Journal of Research in Biology
ISSN No: Print: 2231 –6280; Online: 2231- 6299
An International Scientific Research Journal
Original Research
Journal of Research in Biology
Application of multivariate principal component analysis on dimensional reduction of milk composition variables 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, Shika-Zaria 3. Kabba College of Agriculture, Ahmadu Bello University, Kabba, Nigeria
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 multi-collinear 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.
Corresponding author: Alphonsus C
Keywords: Principal component analysis, eigenvalues, communality
Email Id:
Article Citation: Alphonsus C, Akpa GN, Nwagu BI, Abdullahi I, Zanna M, Ayigun AE, Opoola E, Anos KU, Olaiya O and Olayinka-Babawale OI Application of multivariate principal component analysis on dimensional reduction of milk composition variables Journal of Research in Biology (2014) 4(8): 1526-1533
Web Address: http://jresearchbiology.com/ documents/RA0489.pdf
Dates: Received: 27 Oct 2014
Accepted: 15 Nov 2014
Published: 03 Dec 2014
This article is governed by the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0), which gives permission for unrestricted use, non-commercial, distribution and reproduction in all medium, provided the original work is properly cited. Journal of Research in Biology An International Scientific Research Journal
1526-1533| JRB | 2014 | Vol 4 | No 8
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Alphonsus et al., 2014 used to reduce the number of predictive variables as well
INTRODUCTION In recent times, many scientist, especially in the
as solving the problem of multicollinearity (Bair et al.,
field of dairy science have postulated the use of milk
2006). It transforms the original independent variables
composition variables as a tool for monitoring and
into newly uncorrelated variables called Principal
evaluation of energy balance (Friggens et al., 2007;
Components (PCs) (Lafi and Kaneene, 1992), so that
Lovendahl et al., 2010; Alphonsus, 2014), health
each PC is a linear combination of all the original
(Hansen et al., 2000; Pryce et al., 2001; Invartsen et al.,
independent variables. It looks for a few linear
2003; Cejna and Chiladek, 2005), fertility (Harris and
combinations of variables that can best be used to
Pryce, 2004; Fahey, 2008) and nutritional status
summarize the data without loosing information of the
(Kuterovac et al., 2005; Alphonsus et al., 2013) of dairy
original variables (Lafi and Kaneen, 1992; Bair et al.,
cows. One way of validating this hypothesis is to assess
2006)
the relationship between the milk composition variables
This study therefore attempted to apply the
and the parameters in question through multiple
principle of Principal Component Analysis (PCA) on
regression analysis. However, the drawback in applying
variable selection and dimension reduction of milk
multiple regression analysis to the milk composition
composition variables
variables is that most of the milk composition variables are highly correlated (Lovendahl et al., 2010; Alphonsus
MATERIALS AND METHODS
and Essien, 2012).
Experimental site
A high degree of correlation among the
Data for this study were collected from 13
predictive variables increases the variance in estimates of
primiparous and 47 multiparous Friesian x Bunaji dairy
the regression parameters (Yu, 2010). This problem is
cows, at the dairy herd of National Animal Production
known as multicollinearity (Kleinbaum et al., 1998;
Research
Fekedulegn et al., 2002; Leahy, 2001;
between latitude 11° and 12°N at an altitude of 640m
The
problem
with
multicollinearity
Yu, 2008). is
that
Institute
(NAPRI)
Shika-Zaria,
located
it
above sea level, and lies within the Northern Guinea
compromises the basic assumption of multiple regression
Savannah Zone (Oni et al., 2001). The cows were
that state that “the predictive variables are uncorrelated
managed during the rainy season on both natural and
and independent of each other” and parameters estimated
paddock–sown pasture, while during the dry season they
using multi-collinear data are unstable and can change
were fed hay and /or silage supplemented with
with slight change in data, hence are unreliable for
concentrate mixture of undelinted cotton seed cake and
predicting the future. When predictors suffer from
grinded maize. They had access to water and salt lick ad-
multicollinearity, using multiple regressions may lead to
libitum. Unrestricted grazing was allowed under the
inflation of regression coefficients. These coefficients
supervision of herdsmen for 7 – 9 hours per day
could fluctuate in signs and magnitude as a result of a
(Alphonsus et al., 2013)
slight change in the dependent variables (Fekedulegn
Milk composition measures
et al., 2002).
Cows were milked twice daily (morning and
Therefore, the first step to counteract this
evening) and milk yield was recorded on daily basis. The
problem of multicollinearity is the use of Principal
milk sampled for the determination of fat, protein and
Component
Analysis (PCA). Principal component
lactose percentages were taken once per week starting
analysis is a multivariate statistical tool that is commonly
from 4 days postpartum to the end of each lactation.
1527
Journal of Research in Biology (2014) 4(8):1526-1533
Alphonsus et al., 2014 The milk samples were frozen immediately after o
The principal component analysis was run using
collection and stored at -20 C until analysed (Alphonsus
PROC Factor SAS software (SAS, 2002).
et al., 2013). The milk composition analysis was carried
RESULTS AND DISCUSSION
out at the Food Science and Technology Laboratory of
Correlation matrix of the milk composition variables
Institute of Agricultural Research (IAR) in Ahmadu
The correlation matrix shows high degree of
Bello University, Zaria-Nigeria. The yield values and the
correlation among the milk composition variables (Table
ratios were derived from the percentage values of fat,
1). This strong correlation among the measured variables
protein and lactose (Friggens et al., 2007 Lφvendahl et
is called multicollinearity (Kleinbaum et al., 1998;
al., 2010). The following milk composition measures
Vaughan and Berry, 2005). Multicollinearity is a serious
were calculated: Milk Fat Content (MFC), Milk Protein
problem in multiple regression analysis because it
Content (MPC), Milk Lactose Content (MLC), Milk Fat
violates the basic assumption of regression that requires
Yield (MFY), Milk Protein Yield (MPY), Milk Lactose
the predictors to be independent and uncorrelated with
Yield (MLY), Fat-Protein Ratio (FPR), Fat-Lactose
each others. It also compromise the integrity and
Ratio (FLR), Protein - Lactose Ratio (PLR), change in
reliability of the regression models (Kleinbaum et al.,
Milk Yield (dMY), change in Milk Protein Content
1998; Maitra and Yan, 2008).
(dMPC), change in Milk Fat Content (dMFC), change in
The problem of multicollinearity is as a result of
Milk Lactose Content (dMLC), change in Fat Protein
redundancy of some variables. Redundancy in this case
Ratio (dFPR), change in Fat Lactose Ratio (dFLR) and
means that some of the variables are strongly correlated
change in Protein-Lactose Ratio (dPLR).
with one another, possibly because they are measuring
Statistical Analysis
the
The
correlation
matrix
of
all
the
milk
same
characteristic
(http://support.sas.com/
publishing/publicat/chaps/55).
For
example,
milk composition
the
composition variables was first run using PROC CORR
correlations between the
yield
procedure of SAS (2000) to determine the level of the
variables (MFY, MPY, MLY) were very strong (r =
collinearity among milk composition variables.
0.943 to 0.989). Likewise, the correlations between the
Principal component analysis
rate of change „d‟ in milk composition variables (dMY,
Principal component analysis is a method for
dMFC, dMPC, dMLC) were very strong ranging from
transforming the variables in a multivariate data set
0.980 to 0.992, and a lot of others. Therefore, given this
X2, X2,…….Xn, into new variables, Y1, Y2,……..Yn,
apparent redundancy, it is likely that these correlated
which are uncorrelated with each other and account for
variables are measuring the same construct or have the
decreasing proportions of the total variance of the
same characteristics. Therefore, it could be possible to
original variables, defined as
reduce these collinear variables into smaller number of
Y1 = P11X1 + P12X2 +………………. +P1nXn
composite variable (artificial variables) called Principal
Y2 = P21 X1 + P22X2 + ……………… + P2nXn
Components (PCs) that are independent and account for
Y3 = Pn1X1 + Pn2X2 + ………………. + PnnXn
most of the variation in the milk composition variables.
With
the
coefficient
being
chosen
so
that
The PCs can then be used for subsequent multiple
Y1, Y2, …….. Yn account for decreasing proportion of
regression analysis. One way of achieving this is the use
the total variance of the original variables X1, X2 …..Xn
of Principal Component Analysis (PCA).
(Lafi and Kaneene, 1992).
Principal Component Analysis The measured milk composition variables were
Journal of Research in Biology (2014) 4(8): 1526-1533
1528
1529
0.070
dPLR
-0.061
-0.433
-0.388
0.021
0.000
-0.056
0.037
-0.669
0.044
0.853
0.019
-0.189
0.025
0.853
0.352
-
MFC
-0.391
-0.336
-0.002
0.187
0.117
0.120
0.232
0.162
0.169
-0.079
-0.089
-0.029
-0.054
0.305
-
MPC
0.147
0.017
-0.154
-0.084
-0.068
-0.085
-0.058
-0.889
-0.484
0.773
0.014
-0.275
-0.078
-
MLC
0.038
-0.254
-0.279
-0.671
-0.695
-0.714
-0.681
0.056
0.203
0.063
0.988
0.966
-
MFY
-0.005
-0.196
-0.183
-0.634
-0.666
-0.668
-0.645
0.272
0.218
0.191
0.943
-
MPY
0.115
-0.126
-0.220
-0.723
-0.734
-0.742
-0.728
-0.057
0.016
0.078
-
MLY
0.165
-0.284
-0.352
0.183
-0.118
-0.133
-0.095
-0.841
-0.048
-
FPR
-0.385
-0.691
-0.352
0.183
0.118
0.056
0.161
0.579
-
FLR
-0.363
0.171
-0.152
0.176
0.123
0.144
0.171
-
PLR
-0.297
-0.021
0.240
0.992
0.985
0.980
-
DMY
-0.23
0.142
0.345
0.983
0.989
-
dMFC
0.459
0.063
0.212
0.985
-
dMPC
-0.321
-0.036
0.246
-
dMLC
-0.427
0.605
-
dFPR
0.459
-
dFLR
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. 2 cummulative percentages of variation explained with increasing number of PC indicated
*
-0.129
dFLR
0.240
PLR
-0.182
0.183
FLR
dFPR
-0.177
FPR
-0.653
0.939
MLY
dMLC
0.986
MPY
-0.671
0.956
MFY
dMPC
-0.321
MLC
-0.674
-0.195
MPC
dMFC
-0.264
MFC
-0.669
-
ADMY
dMY
ADMY
*Milk variables
Table 1: Correlation co-efficients among milk yield and milk composition variables used for prediction of Energy Balance (EB)
Alphonsus et al., 2014
Journal of Research in Biology (2014) 4(8): 1526-1533
Alphonsus et al., 2014 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 -0.34 -0.02 0.00 -0.00 99.81 0.93 (ADMY) Milk Fat Content (MFC) 0.02 0.13 -0.23 0.42 99.96 0.85 Milk Protein Content (MPC) -0.04 0.15 0.05 0.07 99.96 0.98 Milk Lactose Content (MLC) 0.05 0.13 0.08 0.37 99.89 0.82 Milk Fat Yield (MFY) -0.33 0.17 -0.05 0.08 99.88 0.92 Milk Protein Yield (MPY) -0.34 -0.01 -0.01 0.14 99.90 0.93 Milk Lactose Yield (MLY) -0.34 0.19 0.01 0.07 99.83 0.91 Fat-Protein Ratio (FPR) 0.05 0.04 -0.27 -0.25 99.97 0.90 Fat-Lactose Ratio (FLR) 0.10 -0.04 -0.15 -0.48 -0.05 99.97 Protein-Lactose Ratio (PLR) 0.03 -0.06 -0.86 -0.09 0.17 99.86 dMY -0.33 0.01 -0.06 0.01 99.40 0.94 dMFC -0.32 -0.02 0.06 -0.03 99.77 0.94 dMPC -0.32 0.02 -0.06 -0.04 88.81 0.94 dMLC -0.31 -0.01 -0.07 -0.02 99.77 0.95 dFPR -0.04 0.03 -0.25 0.04 99.94 0.81 dFLR -0.04 -0.05 -0.08 -0.03 99.95 0.92 dPLR -0.01 -0.09 0.22 0.02 -0.09 99.96 3 % variance 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. 2 cummulative 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 subjected to Principal Component Analysis (PCA) using
component if the factor loading was 0.50 or greater.
„one‟ as a prior communality estimate. The principal axis
Using these criteria, it was obvious that the change “d”
method was used to extract the components, and this was
in milk composition variables (dMY, dMFC, dMPC,
followed by varimax (orthogonal) rotation. Only the first
dMLC) loaded heavily on the first Principal Component
five components accounted for a meaningful amount of
(PC)
the total variance (92.38%) in the milk composition
component”. Also, the four milk composition yield
variables. Also using eigenvalue criteria of one,
variables (ADMY, MFY, MPY, MLY) loaded heavily on
it was obvious that the first five components displayed
the second PC and were labeled “yield component”.
eigenvalues equal to or greater than one. Therefore, the
Other variables like MFC, MLC, FPR and FLR loaded
first five principal components were retained and used
heavily on the third PC and were labeled “mixed
for rotation and interpretation. The milk composition
component”. Change in Fat-Protein Ratio (dFPR) and
variables and the corresponding factor loadings are
Fat-Lactose Ratio (dFLR) loaded heavily on the fourth
presented in Table 2. In interpreting the rotated factor
PC and were labeled “change in fat ratio component”.
pattern, an item was said to load heavily on a given
The last PC had only one variable (MPC) heavily loaded
Journal of Research in Biology (2014) 4(8): 1526-1533
which
were
subsequently
labeled
“change
1530
Alphonsus et al., 2014 Table 3: Pearson correlation between the Principal components and milk composition variables Variables i Average daily milk yield (ADMY) Milk fat content (mFc) Milk protein content (mPc)
Principal Components (PCs) PC1 PC2 PC3 -0.340 0.938** -0.016
PC 0.000
PC5 0.004
0.018 -0.043
0.135 0.146
0.853** 0.059
-0.233 0.015
0.317 0.981**
Milk lactose content(mLc) Milk fat yield (mFy)
-0.052 -0.327
0.128 0.923**
0.825 0.168
0.085 -0.053
0.373 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
Fat-protein ratio (FPR) 0.046 0.039 0.900** -0.267 -0.246 Fat-lactose ratio (FLR) 0.098 -0.036 -0.153 -0.476 -0.046 Protein-lactose 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 â&#x20AC;&#x153;dâ&#x20AC;? 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 on it, suggesting that MPC is not strongly correlated with
on the PCs (the best loading of each variable is indicated
any of the measured milk composition variables (as can
by the bolded values). Each variable loaded only on one
be verified in Table 1) and could therefore be treated as
component. No variable loaded heavily on more than one
independent variable in subsequent multivariate analysis.
PC. This suggested that the milk composition variables
Since PCs are labeled according to the size of
can be reduced into smaller composite variable without
their variances, the first Principal Component (PC) explained larger amount of variation (38.88%) among
losing much of the information. The
PCs
displayed
varying
degrees
of
the variables, while the last PC explained the least
correlations with the
(07.08%). Also, the eigenvalues followed the same trend
(Table 3) and the correlation structure was similar to the
as the percentage variance explained by each of the PCs.
loading pattern of the milk composition variables on the
The communality estimates, which tells us how much of
PCs. Thus, confirming the loading pattern of the
the variance in each of the original variables is explained
principal component analysis (Table 2). However, the
by the extracted PC was very high ranging from 83.30 to
correlation among the PCs was zero. This shows that the
99.71%. There was a clear grouping of the measured
Principal component analysis resulted in orthogonal
variables evident by the loading pattern of the variables
solution whereby the PCs extracted were completely
1531
milk composition variables
Journal of Research in Biology (2014) 4(8): 1526-1533
Alphonsus et al., 2014 Table 4: Descriptive statistics of the principal components Principal components N Means S.D Min (PCs) PC1 60 0.00 1.000 -2.544 PC2 60 0.00 1.000 -3.001 PC3 60 0.00 1.000 -2.626 PC4 60 0.00 1.000 -5.045 PC5 60 0.00 1.000 -4.104
Max 3.102 2.573 2.036 2.563 2.085
N= animals, S.D = standard deviation, Min =minimum, Max = maximum uncorrelated and independent of each other. Also, the
Composition Analysis .Journal of
PCs were standardized to have a mean of zero and
Advances. 3(5): 219-225.
standard deviation of one (Table 4)
Animal
Science
Bair Eric, Trevor Hastie, Paul Debashis and Robert Tibshirani. 2006. Prediction by supervised Principal
CONCLUSION The Principal Component Analysis (PCA) successfully reduced the dimensionality of the milk
Components. Journal of the American Statistical Association. 473 (19): 119-137.
composition variables, by grouping the 17 milk
Čejna V and Chládek G. 2005. The importance of
composition variables into five Principal Components
monitoring changes in milk fat to protein ratio in
(PCs) that were uncorrelated and independent of each
Holstein cows during lactation. Journal of
other, and explained about 92.38% of the total variation
European Agriculture. 6: 539-545.
in the milk composition variables. Therefore, PCA can be used to solve the problem of multicollinearity and variable reduction in multiple regression analysis
Fahey J. 2008. Milk protein percentage and dairy cow fertility. University of Melbourne, Department of Veterinary Science, VIAS, Sneydes Road 600, Werribee, Victoria,
REFERENCES Alphonsus C. 2014. Prediction of energy balance and post-partum
reproductive
function
using
Central
milk
Australia.P.12.
Web
link:
http://
w w w . n h i a . o r g . a u / h t m l / body_milk_protein__fertility.html 30/10/2008.
composition measures in dairy cows. PhD Dissertation
Fekedulegn DB, Colbert JJ, Hicks Jr RR and
Submitted to Department of Animal Science, Ahmadu
Schuckers ME. 2002. Coping with multicollinearity: An
Bello University, Zaria-Nigeria.
example on application of Principal Components
Alphonsus C and Essien IC. 2012. The relationship estimates amongst milk yield and milk composition characteristics of Bunaji and Friesian × Bunaji cows. African Journal of Biotechnology. 11(36): 8790-8793. Alphonsus C, Akpa GN, Nwagu BI, Barje PP, Orunmuyi M, Yashim SM, Zana M, Ayigun AE and Opoola E. 2013. Evaluation of Nutritional Status of Friesian
x Bunaji
Dairy Herd
Based
on Milk
Journal of Research in Biology (2014) 4(8): 1526-1533
Regression in Dendroecology. Research Paper NE-721, Newton Square PA: United States Department of Agriculture.
Forest
service.
1-48p
Web
link:
www.fs.fed.us/ne/morgantown/4557/dendrochron/ rpne721.pdf. Friggens NC, Ridder C and Løvendahl P. 2007. On the use of milk composition measures to predict the energy balance of dairy cows. Journal of Dairy Science. 90(12): 5453-5467 1532
Alphonsus et al., 2014 Hansen LB. 2000. Consequences of selection for milk yield from a geneticist's viewpoint. Journal of Dairy Science. 83(5): 1145-1450.
reduction techniques for regression. Casualty Actuarial Society, Discussion paper program. pp.79-90 Oni OO, Adeyinka IA, Afolayan RA, Nwagu BI Malau-Aduli AEO, Alawa CBI and Lamidi OS. 2001.
Harris BL and Pryce JE. 2004. Genetic and Phenotypic
Relationships between milk yield, post partum body
relationships
percentage,
weight and reproductive performance in Friesian x
reproductive performance and body condition score in
Bunaji Cattle. Asian–Australian Journal of Animal
New Zealand dairy cattle. Proceeding of the New
Science. 14(11): 1505 – 1654.
between
milk
protein
Zealand Society of Animal Production. 64: 127-131. Ingvartsen KL, Dewhurst RJ and Friggens NC. 2003. On the relationship between lactational performance and health: Is it yield or metabolic imbalance that cause production diseases in dairy cattle? A position paper. Livestock Production Science. 83: 277–308. Klainbaum DG, Kupper LL and Muller KE. 1998. Applied
Regression
Analysis
and
Multivariable
Methods. 3rd Edition (Colle Pacific Grove, CA). Kuterovac K, Balas S, Gantner V, Jovanovac S, Dakic A. 2005. Evaluation of nutritional status of dairy cows based on milk analysis results. Italian Journal of Animal Science. 4(3): 33-35. Lafi SQ and Kaneene JB. 1992. An explanation of the use of principal component analysis to detect and correct for multicollinearity. Preventive Veterinary Medicine. 13 (4): 261-275. Leahy K. 2001. Multicollinearity: When the solution is the problem. In Olivia Parr Rud (Ed.) Data Mining
Principal Component Analysis http://support.sas.com/ publishing/publicat/chaps/55 ) Pryce JE, Royal PC, Garnsworthy and Mao IL. 2001. Fertility in the high-producing dairy cow. Livestock Production Science. 86(1-3): 125-135. SAS. 2000. SAS User‟s Guide Version 8.1. Statistical Analysis system institute Inc, Cary, Nc, USA. Vaughan TS and Berry KE. 2005. Using Monte Carlo techniques to demonstrate the meaning and implications of multicollinearity. Journal of Statistics Education, 13 (1): 1-9. Web link: www.amstat.org/publications/jse/ v13n1/vaughan.html Yu CH. 2008. Multi-collinearity, variance inflation, and orthogonalization in regression. Web link: http:// www.creative-wisdom.com/computer/sas/collinear.html Yu CH. 2010. Checking assumptions in regression. Web link:
http://www.creativewisdom.com/computer/sas/
regression_assumption.html
Cookbook (pp. 106 - 108). New York: John Wiley &
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Løvendahl P, Ridder C and Friggens NC. 2010. Limits to prediction of energy balance from milk composition measures at individual cow level. Journal of Dairy Science. 93(5): 1998–2006.
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