© The Authors, 2022, Published by the Universidad del Zulia
*Corresponding author: gdgonzalezs@unal.edu.co
Nelson Bernal-Margfoy
Enrique Darghan
Luis Ernesto Rodríguez
German Gonzalez
*
Rev. Fac. Agron. (LUZ). 2022, 39(4): e223949
ISSN 2477-9407
DOI: https://doi.org/10.47280/RevFacAgron(LUZ).v39.n4.04
Crop Production
Associate editor: Dr. Jorge Vilchez-Perozo
University of Zulia, Faculty of Agronomy
Bolivarian Republic of Venezuela
Keywords:
Zero-inated model
Size tubers
Negative binomial regression
Evaluation size and number of yellow potato tubers under different planting densities using
zero-inated models
Evaluación del tamaño y número de tubérculos de papa amarilla bajo diferentes densidades de
siembra usando modelos cero inados
Avaliação do tamanho e número de tubérculos de batata amarela sob diferentes densidades de plantio
usando modelos inados a zero
Facultad de Ciencias Agrarias, Universidad Nacional
de Colombia, sede Bogotá, www.unal.edu.co, Bogotá,
Colombia.
Received: 09-08-2022
Accepted: 27-09-2022
Published: 25-10-2022
Abstract
A eld study was carried out on the cultivation of the yellow diploid
potato (Solanum tuberosum Phureja Group) to evaluate the inuence of the
planting density associated with distances between plants of 30, 40, and 50
cm and distances between the rows and paths of 100 cm on the tuber count
with sizes less than 2 cm, 2-4 cm, 4-6 cm, and more than 6 cm. At the time
of the harvest of the tubers, they were classied by size and respective count
was made. The modelling of the counts was done by means of the usual
negative binomial regression and by the inated zeros option. The zero-
inated negative binomial regression models showed a signicant effect of
the sowing density on the tuber count in the sizes that were superior to 4 cm
while the negative binomial model showed a signicant effect it in the case
of the sizes lower than 4 cm. The results on size and density are attributes of
interest in both the agronomic management of this crop and in the industrial
management of the tubers, so the relationship that we found can be adopted
in both areas to generate the desired attributes of the crop for improving the
production and guiding the process of industrialization.
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2-5 |
Resumen
Se realizó un estudio de campo en el cultivo de papa diploide
amarilla
(Solanum tuberosum Grupo Phureja) para evaluar la
inuencia de la densidad de siembra asociada a distancias entre plantas
de 30, 40 y 50 cm y distancias entre hileras y caminos de 100 cm en el
número de tubérculos con tamaños menores de 2 cm, 2-4 cm, 4-6 cm
y mayores de 6 cm. Al momento de la cosecha de los tubérculos, se
clasicaron por tamaño y se realizó el conteo respectivo. El modelado
de los conteos se realizó mediante la habitual regresión binomial
negativa y mediante la opción de ceros inados. Los modelos de
regresión binomial negativa con ceros inados mostraron un efecto
signicativo de la densidad de siembra sobre el conteo de tubérculos
en las tallas superiores a 4 cm mientras que el modelo binomial
negativo mostró un efecto signicativo en el caso de las a 4 cm. Los
resultados sobre tamaño y densidad son atributos de interés tanto en
el manejo agronómico de este cultivo como en el manejo industrial de
los tubérculos, por lo que la relación en contrada puede ser adoptada
en ambas áreas para generar los atributos deseados del cultivo para
mejorar la producción y orientar el proceso de industrialización.
Palabras clave: modelo inflado con ceros, tamaño de tubérculos,
regresión binomial negativa.
Resumo
Foi realizado um estudo de campo no cultivo da batata diplóide
amarela (Solanum tuberosum Grupo Phureja) para avaliar a
inuência da densidade de plantio associada a distâncias entre
plantas de 30, 40 e 50 cm e distâncias entre linhas e caminhos de
100 cm na contagem de tubérculos com tamanhos inferiores a 2 cm,
2-4 cm, 4-6 cm e superiores a 6 cm. No momento da colheita dos
tubérculos, os mesmos foram classicados por tamanho e foifeita
a respectiva contagem. A modelagem das contagens foi feita por
meio da usual regressão binomial negativa e pela opção de zeros
inados. Os modelos de regressão binomial negativa inados a zero
mostraram efeito signicativo da densidade de semeadura sobre a
contagem de tubérculos nos tamanhos superiores a 4 cm enquanto
o modelo binomial negativo mostrou efeito signicativo no caso dos
tamanhos inferiores a 4 cm. Os resultados de tamanho e densidade
são atributos de interesse tanto no manejo agronômico desta cultura
quanto no manejo industrial dos tubérculos, de modo que a relação
que encontramos pode ser adotada em ambas as áreas para gerar os
atributos desejados da cultura para melhorar o produção e orientando
o processo de industrialização.
Palavras-chave: modelo inflado de zero, tamanho de tubérculos,
regressão binomial negativa.
Introduction
The Colombian yellow diploid potato is an important genetic
resource, consisting primarily of diploid genotypes with short-day
adaptation. These potatoes are of economic importance in Andean
countries, especially since they are cultivated by small farmers.
Colombia is the largest producer, exporter and consumer of the yellow
diploid potatoes (Barragán, 2019). The production of yellow potatoes
is classified in different categories according to the diameter of the
tuber, for example, diameters less than 2 cm, 2-4 cm and diameters
greater than 4 cm (Tabares et al. 2009).
There are various investigations relating the size and quality of
yellow potato tubers to planting density. In the Bogotá savannah
in Colombia, under these agro-climatic conditions, high planting
densities (0.7 m between rows and 0.2 m between plants) favour the
increase of small tubers (Arias et al. 1996). The same authors relate
the stem density with the number of tubers produced in different
planting densities and conclude that the stem density increases with
an increase in the planting density. A higher planting density is related
to a higher yield per unit area, but lower yield per plant, showing
consistency between the planting densities tested for all trials in a
study in subtropical China (He et al. 1998).
According to Rodríguez et al. (2003), it is important to determine
the effects of the density components (represented mainly by the size
of the tuber-seed, the spacing, the number of stems per plant and, in
general, the spatial arrangement) on the morphology, physiology and
the agronomic performance of the cultivar in different environmental
conditions.
Appropriate modelling can yield contrasting results with those
analyses where the data has been intervened to nd the assumptions
for the application of a specic technique, so that the choice of an
appropriate methodology could yield different results from those
where the initial distribution has been modied; and, therefore,
the absence of statistical signicance due to planting density can
not only be attributed to the factors involved but to the model used
(Wasserstein and Lazar, 2016)
In some agronomic research it is possible to nd count data
(enumerated units) with additional zeros from those that would be
expected in those situations. This is common in the modelling of
counts with excess zeros or inated by zeros (Hilbe 2014).
In this research, the modelling of the number of tubers per size
(four categories) is proposed using regression models by number,
specically the zero-inated negative binomial model using three
planting densities as the only factor in two repetitions per density and
with sufcient experimental units to avoid problems in estimating the
over-dispersion parameters due to the sample size.
Materials and methods
This research was carried out at the Marengo Agricultural Centre
of the Universidad Nacional de Colombia, Mosquera municipality,
Cundinamarca department (74°12’58.51 W; 4°40’52.92 N), at an
altitude of 2516 meters above sea level. The mean temperature is 14
°C in a range from 12 °C to 18 °C and mean rainfall of 500 mm to
1000 mm. The soils are moderately deep and well drained, and the
water table is less than 0.5 m below the surface with a 15% moisture
content. According to the characteristics of precipitation, temperature
and evapotranspiration, the area is classied as low montane dry
forest.
Yellow diploid potato seeds (Solanum tuberosum Phureja
Group) cv. Criolla Colombia are sown one per planting site, four
centimetres in diameter, without rotting or skin defects. This cultivar
is yellow, has a medium plant size, with slightly light green foliage
and abundant owering. Tuber production shows a size distribution
of diameters between 1 and 10 cm. The tubers are harvested at 120
days and counted according to their diameter in the categories 2, 2-4,
4-6 and > 6 cm. To study the effect of planting density, the plants
were planted at 30, 40 and 50 cm between plants, all with 100 cm
separation between rows (gure 1).
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Bernal-Margfoy et al. Rev. Fac. Agron. (LUZ). 2022, 39(4): e2239493-5 |
Figure 1. Harvested tubers of different sizes.
Sowing was carried out in rows precisely aligned according to
the planting density, using three successive rows according to the
geometry of the lot for each density with two repetitions per density
for a total of 18 rows and 2841 plants. The experimental unit was the
plant (tubers), sown on different plots within the same lot (blocks),
recording the data for each plant. Under these conditions, the design
turned out to be a simple factorial in a generalized randomized block
(complete) design, taking the distances between plants as the factor
levels associated with the plant density.
The statistical analysis initially involved the descriptive
component, generating cross tables and plots for the number of tubers
by density and size. For the inferential component various models
of the generalized linear type were adjusted, nally presenting
those with the best descriptors of the adjustment and fullment of
assumptions. From the exploratory analysis, it was already known
that the negative binomial distribution seemed to generate the best
t, so the following regression models used the negative binomial
distribution for each size, using the planting density as a factor. The
analysis was carried out in the R software and the package used was
Hilbe JM (2014). COUNT: Functions, Data and Code for Count Data.
R package version 1.3.4.
Results and discussion
The size of the tubers seems to be regulated by various
mechanisms and Struik et al. (1990) consider the planting density,
the number of stems per plant, and the number of tubers per stem
as important variables in the manipulation and modelling of crop
yield. Cotes et al. (2000) recommend for this cultivar a size of less
than 2 cm but without the effect of the planting density. However,
they recommend a distance between plants of 30 cm with 130 cm
between rows. The study of the size of the tubers plays an important
role in achieving a presentation and conservation of the tubers that
is acceptable to consumers and industry. In the case of potatoes for
freezing or precooking, low sizes are desirable; and in the case of
fried tubers or strips, sizes greater than 4 cm are desirable. In another
study, the decrease in planting density increased the number of tubers
per seedling for all sizes. In fact, a reduction from approximately 146
to 25 plants.m
2
doubled the number of usable tubers (Van der Veeken
and Lommen 2009).
Considering these ideas, the results shown below are associated
with modelling the number of yellow potato tubers for three different
planting densities and for four sizes of tubers. Table 1 represents
the number of tubers by size and the total number for each planting
density, the highest number is in the distances between plants of 30
cm and 40 cm. However, the percentages of tubers conditioned by
density are similar in the three densities, which at rst glance suggests
that there are no important differences in the number of tubers
obtained by density.
Table 1. Distribution of the number of tubers by density and size
class marks.
Density Sizes (cm) Sub-total
≤ 2 (2 - 4] (4 - 6] > 6
d1:(30*100cm) 8258 8337 4152 159 20906
d2:(40*100cm) 8088 8538 4254 215 21095
d3:(50*100cm) 7291 7064 4406 220 18981
Sub-total 23637 23939 12812 594
Total (60982)
In gure 2 the behaviour of the number of tubers is illustrated
discriminated by density, to show the excesses of zeros present in the
two largest sizes for all the planting densities considered. The sizes
were labelledS1, S3, S5, and S9 and only indicate the midpoint of the
sizes used.
Figure 2. Comparison of observed and predicted values by size
and density.
Several studies include the two most used distributions for this
type of data, namely the Poisson and the negative binomial or its
zero-inated option (Henne, 2012). For this reason, below, the results
obtained by some test models and the models nally proposed are
illustrated to relate the number of tubers with relation to the planting
density using the negative binomial regression model, which, through
the graphs previously shown seemed to be adequate in the case of
modelling the sizes for each density. Modelling was done for sizes,
since the presence of excessive zeros in the largest sizes was evident,
therefore, from now on, the results of the modelling with negative
binomial regression and zero inated negative binomial are presented.
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4-5 |
Table 4. Zero Inated negative binomial(log link) for density for
the sizes 4-6 cm.
Coefcients Estimation
Standard
Error
Z Value Probability
Intercept 1.445 0.026 56.43
2.00e
-16
Density (40) 0.163 0.036 4.51
6.50e
-6
Density (50) 0.349 0.037 9.53
2.00e
-16
Theta 1.180 0.062 19.14
2.00e
-16
Table 5. Zero Inated negative binomial (log link) in the sizes of
more than 6 cm.
Coefcients Estimation
Standard
Error
Z Value Probability
Intercept -0.870 0.266 -3.26
0.0011
Density (40) 0.792 0.219 3.61
0.0003
Density (50) 0.641 0.219 2.93
0.0034
Log(theta) 0.217 0.515 0.42
0.6740
With all these results, it only remains to graphically show
the results of the adjustment of the rst two sizes in the negative
binomial models and the two upper sizes in the negative binomial
models inated by zeroes. Figure 3 corroborates the adjustment
obtained in the Poisson, negative binomial (Neg Bin) and negative
binomial adjustment modalities inated by zeros (Zinb). In all cases
observed for each size and density (Observed 1: sizes ≤ 2 cm (A);
Observed 3: sizes (2-4) cm (B); Observed 5: sizes (4-6) cm (C) and
Observed 9: size > 6 cm (D) the models in the negative binomial
distribution were superior; however, the inated option by zeros
was the most convenient in the two larger sizes.
Figure 3. Comparison of observed and predicted values by size
and density.
In the negative binomial model, the variance is not restricted to
being equal to the mean and facilitating the modelling like the data
we are evaluating, where the variance is a little more than the triple
the average. The negative binomial allowed the manipulation of
excesses of zeros as they are found in the larger diameters or sizes,
so this distribution is quite justied for modelling the numbers of
tubers by density. The data collected showed an excess of zeros
especially in the two upper sizes, so it was necessary to contrast
this by means of a test, the negative binomial models with their
counterpart that accepts excess zeros. Next, the results obtained
using the R software (pscl package) for the construction of the
outputs associated with the estimation and adjustment process of
the zero-inated negative binomial models are described using
density as a factor and modelling the response by size.
Tables 2 and 3 show the results of the analysis, and these two
cases are shown initially since they are the ones that did not have
an evident excess of zeros in the number of tubers associated with
the zero value, in this case the usual negative binomial model was
used. In both cases the signicant effect of planting density on size
is evident, as found in various studies that evaluate planting density
as a factor that presumably affects yield. However, we cannot assert
anything with respect to the similarity with other research, since
this is not usually the form of modelling found in the literature for
counts.
Table 2. Negative binomial model (log link) by density at size ≤
(2 cm).
Coefcient Estimation
Standard
Error
Z-value Probability
Intercept 1.985 0.019 103.87 2.00e
-16
Density (40) 1.183 0.028 6.51 7.50e
-11
Density (50) 0.253 0.029 8.64 2.00e
-16
Table 3. Negative binomial model (log link) by density in size
(2-4) cm.
Coefcient Estimation
Standard
Error
Z-value Probability
Intercept 1.994 0.020 101.36 2.00e
-16
Density (40) 0.227 0.029 7.88 3.20e
-15
Density (50) 0.212 0.030 7.00 2.70e
-12
The results of tables 4 and 5 show again the signicant effect
of the terms associated with the planting density on the number
of tubers. In both tables the negative binomial was used with
the logarithmic link function. The two parts of an inated model
represent a binary model, generally a logit model for which of the
two processes the zero result is associated and a count model (in
this case, a negative binomial model). In both cases the statistical
relationship between the density of seeding and the count of potato
tubers was evident.
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Once the modelling process has concluded and the adequacy
of the modelling is clearly appreciated respecting the nature of the
variables, it can be statistically asserted that the planting density
factor explains the numbers for all the sizes evaluated, corroborated
by different authors like Escobar and Zaag (1988), where an increase
in the sowing density from 40,000 to 100,000 plants per hectare
increased the yield by 50%; but smaller tubers were generated,
interpreted as an effect of the density on the size of the tubers.
The results of table 6 suggest the need to use low planting
densities (fewer plants per hectare) if the market requires larger
tubers. However, this is accompanied by low yields that will most
likely be offset by higher costs when selecting tubers by size. Note
that average tubers per plant is reduced by almost 130% in the two
extreme densities. Additionally, in the rst density for every 27 tubers
of the two largest sizes approximately 104 of the smaller sizes are
generated (almost 4:1), whereas in the lowest density the ratio is
approximately 3:1 ((33+32).(20+1))
-1
.
Table 6. Distribution of fresh weight (t.ha
-1
), mean of tubers
per plant and ratios of generated tuber by size in each
density.
Density
Fresh weight
(t.ha
-1
)
Tubers.plant
-1
Ratio
d1:(30cm*100cm) 10.77 18.4 52:52:26:1
d2:(40cm*100cm) 8.20 11.4 38:40:20:1
d3:(50cm*100cm) 5.37 8.1 33:32:20:1
Tuberization in potatoes involves the differentiation of stolon
in young tubers (initiation) and the collection of young tubers (Dutt
et al. 2017). Competition for resources at high densities can affect
tuberization by reducing the number of starting tubers (Mackerron et
al. 1988). In addition, these resourced-related stresses (for example
water) can reduce tuber lling with assimilated tubers in the plant’s
growth phase (Lahlou et al. 2003). In both cases, the result in a
reduction in tuber yield.
Marketable tuber yield depends on the average tuber size, that is,
both the total tuber weight and the total number of tubers. Therefore,
cultivars that produce fewer tubers in drought-prone areas are
recommended. If you have a smaller number of tubers, it is more
likely that they are larger when the photo-assimilated are limited
during drought, thus increasing their average size (Aliche et al. 2019)
The negative binomial distribution or zero-inated negative
binomial model can provide information on the marketable proportion
of tuber yields. However, not much research has been conducted
towards understanding the underlying reason for the model
parameters that describe total and marketable tuber size distribution,
although it seems to be associated with the number and size of tubers
under quantitative inheritance (Celis-Gamboa, 2002). Despite this,
the relationship between the density of seeding and the count of
tubers by size was evident and can be used to direct the production in
favour of generating the sizes required by the market.
Conclusions
In sizes less than 4 cm adjusting negative binomial models without
excess zeros found that the terms associated with the planting density
were more appropriate to show the statistical relationship between the
density of the seedlings and the number of tubers. Similarly, in sizes
greater than 4 cm adjusting negative binomial models with excess
zeros showed the terms associated with the sowing density were the
ones with the best statistical adjustment. So, it can be statistically
asserted that sowing density inuences the number of tubers in larger
sizes.
Larger tuber sizes were associated with lower planting density,
but this was associated with lower yields, suggesting that there is a
yield penalty in the interest of improving tuber sizes.
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