Aquiles Enrique Darghan Contreras

Carlos Armando Rivera Moreno

Nair Jose González Sotomayor

Jose Luis Castellanos Coronel

Rev. Fac. Agron. (LUZ). 2022, 39(1): e223918

ISSN 2477-9407

DOI: https://doi.org/10.47280/RevFacAgron(LUZ).v39.n1.18

Crop Production

Associate editor: Dr. Jorge Vilchez-Perozo

Keywords:

Border effect

Competition coefcient

Reparameterization,

Monte Carlo Simulation

Estimation and computational evaluation of the coefcient of intraspecic competition in

edges in the context of linear models

Estimación y evaluación computacional del coeciente de competencia intraespecíca en bordes en

contexto de modelos lineales

Estimativa e avaliação computacional do coeciente de competição intraespecíca em arestas no

contexto de modelos lineares

Department of Agronomy, Universidad Nacional de

Colombia, Bogotá, Colombia. Postal code: 111321

Received: 24-09-2021

Accepted: 13-11-2021

Published:

20-02-2022

Abstract

In experimental trials, it is usually of interest to give special regard to

the response of experimental units at the edges, since it is well known that

the performance of these can be greater than that of the rest of the units

due to having less competition from neighboring units. When treatments

are available, it is possible that the differences in the mean crop response

are attributable to the edge effect. Therefore, it is important to consider the

edge or not in the modeling process. In this case, using the Kempton-Besag

model and the reparameterization of the model, the intraspecic competition

coefcient was estimated through least quadratic estimation that in this case

was associated with the edge effect. Its distributional pattern was studied

using Monte Carlo simulation. Simulated variance analyses were carried out

to see the distributional effect of the F-statistic in the presence of the edge

effect as a form of spatial dependence that was evaluated with the Moran

index. The coefcient associated with the edge effect showed a clear normal

distribution in all the considered edge scenarios. The sign of the coefcient

and the condence intervals generated made it possible to discriminate the

presence/absence of edge effect. In addition, a method was proposed to allow

a user to mitigate the fuzziness that may result from the point estimate of the

coefcient. This procedure can be used in other neighborhood patterns and

other design models of importance in agricultural research.

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Resumen

En los ensayos experimentales suele ser de interés prestar especial

atención a la respuesta de las unidades experimentales en los bordes,

ya que es bien sabido que el rendimiento de estas puede ser superior

al del resto de unidades por tener menor competencia de unidades

vecinas. En un experimento, donde se tienen tratamientos, es posible

que las diferencias en la respuesta media de las variables medidas

sean atribuibles al efecto de borde. Por lo tanto, es importante

considerar el borde o no en el proceso de modelado. En este caso,

utilizando el modelo de Kempton-Besag y la reparametrización

del modelo, se estimó el coeciente de competencia intraespecíca

mediante estimación mínima cuadrática que en este caso se asoció

con el efecto de borde. Su patrón de distribución se estudió mediante

simulación Monte Carlo.De cada análisis de varianza con datos

simulados se extrajo el valor del estadístico F y se representó su

comportamiento distribucional para relacionarlo con la ausencia

o presencia del efecto del borde como una forma de dependencia

espacial, la cual se conrmó desde el punto de vista estadístico con

el índice de Moran. El coeciente asociado con el efecto de borde

mostró una clara distribución normal en todos los escenarios de borde

considerados. El signo del coeciente y los intervalos de conanza

generados permitieron discriminar la presencia/ausencia de efecto de

borde. Además, se propuso un método para permitir al usuario mitigar

la falta de claridad que puede resultar de la estimación puntual del

coeciente. Este procedimiento se puede utilizar en otros patrones de

vecindad y otros modelos de diseño de importancia en la investigación

agrícola.

Palabras clave: efecto borde, coeciente de competición,

reparametrización,simulación Monte Carlo.

Resumo

Em ensaios experimentais, geralmente é de interesse dar atenção

especial à resposta das unidades experimentais nas bordas, uma vez

que é bem conhecido que o desempenho destas pode ser maior do

que o do resto das unidades devido à menor competição de unidades

vizinhas. Quando os tratamentos estão disponíveis, é possível

que as diferenças na resposta média da cultura sejam atribuídas ao

efeito de borda. Portanto, é importante considerar a borda ou não

no processo de modelagem. Nesse caso, utilizando o modelo de

Kempton-Besag e a reparametrização do modelo, o coeciente de

competição intraespecíco foi estimado por meio da estimativa do

mínimo quadrático que, neste caso, foi associada ao efeito de borda.

Seu padrão de distribuição foi estudado usando simulação de Monte

Carlo. Análises de variância simuladas foram realizadas para vericar

o efeito distributivo da estatística F na presença do efeito de borda

como uma forma de dependência espacial que foi avaliada com o índice

de Moran. O coeciente associado ao efeito de borda apresentou uma

distribuição normal clara em todos os cenários de borda considerados.

O sinal do coeciente e os intervalos de conança gerados permitiram

discriminar a presença / ausência do efeito de borda. Além disso, foi

proposto um método para permitir ao usuário mitigar a imprecisão

que pode resultar da estimativa pontual do coeciente. Este

procedimento pode ser usado em outros padrões de vizinhança

e outros modelos de projeto importantes na pesquisa agrícola.

Palabras chave: efeito de borda, coeciente de competição,

reparameterização,Simulação de Monte Carlo

Introduction

It has been 100 years since the publications of Arny (1922),

describing the edge effect in agricultural experiments and where

proposing ways to avoid it. Even the term competition was used

to associate it with the performance of off-edge row performance

with the rows at the edge, recognizing since then that removing the

rows from the edge could in certain situations be an unnecessary

operation, something that recent studies endorse (Romani et al.,

1993; Kuemmel, 2003). However, others recommend the elimination

of edges depending on the crop, with the purpose of making a better

estimation of the yield (Gałęzewski et al., 2013). According to Keddy

(2001). The denition presents a challenge since it is an example of

a generalized phenomenon that can occur in quite diverse conditions.

Current research focuses on an autoregressive competition model

(Ord, 1975) used by several authors to consider such an effect and for

which various proposals are put forth (Connolly et al., 1993). But this

time, rather than estimating the parameters of the model associated

with the effects considered, computational analysis was done on the

least-squared estimator of the Kempton-Besag competition model.

These generated scenarios associated with situations described for

performance when there is no edge effect and when it is perceived not

only at the most exposed edge but also at the two outermost edges of

the batch, as well as in partial edge situations (Besag and Kempton,

1986; Shukla and Subrahmanyam, 1999). With the calculation of the

Moran index, the spatial dependence of the residuals of the model was

studied. Computationally the effect obviating the spatial dependence

of the residuals on the analysis of variance for the two-way model

was shown. Finally, with the simulated construction of the condence

interval for the competition estimator, the presence/ absence of

competition could be inferred. This is something that many research

usually describe in their trials (Phillips et al., 2020).

Materials and methods

This section considers a two-way classication model including

its reparametrization from the perpendicular projection operators

approach using the estimator obtained through least-squared

estimation of the coefcient of competition associated with the edge

effect while considering a series of closed-edge scenarios as partial

edges and off-edge. This suggests a greater response in units located

in any edge modality.

Model Structure and estimation approach

The Besag-Kempton autoregressive model based on the argument

that the random response of neighboring units affects or competes is

written as:

Y = Xβ + κWY + ε, (1)

where Y is a random vector of length and denotes the response of

the n units, X is a known design matrix of dimension n x p, β is the

vector of unknown parameters of length p and consists of the effects

of the model (treatments and blocks), κ is the unknown competition

coefcient that was associated with the edge effect, W is the matrix

of weights of dimension n x n that in this case were associated with

the inverses of the Euclidean distances between the units with the

assumption that the non zero weights of the neighbors on a particular

unit yield a Markov matrix, and the effect of unit on itself is zero,

yielding a square matrix of the Hollow type (In our case, the weights

matrix was standardized). Finally, independence and identical normal

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3-6 |

distribution with zero mean and variance σ

I was assumed for the

vector of errors, I being an identity matrix of dimension n x n. The

design matrix was associated with a two-way model, to which lateral

conditions were imposed. The model is written as follows:

(2)





=  + 



+ 



+ 



 = 1,2, … ,  ;  = 1,2, … , ,

with τ

as the effect of the i-th treatment and τ

as the effect of the

j-th block that in matrix form is written as Y=Xβ+ε . In model (2)

there is a deciency in the range of magnitude p-q=2, with p=t+b+1

and q=t+b-1. To cover this deciency, the usual non-estimable

functions in this design model were used, that is, and

. Under the hypotheses H

:k=0 of the model in (1) we have the linear

model Y=Xβ+ε and solving the problem of the range of X with the

imposition of lateral conditions we proceeded to the estimation of the

parameters. For model (1), the logarithm of the likelihood is given by.



(

, 

, 

)

= −



(

2

)

−



(



)

+ ln

 − 

−

‖

(

−

)

−

‖

2

(3)

with |I-κW| like the Jacobian transformation.

One use of the lateral conditions is the reparameterization of

the model, where the Besag- Kempton incomplete-range model is

transformed into a full-range model. This simplies the estimation

procedure and allows obtaining unique estimators (Chistensen, 2011).

The method on model (1) is described below:

Let Y = Xβ +κWY + ε be the model and Lβ = 0 the matrix

expression associated with the lateral conditions, where L is a matrix of

dimension l×p, which is linked to the two-way model l=p-q. Lβ is a set

of non-estimable functions. If Lβ = 0, then β'L' = 0, so that β' belongs

to the orthogonal complement onto column space of L' (β' ϵC(L' )). To

identify the reparametrized model it is necessary to select a matrix Z

(Supplemental Material) such that it has the same C(L'); and, in this

way, the vector β belongs to C(Z), that is, β = Zγ, for some vector

γ. By substituting β = Zγ in model (1) the reparameterized model is

obtained:

Y = XZγ +κWY + ε , (4)

and doing the reparameterized model of Besag-Kempton is

nally written as:

Y = Pγ +κWY + ε , (5)

From model (5) we can obtain the logarithm function of the

likelihood as well as the estimators for γ and σ

for a restricted model

under H

. However, as the estimator for κ in this estimation mode did

not present a closed solution (6), the least square estimation was used.

(6)

where M = P(P'P)

-1

P' is the perpendicular projection matrix

on C(P) and trz represents the trace. In the case of the least-square

estimation, for which no distributional assumption is required, we

seek to minimize ε'ε, that is,

(7)

Note that the predicted value estimates E(y

) and not y

, however,

it is usually written as . To nd the estimator for from (7) we

differentiated with respect to and set it equal to zero and obtained

the closed solution for that was used to detect the edge effect of

agronomic trials and computationally studied its distributional pattern

based on simulation Monte Carlo. The estimator turned out to be:

(8)

Once the design matrix X, the matrix Z for the reparameterization

of the model, and the neighborhood pattern W were dened, the

functions were created in the RStudio software to perform the Monte

Carlo simulations of all the scenarios. This involved randomizing the

treatments so that the response vector Y associated with performance

did not show the edge effect. Higher response measures were

generated in the treatments that resulted in the complete edge, and

a double row of higher response was generated in the two outer and

complete edges. Unilateral, bilateral, and trilateral partial borders

with edge effect were generated. For the blocking reason, the reader

can abstract to some situation that can be linked with simplicity and

that causes some randomization restriction to create the blocks, such

as the origin of the seed.

Finally, whatever the estimation methodology is used in the

estimation of parameters, for example, the lateral conditions, the

current interest is related to the competition coefcient associated

with the edge effect in this case, so there is no It is of interest to

search for information on the edge effect with different proposals for

estimating the competition coefcient, since it is unprecedented as an

alternative to study the edge effect.

Scope of the methodology

The usefulness of the statistical-computational approach is subject

to a previous exploration of the data set to detect the possible presence

of the edge effect, since the simple obtaining of the statistic is not

directly associated to the absence or presence of the edge effect but

rather to the presence or absence of the competition effect such as the

nature of the model used for the construction of the statistic.

Motivating dataset

Figure 1 illustrates the arrangement of the treatments and blocks

in the study area. The label associated with the treatment, block,

and repetition appears. The number above each label represents the

response as it was recorded in the eld and the number to the right

represents the position in the response vector due to randomization.

In the case of the simulated responses, they were generated by Monte

Carlo method to obtain all the scenarios described above. For all

the scenarios where the response was generated it was guaranteed

that the treatment means were not different. Therefore, once the

analysis of variance (aov) of the two-way model without competition

or edge effect was run (standard linear model in 2), no differences

in the treatment means were seen. Therefore, the distribution of

the F-statistic of the 3000 simulations followed the characteristic

F-distribution under the null hypothesis and thus could detect any

departure from this distribution not because of the differences in

means but because of the spatial dependence induced by the edge

effect.

Figure 1. Distribution of blocks and treatments on the

experimental units simulated.

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The following gures show different patterns of the different

scenarios developed that are associated with edge effect in

agricultural experiments. The rst group of patterns (Fig. 2)

distributes the response (value) causing 1) absent edge effect (left),

2) complete outer edge around the entire convex contour (center)

and 3) two outer edges for the two successive convex contours

(right).

Figure 2. Distribution of responses in the three main border

patterns.

Figure 3 shows all possible cases to illustrate the unilateral

edge effect, for which a higher yield was associated with the more

accentuated gray points.

Figure 3. Distribution of responses on one-sided edges.

Figure 4 incorporates the successive or separate bilateral border

structures with six possible cases. It would be expected that if the

statistics detects the edge effect, these structures show a clearer

effect than those of the previous gure.

Figure 4. Distribution of responses on the bilateral border with

the highest yields

Finally, the trilateral border patterns that are the most similar

structures to the usual simple border in agricultural trials, thus, a

behavior in the statistic is expected to be much more like that

obtained in the presence of a simple edge (Fig. 5).

Figure 5. Distribution of responses at the trilateral borders.

Mathematical classication and construction of simulated

density

Let Ψ the space with points associated with

the combination of treatments and blocks in an experimental design

and let S

(i=1,2) the i-th subset of Ψ containing

the points identifying the outer single edge or outer double edge in

a two-dimensional spatial array. The indicator function S

denoted

by I

(.), is the function with domain Ψ and codomain equal to the

set consisting of two real numbers 0 and 1 dened by if

∈

or =0 if

∉

i ;

(.) clearly indicates S

With the previous description, the two situations are

mathematically formalized and will be illustrated when a user

wants to know if the data of his experiment show the edge effect.

The selection of S

depends on the distribution of the units in

the eld and as in our case it was a staggered planting with a

rectangular lot with a dened number of rows and columns, the

points with the minimum common x-coordinate and maximum

common x-coordinate as well as the minimum and maximum

common y-coordinate were identied. For the single edge effect

case if

∈

indicates the combination of blocks

and treatments on this, the outermost edge. For the case of two

outer edges, for the purposes of the algorithm (see supplementary

material) all elements of S

in Ψ, i.e., we obtain Ψ-S

=Ψ

(Ψ\S

) and

again identify the points with the minimum common x-coordinate

and maximum common x-coordinate as well as the minimum and

maximum common y-coordinate but inΨ

, being Ψ

the selected

points on this inner edge. To count the two outer edges we do

, in this way, if Ψ

∈

Once the single and double external edges have been indicated,

we proceed to propose a computational method that the user can

use to have clarity of the edge effect. It should be remembered that

an experimental test only generates a measure of the coefcient

that we have associated with the edge effect, so it would not be

simple to clearly discriminate the presence of the edge effect. To

avoid this problem, we use RStudio’ssimulate() function (R Core

Team, 2021) to generate a simulated density for (found by the

researcher) using RStudio’s lm() function for the formula: response

~ S

and do the desired simulations to construct the density of the

outermost edge. The element vector incontaining zeros and ones

of length equal to the evaluated response is simulated over a user-

dened number of simulations, in our case 500 simulations were

used and then we plotted the density, and we can notice the pattern

that certainly tells us if it is contained to zero or not to decide if

we have the effect associated with the outer edge. If the formula

response ~ S

+ S

is used and the linear model is tted, we now

have the possibility to obtain the density for the outermost edges

and again the edge effect considered will be clear (supplementary

material).

Results and discussion

When the usual aov is used in linear modeling the technique

uses spatial dependence to modify the usual inferential procedures

(Gotway and Cressie 1990). In fact, the distributional assumption

based on the F-curve may not be valid. By maintaining the null

hypothesis of equality of means of the treatments, it is possible

by simulation to see the effect in the F-distribution. Figure 6-left

shows the distribution that is expected for two degrees of freedom

associated with the treatments and 92 degrees of freedom for the

error in the two-way model. Figures 6-center and 6-right show the

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Darghan et al. Rev. Fac. Agron. (LUZ). 2022, 39(1): e223916

5-6 |

departure from this distribution when there is a clear edge effect

generating spatial dependence. Some might confuse the result of

these two nal gures if they do not realize that the samples did not

provide statistical evidence against the null hypothesis of equality

of means in the simulations generated. The gure also shows the

results of the Kolmogorov-Smirnov test for the adjustment for the

distribution and its D statistic. Only in the case of no edge (Fig.

6-left) the distribution is precisely the F.

Figure 6. Distribution of the F statistic generated in the 3000

Monte Carlo simulations.

In Figure 7 the distribution of the p values of the Moran index

of the cases associated with gure 2 (no border, single border,

and double border) is illustrated. As can be seen in 7 (left), most

of the simulations did not provide evidence against the hypothesis

of spatial independence of the residuals of the analysis of variance

model, with which the results of the technique are applicable if

the assumptions are met. In Figures 7-center and 7-right most

simulations show spatial dependence, especially in the case with

two edges. The vertical lines illustratively demarcate the cut for

α=5% and the simulation count is also printed where the hypothesis

of spatial independence is rejected. Undoubtedly, the presence of an

edge is associated with a modality of spatial dependence, so results

of an aov to the standard linear model may not be valid, and hence

the importance of taking some remedial measure in the presence

of an edge effect. This is done in the study of the speed of the root

front of corn and soybeans (Ordóñez et al., 2018).Although an edge

effect could not always be present in our research as was the case of

pasture and gramineous, some authors have concluded that both the

vegetation and soil properties are inuenced by the proximity to the

edge (Ruwanza, 2018) .

Figure 7. Distribution of the p value of the Moran index

generated in the 3000 simulations.

From (5), the coefcient associated with the edge effect in

model (1) was obtained by the least-squared estimation for all

scenarios, showing a clear normal distribution by simulation, that

was corroborated with the Shapiro-Wilks Normality test (p-values

within each image in gure 8). With the mean for in the distribution

of the simulations, a 95% condence interval was constructed for

, specically for the cases without edge effect, single and double

edge effect, and it is here where one of the most important results.

Clearly the rst interval (Fig. 8-left) contains zero, with which we

assert the absence of edge effect, while Figures 8-center and 8-right

not only do not contain zero, but also obtain negative lower and

upper limits, that is, the negative coefcient associated with the

edge effect shows the effect of interest.

Figure 8. Distribution of the coefcient associated with the

competition model or edges.

It is reasonable to think that even with many simulations for the

rst three cases it is possible to detect partial edges effect with the

same statistic. The cases of unilateral edge effect (Fig. 9) were run,

nding again the detection of unilateral partial edge but with a mean

for a normal distribution closer to zero on the left, especially at the

edges of shorter length (lower and upper). So, a greater distance

from zero on the left shows a more dened edge effect.

Figure 9. Distribution of the coefcient associated with the one-

sided edges model.

In the cases of more dened but partial edges effect such as

bilateral ones, the distribution of the statistic is maintained but its

means are shifted further to the left. This denes the partial edge

more clearly, especially when the edges are those with the greatest

perimeter (Fig. 10). This result, which is becoming systematic, is

validated with the trilateral cases of Figure 11. When the edges

involve the widest perimeters, the coefcient associated with the

most negative edge is obtained.

Figure 10. Distribution of the coefcient associated with the

bilateral edges model.

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Figure 11. Distribution of the coefcient associated with the

trilateral edges effect.

Application example

Using the same design structure and creating a single external

edge condition in principle the researcher can obtain the

estimator. As this value is not easy to use to discriminate the edge

effect, we adjusted the models lm(response ~ S

) and lm(response

~ S

+ S

) and using simulate() the simulations were generated.

The plots of the respective densities and the simultaneous display

of the estimated value allow us to conclude the presence of a

single edge and not a double edge, thus, the computational strategy

allows for edge detection and simplies the analytical operations as

described in (Paolella, 2019) to nd the distribution of the quotient

of quadratic forms associated with the proposed estimator.

Figure 12. Simulated density plot for application example the

line indicate the experimental kappa.

Conclusions

Current research uses the Besag-Kempton model and the

least-quadratic estimation of parameters. From this model was

possible to obtain the estimator for the competition coefcient that

was associated with the edge effect. The results made it possible

to visualize the effect on the distribution of the F statistic of the

analysis of variance in the presence of spatial dependence evaluated

using the Moran index, evident in the case of the synthetically

generated edges effect.

Once a series of Monte Carlo simulations had been made for

a normally distributed response, the distributional pattern of

statistic associated with the edge effect was studied. In all cases, the

normal distribution was evident in the scenarios of single, double,

no border, partial borders, and the contrasting pattern of opposite

borders effect. We observed that in the absence of the edge effect,

this interval contained zero, while in the case of the presence of

partial unilateral, bilateral, and trilateral borders.

The use of the sets of points associated with the edges of interest

and the linear modeling allowed the construction of a simulated

density for , which eliminated the ambiguity that could arise when

making a point estimate for , so that a user of the method can easily

judge the presence or absence of edge effect.

Supplemental material

Z and L matrices are available. Repository on Github of R code:

The code for all the simulations in: CarlosRivera1212/BorderEffect

(github.com)(edge effect coefcients)

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