Article Citation:
Alphonsus C, Akpa GN, Nwagu BI, Abdullahi I, Zanna M, Ayigun AE, Opoola E, Anos KU, Olaiya O and OlayinkaBabawale OI
Application of multivariate principal component analysis on dimensional reduction of milk composition variables
Journal of Research in Biology (2014) 4(8): 15261533
Journal of Research in Biology
Application of multivariate principal component analysis on dimensional reduction of milk composition variables
Keywords:
Principal component analysis, eigenvalues, communality
ABSTRACT:
Variable selection and dimension reduction are major prerequisites for reliable multivariate regression analysis. Most a times, many variables used as independent variables in a multiple regression display high degree of correlations. This problem is known as multicollinearity. Absence of multicollinearity is essential for multiple regression models, because parameters estimated using multicollinear data are unstable and can change with slight change in data, hence are unreliable for predicting the future. This paper presents the application of Principal Component Analysis (PCA) on the dimension reduction of milk composition variables. The application of PCA successfully reduced the dimension of the milk composition variables, by grouping the 17 milk composition variables into five principal components (PCs) that were uncorrelated and independent of each other, and explained about 92.38% of the total variation in the milk composition variables.
15261533 JRB  2014  Vol 4  No 8
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Authors:
Alphonsus C1, Akpa GN1, Nwagu BI2, Abdullahi I2, Zanna M3, Ayigun AE3, Opoola E3, Anos KU3, Olaiya O3 and OlayinkaBabawale OI3
Institution:
1. Animal Science Department, Ahmadu Bello University, Zaria, Nigeria.
2. National Animal Production Research Institute, ShikaZaria
3. Kabba College of Agriculture, Ahmadu Bello University, Kabba, Nigeria
Corresponding author:
Alphonsus C
Email Id:
Web Address:
http://jresearchbiology.com/documents/RA0489.pdf
Dates:
Received: 27 Oct 2014 Accepted: 15 Nov 2014 Published: 03 Dec 2014
Journal of Research in Biology
An International Scientific Research Journal
Original Research
ISSN No: Print: 2231 –6280; Online: 2231 6299
INTRODUCTION
In recent times, many scientist, especially in the field of dairy science have postulated the use of milk composition variables as a tool for monitoring and evaluation of energy balance (Friggens et al., 2007; Lovendahl et al., 2010; Alphonsus, 2014), health (Hansen et al., 2000; Pryce et al., 2001; Invartsen et al., 2003; Cejna and Chiladek, 2005), fertility (Harris and Pryce, 2004; Fahey, 2008) and nutritional status (Kuterovac et al., 2005; Alphonsus et al., 2013) of dairy cows. One way of validating this hypothesis is to assess the relationship between the milk composition variables and the parameters in question through multiple regression analysis. However, the drawback in applying multiple regression analysis to the milk composition variables is that most of the milk composition variables are highly correlated (Lovendahl et al., 2010; Alphonsus and Essien, 2012).
A high degree of correlation among the predictive variables increases the variance in estimates of the regression parameters (Yu, 2010). This problem is known as multicollinearity (Kleinbaum et al., 1998; Fekedulegn et al., 2002; Leahy, 2001; Yu, 2008). The problem with multicollinearity is that it compromises the basic assumption of multiple regression that state that “the predictive variables are uncorrelated and independent of each other” and parameters estimated using multicollinear data are unstable and can change with slight change in data, hence are unreliable for predicting the future. When predictors suffer from multicollinearity, using multiple regressions may lead to inflation of regression coefficients. These coefficients could fluctuate in signs and magnitude as a result of a slight change in the dependent variables (Fekedulegn et al., 2002).
Therefore, the first step to counteract this problem of multicollinearity is the use of Principal Component Analysis (PCA). Principal component analysis is a multivariate statistical tool that is commonly used to reduce the number of predictive variables as well as solving the problem of multicollinearity (Bair et al., 2006). It transforms the original independent variables into newly uncorrelated variables called Principal Components (PCs) (Lafi and Kaneene, 1992), so that each PC is a linear combination of all the original independent variables. It looks for a few linear combinations of variables that can best be used to summarize the data without loosing information of the original variables (Lafi and Kaneen, 1992; Bair et al., 2006)
This study therefore attempted to apply the principle of Principal Component Analysis (PCA) on variable selection and dimension reduction of milk composition variables
MATERIALS AND METHODS
Experimental site
Data for this study were collected from 13 primiparous and 47 multiparous Friesian x Bunaji dairy cows, at the dairy herd of National Animal Production Research Institute (NAPRI) ShikaZaria, located between latitude 11° and 12°N at an altitude of 640m above sea level, and lies within the Northern Guinea Savannah Zone (Oni et al., 2001). The cows were managed during the rainy season on both natural and paddock–sown pasture, while during the dry season they were fed hay and /or silage supplemented with concentrate mixture of undelinted cotton seed cake and grinded maize. They had access to water and salt lick adlibitum. Unrestricted grazing was allowed under the supervision of herdsmen for 7 – 9 hours per day (Alphonsus et al., 2013)
Milk composition measures
Cows were milked twice daily (morning and evening) and milk yield was recorded on daily basis. The milk sampled for the determination of fat, protein and lactose percentages were taken once per week starting from 4 days postpartum to the end of each lactation.
The milk samples were frozen immediately after collection and stored at 20oC until analysed (Alphonsus et al., 2013). The milk composition analysis was carried out at the Food Science and Technology Laboratory of Institute of Agricultural Research (IAR) in Ahmadu Bello University, ZariaNigeria. The yield values and the ratios were derived from the percentage values of fat, protein and lactose (Friggens et al., 2007 Lφvendahl et al., 2010). The following milk composition measures were calculated: Milk Fat Content (MFC), Milk Protein Content (MPC), Milk Lactose Content (MLC), Milk Fat Yield (MFY), Milk Protein Yield (MPY), Milk Lactose Yield (MLY), FatProtein Ratio (FPR), FatLactose Ratio (FLR), Protein  Lactose Ratio (PLR), change in Milk Yield (dMY), change in Milk Protein Content (dMPC), change in Milk Fat Content (dMFC), change in Milk Lactose Content (dMLC), change in Fat Protein Ratio (dFPR), change in Fat Lactose Ratio (dFLR) and change in ProteinLactose Ratio (dPLR).
Statistical Analysis
The correlation matrix of all the milk composition variables was first run using PROC CORR procedure of SAS (2000) to determine the level of the collinearity among milk composition variables.
Principal component analysis
Principal component analysis is a method for transforming the variables in a multivariate data set X2, X2,…….Xn, into new variables, Y1, Y2,……..Yn, which are uncorrelated with each other and account for decreasing proportions of the total variance of the original variables, defined as
Y1 = P11X1 + P12X2 +………………. +P1nXn
Y2 = P21 X1 + P22X2 + ……………… + P2nXn
Y3 = Pn1X1 + Pn2X2 + ………………. + PnnXn
With the coefficient being chosen so that Y1, Y2, …….. Yn account for decreasing proportion of the total variance of the original variables X1, X2 …..Xn (Lafi and Kaneene, 1992).
The principal component analysis was run using PROC Factor SAS software (SAS, 2002).
RESULTS AND DISCUSSION
Correlation matrix of the milk composition variables
The correlation matrix shows high degree of correlation among the milk composition variables (Table 1). This strong correlation among the measured variables is called multicollinearity (Kleinbaum et al., 1998; Vaughan and Berry, 2005). Multicollinearity is a serious problem in multiple regression analysis because it violates the basic assumption of regression that requires the predictors to be independent and uncorrelated with each others. It also compromise the integrity and reliability of the regression models (Kleinbaum et al., 1998; Maitra and Yan, 2008).
The problem of multicollinearity is as a result of redundancy of some variables. Redundancy in this case means that some of the variables are strongly correlated with one another, possibly because they are measuring the same characteristic (http://support.sas.com/publishing/publicat/chaps/55). For example, the correlations between the milk composition yield variables (MFY, MPY, MLY) were very strong (r = 0.943 to 0.989). Likewise, the correlations between the rate of change „d‟ in milk composition variables (dMY, dMFC, dMPC, dMLC) were very strong ranging from 0.980 to 0.992, and a lot of others. Therefore, given this apparent redundancy, it is likely that these correlated variables are measuring the same construct or have the same characteristics. Therefore, it could be possible to reduce these collinear variables into smaller number of composite variable (artificial variables) called Principal Components (PCs) that are independent and account for most of the variation in the milk composition variables. The PCs can then be used for subsequent multiple regression analysis. One way of achieving this is the use of Principal Component Analysis (PCA).
Principal Component Analysis
The measured milk composition variables were
subjected to Principal Component Analysis (PCA) using „one‟ as a prior communality estimate. The principal axis method was used to extract the components, and this was followed by varimax (orthogonal) rotation. Only the first five components accounted for a meaningful amount of the total variance (92.38%) in the milk composition variables. Also using eigenvalue criteria of one, it was obvious that the first five components displayed eigenvalues equal to or greater than one. Therefore, the first five principal components were retained and used for rotation and interpretation. The milk composition variables and the corresponding factor loadings are presented in Table 2. In interpreting the rotated factor pattern, an item was said to load heavily on a given component if the factor loading was 0.50 or greater. Using these criteria, it was obvious that the change “d” in milk composition variables (dMY, dMFC, dMPC, dMLC) loaded heavily on the first Principal Component (PC) which were subsequently labeled “change component”. Also, the four milk composition yield variables (ADMY, MFY, MPY, MLY) loaded heavily on the second PC and were labeled “yield component”. Other variables like MFC, MLC, FPR and FLR loaded heavily on the third PC and were labeled “mixed component”. Change in FatProtein Ratio (dFPR) and FatLactose Ratio (dFLR) loaded heavily on the fourth PC and were labeled “change in fat ratio component”. The last PC had only one variable (MPC) heavily loaded
on it, suggesting that MPC is not strongly correlated with any of the measured milk composition variables (as can be verified in Table 1) and could therefore be treated as independent variable in subsequent multivariate analysis.
Since PCs are labeled according to the size of their variances, the first Principal Component (PC) explained larger amount of variation (38.88%) among the variables, while the last PC explained the least (07.08%). Also, the eigenvalues followed the same trend as the percentage variance explained by each of the PCs. The communality estimates, which tells us how much of the variance in each of the original variables is explained by the extracted PC was very high ranging from 83.30 to 99.71%. There was a clear grouping of the measured variables evident by the loading pattern of the variables on the PCs (the best loading of each variable is indicated by the bolded values). Each variable loaded only on one component. No variable loaded heavily on more than one PC. This suggested that the milk composition variables can be reduced into smaller composite variable without losing much of the information.
The PCs displayed varying degrees of correlations with the milk composition variables (Table 3) and the correlation structure was similar to the loading pattern of the milk composition variables on the PCs. Thus, confirming the loading pattern of the principal component analysis (Table 2). However, the correlation among the PCs was zero. This shows that the Principal component analysis resulted in orthogonal solution whereby the PCs extracted were completely
uncorrelated and independent of each other. Also, the PCs were standardized to have a mean of zero and standard deviation of one (Table 4)
CONCLUSION
The Principal Component Analysis (PCA) successfully reduced the dimensionality of the milk composition variables, by grouping the 17 milk composition variables into five Principal Components (PCs) that were uncorrelated and independent of each other, and explained about 92.38% of the total variation in the milk composition variables. Therefore, PCA can be used to solve the problem of multicollinearity and variable reduction in multiple regression analysis
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Yu CH. 2008. Multicollinearity, variance inflation, and orthogonalization in regression. Web link: http://www.creativewisdom.com/computer/sas/collinear.html
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Table 1: Correlation coefficients among milk yield and milk composition variables used for prediction of Energy Balance (EB) 

*Milk variables 
ADMY 
MFC 
MPC 
MLC 
MFY 
MPY 
MLY 
FPR 
FLR 
PLR 
DMY 
dMFC 
dMPC 
dMLC 
dFPR 
dFLR 
ADMY 
 















MFC 
0.264 
 














MPC 
0.195 
0.352 
 













MLC 
0.321 
0.853 
0.305 
 












MFY 
0.956 
0.025 
0.054 
0.078 
 











MPY 
0.986 
0.189 
0.029 
0.275 
0.966 
 










MLY 
0.939 
0.019 
0.089 
0.014 
0.988 
0.943 
 









FPR 
0.177 
0.853 
0.079 
0.773 
0.063 
0.191 
0.078 
 








FLR 
0.183 
0.044 
0.169 
0.484 
0.203 
0.218 
0.016 
0.048 
 







PLR 
0.240 
0.669 
0.162 
0.889 
0.056 
0.272 
0.057 
0.841 
0.579 
 






dMY 
0.669 
0.037 
0.232 
0.058 
0.681 
0.645 
0.728 
0.095 
0.161 
0.171 
 





dMFC 
0.674 
0.056 
0.120 
0.085 
0.714 
0.668 
0.742 
0.133 
0.056 
0.144 
0.980 
 




dMPC 
0.671 
0.000 
0.117 
0.068 
0.695 
0.666 
0.734 
0.118 
0.118 
0.123 
0.985 
0.989 
 



dMLC 
0.653 
0.021 
0.187 
0.084 
0.671 
0.634 
0.723 
0.183 
0.183 
0.176 
0.992 
0.983 
0.985 
 


dFPR 
0.182 
0.388 
0.002 
0.154 
0.279 
0.183 
0.220 
0.352 
0.352 
0.152 
0.240 
0.345 
0.212 
0.246 
 

dFLR 
0.129 
0.433 
0.336 
0.017 
0.254 
0.196 
0.126 
0.284 
0.691 
0.171 
0.021 
0.142 
0.063 
0.036 
0.605 
 
dPLR 
0.070 
0.061 
0.391 
0.147 
0.038 
0.005 
0.115 
0.165 
0.385 
0.363 
0.297 
0.23 
0.459 
0.321 
0.427 
0.459 
*milk composition variables indicated by the following: Average Daily Milk Yield (ADMY), Milk Fat Content (MFC), Milk Protein Content (MPC), Milk Lactose Content (MLC), Milk Fat Yield (MFY), Milk Protein Yield (MPY), Milk Lactose Yield (MLY), Fat Protein Ratio (FPR), Fat Lactose Ratio (FLR), Protein Lactose Ratio (PLR). Variable abbreviations starting with “d” are the current minus the previous values of milk measures in question. Yield values are in kilogram per day (kg/day), content values are in percentages (%) and ratios are unitless. The measures used were group mean averages. 2cummulative percentages of variation explained with increasing number of PC indicated 
Journal of Research in Biology (2014) 4(8): 15261533 1530
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Table 2: Relationships among milk composition measures1 expressed as loadings in a principal component analysis. 

Items a 
Principal components (PCs) 
h 

PC1 
PC2 
PC3 
PC4 
PC5 

Variable explained2 
38.88 
60.01 
75.00 
85.30 
92.38 
 
Average Daily Milk Yield (ADMY) 
0.34 
0.93 
0.02 
0.00 
0.00 
99.81 
Milk Fat Content (MFC) 
0.02 
0.13 
0.85 
0.23 
0.42 
99.96 
Milk Protein Content (MPC) 
0.04 
0.15 
0.05 
0.07 
0.98 
99.96 
Milk Lactose Content (MLC) 
0.05 
0.13 
0.82 
0.08 
0.37 
99.89 
Milk Fat Yield (MFY) 
0.33 
0.92 
0.17 
0.05 
0.08 
99.88 
Milk Protein Yield (MPY) 
0.34 
0.93 
0.01 
0.01 
0.14 
99.90 
Milk Lactose Yield (MLY) 
0.34 
0.91 
0.19 
0.01 
0.07 
99.83 
FatProtein Ratio (FPR) 
0.05 
0.04 
0.90 
0.27 
0.25 
99.97 
FatLactose Ratio (FLR) 
0.10 
0.04 
0.15 
0.48 
0.05 
99.97 
ProteinLactose Ratio (PLR) 
0.03 
0.06 
0.86 
0.09 
0.17 
99.86 
dMY 
0.94 
0.33 
0.01 
0.06 
0.01 
99.40 
dMFC 
0.94 
0.32 
0.02 
0.06 
0.03 
99.77 
dMPC 
0.94 
0.32 
0.02 
0.06 
0.04 
88.81 
dMLC 
0.95 
0.31 
0.01 
0.07 
0.02 
99.77 
dFPR 
0.04 
0.03 
0.25 
0.81 
0.04 
99.94 
dFLR 
0.04 
0.05 
0.08 
0.92 
0.03 
99.95 
dPLR 
0.01 
0.09 
0.22 
0.02 
0.09 
99.96 
% variance3 
38.88 
21.20 
14.92 
10.30 
07.08 
 
Eigen values 
6.610 
3.604 
2.536 
1.751 
1.204 
 
a Variable abbreviations starting with “d” are the change variables signifying current minus the previous values of milk measures in question. Yield values are in kilogram per day (kg/day), content values are in percentages (%) and ratios are unitless. 2cummulative percentages of variation explained with increasing number of PC indicated 3 percentage variance explained by each principal components h= communality estimates is a variance in observed variables acounted for by a common factor 
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Table 3: Pearson correlation between the Principal components and milk composition variables 

Variables i 
Principal Components (PCs) 

PC1 
PC2 
PC3 
PC 
PC5 

Average daily milk yield (ADMY) 
0.340 
0.938** 
0.016 
0.000 
0.004 
Milk fat content (mFc) 
0.018 
0.135 
0.853** 
0.233 
0.317 
Milk protein content (mPc) 
0.043 
0.146 
0.059 
0.015 
0.981** 
Milk lactose content(mLc) 
0.052 
0.128 
0.825 
0.085 
0.373 
Milk fat yield (mFy) 
0.327 
0.923** 
0.168 
0.053 
0.079 
Milk protein yield (mPy) 
0.341 
0.928** 
0.012 
0.005 
0.137 
Milk lactose yield (mLy) 
10.342 
0.909** 
0.193 
0.014 
0.078 
Fatprotein ratio (FPR) 
0.046 
0.039 
0.900** 
0.267 
0.246 
Fatlactose ratio (FLR) 
0.098 
0.036 
0.153 
0.476 
0.046 
Proteinlactose ratio (PLR) 
0.029 
0.061 
0.859** 
0.085 
0.172 
dmy 
0.939** 
0.328 
0.008 
0.057 
0.009 
dmFc 
0.942** 
0.322 
0.021 
0.069 
0.030 
dmPc 
0.941** 
0.316 
0.021 
0.062 
0.039 
dmLc 
0.944** 
0.307 
0.008 
0.069 
0.022 
dFPR 
0.041 
0.027 
0.254 
0.811** 
0.043 
dFLR 
0.044 
0.055 
0.082 
0.921** 
0.028 
dPLR 
0.005 
0.091 
0.221 
0.049 
0.087 
PC1 
1.000 
0.000 
0.000 
0.000 
0.000 
PC2 
0.000 
1.000 
0.000 
0.000 
0.000 
PC3 
0.000 
0.000 
1.000 
0.000 
0.000 
PC4 
0.000 
0.000 
0.000 
1.000 
0.000 
PC5 
0.000 
0.000 
0.000 
0.000 
1.000 
I Variable abbreviations starting with “d” are the current minus the previous values of milk measures in question. Yield values are in kilogram per day (kg/day), content values are in percentages (%) and ratios are unitless. The measures used were group mean averages. ** = P < 0.001

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Table 4: Descriptive statistics of the principal components 

Principal components (PCs) 
N 
Means 
S.D 
Min 
Max 
PC1 
60 
0.00 
1.000 
2.544 
3.102 
PC2 
60 
0.00 
1.000 
3.001 
2.573 
PC3 
60 
0.00 
1.000 
2.626 
2.036 
PC4 
60 
0.00 
1.000 
5.045 
2.563 
PC5 
60 
0.00 
1.000 
4.104 
2.085 
N= animals, S.D = standard deviation, Min =minimum, Max = maximum 
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