2 Modelling the Scatter

We model the scatter at a fixed development stage by studying the occurrence of roots in the different observation windows of the minirhizotron tubes. Denote by Y_tkz^[d] the random variable representing the number of windows where a root is present in the z^th soil depth zone (z = A, B, C representing the soil horizons) at the k^th tube (k = 1, . . . ,6) exposed to the t^th treatment (t = 1, . . . ,4), observed at thed^th development stage (d= 1,2,3). We keep the development stage fixed in this section and in Section III.2.3. Moreover, following the same convention for the sub-indices used above, denote the number of observation windows at the tkz^th observation by n_tkz. Note that by design, the number of observation windows does not change in the different observation times.

Denote, fort= 1, . . . ,4 andk = 1, . . . ,6, byU_tk^[d]an unobservable random variable taking the same value for all observations arising from the tk^th tube. We assume that those random variables, corresponding to the 24 tubes used in the experi- ment, are independent and normally distributed with expectation zero and variance σ_U[d]² . According to the model in discussion, the random variables Y_11A^[d], . . . , Y_46C^[d] , representing the observations, are conditionally independent given the random com- ponents U₁₁^[d], . . . , U₄₆^[d]. Moreover, we assume that, for t= 1, . . . ,4, k = 1, . . . ,6 and z = A, B, C, the random variable Y_tkz^d is conditionally binomial distributed given U_tk^[d], withY_tkz^[d]|U_tk^[d]=u∼Bi(n_tkz, p^[d]_tkz), where

logit(p^[d]_tkz) =β_tz,[d]+u, for all u∈R. (III.1) The model described above coincides with a generalised linear mixed model (GLMM) defined with the binomial distribution, the logistic link function, a fixed effect repre- senting the interaction of treatment and soil depth zone, and a random component representing the tubes.

The parameter β_tz,[d] in (III.1) is clearly related to the scatter. Indeed, according to the model above, the probability of finding a root which is visible in an observation

window at thez^th soil depth of the plots that received thet^th treatment (z =A, B, C and t= 1, . . . ,4) at the d^th development stage is

E

Y_tkz^[d]

n_tkz

=

Z

R

exp(β_tz,[d]+u)

1 + exp(β_tz,[d]+u)φ(u; 0, σ_U[d]² )du^def=α_tz(β_tz,[d], σ_U[d]² )^def= ˜α^[d]_tz . (III.2) Here φ(·; 0, σ_U[d]² ) denotes the density of a normal distribution with expectation 0 and variance σ_U[d]² , which is the distribution of the random component U_tk^[d]. The quantity ˜α^[d]_tz can easily be evaluated once we have estimated the parameters β_tz,[d]

and σ²_U[d] by numerically integrating the integral in (III.2) or using a straightforward Monte Carlo integration.

III.2.3 Modelling the Intensity

Let C_tkz^[d] be a random variable representing the total number of times the roots cross the reference lines in all the observation windows at the z^th soil depth zones (z = A, B, C) at the k^th tube (k = 1, . . . ,6) exposed to the t^th treatment (t = 1, . . . ,4), observed at the d^th development stage (d= 1,2,3, fixed along this section).

Denote, fort= 1, . . . ,4 andk = 1, . . . ,6, byV_tk^[d]an unobservable random variable taking the same value for all observations arising from the same tube. Those random variables are assumed to be independent and normally distributed with expectation zero and variance σ²_V[d]. The random components defined above are analogous to the random components used for modelling the scatter. According to the model, the random variables C_11A^[d] , . . . , C_46C^[d] , representing the observations of numbers of crosses, are conditionally independent given the random componentsV_{1 1}^[d], . . . , V_{4 6}^[d]. Moreover, we assume that, fort = 1, . . . ,4, k = 1, . . . ,6 and z =A, B, C, the random variable C_tkz^[d] is conditionally Poisson distributed given V_tk^[d], with conditional expectation given by

log(E[C_tkz^[d]|V_tk^[d] =v]) = θ_tz,[d]+v + log (n_tkz) for all v ∈R. (III.3) The model above allows us to estimate the local length of the roots visible in the observation windows, characterising in this way the root intensity, as described below. Exponentiating both sides of (III.3) and taking expectations with respect to the distribution of the random components yields for t= 1, . . . ,4,k = 1, . . . ,6 and z =A, B, C, that

E

C_tkz^[d]

n_tkz

=

Z

R

exp(θ_tz,[d]) exp(v)φ(v; 0, σ_V²_[d])dv= exp(θ_tz,[d]) exp(σ_V²_[d]/2)^def=ω^[d]_tkz. (III.4) The factor exp(σ_V²_[d]/2) in the right side of (III.4) is the expectation of the corre- sponding log-normal distribution (see Aitchison & Brown 1957).

The quantity ω^[d]_tkz defined in (III.4) is straightforwardly related to the intensity since the more intense the root colonisation process in a region around the tube is,

the more likely will be the occurrence of roots crossing the reference lines in the observation windows. Additionally, ω^[d]_tkz can be interpreted as an estimate of the length of the roots that are visible in an observation window using the argument sketched below. A classic argument for the Buffon’s needle problem allows one to calculate the length of a rigid straight needle by randomly throwing the needle in a surface with parallel reference lines (see Klain & Rota 1997), the length of the needle being proportional to the probability of the needle cross a line. The Buffon’s needle problem can be extended by dropping the assumption that the needle has a perfect straight form, yielding the so called Buffon’s noodle problem. According to Ramaley (1969) the length of a one dimensional structure (the "noodle" replacing the needle) is proportional to the mean of the number of times the structure crosses the reference lines. Taking this approach, the left size of (III.4) is interpreted as the expected value of the Buffo’s noodle estimate of the length of the roots that are visible in the observation windows.

The model described above coincides with a generalised linear mixed model (GLMM) defined with the Poisson distribution, the logarithm link function, a fixed effect representing the interaction of treatment and soil depth zone, an offset repre- senting the logarithm of the number of observation windows and a random component representing the tubes.

III.3 Multivariate Simultaneous Models for the