Application of multivariate principal component analysis on dimensional reduction of milk compositio

Journal of Research in Biology

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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

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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.

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Advantages

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