Analysis of Variables Affecting Carcass Weight of White Turkeys by Regression Analysis Based on Factor Analysis Scores and Ridge Regression

In this study, the influence of carcass parts weights (thigh, breast, wing, back weight, gizzard, heart, and feet) on whole carcass weight of white turkeys (Big-6) was analyzed by regression analysis based on ridge regression and factor analysis scores. For this purpose, a total of 30 turkey carcasses of 15 males and 15 females with 17 weeks of age, were used. To determine the carcass weight (CW), thigh weight (TW), breast weight (BRW), wing weight (WW), back weight (BW), gizzard weight (GW), heart weight (HW), and feet weight (FW) were used. In the ridge regression model, since the Variance Inflation Factor (VIF) values of the variables were less than 10, the multicollinearity problem was eliminated. Furthermore, R2=0.988 was obtained in the ridge regression model. Since the eigenvalues of the two variables predicted by factor analysis scores were greater than 1, the model can be explained by two factors. The variance explained by two factors constitutes 88.80% of the total variance. The regression equation was statistically significant (p<0.01). In the regression equation, two factors obtained by using factor analysis scores were independent variables and standardized carcass weight was considered as dependent variable. In the regression model created by factor analysis scores, the Variance Inflation Factor values were 1 and R2=0.966. Both regression models were found to be suitable for predicting carcass weight of turkeys. However, the ridge regression method, which presented higher R2 value, has been shown to better explain the carcass weight.


Analysis of Variables Affecting Carcass Weight of White Turkeys by Regression Analysis Based on Factor Analysis Scores and Ridge Regression
reported significant carcass trait differences in turkeys between 12 and 21 weeks of age.Sarıca & Camci (1993) reported that the carcass yield and breast weight increased from 72% to 79%, and 27% to 35%, respectively, between 12-24 weeks of age.Waldroup et al. (1997) observed increasing live weight and carcass yield as turkeys aged.Ogah (2011) determined average body weight, body length, and breast circumference as 3. 38 and 2.65 kg, 35.05 and 31.86 cm, and 47.38 and 36.62 cm in males and females of 22-wk-old Nigerian indigenous turkeys, respectively.Ramkrishna et al. (2012) studied three different turkey breeds, and determined average body weights as 3570.35 and 2521.89g in 16-wkold male and female turkeys, respectively, as well as average breast, back, thigh, and wing yields of 19.61-19.89%, 11.07-11.11%, 11.35-11.56%, and 11.61-11.55% in males and females, respectively. Shamseldin et al. (2014) showed average slaughter weight (kg) and carcass weight (kg) in 16-wk-old male and female turkey reared in semi-intensive and extensive systems as 7.3-6.5 and 6.6-6.1 kg, respectively.In a study with 17-wk-old Converter turkeys, males presented higher live weight and carcass weight than females (Chodová et al., 2014).Ribarski & Oblakova (2016) evaluated the slaughter and carcass weight of wild turkeys, and also obtained higher values in males than females.Roberson et al. (2003), evaluating British United Turkeys (BUT), Hybrid and Nicholas turkeys, determined that average body weight and carcass yield were 17.12 kg and 75.9% at 18 weeks of age, with no differences among strains.On the other hand, Werner et al. (2008) reported lower carcass weight in slowgrowing compared with fast-growing commercial turkey strains.
The aim of the current study was to investigate the influence of carcass parts (breast, thighs, back, wing, heart, gizzard) weights on the whole carcass weight of White turkeys using Multiple Regression Analysis Techniques obtained with ridge Regression and Factor Analysis Scores.

MATeRIAlS And MeThodS
This study was carried out in poultry facilities of Agriculture Faculty in Bingol University, in Turkey.Recorded average environmental temperature and relative humidity were of 27.5 ºC and 44%, respectively.
A total of 30 white males and females turkeys (Big-6), 15 males and 15 females, were reared in an opensided house on litter until 17 weeks of age together.
Both males and females received the same diet.A three-phase feeding program was applied.The starter (0-8 weeks), grower (9-14 weeks), and finisher (15-17 weeks of age) diets were formulated to meet the birds' nutritional requirements according to the NRC (1994).Feed and water were offered ad libitum.The composition of the diets manufactured in the experimental facilities is presented Table 1.Turkeys were slaughtered at 17 weeks of age, and the following parameters were measured: carcass weight (CW), thigh weight (TW), breast weight (BRW), wing weight (WW), back weight (BW), gizzard weight (GW), heart weight (HW), and feet weight (FW).These measurements were taken out in Facility of Agriculture in Bingol University, in Turkey.
Regression analysis is a statistical technique for research and modeling the relation among variables.These variables are dependent and independent variables (Montgomery et al., 2012).
Multiple linear regression model is as in Equation 1.
Where Y is an (n x 1) column vector of observations belonging to the dependent variable, X is an (nxp) fixed matrix of observations if the variables and is of full rank p (p ≤ n), b is a (px1) unknown column vector of regression coefficients, and e is an n x1 vector of random errors;

Analysis of Variables Affecting Carcass Weight of White Turkeys by Regression Analysis Based on Factor Analysis Scores and Ridge Regression
E(e) = 0; E(e e 1 ) = s 2 I n , where I n denotes the n x n identity matrix and the prime denotes the transpose of a matrix (Draper and Smith, 1998).The ordinary least squares (OLS) estimator, b of the parameters is obtained by equation 2 (Draper & Smith, 1998) e's are independently and identically distributed as normal with mean 0 and variance s 2 (Montgomery et al., 2012).
Ridge regression is a statistical tool used to deal with multicollinearity and to avoid problems related to small sample size and/or a large number of predictor variables (Gruber, 1998;Hastie et al., 2001).Ridge regression is also known as Tikhonov regularization (Tikhonov et al., 1977).
The ridge estimator is shown by (equation 3).
where I denotes an identity matrix and k is a positive number determined as ridge parameter.Alkhamisi & Shukur (2007) proposed new estimators by adding (equation 4) to some well-known estimators to estimate the ridge parameter, where l max is the highest eigenvalue of X'X.
Multiple regression was used to estimate carcass weight from different carcass part measurements.Factor scores derived from factor analysis were used for multiple regression analysis in order to remove multicollinearity problem (Eyduran et al., 2010).Factors with eigenvalues >1 were employed in multiple regression analysis (Tabachnick & Fidell, 2001;Johnson and Wichern, 2002).
Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy and Bartlett's test of sphericity were applied to determine whether the correlation matrix is an identity matrix, indicating if the factor model is unsuitable (Sharma, 1996).In order to simplify the interpretation of factor loading (λ), Varimax rotation was used.Factor coefficients (F) were used to obtain factor scores for selected factors (Eyduran et al., 2010).
The factor analysis equation can be written in matrix form (Equation 5) Where Z is a px1 vector of variables, l is a (pxm) matrix of factor loadings, F is an (mx1) vector of factors, and e is a (px1) vector of error or residual factors (Sharma, 1996).
Score values of selected factors were considered as independent variables for predicting of carcass weight.The regression equation fitted to standardize carcass weight and factor score values are given Equation 6: Where a is the regression constant (the value of the intercept and its value is zero); b 1 , b 2 and b k are the regression coefficients of factor scores (FS).FS is factor score and e is the error term.Regression coefficients were tested by a t-statistics.The quality of the regression was determined by the coefficient of determination (R 2 ) (Draper & Smith, 1998).

ReSulTS
As shown in Table 2, the linear relationship between carcass weight and carcass values was 99.8% in the multiple regression equation obtained by the least squares (LSM) method.The measured weights explain 99.5% of the carcass weight variation.There were no autocorrelation problems in the regression model, where Durbin-Watson statistics were observed as 2.101.The regression model was statistically significant (p<0.001).In this study, the standard error of the constant term was found to be high, according to the least squares coefficient results, as shown in Table 3.When Table 3 was examined, the regression relation obtained as the result of LSM was statistically significant (p<0.01).The standard error of the parameters of the regression model shown in Table 3 was high and the coefficients of some parameters were statistically insignificant.Also, some variables (thigh and wing weights) have multiple connection problems because VIF values were greater than 10.The correlation matrix presented in Table 4 has correlation coefficients higher than 0.90.For these reasons, the ridge regression method was applied to estimate the carcass weight of turkeys.Using the equation ( 5), the value of the k bias constant was approximately k = 0.204.Table 6 shows the ridge regression goodness of fit and standard error values with k = 0.204 bias constant.Using the ridge regression technique and k bias of 0.204, the relationship between carcass weight and carcass parts was 99.4%.Table 4 shows that 98.40% of the variation in carcass weight is explained by the carcass parts.It is shown in Table 5 that the correlation with ridge regression was statistically significant (p<0.01).The predicted parameters of the ridge regression result differ from those obtained with the LSM method.There was a significant decrease in the standard errors and VIF values of the parameters of the ridge regression equation.Thus, a reliable and accurate regression prediction equation was obtained (Table 7).Another way to estimate carcass weight and to determine the factors that affect carcass weight is the regression model, which is created using factor analysis scores.Bartlett's test for sphericity was performed to check the separability of the correlation matrix to the factors.Since the Bartlett test results were p<0.001, we found that the data presented multiple normal distributions.Since the estimated KMO (Kaiser-Meyer-Olkin) coefficient was 0.826, the sample size in the study was sufficient (Table 8).In order to determine the number of significant factors in the application of factor analysis, the variance explanatory percentages of total variance and factors is given in Table 9.According to the factor analysis results presented in Table 8, the eigenvalues of the first 2 out of 7 predicted factors were higher than 1, and therefore, can be used as independent variables in multiple regression analysis.At the beginning, 88.8% of the total variance was explained with 2 factors instead of 7 variables.This value is suitable for determining the optimum number of factors.The total variance ratio described should be at least 2/3 (67%) (Tabachnick & Fidell, 2001).Considering that the total variance value described here was higher than 2/3 (0.888), the factors considered account for the total variance was at a sufficient level.

Analysis of Variables Affecting Carcass Weight of White Turkeys by Regression Analysis Based on Factor Analysis Scores and Ridge Regression
Turned factor loads indicate the relationship between independent variables and factors examined.The highest correlations were estimated between thigh, wing, heart, back weight, feet and breast weights in Factor 1 (0.972, 0.946, 0.909, 0.896, 0.864 and 0.862), and gizzard in Factor 2. In addition, the variance of the variables was reflected effectively, as the amounts of commonality were high.The first two factors considered explained 70.8% and 17.9% of the total variance in all variables, respectively.Factor score coefficients were given in Table 10.According to these coefficients, the factor score for each animal was established according to 2 factors.The factor scores obtained from the factor analysis were used as independent variables to estimate the carcass weight of turkeys and the findings obtained to determine the important factors in carcass weighting are given in Table 11.According to the results of the regression analysis with the factor scores shown in Table 11, the effect of the two factors used as independent variables to estimate the carcass weight of turkeys was statistically significant (p<0.01).With the use of factor scores in the model, the multicollinearity problem was solved, and VIF=1 was found.Factor scores used in the model explained 96.6% of the total variation of white turkey carcass weights.The model was found to be generally significant (F=386.780and p<0.01).There was no autocorrelation problem since the Durbin-Watson d statistic was 1.854.After these results, the carcass weight estimation equation can be established and, it is expressed as: CW= 0.959 FS1-0.214FS2 It is expected that white turkeys with higher thigh, wing, heart, back, feet, and breast weight values have higher carcass weight because of similar signs of rotated factor loads and regression coefficients of factor scores.Here, thigh weight, wing weight, heart weight, back weight, feet weight, and breast weight were positively related with carcass weight.However, gizzard weight was negatively related with carcass weight.In other words, the carcass weight was positive in FS1 and negative in FS2.

Table 1 -
Ingredients and nutritional composition of the diets.

Table 2 -
Multiple regression analysis of variance, goodness of fit and standard error values obtained from least squares means.

Table 5 -
The goodness of fit results in ridge regression (k = 0.204).

Table 6 -
ridge regression variance analysis results.
df: degrees freedom, SS: sum of squares, MS: mean square, F: F test, p: significance level

Table 7 -
Ridge regression parameters, standard errors and VIF values

Table 3 -
Estimated regression parameter, significance level, tolerance and VIF values obtained from least squares means.TW: thigh weight, BRW: breast weight, WW: wing weight, BW: back weight, GW: gizzard weight, HW: heart weight, FW: feet weight.SE: standard error.VIF: variance inflation factor, B: regression parameter, t: t-test statistics

Table 4 -
Correlations between independent variables.

Table 10 -
Component Score Coefficient Matrix.