This study was conducted to determine the inner quality characteristics of
eggs using external egg quality characteristics. The variables were selected in
order to obtain the simplest model using ridge, LASSO and elastic net
regularization methods. For this purpose, measurements of the internal and external characteristics of 117 Japanese quail eggs
were made. Internal quality characteristics were egg yolk
weight and albumen weight; external quality characteristics were egg width,
egg length, egg weight, shape index and shell weight. An ordinary
least square method was applied to the data. Ridge, LASSO and elastic net
regularization methods were performed to remove the multicollinearity of the data.
The regression estimating equations of the internal egg
quality were significant for all methods (

The egg production industry has significant economic value as well as being a remarkable source of employment. Consequently, it has an important place in the development of countries' economies and in meeting the nutritional needs of people worldwide. Determination of egg quality is a requirement for both edible eggs and for the production of hatching eggs. Egg quality is examined in two parts in this study, with focus on both internal and external quality characteristics. Previous research has pointed out that egg weight, shell weight, shell thickness, egg yolk weight, albumen weight, the albumen index, the egg yolk index and the Haugh units are all significant factors affecting egg quality (Uluocak et al., 1995; Khurshid, 2003; Alkan et al., 2010). These egg characteristics are highly correlated and are used for the determination of the relationship between internal and external quality of eggs (Khurshid et al., 2003; Kul and Şeker, 2004; Abanikannda et al., 2007; Üçkardeş et al., 2012).

In multiple linear regression analysis based on the ordinary least squares (OLS) method, this high correlation between independent or predictor variables can lead to the issue of multicollinearity (MC) (Montgomery et al., 2001; Şahinler, 2000). It has been reported that this MC problem causes a reduction in the reliability of estimates, as it expands the standard errors of the regression coefficients (Montgomery et al., 2001, Albayrak, 2005; Yakubu, 2010). As a result of this, although the OLS estimates are still unbiased in the model with the MC issue, it is not clear how the various egg weight measurements are affected by the egg components.

Various methods to overcome the MC problem are discussed in the literature.
One of the methods used in such cases is ridge regression (Hoerl and
Kennard, 1970), which is a regularization method that has been used by a number of researchers (Topal
et al., 2010; Üçkardeş et al., 2012; Shafey et al., 2014; Orhan
et al., 2016). Another regularization method is the least absolute shrinkage and
selection operator, “LASSO” (Tibshirani, 1996). LASSO is a successful
continuous procedure for estimating and selecting variables (Tibshirani,
1996; Efron et al., 2004; Hastie et al., 2007). This method has been successfully
used by Kominakis et al. (2009), Ogutu et al. (2012), Acharjee et al. (2013)
and Amin et al. (2014). However, LASSO has two important limitations which
emerge in cases where the number of variables is too large for the
number of observations

Descriptive statistics of egg quality characteristics.

Min: minimum value; Max: maximum value; SE: standard error; CV: coefficient of variation; EYWT: egg yolk weight; EAWT: egg albumen weight; EWI: egg width; ELE: egg length, EWT: egg weight; SI: shape index; and ESWT: egg shell weight.

Correlation coefficients between internal and external quail egg quality characteristics and between variance inflation factors and tolerance values.

Eigenvalues and conditional index values of external egg quality characteristics predicting EYWT and EAWT.

E: eigenvalue and CI: conditional index.

Therefore, the aims of this study were to determine egg yolk weight and albumen weight from external egg quality characteristics using the ridge, LASSO and EN regression models and to select the variables in order to reduce model complexity.

The materials utilized in this study were 117 eggs taken from Japanese
quails; the eggs were obtained from the Van Yuzuncu Yil University Research and
Application Farm. Egg weight (EWT), egg yolk weight (EYWT), egg albumen
weight (EAWT) and shell weight (ESWT) (in grams) and egg width (EWI) and egg
length (ELE) (in mm) were the variables measured, with the eggs collected daily.
Shape index (SI) is a value that depends on EWI and ELE; SI was calculated
using the following equation: SI

For the multiple linear regression model with as many independent variables
as

Ridge, a biased prediction method, is based on the principle of minimizing
the sum of the residual squares (RSS) in order to obtain the

In this method, it is possible to obtain

Elastic net is an extension of the LASSO method that is robust to extreme correlations
among the predictors (Friedman et al., 2010). The method uses a mixture of
the ridge (

The statistical analyses were performed using the GLMSELECT procedure in SAS/STAT (SAS, 2014).

The descriptive statistics of the egg quality characteristics are shown in Table 1. EYWT, EAWT, EWI, ELE, EWT, SI and ESWT averaged 3.74 g, 6.20 g, 25.38 mm, 32.15 mm, 11.39 g, 79.03 % and 1.46 g, respectively.

The estimation of coefficients obtained using the OLS, ridge, LASSO and EN methods in the multiple linear regression analyses (standard errors in parentheses) for EYWT and EAWT.

Goodness of fit measurements of OLS, ridge, LASSO and EN methods in multiple linear regression analyses.

The Pearson correlation coefficient between internal and external quality
characteristics of quail eggs and MC diagnostics, variance inflation factors
(VIFs) and tolerance values (TVs) are given in Table 2. Eigenvalues and
conditional index (CI) values, the other criteria used to determine MC,
are presented in Table 3. The respective correlations between EWI and EWT and EWI and SI were 0.371 and 0.806 (

The prediction equations of the internal quality characteristics obtained
using the OLS,
ridge, LASSO and EN methods in the multiple linear regression analyses are
given in Table 4. For all of the methods, the prediction equations are found
significant (

The goodness of fit measurements of the prediction equations for the OLS, ridge, LASSO and EN methods and the number of predictors in the prediction are presented in Table 5. There are five predictor variables in OLS and ridge and two in LASSO and EN both for EYWT and EAWT.

Table 5 shows that the

When the data used in the study were evaluated in terms of basic statistics,
EYWT, EAWT, EWI, ELE, EWT and SI were found to be similar to the findings of Kul and
Şeker (2004) (Table 1). However, the mean value of ESWT was 1.46

The results of the correlation analyses showed that high and significant
correlations were obtained between the predictor variables: the correlation between EWI and SI was 0.806 (

In order to investigate the MC problem, the VIFs and TVs in Table 2,
the eigenvalues and CI values in Table 3 were calculated using the OLS
method. This was undertaken because it is known that the correlation between the predictor variables is not
sufficient to define the MC issue (Albayrak, 2005; Shafey et al.,
2014). The OLS results showed that VIF values were greater than 10 in 3
variables: 872.7, 416.4 and 1197.2 for EWI, ELE and SI, respectively. The
TVs values were found to be small, depending on the VIFs due to the
relationship between the two. The high VIF values were caused by the small
tolerance value, as reported by Albayrak (2005). The eigenvalues were very
close to zero (down to 6.18

The aims of this study were to determine the internal quality characteristics
of eggs and to choose variables using the external quality
characteristics of eggs. As previous studies have proven that OLS estimates are
less reliable if the data has an MC problem (Hoerl and Kennard, 1970;
Montgomery et al., 2001; Albayrak, 2005; Yakubu, 2010), ridge regression was applied to the data to eliminate the MC issue (Table 4). The results of the regression
analyses for both EYWT and EAWT were found to be significant (

The goodness of fit statistics used in order to find the best models are only given for OLS and the regularization methods (Table 5). Since the number of
parameters in the prediction equations obtained by the regularization methods
were different from one another,

The determination of internal egg quality characteristics is important in terms of edible eggs and the production of hatching eggs. In this study the ridge, LASSO and EN regularization methods were used in order to perform prediction equations and variable selection for both EYWT and EAWT. It was revealed that LASSO, including two predictors in the prediction equation, was the best model with regard to high predictive accuracy. It was concluded that ELE and EWT were included in the prediction equation for EYWT, while EWT and ESWT were included for EAWT.

Regularization methods are superior to OLS in data with a MC problem because, when these methods are used, more accurate and reliable prediction equations are obtained. In this study we introduced the LASSO and EN methods for prediction and variable selection in agricultural research. It is concluded that LASSO and EN techniques may be utilized to develop the best and most stable models for internal egg quality characteristic prediction using external egg quality characteristics because they overcome the MC problem. These techniques also enable the selection of sufficient variables in order to obtain models that are easily interpreted by researchers.

A total of 117 Japanese layer quails (Coturnix coturnix
japonica) being raised on the Van Yuzuncu Yil University Research and Application Farm
were used in the study. All quails were fed on a basal diet that
contained 2679 kcal ME kg

The authors declare that they have no conflict of interest.

This study based on the first author's master's thesis (Çiftsüren, 2017) and was financially supported by the Van Yuzuncu Yil University Scientific Research Projects Directorate (project no. FYL-2016-5034). Edited by: Manfred Mielenz Reviewed by: Nazire Mikail and one anonymous referee