Estimation of central retinal vascular equivalent canonical correlation analysis

Multiple linear regression analysis and canonical correlation analysis CCA hadbeen applied to quantify CRAE such that the Pearson correlations between CRAE and... 27 3.4 The Coefficients

CANONICAL CORRELATION ANALYSIS

WANG LING(Master of Public Health, University of Texas)

A THESIS SUBMITTEDFOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY

NATIONAL UNIVERSITY OF SINGAPORE

2010

Acknowledgement

I would like to take this opportunity to express my deep and sincere gratitude to mysupervisor Assistant Professor Li Jialiang I do appreciate his valuable advice, guid-ance, endless patience, kindness and encouragement during my graduate period I havelearned many things from him, especially regarding academic research and characterbuilding I would sincerely like to thank Professor Wong Tien Yin from Singapore EyeResearch Institute for providing the data set and sharing his knowledge and experiences

on eye diseases

I would also like to thank all my dear fellow students: Ms Jiang Qian, Ms Li Hua,

Ms Luo Shan, Mr Jiang Binyan and Mr Liang Xuehua, who helped me in studyingstatistical theory My special thanks to Ms Zhao Wanting and Ms Zhang Rongli forteaching me Latex in my thesis writing Sincere thanks to all my friends who help me

in one way or another in my study

Further more, I especially would like to thank my husband Chen Yahua for his love,patience and support during my graduate period I also feel a deep gratitude to my

dearest family for their support in my study.

Finally, my gratitude goes to the National University of Singapore for awarding me

a research scholarship, and the Department of Statistics and Applied Probability for theexcellent research environment I would like to thank all the staffs from General Office

in the Department for their all kinds of help

CONTENTS iii

Contents

1.1 Biological background 1

1.2 Statistical background 5

1.3 Aims and Organization of The Thesis 7

2 Methods 9 2.1 The Singapore Malay Eye Study Data Review 9

2.2 Multiple Linear Regression Analysis 11

2.2.1 Estimation of the parameters 11

2.2.2 Estimation of sample correlation 15

2.2.3 Box-Cox Power Transformation 15

2.2.4 Model Adequacy Checking 16

2.3 Canonical Correlation Analysis (CCA) 17

2.3.1 Canonical Correlations and Variates in the Population 17

2.3.2 Estimation of Canonical Correlation and Variates 20

2.4 Statistical Analysis 21

3 Results 23 3.1 Data Review 23

3.1.1 Baseline Characteristics 23

3.1.2 Frequency of the Predictor Variables 25

3.2 Sample Correlation Coefficients 25

3.3 Multiple Linear Regression Analysis 27

3.3.1 bmi as the response variable 27

3.3.2 glucose as the response variable 28

3.3.3 dbpdia as the response variable 29

3.3.4 dbpsys as the response variable 30

CONTENTS v

3.3.5 Conclusion of multiple linear regression analysis 31

3.4 Canonical Correlation Analysis (CCA) 32

3.4.1 Case I: two response variables dbpdia and bdpsys 32

3.4.2 Case II:two response variables bmi and glucose 32

3.4.3 Case III:three response variables dbpdia, dbpsys, and bmi 35

3.4.4 Case IV: four response variables dbpdia, dbpsys, bmi and glucose 35

3.4.5 Comparison of the four cases 38

4 Discussion 40 4.1 Conclusion 40

4.2 Similar Application of CCA for CRVE 42

4.3 Further Improvement 44

Hypertension, obesity, and diabetes are three common health problems in the world.Retinopathy usually refers to an ocular manifestation of systemic disease and are com-mon in older people There existed direct and indirect associations between these fourhealth problems a quantitative assessment in retinal microvascular caliber may provideinformation to the risks of these systematic health problems

The Singapore Malay Eye Study (SiMES) was a population-based cross-sectionalstudy in Singapore 3280 participants were sampled in the study Diastolic blood pres-sure (dbpdia), systolic blood pressure (dbpsys), body mass index (BMI) and glucosewere measured The diameters of all retinal arterioles and all retinal venules were mea-sured The purpose of this study is to use the statistical methods to quantify the centralretinal arteriole equivalent (CRAE) using all diameters of all retinal arterioles and thecentral retinal venule equivalent (CRVE) with all retinal venule diameters

Multiple linear regression analysis and canonical correlation analysis (CCA) hadbeen applied to quantify CRAE such that the Pearson correlations between CRAE and

SUMMARY vii

dbpdia, dbpsys, BMI and glucose were maximized, respectively The results showedthat the CCA is more appropriate to quantify CRAE in this study

List of Tables

3.1 Participants Characteristics in Singapore Malay Eye Study(N = 3280) 24

3.2 The Counts of the Response and Predictor variables (N = 3280) 26

3.3 The Pearson Correlation Coefficients between the Response and Pre-dictor variables 27

3.4 The Coefficients in Multiple Linear Regression Model using log(bmi) as a response variable 28

3.5 The Coefficients in Multiple Linear Regression Model using 1/glucose as a response variable 29

3.6 The Coefficients in Multiple Linear Regression Model using 1/dbpdia as a response variable 30

3.7 The Coefficients in Multiple Linear Regression Model using 1 3 √ dbpdia as a response variable 31

3.8 The First Standardized Canonical Coefficients in Case I 33

3.9 The First Standardized Canonical Coefficients in Case II 34

3.10 The First Standardized Canonical Coefficients in Case III 36

3.11 The First Standardized Canonical Coefficients (corrected) in Case IV 37

3.12 The Maximum Canonical Correlations 39

3.13 The Pearson Correlation Coefficients between CRAE Estimated in Each Case and the Response Variables 39

LIST OF TABLES ix

4.1 The Pearson Correlation Coefficients between the Response and dictor variables 424.2 The Maximum Canonical Correlations 434.3 The Pearson Correlation Coefficients between CRVE Estimated in EachCase and the Response Variables 44

Pre-List of Figures

4.1 Histograms of the Response and Predictor Variables 45

Hypertension or high blood pressure for adults is defined as a systolic blood pressure

of 140 mmHg or higher or a diastolic blood pressure of 90 mmHg or higher Normalblood pressure is a systolic blood pressure of less than 120 mmHg and a diastolic bloodpressure of less than 80 mmHg Having high blood pressure increases one’s chancefor developing heart disease, a stroke, and other serious conditions (www.cdc.gov) Asystematic review on the worldwide prevalence of hypertension reported varied preva-lence from the lowest in rural India (3.4% in men and 6.8% in women) to the highest

in Poland (68.9% in men and 72.5% in women) in the period from 1980 to 2003 InSingapore, the 2004 National Health Survey showed that the crude prevalence of hy-pertension was decreasing among adults aged between 30 and 69, from 27.3% in 1998

to 24.9% in 2004 but still in a relatively high level (www.openclinical.org)

Obesity and overweight are defined as abnormal or excessive fat accumulation thatpresents a risk to health A crude population measure of obesity is the body mass index(BMI), a person’s weight (in kilograms) divided by the square of his or her height (inmetres) A person with a BMI of 30 or more is generally considered obese A personwith a BMI equal to or more than 25 is considered overweight Overweight and obesityare major risk factors for a number of chronic diseases, including diabetes, cardiovas-cular diseases and cancer Once considered a problem only in high-income countries,overweight and obesity are now dramatically on the rise in low- and middle-incomecountries, particularly in urban settings (www.who.int) According to the WHO, theavailable global database on BMI in 2004 showed that the prevalence of obesity rangedfrom more than 20% in the USA, Seychelles, and New Zealand to less than 10% in Sin-gapore and some other countries The major concern is the increasing trend of obesityprevalence with age among adult people The peak prevalence was reached at around

50 to 60 years old in most developed countries and earlier at around 40 to 50 years old

in many developing countries (Low et al., 2009).

Diabetes is a chronic disease, which occurs when the pancreas does not produceenough insulin, or when the body cannot effectively use the insulin it produces This

Chapter1: Introduction 3

leads to an increased concentration of glucose in the blood (hyperglycaemia) (www.who.int).According to International Diabetes Federation (IDF), there were 246 million peoplewith diabetes in the seven regions of IDF in 2007 while 194 million in 2003 (www.idf.org)

In Singapore, Males had higher proportion (8.9%) of diabetes than females (7.6%).Among different ethnic groups, Indian had highest prevalence of diabetes (15.3% com-pared to 7.1% in Chinese and 11% in Malays) (www.moh.gov.sg)

Retinopathy frequently refers to an ocular manifestation of systemic disease, such assingle microaneurysm, retinal haemorrhage, soft exudates, cotton-wool spots, venularbeading, neovessel formation These abnormalities are common fundus findings in

older people (Wong et al., 2001 and Wong et al., 2003) The Atherosclerosis Risk

in Communities (ARIC) Study found that the prevalence of retinopathy was 7.7% inAfrican Americans and 4.1% in Whites aged 49 years or over The Australian DiabetesObesity and Lifestyle Study reported that retinopathy was common (6.7%) in persons

aged 50 years or up with impaired glucose metabolism (Wong et al., 2005).

There are complicated relationships found between retinopathy and hypertension,diabetes, or obesity Hypertension is a significant risk factor for retinal abnormalities

In The Atherosclerosis Risk in Communities (ARIC) Study, higher blood pressure wasfound to be associated with some retinal changes (including focal arteriolar narrowing(FAN), arteriovenous (AV) nicking, and retinopathy) controlling for age, race, gender,and smoking status When mean arteriolar blood pressure (MABP) increased every10-mmHg, FAN had an odds ratio (OR) of 2.00 with 95% confidence interval (CI) of

1.87-2.14, AV nicking had an OR of 1.25 with 95% CI of 1.16-1.34, and retinopathy had

an OR of 1.25 with 95% CI of 1.15-1.37 (Mimoun et al., 2009; Hubbard et al., 1999).

A prospective cohort study also found that incident retinopathy was related to higher

MABP with OR of 1.5 (95% CI = 1.0-2.3) (Wong et al., 2007) Apart from

hyperten-sion, diabetes is also a risk factor of retinopathy Retinopathy signs are common duringdiabetes, but the earliest stages of some abnormalities such as retinal haemorrhages,microaneuryms and cotton wool spots can be observed within non-diabetes (Mimoun

et al., 2009) The prospective population-based cohort study reported the three-year

retinopathy incidence of 10.1% and cumulative prevalence of 27.2% among personswith diabetes while of incidence of 2.9% and cumulative prevalence of 4.3% among

persons without diabetes (Wong et al., 2007) The relationship between obesity and

retinopathy is not clear Although obesity has been linked with diabetic retinopathy,age-related cataract, and other different eye diseases, there are inadequate evidences to

support any convincing associations for many ocular conditions (Cheung et al., 2007).

Retinal microvascular abnormalities include focal arteriolar narrowing,

arteriove-nous (AV) nicking, and retinopathy (Hubbard et al., 1999) The findings mentioned

above have shown the strong correlation between retinopathy signs and hypertensionand diabetes or the possible linkage with obesity The progress of computerized reti-nal imaging technology has allowed more accurate and reproducible analyses to studyretinal microvascularisation or their early structural changes through non-invasive mea-

surement (Mimoun et al., 2009; Hubbard et al., 1999; Sherry et al., 2002 and Leung et al., 2003) The ARIC study reported that higher blood pressure were significantly asso-

Chapter1: Introduction 5

ciated with several microvascular changes Focal arteriolar narrowing had an OR of 2

(95% CI = 1.87-2.14) for every 10-mmHg MABP increase (Hubbard et al., 1999) The

Beaver Dam Eye Study (BDES) showed the reverse association between retinal

arteri-olar diameters and higher blood pressure (Wong et al., 2004) In addition to that, both

diabetes and retinopathy were associated with larger retinal arteriolar caliber

(Tikel-lis et al., 2007) Participants with diabetes had larger caliber (178.9 um) compared

to the ones with newly diagnosed diabetes (175.6 um, p=0.047), IGT/IFG (175.5 um,p=0.02), or NGT (174.6 um, p=0.02) after multivariable adjustment Besides that, withdiabetes or IGT/IFG, people with each SD increase of venular caliber had higher odds todevelop retinopathy (OR=1.68, 95% CI=1.23-2.29 or OR=1.78, 95% CI=1.36-2.34, re-

spectively) (Tikellis et al., 2007) A further support was given by the Multiethnic Asian

Population-based cross-sectional study, which showed the positive association betweenretinal arteriolar diameters and diabetes or between venular diameters and glucose level

(Jeganathan et al., 2009) These findings suggest that a quantitative assessment in

reti-nal microvascular caliber may provide information to the risks of certain systematichealth problems such as hypertension, diabetes, obesity or retinopathy caused by these

problems (Wong et al., 2004).

1.2 Statistical background

Canonical correlation analysis (CCA) is a statistical method that has been used in thisthesis This method was developed by Hotelling (1935) and is an extension of principal

components analysis (PCA) (Poore and Mobley, 1980) PCA is a multivariate dataanalysis procedure that transform a large group of possibly correlated variables to asmaller number of uncorrelated variables known as principle components (Hardoon,Szedmak and Shawe-Taylor, 2004) CCA is a statistical method to study the linearrelationships between two sets of variables with two or more variables in each set and

to determine the particular variables which attribute to this relationships This methodcan be seen as the problem to select the linear functions of the two sets of variables suchthat the correlation between the two linear functions is maximized

There are two important applications for canonical correlation analysis One is todetermine the partical attributes which are responsible for the relationships between twosets of variables Canonical correlation analysis has been useful in many areas Pooreand Mobley (1980) concluded CCA as an effective analysis tool in studying the marinebenthic survey data Meer (1991) presented CCA method to explore macrobenthos -environment relationship Young and Matthews (1981) investigated the relationshipbetween plant growth and environmental factors and concluded CCA as a powerfultool to analyze the multivariate field data Besides in the ecology study, CCA has also

been used in psychological area (Wade et al., 1992; Han et al., 1996; Philippaerts et al.,

1999) In the studies of food fraud, CCA can be used to detect the orange juice dilutions

masked by adding citric acid and sugars (Capilla et al., 1988) Besides these, CCA

has been used to identify the hydrological neighborhoods in regional flood frequency

analysis (Ribeiro-Correa et al., 1995) Another application of CCA is to estimate a new

resource which can summarize the set of known variables Wasimi (1993) proposed

Chapter1: Introduction 7

canonical correlation model to estimate the channel depth during floods

1.3 Aims and Organization of The Thesis

As mentioned earlier, a quantitative assessment in retinal microvascular caliber mayprovide information to the risks of certain systematic health problems such as hyper-tension, diabetes, obesity or retinopathy caused by these problems These three healthproblems are usually defined by diastolic blood pressure, systolic blood pressure, bodymass index (BMI) and glucose, which constitute the response variable set The diam-eters of all retinal arterioles were measured and summarized into a group of variablescalled the predictor variable set The aims of this thesis are to explore the statisticalmethods, such as CCA and multiple linear regression analysis, to estimate a single cen-tral retinal artery equivalent (CRAE) such that the correlation between CRAE and eachresponse variable is maximized Similarly, a single central retinal venular equivalent(CRVE) was estimated using the measurements of diameters of all retinal venules Fur-ther detail on this approach is provided in the Methods section

This thesis is divided into four sections: Introduction, Methods, Results, and cussion

Dis-In Dis-Introduction section, the first part described the relationship between the threecommon health problems and retinopathy and the association existed between the di-ameters of retinal vasculature and those health problems The second part gave the

description of statistical method - CCA and literature review on it’s applications Thethird part described the purpose and organization of the thesis.

In Methods section, the first part described in detail the sample selection and somemeasurements of data in Singapore Malay Eye Study The second part presented thedetail of multiple linear regression analysis The third part presented the method ofcanonical correlation analysis in detail

In Results section, the first part gave the description data review The second partshowed the individual Pearson correlation coefficients The third part showed the ini-tial results from multiple linear regression analysis The fourth part showed the initialresults from canonical correlation analysis

In Discussion section, the conclusion was made based on the Results Then thesimilar application of CCA for CRVE was presented to approve the conclusion Finally,further improvements were proposed in the analysis

Chapter 2: Methods 9

Chapter 2

Methods

2.1 The Singapore Malay Eye Study Data Review

The Singapore Malay Eye Study was a population-based cross-sectional study in

Sin-gapore (Foong et al., 2007; Su et al., 2007 and Cheung et al., 2008) The rationale

and methodology for the study have been described in detail previously The ple frame consisted of all Malays aged 40-79 years residing in 15 residential districtsacross the southwestern part of Singapore An initial list of 10696 Malay names wascomputer-generated from the sample frame through a simple random sampling proce-dure From this initial list of 10696 names provided by the Ministry of Home Affairs,

sam-a finsam-al ssam-ampling frsam-ame of 5600 nsam-ames wsam-as selected by using sam-an sam-age-strsam-atified rsam-andomsampling procedure, which was 1400 people from each decade of 40-49, 50-59, 60-

69, and 70-79 Of 5600 Malay names, 4168 (74.4%) were determined to be eligible

to participate in the study based on the inclusion criteria mentioned previously.13 Ofthese, 3280 participants were examined in the clinic while 888 (21.3%) were remained

Retinal Vessel Caliber Measurement The color retinal photographs were takenfor both eyes of all participants after pupil dilation Then the retinal photographs wereconverted to digital images by a high-resolution scanner The scanned images weredisplayed on monitors The trained graders read the diameters of all retinal vesselsthrough a specific area based on a standard protocol In the data set being currentlystudied, the diameters of all arterioles were recorded as a1, a2, , a14 and of all venules

recorded as v1, v2, ,v14 (Hubbard et al., 2004; Foong et al., 2007 and Wong et al.,

2004)

Chapter 2: Methods 112.2 Multiple Linear Regression Analysis

Let us consider a dataset with n observations The response variable is Y We have

follows:

Y i = β0+ β1X i1+ β2X i2+ · · · + βp−1 X i,p−1+ i

where: β0, β1, , βp−1 are parameters, Y i , X i1 , X i2 , , X i,p−1 are observations, i are

in-dependent normal error terms, with E[] = 0 and variance σ2, i = 1, , n In this study, the univariate Y denotes response variable dbpdia, or dbpsys, or BMI, or glucose; the

set of predictor variables indicates a1 to a14, or v1 to v14

2.2.1 Estimation of the parameters

The parameters in linear regression models are typically estimated by the method ofleast squares let us define

y = xβ + 

The vector of least squares estimators ˆβ will be found to minimize

ˆβ = (x0x)−1x0y

Therefore,the fitted values are expressed as

ˆy = x ˆβ

Properties of Least Squares Estimates We assumed that the errors are unbiased

which means E[] = 0, then the Least Squares Estimates are unbiased since

E[ ˆβ] = (x0x)−1x0E[y] = (x0x)−1x0xβ = β

The consistency property of covariance β is denoted in the covariance matrix asfollows:

Cov( ˆβ) = E[ ˆβ − E( ˆβ)][ ˆβ − E( ˆβ)]0 = σ2(x0x)−1where the unbiased estimator of σ2is given by

ˆ

σ2= S S E

n − p

Chapter 2: Methods 13

Of these, S S E is called the residual sum of squares which can be shown as:

S S E = y0y − ˆβx0yand

Test for Significance of Regression (F - test) This is a test to determine whether

there is association between y and a subset of the predictor variables X1, X2, , X p−1.The hypotheses are:

H0 : β1= β2 = · · · = βp−1= 0

H1 : βj , 0 for at least one j

The test statistic is

y0y − ˆβx0y F0 follows F distribution with degree of freedom p − 1 and n − p If

F0 > F α,p−1,n−p or the P-value for the F0is less than α, then we reject H0

Tests for Individual Regression Coefficients(t - test) The hypotheses for testingany individual coefficient in the regression (βj) are

The test statistic is

p

ˆσ2C j j where C j j is the ( j j)th element of the (x0x)−1 If |t0| > t α/2,n−p , the null hypothesis H0isrejected

Confidence Intervals on the Individual Regression Coefficients Since ˆβ is alinear combination of the observations, ˆβ follows normal distribution with mean vector

β and covariance matrix σ2(x0x)−1 So each of the statistics

ˆβj− βjp

ˆσ2C j j j = 0, 1, , p − 1

is distributed as t with n − p degree of freedom, where C j j is the ( j j)th element of the

(x0x)−1 Thus, a 100(1 - α) percent confidence interval for the regression coefficient

Unlike R2, the adjusted R2 statistic will not always increases when adding terms to the

model In fact, the value of R2

ad j will decrease if unnecessary terms are added

Chapter 2: Methods 15

2.2.2 Estimation of sample correlation

Given a series of n observations of X and Y written as x i and y i where i = 1, 2, , n,

the Pearson product-moment correlation coefficient can be used to estimate the sample

correlation of X and Y The Pearson correlation coefficient is defined as follows:

where S x and S y are the sample standard deviation of X and Y.

The Pearson correlation coefficient is used to describe the linear relationship

be-tween two variables The range of r is bebe-tween -1 and +1 When r < 0, it indicates that two variables are negatively related When r > 0, it indicates that two variables

are positively related The closer the coefficient is to either -1 or 1, the stronger thecorrelation between the variables If the two variables are uncorrelated , the coefficient

is zero

2.2.3 Box-Cox Power Transformation

In this multiple linear regression analysis, the Box-Cox method was used to transformthe response variable to make the data more like normal (Carroll and Ruppert, 1981)

Suppose for data vectors (y1, , y n ) in which each y i > 0 and λ is the power parameter,the power function is defined in the following:

2.2.4 Model Adequacy Checking

After the regression models are built, the following important procedure is model quacy checking which is equally important in building models The regression assump-tions usually needed to be evaluated are constancy of variance and normality of errors.The adequate way to do this is to check the residual plots Three types of residuals arefrequently used in model checking: the residuals (or the ordinary residuals), the stan-dardized residual, and the studentized residuals The ordinary residuals are defined asfollows:

Chapter 2: Methods 17

The studentized residuals are defined as follows:

ˆσ2(1 − h ii) i = 1, 2, , n

where h ii is the ith diagonal element of H, which is an n by n matrix x(x0x)−1x0

The plot of residuals on the y axis against fitted values on the x axis can be used tocheck the assumption of the constancy of variance The plot of standardized residualsagainst the theoretical quantiles will be used to check whether the errors are normally

distributed The details are described in books written by Seber (1977) and Kutner et al.(2004).

Multiple linear regression analysis studies the linear relationship between thesingle response variable and a set of the predictor variables In order to study thecorrelation between two sets of variables, the canonical correlation analysis hasbeen used in this study

2.3 Canonical Correlation Analysis (CCA)

2.3.1 Canonical Correlations and Variates in the Population

Canonical correlation analysis has been used to study the correlations between two sets

of variables (Anderson, 2003) Suppose the random vector Z of p components has the covariance matrix Σ which is assumed to be positive definite Assume E[Z] = 0 since

only variance and covariance are of interest in the analysis

For convenience, assume p1 ≤ p2, we shall partition the Z into two subvectors Y∗

and X∗with p1and p2components, respectively,

1 = E[U2] = E[α0Y∗Y∗0α] = α0Σ11α (2.4)

1 = E[V2] = E[γ0X∗X∗0γ] = γ0Σ22γ (2.5)

note that E[U] = 0 and E[V] = 0 Thus, the goal is to find α and γ to maximize the

correlation between U and V, which is,

where λ and µ are Lagrange multipliers We take partial derivatives of ψ with respect to

α and γ, then set each equation to zero, which are

−λΣ11 Σ12

Σ21 −λΣ22

(1.15) is a polynomial equation with degree p and has p roots, denoted as λ1 ≥ λ2 ≥

· · · ≥ λp since Σ is positive definite and |Σ11| · |Σ22| , 0

From (1.6) we can see that λ = α0Σ12γ is the correlation between U = α0Y∗ and

V = γ0X∗ when α and γ satisfy (1.14) for some value λ λ = λ1 is the maximumcorrelation A solution to (1.14) for λ = λ1is denoted as α(1)and γ(1), then U1= α(1)0Y∗

and V1 = γ(1)0X∗ Thus U1 and V1 are first normalized linear combinations of Y∗and

X∗with the maximum correlation of λ1

2.3.2 Estimation of Canonical Correlation and Variates

Suppose there are N observations, Z1, , ZN, from N(µ, Σ) Zi is partitioned into two

subvectors with p1and p2components,

X∗ i

i − ¯Y∗

i)(Y∗

i − ¯Y∗

i)0 P(Y∗

i − ¯X∗

i)(Y∗

i − ¯Y∗

i)0 P(X∗

X∗with the maximum correlation of λ1

2.3.2 Estimation of Canonical Correlation and Variates... the correlation between U = α0Y∗ and

V = γ0X∗ when α and γ satisfy (1.14) for some value λ λ = λ1 is the maximumcorrelation