EFFECT OF DIFFERENT
BIOCHEMICAL TRAITS ON SEED COTTON YIELD: AN APPLICATION OF LIU LINEAR
REGRESSION
M. Qasim1,4, M. Amin2 and M. K. S. Sarwar3
1Department of Statistics
and Computer Science, University of Veterinary and Animal Sciences, Lahore, Pakistan.
4Department of
Economics, Finance and Statistics, Jönköping University, Jönköping, Sweden.
2Department of Statistics,
University of Sargodha, Pakistan.
3National Institute
for Biotechnology & Genetic Engineering, Faisalabad, Pakistan.
3Cotton research station,
Ayub agricultural research institute, Faisalabad, Pakistan.
Corresponding Author’s Email: muhammad.qasim@uvas.edu.pk;
qasim.stat@gmail.com
ABSTRACT
This article aimed
to study the associations among the biochemical traits and their effects on seed
cotton yield using the regression analysis and to assess the alternative approach
for reducing the impact of multicollinearity problem in estimating the regression
coefficients. The field experiment was conducted where five explanatory variables
(chlorophyll ‘a’, chlorophyll ‘b’, total chlorophyll, total soluble
protein and total soluble sugar) and one dependent variable seed cotton yield were measured.
The correlation matrix of showed that biochemical traits were
significantly correlated. The multicollinearity problem among the biochemical traits
was determined by condition index and correlation matrix. Using the least square
regression analysis, the effects of biochemical traits on seed cotton yield were
not satisfactory since least square regression model has high value of MSE (3352475), AIC (366.7) and
inconsistent estimates of traits. The Liu regression analysis was efficient (MSE
= 57212 and AIC = 363.8) and reliable in reducing the adverse
effects of multicollinearity. The Liu regression results indicated that total
chlorophyll and total soluble protein were contributed a significant (p-value <
0.05) role in seed cotton yield. In contrast, ordinary least square regression analysis
was showed insignificant (p-value > 0.05) effect of total chlorophyll on seed
cotton yield.
Key words: Seed Cotton Yield,
Biochemical Traits; Multicollinearity; Least Squares Regression Analysis; Liu Linear
Regression.
https://doi.org/10.36899/JAPS.2020.6.0174
Published online August 03,2020
INTRODUCTION
Cotton
yield is important for textile industry, oil production and has a significant contribution
to the economy of cotton-growing areas. Cotton production stood at 11.935 million
bales on an area of 2,699 thousand hectares with an increase of 8.4% with a portion
of 1% in the GDP. It paid 5.50 % in agriculture value addition (Anonymous, 2017-18)
which appeals researchers to keep focus for its stability and improvement. Cotton
attains the potential in yield because of its thin genetic base (Rahman et
al., 2008). Abiotic and biotic stresses caused fluctuation in cotton production
drastically during the last decades. Cotton is the primary source of natural fibers
in the world which impacts on the textile industry and is the leading agriculture
economic indicator of Pakistan’s economy (Amin et al., 2015; Anonymous, 2017-18). Cotton production was negatively influenced
by water stress. Water
stress increases total soluble sugars accumulation in different crops such as rice,
wheat and
Soybean. Scarcity of water severely affects cotton cultivation in arid and semi-arid
regions. Development of high yielding and drought-tolerant cotton varieties is necessary
to fulfil the demand of the ever-growing population of the world. Physiological
and biochemical traits are being used to identify the drought-tolerant cotton germplasm
and to understand the mechanism and genetic variability. Drought stress has decreased
the chlorophyll ‘a’, ‘b’ total chlorophyll, a/b ratio and carotenoids. The photosynthetic
processes with photosynthetic pigments, e.g., chlorophyll "a" and chlorophyll "b" have a major share
in increases crop growth and yield (Taiz and Zeiger, 2006). Chlorophyll "a",
chlorophyll "b",
total chlorophyll and "a/b"
ratio
drastically affected by drought and decreased in concentration (Hamayun et
al., 2010; Sarwar
et al ., 2012, Abid et al., 2016). Yuan et al., (2013) studied
the association between biochemical traits (flavonoid, fructose, glucose, sucrose
and total sugar contents) and fiber quality attributes (length, strength, uniformity,
fitness, elongation rate and cellulose 50 DPA) using heterosis breeding in colored
cotton through correlation coefficient (r) and they found a significant association
between two attributes. Abid et al., (2016) were studied the association
of different biochemical traits and enzymatic antioxidants with Bacillus thuringiensis
(Bt) under normal and drought conditions. Malik et al., (2018) were analyzed
fiber quality attributes (independent variables), quality-related biochemical traits
and measured the boll weight (dependent variable) for colored cotton. Coefficient
of correlation results revealed that there was a significant linear relationship
among fiber quality attributes (Malik et al., 2018).
The
study of the linear relationship among biochemical traits is crucial for the improvement
of seed cotton yield (SCY) and identification tolerant genotypes. However, SCY is
a quantitative character, which is mostly affected by biochemical traits chlorophyll
contents (a, b and total chlorophyll), total soluble protein, total soluble sugar. In order to select
for higher SCY, there is a need to analyze the mathematical relationship between
SCY and biochemical traits. Measuring the direct effects of traits on a SCY requires
the estimation of multiple linear regression model (MLRM) and measuring can be negatively affected
by the multicollinearity problem due to the linear relationship between traits.
In the presence of multicollinearity, the statistical inference may be erroneous
or unreliable. To solve the issue of multicollinearity alternative estimation method
can be used. Bizeti et al., (2004) suggested the ridge regression approach
for estimating the path coefficients of soybean traits on grain yield per plant.
Wang et al., (2011) studied the effect of seed yield components on the seed
yield using path and ridge regression methods. Toebe and Filho (2013) analyzed the
effect of multicollinearity by using path analysis for the maize crop. The literature
shows that no information is available on the mathematical relationship by using
biased estimation methods for measuring the impact of biochemical traits on SCY
in the presence of multicollinearity. Therefore, the aims of the current study were:
i) Biochemical traits are linearly correlated to each other and traits are positively
related to SCY; and ii) compare the traditional MLRM and alternative biased estimation
methods (Liu linear regression) of minimizing the adverse effects of multicollinearity
for estimating the effect of biochemical traits on SCY.
MATERIALS AND METHODS
Experiment: The experimental material
consisted of 32 upland cotton accessions collected from different research institutes
located in different ecological regions of Pakistan. Evaluation of accession was
performed under two irrigation regimes in the field at the research area of National
Institute for Biotechnology and Genetic Engineering (NIBGE), Faisalabad, Pakistan.
SCY (kg ha-1) was hand-picked from all the plots at 180 days after planting.
To address the impact of biochemical traits on cotton crop productivity
and its contribution towards yield, we experimented on chlorophyll contents (a,
b and total chlorophyll), total soluble protein, total soluble sugar under two water
regimes well water (one
irrigation at planting and five subsequent irrigations) and limited water (one irrigation at planting and one supplemental
40 days after planting) at
NIBGE and
the SCY considered as a dependent. Chlorophyll contents (a, b and total) were measured
by following Arnon (1949) and calculated according to Davies (1976). Concentrations
of total sugars and total soluble proteins were determined according to Riazi et
al. (1985) and Lowry et al. (1951), respectively. The main aim of this
experiment was to check the effects of biochemical traits on SCY.
Liu Linear Regression:
Regression
analysis was used to estimates the conditional expectation of the dependent variable
given the independent variables (or “explanatory variables”). It considered a valuable
technique in the agriculture field. The unknown regression coefficients of the MLRM
were estimated through the ordinary least squares (OLS) method. Consider the following
form of the multiple LRM:
, (1)
where
represents the estimated SCY and biochemical
traits represent as explanatory variables, i.e., chlorophyll"a", chlorophyll"b", total chlorophyll, total soluble protein and total soluble sugar. In the matrix
form, Equation (1) can be written as
, (2)
where
is an vector of observations on the response
variable, is an fixed design matrix on the explanatory
variables of full rank, is a column vector of unknown regression
coefficients, where p represents the number of explanatory variables and
is vector of random errors which have
distributed to normal with and where is identity matrix.
The
OLS estimator (OLSE) of the unknown parametric vector is defined as
(3)
where
matrix. It is a common assumption
in the LRM that the explanatory variables are not correlated with each other. Although,
in routine, there may be strong or near to strong linear relationship has been found
among the explanatory variables which lead to the problem of multicollinearity.
The problem of multicollinearity may also be called collinearity (the linear relationship
between two explanatory variables) and ill-conditioning. However, in literature,
many studies showed that the OLSE is no longer efficient in the presence of multicollinearity.
Since the OLSE heavily depends on the cross-product of the matrix . If the matrix is ill-conditioned , the performance of the OLSE does not
satisfactory, for instance, the coefficients may be insignificant with a wrong sign
and have large variance, and statistical inference becomes problematic (Kibria, 2003; Qasim
et al., 2018; Qasim et al., 2019b). In the presence of multicollinearity
problem, it is complicated to make the valid statistical inference. Numerous biased
estimation methods are available to overcome the problem of multicollinearity and
Liu estimator (Liu, 1993) is one of them. Liu estimator was defined by Liu (1993)
, (4)
where
d denotes the Liu shrinkage parameter, and it takes the values between zero
and one. When then and in the case which implies that . However, estimator result in biased for a specific
value of d, it exhibits minimum mean squared error (MSE) than the OLSE. Qasim
et al. (2019a) proposed some shrinkage estimators to estimate the value of
d. We were used the following best estimator that suggested by Qasim et
al. (2019a):
,
where
is jth value of and is the matrix whose columns are the
eigenvectors of such that , where , . Since is the jth
eigenvalues of the matrix and is known as the estimated residual
variance. For more knowledge regarding Liu estimator in the LRM (see, e.g., Liu,
1993, Qasim et al. 2019a). The is on average too long in the presence
of multicollinearity. Thus, is considered to be the best choice
instead of in that condition.
This study also makes a comparison between
the OLS and Liu
estimator,
where the MSE, Akaike information criterion (AIC) and standard error (SE) of the
regression coefficients were considered as performance criteria. We also demonstrate
the benefits of the Liu estimator in the LRM by observing the effect of biochemical
traits on SCY experiment. So
the main aim of the experiment to analyze whether chlorophyll
levels ‘a’ and ‘b’, total chlorophyll, total soluble protein
and total
soluble sugar improve the SCY or not. We used liureg{} statistical package
in R for regression analysis.
RESULTS
First,
we computed the correlation matrix between the biochemical traits and SCY given
in Table 1 and Fig. 1. From Table 1, we observed that biochemical traits were positively
correlated with the SCY. This indicated that all biochemical traits increasing the
SCY. Correlation among traits was significant except the
correlation between chlorophyll ‘b’ and total soluble protein (r = 0.3177), and the correlation coefficients
of total soluble sugar with other commercial traits (r = 0.0888, r = 0.2814, r =
0.2385 and r = 0.2199) were non-significant (Table 1). Next, we fitted the
MLRM using traditional OLSE, which were defined in Equation (3), and the results
were reported in Table 2. We observed that the only biochemical trait, i.e. total
soluble protein (p-value = 0.0166) contributing a significant role in SCY. While other biochemical
traits were showed statistically insignificant role in SCY. In practice, we were
considering that the biochemical traits have a positive impact on SCY and showed
a significant role in increasing the yield of seed cotton. The computed results
of the model given in Equation (1) were presented in Tables 2. From Table 2, we
observed that the chlorophyll ‘a’ has a negative impact on SCY, and this result
clearly shows the drawback of the OLSE due to the presence of multicollinearity
problem. We also scattered the residual scatter plot to check the other necessary
assumption of the regression model. For the identification of multicollinearity,
the correlation
matrix and condition index (CI) were considered. From Table 1, we perceived that
the biochemical traits experiment has a serious multicollinearity problem.
Fig. 1. Matrix scatter plot of the variables
Fig 2. Residual plots for seed cotton yield
model.
Table
1. Correlations matrix among study variables.
|
SCY
|
Chlorophyll ‘a’
|
Chlorophyll ‘b’
|
Total Chlorophyll
|
Total Soluble Protein
|
Total Soluble Sugar
|
SCY
|
1.0000
|
0.5341[a]
|
0.5576[a]
|
0.6437[a]
|
0.5862[a]
|
0.3567[a]
|
Chlorophyll
a
|
|
1.0000
|
0.4723[a]
|
0.8501[a]
|
0.4470[a]
|
0.0888
|
Chlorophyll
b
|
|
|
1.0000
|
0.8588[a]
|
0.3177
|
0.2814
|
Total
Chlorophyll
|
|
|
|
1.0000
|
0.4503[a]
|
0.2385
|
Total
Soluble protein
|
|
|
|
|
1.0000
|
0.2199
|
Total
Soluble sugar
|
|
|
|
|
|
1.0000
|
Note:
[a]Significant
at 0.01 level of significance
Table
2. Ordinary least-squares linear regression model summary.
Variables
|
Estimates
|
Standard Errors
|
t-statistic
|
p-value
|
Chlorophyll
"a"
|
-29.53
|
1088.87
|
-0.027
|
0.9784
|
Chlorophyll
"b"
|
47.30
|
1025.48
|
0.046
|
0.9632
|
Total
Chlorophyll
|
308.96
|
1054.60
|
0.293
|
0.7695
|
Total
Soluble Protein
|
115.94
|
48.39
|
2.396
|
0.0166
|
Total
Soluble Sugar
|
31.41
|
26.37
|
1.191
|
0.2337
|
F-Statistic = 6.696 (p-value < 0.05);
AIC = 366.7 and MSE = 3352475
|
Note:
AIC = “Akaike Information Criterion” and MSE = “Mean Square Error”
Table
3. Liu linear regression model summary.
Variables
|
Estimates
|
Standard Errors
|
t-statistic
|
p-value
|
Chlorophyll
"a"
|
73.25
|
104.42
|
0.701
|
0.48301
|
Chlorophyll
"b"
|
120.30
|
104.62
|
1.150
|
0.25022
|
Total
Chlorophyll
|
198.17
|
67.48
|
2.937
|
0.00332[a]
|
Total
Soluble Protein
|
117.22
|
45.08
|
2.600
|
0.00932[a]
|
Total
Soluble Sugar
|
33.32
|
24.55
|
1.357
|
0.17476
|
F-statistic = 6.953 (p-value < 0.05);AIC = 363.8 and MSE = 57212
|
Note:
[a]Significant
at 0.01 level of significance
The problem
of multicollinearity can also be tested by Condition Index (CI) as
which showed the existence
of multicollinearity in the experimental dataset. Therefore, we used a Liu estimator
in the MLRM instead of traditional OLSE for estimating the biochemical traits.
However, in practice, chlorophyll ‘a’ has a positive impact on SCY, and the expected
results show by Liu estimator in Table 3. As we mentioned earlier, the OLSE may
produce the wrong sign of the regression coefficients, and we overcome this problem
using Liu estimator in LRM. We also observe that the standard errors of the OLSE
are higher than the Liu estimator,
which clearly shows the inconsistency of the OLSE. It is noted that total chlorophyll
and total soluble protein have a significant positive impact on SCY
by applying Liu estimator. While total chlorophyll
and total soluble protein have an insignificant effect on SCY, when we were
used the OLSE. So, the Liu estimator with
the propose shrinkage parameter d provides efficient results as compared
to the OLSE. Besides, we can see that the t-statistic values are smaller by using
OLSE than Liu estimator, which also demonstrates
the advantage of our new method. The performance of the Liu estimator is quite well
as compared to the OLSE in the sense of MSE and AIC; for the OLSE, MSE = 3352475
and AIC = 366.7; for the Liu estimator, MSE = 57212 and AIC = 363.8. One can easily
see that Liu estimator gives minimum values of the MSE and AIC as compared to the
MSE and AIC of the OLS estimator. By observing the above problems, we recommend
that the agriculture practitioners should use Liu
estimator, which performs quite well instead of the traditional
OLSE in the presence of multicollinearity.
DISCUSSION
In
the literature, we observed that biochemical traits contributing a significant role
in the yield of different crops (Ansari et al., 1999; Boggs et al.,
2003; Song et al., 2005). To determine the role of these biochemical traits,
most researchers focused on the regression analysis without incorporating the multicollinearity
problem (Ansari et al., 1999; Boggs et al., 2003; Abid et al.,
2016). We
were observed from the correlation matrix (Table 1) that some biochemical traits
were linearly correlated with each other, which cause the issue of multicollinearity.
For diagnosis of multicollinearity, we used another test known as CI, which also
confirmed the existence of multicollinearity.
Multicollinearity
among biochemical traits may give incorrect and unreliable results since the explanatory
variables are linearly correlated, and the inverse of the matrix will be singular
(Carvalho et al., 1999). In this study, the traditional OLSE demonstrated
wrong inference due to the severe issue of multicollinearity. Therefore, we used
an alternative biased method (Liu estimator). Similar biased estimation techniques
were also used by Carvalho et al., (1999), Bizeti et al., (2004),
Wang et al., (2011) and Toebe et al., (2013) for different crops.
These authors were focused on path analysis for estimating direct and indirect effects.
However, the current study was used MLRM with OLSE and Liu estimator. The OLSE demonstrated
the negative impact of chlorophyll ‘a’ (B = -29.53) (Table 2) while the Liu linear
regression analysis showed highly positive effect (B = 73.25) (Table 4). Similar
wrong sign problem results using biased estimation method were also obtained by
Hoerl and Kennard (1970). Another point which was focusable due to multicollinearity,
total chlorophyll showed now a significant role in the SCY. In contrast, Boggs et
al. (2003) showed that total chlorophyll plays a vital role, and it was confirmed
by Liu linear regression. The effects of total chlorophyll and total soluble protein on
SCY were changed from non-significant (p-value = 0.7695 and p-value = 0.0166) to
highly significant (p-value = 0.0033 and p-value = 0.0093) by using Liu linear regression
(Tables 2 and 3) since Liu estimator showed lowest standard errors of the estimates
of total chlorophyll and total soluble protein. These results confirm that OLSE
is inconsistent, and not reliable in the presence of a high degree of collinearity
and these results correspond with the findings of Carvalho et al., (2001)
and Toebe
et al., (2013). A
regression model gives better estimation if it has minimum MSE, AIC and SEs of the
estimated parameters. As we have seen that the Liu estimator have minimum MSE, AIC
and SEs as compared to the OLSE in the MLRM. The lowest MSE and SEs results are
also found by Liu (1993).
The
MSE, AIC, and SE were larger in the presence of multicollinearity. Based on the
above discussion and results, we suggest that researchers give information about
the degree of multicollinearity in the matrix in research publications that
use regression analysis. The information regarding ill-conditioning matrix
has been discussed for soybean traits (Bizeti et al., 2004), seed yield components
(Wang et al., 2011) and maize crop (Carvalho et al., 2001; Mohammadi
et al., 2003; Toebe et al., 2013). Furthermore, we suggest the use
of MLRM with biased estimation method (Liu estimator) instead of traditional OLSE when agriculture researcher
wants to estimate the effects of different factors, such as biochemical traits,
seed yield components, cellulose contents, fiber quality traits and among others
on the agriculture output with multicollinear factors.
Conclusions: The biochemical traits
were significantly correlated with each other and had a severe effect on seed cotton
yield. The condition index and correlation matrix indicated the problem of multicollinearity
among biochemical traits. Traditional least square regression analysis in the presence
of multicollinearity provided inadequate statistical inference regarding seed cotton
yield. The Liu linear regression was an efficient approach in reducing the adverse
effects of multicollinearity and more adequate than the ordinary least square method.
Thus, the Liu linear regression is a more reliable approach for estimating the actual
effects of biochemical traits on seed cotton yield.
Acknowledgements:
We acknowledge the kind support of Dr. Mehboob-ur-Rahman (PP),
Principal Scientist Plant Genomics group and Molecular Breeding NIBGE, and Dr. M.
Yasin Ashraf (T.I), Deputy Chief Scientist NIAB for their assistance in the experimentation
of the research activities of this research project. The authors are also thankful
to the Editorial team and the anonymous reviewers for their careful reading and
suggestions, which certainly improved the quality of the article.
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