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.

1. 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
JournalofResearchinBiology
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 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.
1526-1533| JRB | 2014 | Vol 4 | No 8
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.
www.jresearchbiology.com
Journal of Research in Biology
An International
Scientific Research Journal
Authors:
Alphonsus C1
, Akpa GN1
,
Nwagu BI2
, Abdullahi I2
,
Zanna M3
, Ayigun AE3
,
Opoola E3
, Anos KU3
,
Olaiya O3
and Olayinka-
Babawale 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
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

2. 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 multi-collinear 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) Shika-Zaria, 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 ad-
libitum. 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.
Alphonsus et al., 2014
1527 Journal of Research in Biology (2014) 4(8):1526-1533

3. The milk samples were frozen immediately after
collection and stored at -20o
C 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, Zaria-Nigeria. 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), Fat-Protein Ratio (FPR), Fat-Lactose
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 Protein-Lactose 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
Journal of Research in Biology (2014) 4(8): 1526-1533 1528
Alphonsus et al., 2014

4. Alphonsus et al., 2014
1529 Journal of Research in Biology (2014) 4(8): 1526-1533
Table1:Correlationco-efficientsamongmilkyieldandmilkcompositionvariablesusedforpredictionofEnergyBalance(EB)
*Milk
variables
AD-
MY
MFCMPCMLCMFYMPYMLYFPRFLRPLRDMYdMFCdMPCdMLCdFPRdFLR
ADMY-
MFC-0.264-
MPC-0.1950.352-
MLC-0.3210.8530.305-
MFY0.9560.025-0.054-0.078-
MPY0.986-0.189-0.029-0.2750.966-
MLY0.9390.019-0.0890.0140.9880.943-
FPR-0.1770.853-0.0790.7730.0630.1910.078-
FLR0.1830.0440.169-0.4840.2030.2180.016-0.048-
PLR0.240-0.6690.162-0.8890.0560.272-0.057-0.8410.579-
dMY-0.6690.0370.232-0.058-0.681-0.645-0.728-0.0950.1610.171-
dMFC-0.674-0.0560.120-0.085-0.714-0.668-0.742-0.1330.0560.1440.980-
dMPC-0.6710.0000.117-0.068-0.695-0.666-0.734-0.1180.1180.1230.9850.989-
dMLC-0.6530.0210.187-0.084-0.671-0.634-0.7230.1830.1830.1760.9920.9830.985-
dFPR-0.182-0.388-0.002-0.154-0.279-0.183-0.220-0.352-0.352-0.1520.2400.3450.2120.246-
dFLR-0.129-0.433-0.3360.017-0.254-0.196-0.126-0.284-0.6910.171-0.0210.1420.063-0.0360.605-
dPLR0.070-0.061-0.3910.1470.038-0.0050.1150.165-0.385-0.363-0.297-0.230.459-0.321-0.4270.459
*
milkcompositionvariablesindicatedbythefollowing:AverageDailyMilkYield(ADMY),MilkFatContent(MFC),MilkProteinContent(MPC),Milk
LactoseContent(MLC),MilkFatYield(MFY),MilkProteinYield(MPY),MilkLactoseYield(MLY),FatProteinRatio(FPR),FatLactoseRatio(FLR),
ProteinLactoseRatio(PLR).Variableabbreviationsstartingwith“d”arethecurrentminusthepreviousvaluesofmilkmeasuresinquestion.Yieldvaluesare
inkilogramperday(kg/day),contentvaluesareinpercentages(%)andratiosareunitless.Themeasuresusedweregroupmeanaverages.
2
cummulativepercentagesofvariationexplainedwithincreasingnumberofPCindicated

5. 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 Fat-Protein Ratio (dFPR) and
Fat-Lactose 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
Journal of Research in Biology (2014) 4(8): 1526-1533 1530
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
(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
Fat-Protein Ratio (FPR) 0.05 0.04 0.90 -0.27 -0.25 99.97
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 0.94 -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.
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

6. 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
Alphonsus et al., 2014
1531 Journal of Research in Biology (2014) 4(8): 1526-1533
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
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 “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 rati-
os are unitless. The measures used were group mean averages. ** = P < 0.001

7. 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
REFERENCES
Alphonsus C. 2014. Prediction of energy balance and
post-partum reproductive function using milk
composition measures in dairy cows. PhD Dissertation
Submitted to Department of Animal Science, Ahmadu
Bello University, Zaria-Nigeria.
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
Composition Analysis .Journal of Animal Science
Advances. 3(5): 219-225.
Bair Eric, Trevor Hastie, Paul Debashis and Robert
Tibshirani. 2006. Prediction by supervised Principal
Components. Journal of the American Statistical
Association. 473 (19): 119-137.
Čejna V and Chládek G. 2005. The importance of
monitoring changes in milk fat to protein ratio in
Holstein cows during lactation. Journal of Central
European Agriculture. 6: 539-545.
Fahey J. 2008. Milk protein percentage and dairy cow
fertility. University of Melbourne, Department of
Veterinary Science, VIAS, Sneydes Road 600, Werribee,
Victoria, 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.
Fekedulegn DB, Colbert JJ, Hicks Jr RR and
Schuckers ME. 2002. Coping with multicollinearity: An
example on application of Principal Components
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
Journal of Research in Biology (2014) 4(8): 1526-1533 1532
Alphonsus et al., 2014
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

8. Hansen LB. 2000. Consequences of selection for milk
yield from a geneticist's viewpoint. Journal of Dairy
Science. 83(5): 1145-1450.
Harris BL and Pryce JE. 2004. Genetic and Phenotypic
relationships between milk protein percentage,
reproductive performance and body condition score in
New Zealand dairy cattle. Proceeding of the New
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
Cookbook (pp. 106 - 108). New York: John Wiley &
Sons, Inc.
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.
Maitra S and Yan J. 2008. Principal Component
Analysis and Partial Least Squares: two dimension
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.
Relationships between milk yield, post partum body
weight and reproductive performance in Friesian x
Bunaji Cattle. Asian–Australian Journal of Animal
Science. 14(11): 1505 – 1654.
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
Alphonsus et al., 2014
1533 Journal of Research in Biology (2014) 4(8): 1526-1533
Submit your articles online at www.jresearchbiology.com
Advantages
Easy online submission
Complete Peer review
Affordable Charges
Quick processing
Extensive indexing
You retain your copyright
submit@jresearchbiology.com
www.jresearchbiology.com/Submit.php