Hierarchical Modeling of Biological Rhythm Data: Classical ... · Hierarchical Modeling of Biological Rhythm Data: Classical and Bayesian Approaches by Prakash Dhakal Submitted in

Hierarchical Modeling of Biological Rhythm Data:Classical and Bayesian Approaches

by

Prakash Dhakal

Submitted in Partial Fulfillmentof the Requirements for the Degree of

Master of Science in Mathematicswith Specialization in Operations Research and Statistics

New Mexico Institute of Mining and TechnologyDepartment of Mathematics

Socorro, New MexicoAugust, 2016

ABSTRACT

Biological rhythm data is collected by measuring a physiological variable multi-ple times at different time points. These rhythms exhibit a cyclic pattern over a periodof time which can be modeled as a cosine function of time. The biological rhythmsvary considerably among the subjects. This thesis explains and develops the hierarchi-cal (multilevel) models as a technique to analyze this kind of data from both classicaland Bayesian perspectives. The models are further applied to analyze the circadianrhythms of systolic blood pressure obtained by ambulatory blood pressure monitoringof hemodialysis patients with hypertension. The estimates obtained by using both theapproaches are compared.

Keywords: HIERARCHICAL; MULTILEVEL; ABPM; BLOOD PRESSURE; COSINOR;RANDOM EFFECT; BAYESIAN

ACKNOWLEDGMENTS

I would like to thank my adviser, Dr. Anwar Hossain, for his support and guid-ance in all aspects of my degree at New Mexico Tech. Special thanks to Dr. BrianBorchers and Dr. Oleg for their support on my committee. Also, I would like to thankDr. Gabriel Huerta from University of New Mexico for providing us the data. Finally, Icannot forget all the friendly faculty and staffs of the Mathematics department.

This thesis was typeset with LATEX1 by the author.

1The LATEX document preparation system was developed by Leslie Lamport as a special version ofDonald Knuth’s TEX program for computer typesetting. TEX is a trademark of the American MathematicalSociety. The LATEX macro package for the New Mexico Institute of Mining and Technology thesis formatwas written by John W. Shipman.

ii

CONTENTS

LIST OF TABLES v

LIST OF FIGURES vi

LIST OF ABBREVIATIONS viii

1. INTRODUCTION 11.1 Biological Rhythm Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Properties of a Biological Rhythm . . . . . . . . . . . . . . . . . . . 21.2 Ambulatory Blood Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. HIERARCHICAL MODELS 42.1 Repeated Measures Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Historical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Hierarchical Models for Repeated Measures . . . . . . . . . . . . . . . . . . 5

2.3.1 Random Intercept Model . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Random Intercept and Slope . . . . . . . . . . . . . . . . . . . . . . 82.3.3 Parameter Estimation: Classical Approach . . . . . . . . . . . . . . 11

2.4 Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.1 Hierarchical Modeling in Bayesian Framework . . . . . . . . . . . . 14

3. ANALYSIS OF AMBULATORY BLOOD PRESSURE MONITORING (ABPM)DATA 173.1 Study Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Data Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Cosinor Model for Rhythmic Data . . . . . . . . . . . . . . . . . . . . . . . 193.4 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.2 Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Interpretation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iii

4. DISCUSSION 414.1 Comparison of Classical and Bayesian Approaches . . . . . . . . . . . . . . 41

4.1.1 Single Cosinor Model . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Random Intercept Model . . . . . . . . . . . . . . . . . . . . . . . . 424.1.3 Random Intercept and Slope Model . . . . . . . . . . . . . . . . . . 42

4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Appendices 45

A. SAS CODES 46A.1 Individual Means with Standard Error Bars . . . . . . . . . . . . . . . . . . 46

B. R CODES 48B.1 Individual Patient Profiles of 3 Subjects . . . . . . . . . . . . . . . . . . . . 48B.2 Parameter Estimation using Maximum Likelihood . . . . . . . . . . . . . . 48

C. OPENBUGS CODES 51C.1 Single Cosinor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51C.2 Random Intercept Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51C.3 Random Intercept and Slope Model . . . . . . . . . . . . . . . . . . . . . . 52

REFERENCES 55

iv

LIST OF TABLES

3.1 Variable Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Comparison of Model Parameters . . . . . . . . . . . . . . . . . . . . . . . 283.3 Comparison of Bayesian Model Parameters . . . . . . . . . . . . . . . . . . 40

4.1 Comparison of Single Cosinor Model . . . . . . . . . . . . . . . . . . . . . . 424.2 Comparison of Random Intercept Model . . . . . . . . . . . . . . . . . . . 424.3 Comparison of Random Intercept and Slope Model . . . . . . . . . . . . . 43

v

LIST OF FIGURES

1.1 Properties of a Sinusoidal Biological Rhythm . . . . . . . . . . . . . . . . . 2

2.1 Directed Acyclic Graph of Random Intercept Model . . . . . . . . . . . . . 152.2 Directed Acyclic Graph of Random Intercept and Slope Model . . . . . . 16

3.1 Ambulatory Systolic Blood Pressure Profiles of 3 Different Subjects Ap-proximately 44 Hours Post-dialysis . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Mean Ambulatory Systolic Blood Pressure Profiles for all 49 Subjects Ap-proximately 44 Hours Post-dialysis . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Single Cosinor Model Fit for 3 Subjects . . . . . . . . . . . . . . . . . . . . . 233.4 Serial Autocorrleation of Residuals Using Single Cosinor Model . . . . . . 233.5 Random Intercept Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 243.6 Serial Autocorrleation of Residuals Using Random Intercept Model . . . . 253.7 Random Intercept and Slope Model Fit . . . . . . . . . . . . . . . . . . . . 273.8 Serial Autocorrleation of Residuals from Random Intercept and Slope

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.9 Convergence History of Two MCMC Chains for the Parameters in Single

Cosinor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.10 Summary of Posterior Distribution Estimates of Single Cosinor Model Pa-

rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.11 Convergence History of Two MCMC Chains for the Parameters in Ran-

dom Intercept Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.12 Summary of Posterior Distribution Estimates of the Random Intercept

Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.13 Caterpillar Plot of the Mean of the Posterior Distribution of Varying In-

tercept(M) of 49 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.14 Convergence History of Two MCMC Chains for the Parameters in the

Random Intercept and Slope Model . . . . . . . . . . . . . . . . . . . . . . 353.15 Summary of Posterior Distribution of the Random Effect Parameters . . . 353.16 Posterior Distribution and Trace Plot of Convergence of Within-subject

Variation σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.17 Caterpillar Plots of Random Intercept M and Random Slope β . . . . . . . 363.18 Caterpillar Plots of Random Slopes : γ and α . . . . . . . . . . . . . . . . . 37

vi

3.19 Correlations Between Slopes and Intercept . . . . . . . . . . . . . . . . . . 38

vii

LIST OF ABBREVIATIONS

ABPM Ambulatory Blood Pressure Monitoring

ACCORD Action to Control Cardiovascular Riskof Diabetes

AIC Akaike Information Criterion

ANOVA Analysis Of Variance

BIC Bayesian Information Criterion

BLUE Best Linear Unbiased Estimator

BLUP Best Linear Unbiased Predictor

BP Blood Pressure

DIC Deviance Information Criterion

GLS Generalized Least Squares

MANOVA Multivariate Analysis Of Variance

MESOR Midline Estimating Statistic of Rhythm

MCMC Markov Chain Monte Carlo

MLE Maximum Likelihood Estimation

mm Hg millimeter of mercury

MVN Multivariate Normal

N Normal

NIH National Institute of Health

NLME Nonlinear Mixed Effects

OpenBUGS Open Bayesian inference Using Gibbs Sam-pling

R R Statistical Software

REML Restricted Maximum Likelihood

SAS Statistical Analysis System

viii

SBP Systolic Blood Pressure

SD Standard Deviation

SE Standard Error

SPRINT Systolic Blood Pressure Intervention Trial

WHO World Health Organization

ix

This thesis is accepted on behalf of the faculty

of the Institute by the following committee:

Dr. Anwar Hossain

______________________________________________________

Academic Advisor

Dr. Anwar Hossain

______________________________________________________

Research Advisor

Dr. Anwar Hossain

______________________________________________________

Committee Member

Dr. Brian Borchers

_______________________________________________________

Committee Member

Dr. Oleg Makhnin

______________________________________________________

Committee Member

I release this document to New Mexico Institute of Mining and Technology

Prakash Dhakal 07/01/2016

________________________________________________________

Student Signature Date

CHAPTER 1

INTRODUCTION

There are various biological phenomena prevalent in both the plant and animalkingdoms that exhibit rhythmic patterns. The temporal characteristics of biological datahave been of great research interest for clinicians and bio-statisticians in recent years. Afield of biology that studies biological data and their association with time or cycle isknown as ”Chronobiology” [28]. The main purpose of this thesis is to present statisticalmethodologies to study the rhythmic characteristics of biological data repeatedly mea-sured from subjects over a period of time. A real dataset of ambulatory blood pressure[45] that exhibits rhythmic patterns is analyzed in Chapter 3 by using the hierarchicalmodels discussed in Chapter 2.

1.1 Biological Rhythm Data

The rhythms exhibited by different biological phenomenon have different fre-quencies. There are considerable variations in the values of the measurements amongthe individuals but the periodicity of these rhythms are consistent. For example, humanbody temperature, blood pressure and other many biological phenomena exhibit an ap-proximate period of 24 hours. These rhythms are called circadian rhythms. The earlywork on the study of circadian rhythm was done by Halberg [28].

The study of biological rhythms has been influential in the treatment of hyperten-sion [10], treatment of cancer[2], sleep-wake phenotype and disorders [16] [62], obesityand type 2 diabetes [11] [27], major depressions and bipolar disorders [43]. The stud-ies are usually conducted by analyzing the data measured from subjects over a periodof time. The inclusion of time in chronobiological investigations makes it possible forquantitative analysis of biological rhythms with respect to time [12]. The inferences fromthe parameter estimation of the rhythmic characteristics of biological data can servemany research purposes. Early detection of alteration in rhythm characteristics of bio-logical data can be indicative of a heightened health risk [28].With the help of statisticaltools, not only within subject rhythmic variability but also, between subject variabilitycan be studied. A study of such rhythms alongside the influence of treatment drugs canbe helpful in the optimization of treatment by timing [12].

With advances in mobile technology, the experimental endpoints can be moni-tored more continuously over time. More repeated measurements can give us deeperinsights in understanding biological rhythm data.

1

1.1.1 Properties of a Biological Rhythm

The full characterization of a biological rhythm requires five important attributes:mean, period, amplitude, phase and waveform.

Figure 1.1: Properties of a Sinusoidal Biological Rhythm

• Mean Level(MESOR): The dotted line represents the central value around whichthe cycle oscillates. It is also called the Midline Estimating Statistic of Rhythm(MESOR).

• Period: It is the duration of a full cycle i.e. the time between two consecutive peaks(or troughs).

• Amplitude: It is half the range of oscillation with the given period.

• Acrophase: It is the measure of the time at which the cycle reaches its peak. It mayvary for each cycle [12].

• Waveform: The waveform describes the shape of the rhythm. As in Figure 1.1, thebiological rhythm is sinusoidal. It is not necessary that all the biological rhythmshave sinusoidal waveforms.

2

1.2 Ambulatory Blood Pressure

Blood pressure is the pressure exerted on the walls of blood vessels during circu-lation [56]. It is measured in terms of systolic pressure (when the heart muscle contracts)and diastolic pressure (when the heart muscle is resting between beats) in millimetersof mercury (mmHg). It is a common and important prognosis tool. According to WHO,the normal adult blood pressure is 120/80 mm Hg. When a systolic blood pressure isequal or above 140 mm Hg and/or a diastolic blood pressure is equal to or above 90 mmHg, it is considered to be hypertension or high blood pressure. It is one of the leadingcauses for 9.4 million deaths worldwide every year [47].

Mercury sphygmomanometers have been the popular method of measuring bloodpressure for many years. The use of Ambulatory (or Automated) Blood Pressure Moni-tors (ABPM) has been becoming popular in both clinical research and practice recently[8]. An ABPM device is a portable device that can be easily worn over the shoulder withthe help of a strap or a belt. It collects the patient 's blood pressure readings at pre-settimes. Hence, it becomes possible to investigate the blood pressure profile over a con-tiuious period of 24-hour period or longer. Following ABPM data analysis, it has beenfound that the blood pressure dips at night [55]. Many studies have associated dippingin blood pressure with diseases like chronic renal disease and pre-eclampsia [51].

Blood pressure measurements obtained in an office, hospital, or clinic are prone toerrors and variations in readings [53]. The use of ABPM has significant benifits over con-ventional sphygmomanometry. Since ABPM is automated, there is little risk of the ob-server errors [50] and faulty equipment errors[5]. Another common problem is knownas ’white-coat syndrome’. It has been found that some individuals’ blood pressure in-creases in the presence of a health professional [39]. The use of automated techniquesfor measuring blood pressure can reduce such a problem [53]. Another advantage of us-ing ABPM is that it can reduce sampling variation since it involves an increased numberof readings compared to sphygmomanometry, where often clinical decisions are basedon only one reading [7]. Mercury contamination is another possible risk associated withthe conventional sphygmomanometer [40].

The data analysis of ABPM collected measurements is challenging as there are alarge number of observations per subject. A mere descriptive statistic of the data is notenough to describe the overall profile of the subject or the population. More advancedmodels like hierarchical models(which are described in Chapter 2) do more justice toanalysis of this kind of data.

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CHAPTER 2

HIERARCHICAL MODELS

In this chapter, the theoretical aspects of hierarchical models are discussed. Theimportance and appropriateness of the models in longitudinal data will be elaborated.The application of the models on ABPM data is discussed in Chapter 3.

2.1 Repeated Measures Data

When multiple measurements are made on an experimental unit over a periodof time, the data obtained is called repeated measures data. This design of data is alsocalled longitudinal data or panel data. In clinical studies, the measurement might be vi-tal signs like blood pressure, body temperature or lab test parameters like cholesterollevel, serum glucose level, drug blood concentrations, etc. from each individual, at dif-ferent times and under different conditions. The primary interest of repeated measuresanalysis is to determine the variation between experimental units and the variation inmeasurements within an experimental unit. Since the repeated measures are nestedwithin the experimental units, this type of data is also called hierarchical data.

The data of this form is analyzed by using some variant of a two-level model. Inthis formulation, the key idea is to assume that the probability distribution for the multi-ple measurements is the same for each experimental unit but the distribution parametersvary over the experimental units [32]. The first level of heirarchy is the distribution ofthe repeated measurements over time in each experimental unit. The second level is thedistribution of the parameters that vary over the units [32]. These are the random effectsthat can be used to quantify and explore the nested structure of the data to study thewithin-unit and between-unit variation.

2.2 Historical Approaches

One of the cumbersome aspects in analyzing repeated measures data is that themeasurements are correlated over time. A simple statistical technique of analysis canbe reducing the vector of responses for each subject to a single value by a descriptivestatistic. Then, standard univariate approaches can be applied to test the correspondingsummary measure[42][46]. This method is simple and efficient when the sample size isnot sufficiently large. Although this approach is simple, it is sensitive to missing data

4

or irregularly spaced repeated measures [46]. When n measurements are replaced by asingle summary statistic, it results in loss of some information [18].

Other statistical techniques for the analysis of repeated measures data are Anal-ysis of Variance (ANOVA) and Multivariate Analysis of Variance (MANOVA) [15]. Theunivariate ANOVA model can be expressed as

Yij = Xᵀijβ + bi + eij (2.1)

where Yij is the vector of response variable, Xij is a design vector of indicator variables,bi is a random effect specific to subject, bi ∼ N(0, σ2

b ), eij is the within-subject error ,eij ∼ N(0, σ2

e ). This assumption induces a compound symmetry structure for the co-variance with the diagonal elements representing the variance as Var(Yi) = σ2

b + σ2e and

covariance as Cov(Yij, Yik) = σ2b [18].

ANOVA regards time as a factor of n levels in a hierarchical design with units assub-plots. This is also called split-plot design. The first limitation of ANOVA techniqueis that it fails to account for the covariance among repeated measures and assumes con-stant variance across time. The second limitation is that this method only works forbalanced array of data i.e. the number of observations should be same for each unit atthe same time points. MANOVA is an extension of ANOVA with several dependentvariables. Both ANOVA and MANOVA focus on comparing groups in terms of theirmean response trend over time but provide little information about how the individualprofiles change over time [18]. With MANOVA, the modeling of covariance structurefurther becomes complicated [15].

2.3 Hierarchical Models for Repeated Measures

The term hierarchical models was introduced by Lindley and Smith in 1972 in thecontext of Bayesian estimation of the linear model [34]. In the literature, the modelingof hierarchical data has been referred to by various names such as: multilevel models[24][22], random effects models [32], mixed effects models [4], random coefficients mod-els [38] and variance component models [37]. The introduction of the EM algorithmgreatly favored the hierarchical modeling of data [13] [14]. The EM algorithm is aniterative algorithm to compute maximum-likelihood estimates. The advancement andaffordability of computational resources made the EM algorithm popular. Its implemen-tation in analyzing hierarchical data also became popular[32].

Let us discuss the different hierarchical models that we can consider for the dataanalysis. Let Yij denote the response variable for the ith subject (i = 1, .., N) at the jth

timepoint (j = 1, .., ni). Given that we have ni repeated measures on the same subjectobserved at time tij, the response variables can be grouped as an ni × 1 vector.

Yij =

Yi1Yi2...

Yini

5

With each response variable, Yij, there is a p× 1 vector of covariates or predictor vari-ables.

Xij =

Xij1Xij2

...Xijp

There is a corresponding vector of covariates associated with each of the ni repeatedmeasurements on the ith subject. We can group the vectors of covariates into an ni × pmatrix of covariates as:

Xi =

Xi11 Xi12 · · · Xi1pXi21 Xi22 · · · Xi2p

...... . . . ...

Xini1 Xini2 · · · Xini p

where each row of Xi corresponds to the Yi response at the ni different measurementsand the columns of Xi correspond to the p distinct covariates. Let us consider a linearregression for the mean response relating the changes to covariates over time[18],

Yij = β1Xij1 + β2Xij2 + ... + βpXijp + eij (2.2)

where β1, ..., βp are unknown coefficients relating the mean of Yij to its correspondingcovariates in Xi. Thus, there are ni separate regression equations to be solved for theresponse variable at each of the ni measurement. The eij are the random errors withmean zero:

E(Yij|Xij) = β1Xij1 + β2Xij2 + ... + βpXijp

Generally, Xij1 = 1 for all i and j, also called the intercept in the model.The matrix representation of regression equation (2.2) is:

Yi = Xiβ + ei (2.3)

where ei = (ei1, ei2, ..., eini)ᵀ is an ni × 1 vector of random errors associated with the

corresponding ith subject’s response vector. It is assumed that ei ∼ N(0, σ2). For acontinuous response variable, the response vector is assumed to be multivariate normaldistribution, with mean response vector:

E(Yi) = µi = Xiβ

and covariance matrix,Σi = Cov(Yi)

The serial correlation between the repeated measurements is captured by the off-diagonalelements of the covariance matrix, Σi [18].

To address the hierarchy of the data and the sources of natural heterogeneityamong the subjects, the hierarchical models consider some of the regression coefficients,β, to vary randomly between or within the subjects. The idea here is to model the meanresponse as a combination of population characteristics (estimates), β, that are commonto all subjects (fixed effects) and the specific effects of a particular subject (random effects)[18]. Thus, this type of model is also called mixed model.

6

2.3.1 Random Intercept Model

Let us consider data with two levels of hierarchy. The repeated measurementscollected over time (Level 1) are nested within the subjects(Level 2). Assuming thateach subject has some effect on the response in the population, we can rewrite the linearregression model (2.3) as:

Yij = Xᵀijβ + bi + eij (2.4)

where Yij is the response at the jth time point for the ith subject, bi is the random subjecteffect and eij is the sampling error at the jth time point. Both the bi and eij are assumedto be normal random variables with the following parameters

bi ∼ N(0, σ2b ), eij ∼ N(0, σ2)

Also, bi and eij are assumed to be independent of one another. In general, the model tellsus that the response at the jth time point for the ith subject differs from the populationmean, Xᵀ

ijβ by a random effect induced by subject level, bi, and a within-subject error,eij. Hence, we can write the mean response of any subject during j = 1, .., ni period tobe:

E(Yij|bi) = Xᵀijβ + bi

Let us examine equation (2.4) further [18]

Yij = Xᵀijβ + bi + eij

= β1Xij1 + β2Xij2 + ... + βpXijp + bi + eij= β1 + β2Xij2 + ... + βpXijp + bi + eij= (β1 + bi) + β2Xij2 + ... + βpXijp + eij

With Xij1 = 1 for all i and j, β1 represents the intercept which is the fixed effect. Sincebi is random, the intercept for the ith subject, β1 + bi, varies randomly between subjects.For simplicity, the two levels of effects can be written as:

Yij = β1i + β2Xij2 + ... + βpXijp + eijβ1i = β1 + bi

The marginal variance of each response, Yij, is given by:

Var(Yij) = Var(Xᵀijβ + bi + eij)

= Var(bi) + Var(eij)= σ2

b + σ2

The marginal covariance between any two responses, Yij and Yik , is

Cov(Yij, Yik) = Cov(Xᵀijβ + bi + eij, Xᵀ

ikβ + bi + eik)

= Cov(bi + eij, bi + eik)= Cov(bi, bi)= σ2

b

7

The correlation between any pair of responses is:

Corr(Yij, Yik) =Cov(Yij, Yik)√

Var(Yij)√

Var(Yik)(2.5)

ρ =σ2

bσ2

b + σ2(2.6)

The consideration of between subject variability, σ2b , induces the correlation among the

repeated measurements. Without the σ2b term, the correlation becomes zero. The ran-

dom intercept model assumes only intercept to vary whereas the slopes or other regres-sion coefficients are the same for all the subjects. The major drawback of this modelis that it assumes the correlation between measurements to be constant throughout allthe time-points. A more flexible model is necessary that can accommodate the varyingcorrelations among the covariates.

2.3.2 Random Intercept and Slope

A generalized form of hierarchical model discussed till now is a model that con-siders that the intercepts and regression coefficients vary randomly between subjectsinducing random effects covariance structure. Let us consider a response Yij measuredat time tij of the ith subject at the jth measurement occurrence j = 1, · · · , ni. The re-peated measurements can be inherently unbalanced over time. Using vector and matrixnotation, this model can be expressed as

Yi = Xiβ + Zibi + ei (2.7)

where β is a p× 1 vector of fixed effects, bi is a (q× 1) vector of random effects, Xi is a(ni × p) matrix of covariates, and Zi is a (ni × q) matrix of covariates, with q ≤ p [18].Zi is the design matrix to facilitate β components to vary randomly by including thecorresponding columns of Xi in Zi.

The vector of random effects, bi, are assumed to be multivariate normal withE(bi) = 0 and Cov(bi) = G.

The vector of errors, ei, are assumed to be multivariate normal with E(ei) = 0and Cov(ei) = Ri. Assuming a pair of errors eij and eik uncorrelated, we can denote Rito be a diagonal matrix, σ2 Ini , where Ini denotes an ni × ni identity matrix. We can selecta covariance pattern matrix for Ri that better fits the data [36]. However, modelingeij with a covariance pattern assigned to Ri, the eij’s may not be well interepreted asmeasurement errors [18]. Also, it may not be feasible all the time to estimate both Gand Ri from the available data. We assume that the random effects (bi) and the randomerrors(ei) are independent of each other.

The mean response profile of a subject or conditional mean, Yi, given bi is

E(Yi|bi) = Xiβ + Zibi (2.8)

8

The population averaged or marginal mean of Yi, taken mean over the random effects, bi,is

E(Yi) = E{E(Yi|bi)}= E(Xiβ + Zibi)= Xiβ + ZiE(bi)= Xiβ ∵ E(bi) = 0

(2.9)

Hence, the vector of regression parameters β (the fixed effects), are assumed to be thesame for all the subjects [18]. The vector bi (when combined with the corresponding β)are the random effects. The conditional covariance of Yi, given bi, is

Cov(Yi|bi) = Cov(ei) = Ri (2.10)

while the marginal covariance of Yi, averaged over the distribution of the randomeffects, bi, is

Cov(Yi) = Cov(Zibi) + Cov(ei)= ZiCov(bi)Zᵀ

i + Cov(ei)= ZiGZᵀ + Ri

Even when we assume the pair of measurements to be uncorrelated i.e. Ri =σ2 Ini , the Cov(Yi) = ZiGZᵀ + σ2 Ini , have non-zero off-diagonal elements, thereby ac-counting for the correlation among the repeated observations on the same subjects [18].Thus, between-subject(G) and within-subject(Ri) variations are effectively accounted forthis model.

To explain the hierarchy, let us consider

Xi = Zi =

1 ti11 ti2...

...1 tini

The equation 2.7 can be written as :

Yij = β1 + β2tij + b1i + b2itij + eij (2.11)

Level 1 of hierarchy can be thought of the subject level mean

Yij = β1i + β2itij + eij

where eij ∼ N (0, σ2)

Level 2 of hierarchy consists of random βi which can be expressed as a linearfunction of between-subject covariates,Ai ,

E(βi) = Aiβ and Cov(βi) = G

orβ1i = a11 + b1i

9

β2i = a21 + b2i

where (b1i

b2i

)= bi ∼ N

((00

),(

g11 g12g21 g22

))The variance of Yij

Var(Yij) = Var(β1 + β2tij + b1i + b2itij + eij)= Var(b1i + b2itij + eij)= Var(b1i) + 2tijCov(b1i, b2i) + t2

ijVar(b2i) + Var(eij)

= g11 + 2tijg12 + t2ijg22 + σ2

The variance is a function of time, tij and is not constant over tij. Similarly, the covarianceof measurements within-subject can be shown to be

Var(Yij, Yik) = g11 + (tij + tik)g12 + tijtikg22

In general, a two level model can be written as:Yi1Yi2...

Yini

=

1 ti11 ti2...

...1 tini

(

a11a21

)+

1 ti11 ti2...

...1 tini

(

bi1bi2

)+

ei1ei2...

eini

Yi = Xiβ + Zibi + ei

The elements on Zi is the subset of the columns of Xi. Hence, we can divide the columnsof Xi into a set of columns that corresponds to the fixed effects X(F)

i and a set of columns

that corresponds to the random effects X(R)i as [18].

Yi = X(F)i β(F) + X(R)

i β(R)i + ei (2.12)

where β has been partitioned into fixed effects β(F) and random β(R)i effects. As we

can see from equation (2.12) that the hierarchical model can be designed by accountingthe fixed effects and the random effects. The main goal here is to estimate the fixed ef-fects parameters and the random effects parameters. In practice, we find the MaximumLikelihood Estimation or Restricted Maximum Likelihood Estimation of the parameters.We, then, optimize the function using one of the algorithms like Expectation Maximiza-tion algorithm [35], Iterative Generalized Least Squares [24] and Newton-Raphson algo-rithm [35]. From Bayesian approach, the parameter estimation/prediction is done usingMarkov Chain Monte Carlo (MCMC) methods like Gibbs Sampling [64].

10

2.3.3 Parameter Estimation: Classical Approach

From the assumptions made in equation (2.7) and equation(2.8), the conditionalmodel can be written as multivariate normal distribution(MVN )

Yi|bi ∼MVN (Xiβ + Zibi, Ri) (2.13)

Similarly, from equation (2.9), the corresponding marginal model can be written as

Yi ∼MVN (Xiβ, Vi) (2.14)

where Vi = ZiGZᵀ + Ri

If Vi are assumed known, we can obtain the maximum likelihood estimate of fixedeffect,β, by finding β that maximizes log-likelihood function of multivariate marginalfunction equation (2.14) which is

l = −K2

log(2π)− 12

N

∑i=1

log|Vi| −12

(N

∑i=1

(Yi − Xiβ)ᵀV−1

i (Yi − Xiβ)

)(2.15)

where K = ∑Ni=1 ni and N=total number of subjects

Under the multivariate model assumptions, maximizing l, yields

β =

(N

∑i=1

Xᵀi V−1

i Xi

)−1 N

∑i=1

(Xᵀi V−1

i Yi) (2.16)

We can also show that β is unbiased [18], i.e. E(β) = β. Also, we can show that asymp-totically, the sampling distribution of β is multivariate normal distribution with mean,β, and covariance

Cov(β) =

(N

∑i=1

Xᵀi V−1

i Xi

)−1

(2.17)

Since the random effects, bi, are random variables with multivariate normal dis-tribution, we ”predict” the random effects rather than estimating them. The prediction ofa random variable is done by predicting its conditional mean, given the available data.i.e.

bi = E(bi|Yi) = GZᵀi V−1

i (Yi − Xi β) (2.18)

This is known as the Best Linear Unbiased Predictor (or BLUP). As noted by Liardand Ware [32], rather than Cov(bi), the variability of the random variable bi is addressedby finding Cov(bi − bi). The standard errors for the prediction can be obtained as [29][61]

Cov(bi − bi) = G− GZᵀi V−1

i ZiG + GZᵀi V−1

i Xi

(N

∑i=1

Xᵀi V−1

i Xi

)−1

Xᵀi V−1

i ZiG

11

In actual practice, Vi is unknown. Therefore, it must be estimated before obtain-ing β and βi. The maximum likelihood estimation (ML) or Restricted maximum likeli-hood estimation (REML) of Vi can be done by maximizing the log-likelihood function ofresiduals with respect to Vi. Suppose the estimated residual vector is εi = Yi − Xi β. Thelog-likelihood function of residuals function can be written as:

l = −12

N

∑i=1

log|Vi| −12

(N

∑i=1

εᵀi V−1i εi

)− 1

2log

∣∣∣∣∣ N

∑i=1

Xᵀi V−1

i Xi

∣∣∣∣∣ (2.19)

The additional term in equation (2.19) when compared to ML likelihood function is:

−12 log

∣∣∣∑Ni=1 Xᵀ

i V−1i Xi

∣∣∣ = 12 log

∣∣∣∣(∑Ni=1 Xᵀ

i V−1i Xi

)−1∣∣∣∣ = log

∣∣cov(β)∣∣ 1

2 The REML likeli-

hood multiplies the ML likelhood by the factor of the square root of the covariace of β.This is correction adjustment to ML likelihood. The use of REML is often recommendedover ML for estimation of Vi since, the REML estimator are less biased than ML estima-tors [61] [49]. The estimates obtained from ML and REML are similar when the sam-ple size is large enough [49]. The maximization becomes challenging to be performedanalytically using calculus. It is performed numerically using iterative algorithm likeExpectation Maximization algorithm [35], Iterative Generalized Least Squares [24] andNewton-Raphson algorithm[35]. After the estimates of V−1

i and G are obtained, we can

estimate the fixed effects and the random effects by substituting Vi, Vi−1

and G for Viin the above equations. These are known as eBLUEs for the fixed effects and eBLUPsfor the random effects. The predicted response profile of the ith subject given bi , can beestimated as:

Yi = Xi β + Zibi (2.20)

2.4 Bayesian Approach

Let us explore the hierarchical models from Bayesian perspective without takinga side in the long going debate in the field of statistics about the superiority of Bayesianapproach over frequentist approach or other way round. This debate seems to be morefrom philosophical and pedagogical point of view rather than in practicality[1].

In classical approach, the sample of observations y = y1, y2, · · · , yn are taken asrandom while the unknown parameter, θ, are considered fixed. In the Bayesian ap-proach, the focus is on updating the unknown parameters, θ, by considering them tofollow a probability distribution based on the sample of observations y. Initially, wedefine a prior distribution for θ i.e. p(θ). We, then update our belief about θ based on theobserved data, y. This is defined by p(θ|y) and is known as posterior distribution. Bayes’rule establishes a relationship between prior and posterior distribution as follows:

p(θ|y) = p(y|θ)p(θ)p(y)

(2.21)

12

p(y|θ) is the likelihood, the probability of the observed data y, given the parameters θ.The denominator p(y) is the marginal likelihood of the data which can be found by

p(y) =∫

p(θ)p(y|θ)dθ

Since p(y) in equation(2.21) is a constant value ensuring p(θ|y) integrates to 1, we canwrite

p(θ|y) ∝ p(y|θ)p(θ) (2.22)

The point estimates of the posterior distribution are of primary interests. Typi-cally, we find the mean or mode of the posterior. The posterior mean is

θ =∫

θp(θ|y)dθ =

∫θp(y|θ)p(θ)dθ∫p(y|θ)p(θ)dθ

(2.23)

Predictive inference is another goal of Bayesian analysis. After observing the datay, we might be interested in predicting unobserved data, y, from the same process. Thedistribution of y, called posterior predictive distribution, is:

p(y|y) =∫

p(y, θ|y)dθ=

∫p(y|θ, y)p(θ|y)dθ

=∫

p(y|θ)p(θ|y)dθ(2.24)

Consider θ = (θ1, θ2, · · · , θk). We might be interested in the posterior for onecomponent e.g. θ1. The marginal posterior density is [63]

p(θ1|y) =∫ ∫

· · ·∫

p(θ1, θ2, · · · , θk|y)dθ2dθ3 · · · dθk

To evaluate this high dimensional integral, we need to integrate out all the pa-rameters except θ1. It is infeasible to perform this integration analytically when thereare many parameters. It is one of the reason why Bayesian methods were unpopularuntil the last few decades.

With the advent of Markov Chain Monte Carlo (MCMC) algorithms for estima-tion [19] [23] and improvements in computing technology and speed to implement thesealgorithms, the application of Bayesian methods has increased significantly. In chapter3, I use Gibbs sampling to estimate parameters of hierarchical models by making use ofstatistical tool like OpenBUGS / BUGS(Bayesian Inference Using Gibbs Sampling) [59] andR package BRugs [60].

Gibbs sampling or a Gibbs sampler is one of the MCMC algorithms. The keyidea of Gibbs sampling is that for a given multivariate distribution that can be isolatedas the conditional distribution of each parameter given all of the other parameters, it issimpler to obtain samples from each conditional distribution than from the joint distri-bution. For instance, consider the random parameters θ1, θ2 and θ3. Choose the initialvalues θ

(0)1 , θ

(0)2 and θ

(0)3 from a prior distribution q. At iteration i, we sample θ

(i)1 from

13

p(θ1|θ(i−1)2 , θ

(i−1)3 , y), θ

(i)2 from p(θ2|θ(i)1 , θ

(i−1)3 , y) and θ

(i)3 from p(θ3|θ(i)1 , θ

(i)3 , y). This pro-

cess is repeated until the convergence is achieved. The convergence is not to a fixedvalue rather it is to the probability distribution i.e. the samples obtained after manyiterations have the distribution as if they were sampled from the full conditional jointdistribution on the data and from all other parameters in the model. The steps involvedcan be formulated as Algorithm 1.

Algorithm 1: Gibbs Sampling

Initialize θ(0) ∼ q(θ)For i=1,2,3,...,M generate

θ(i)1 ∼ p(θ1|θ

(i−1)2 , θ

(i−1)3 , · · · , θ

(i−1)k , y)

θ(i)2 ∼ p(θ2|θ(i)1 , θ

(i−1)3 , · · · , θ

(i−1)k , y)

...θ(i)k ∼ p(θk|θ

(i)1 , θ

(i)2 , · · · , θ

(i)k−1, y)

After many draws, say M, with each draw being a vector θ(i), we would have generateda Markov chain. If the generated Markov chain is aperiodic, irreducible and positiverecurrent, g(θ) is any integrable function and E[g(θ)] < ∞, then the ergodic theorem [57]implies that the Markov chain will have a stationary distribution. Monte Carlo integra-tion can be written as

limM→∞

1M

M

∑i=1

g(θi)→∫

g(θ)π(θ)dθ

where π is the stationary distribution as M → ∞. The stationary distribution, π, isapproximately close to the posterior distribution p(θ|y). To make the draws closer to theposterior distribution and less dependent on the initial values, we discard the samplesgenerated from the early draws. This is known as the burn-in. Averaging or quantileestimation can be performed on the draws from stationary distribution to estimate theparameters of interest of posterior distribution.

2.4.1 Hierarchical Modeling in Bayesian Framework

For simplicity in explaining hierarchical modeling from Bayesian perspective, letus consider a two-level hierarchy of repeated measurements from the ith subject at jth

time-point, yij, (Level 1) nested within the subjects (Level 2) with one predictor tij.

A random intercept model can be written as

yij = µij + eij (2.25)

where µij = bi + tijβ

14

Let us assume that yij and bi have the following probability distributions

yij ∼ N(µij, σ2y ) (2.26)

bi ∼ N(µb, σ2b ) (2.27)

Model (2.26) has hyperparameters : β, σy, µb and σb. The hyperparameters of the popu-lation distribution i.e. β and σy can be estimated by assuming reasonable non-informativeprior distributions. Various prior distributions for µb and σb have been suggested in theliterature [20]. In practice, an improper Uniform distribution or proper Inverse-Gammadistribution on σb and Normal distribution on µb are commonly used[21].

This model can be shown graphically as in Figure (2.1). Each component of the

Figure 2.1: Directed Acyclic Graph of Random Intercept Model

model appears in the graph as a node. The dashed arrows represent deterministic de-pendencies between the parameters (e.g. from β to µij) whereas the solid arrows rep-resent probabilistic dependencies (e.g. from σ2

y to yij). A random intercept and slopemodel can be written as equation(2.25) with

µij = bi + tijβi

We assume yij, bi and the random slope βi to have the following probability distributions

yij ∼ N(µij, σ2y ) (2.28)

bi ∼ N(µb, σ2b ) (2.29)

βi ∼ N(µβ, σ2β) (2.30)

15

The model can be represented graphically as Figure (2.2). Compared to Figure (2.1), β

Figure 2.2: Directed Acyclic Graph of Random Intercept and Slope Model

is a probabilistic node now. Thus, we have random slopes for each subject along withrandom intercepts.

A more compact way of writing the model by addressing the correlation betweenparameters is as follows [22]:

yij ∼ N(bi + tijβi, σ2y ) (2.31)

[bβ

]∼ MVN

((µbµβ

),

(σ2

b ρσbσβ

ρσbσβ σ2β

))The hyperparameters (µb, µβ, σb, σβ and ρ) can be assumed to have uniform hyper priordistribution with appropriate assumptions [22].

In matrix form, equation (2.31) can be related to random-effects model (2.14) inits marginal form as

Yi ∼ MVN(Xiβ, ZiGZᵀi + Ri)

More discussion about the Bayesian model is done in chapter 3 with its application torepeated measures data.

16

CHAPTER 3

ANALYSIS OF AMBULATORY BLOOD PRESSURE MONITORING(ABPM) DATA

The data that has been used for analyses in this thesis is from an open label,randomized control trial to study the blood pressure pattern in dialysis patients withhypertension. The study was sponsored by University of New Mexico. It is registeredin ClinicalTrials.gov with identifier: NCT01421771 [45].

3.1 Study Rationale

It has been found that the mortality and morbidity among hemodialysis patientsis high due to hypertension. Studies conducted on such patients have shown a con-tinuous reduction in cardiovascular risk with each mmHg drop in the systolic bloodpressure(SBP). National Institute of Health (NIH) has funded two studies Action to Con-trol Cardiovascular Risk in Diabetes (ACCORD) and Systolic Blood Pressure Interven-tion Trial (SPRINT) to study the impact of intensive (SBP < 120mmHg) and standard(SBP < 160mmHg) lowering of blood pressure to control hypertension among patientswith type 2 diabetes mellitus and chronic kidney disease respectively [45].

The primary purpose of this study (NCT01421771) was to understand the feasibil-ity and safety of randomizing patients to a low (100-140 mmHg) and standard (155-165mmHg) pre-dialysis SBP goal. Since this study was not completed, due to nonavailabil-ity of enough data, the primary purpose of the study was not achieved. However, aneffort is made in this thesis to analyze this incomplete data to model the SBP patternsof the patients collected over approximately 44-hours post-dialysis. The study of bloodpressure recording patterns over an interdialytic period can give valuable insights aboutthe treatment effects[31].

More details about the purpose of the study and the inclusion/exclusion criteriafor the patient enrollement can be obtained from [45].

3.2 Data Exploration

As described in Section 1.2, Ambulatory BP monitoring (ABPM) is an efficientway of collecting the measurements over the time. Although the initial plan of the study

17

was to recruit at least 120 patients, the study was discontinued as the patient recruitmentwas slow and for various other reasons [45].

The data used in this thesis has ABPM measurements of only 49 subjects per-formed over a single session of approximately 44-hours mid-week inter-dialytic period.ABPM was performed using a SpaceLabs 90207 monitoring device [30]. The subjectswore the device on their non-access arms post-dialysis. During ABPM, BP measure-ments were obtained every 30-minutes between 6:00 AM and 10:00 PM and every 60-minutes between 10:00 PM and 6:00 AM. A total of 3397 repeated measurements wasavailable in the data for analysis [26]. A similar study design has been explained in [31].The primary response variable of interest is Systolic Blood Pressure (SBP)[45].

The data consists of the following variables:

Table 3.1: Variable DescriptionVariableName

Label Type Description

subj Subject ID Numeric Unique identifier for each sub-ject in the study

SYSTOLIC Systolic Blood Pressure(mmHg)

Numeric Systolic ambulatory blood pres-sure collected by the device

tdhr Time Since Dialysis Ses-sion (Hrs)

Numeric Time elapsed post dialysis

ts Session Time (Hrs) Numeric Actual clock time when the datais collected

cos24 Cosine Component (24Hrs)

Numeric Cosine of (2π24 ts) with 24 hours

periodicitysin24 Sine Component (24 Hrs) Numeric Sine of (2π

24 ts) with 24 hours pe-riodicity

subj is the unique patient ID of each subject. Since the collected IDs were notin sequence, they were renumbered from 1 to 49 for easy interpretation. SYSTOLICis the collected reading of systolic blood pressure . tdhr is the time of measurementwith reference to dialysis. ts is the actual clock time at which the measurements aretaken. cos24 and sin24 are calculated as cosine and sine function of ts by assuming 24hours periodicity respectively. As literature suggests the rhythm is circadian in natureduring interdialytic period, a constant periodicity of 24 hours is assumed in the modelsdiscussed in this thesis [48] [9] [31].

Let us analyze the pattern of data for three randomly selected subjects from thestudy. We can see in Figure (3.1) that the systolic blood pressure doesn’t remain con-stant over the period of measurements. Clearly, there is no linear relationship betweentime and blood pressure measurements. There’s a rhythmic pattern exhibited by thedata. Various literature suggest that this rhythmic pattern can be better described usingcosinor model[10] [12] [3].

A plot of mean systolic BP per subject in Figure (3.2) gives us an idea how eachsubject has different mean value of BP than the other. Clearly, the means vary among

18

Figure 3.1: Ambulatory Systolic Blood Pressure Profiles of 3 Different Subjects Approx-imately 44 Hours Post-dialysis

the subjects. This makes us cautious to design a model that can address this inter-subjectvariability. Within-subject variability can be seen as the error bars in the plot.

3.3 Cosinor Model for Rhythmic Data

One of the popular method for performing the analysis of biological rhythm is us-ing Cosinor Rhythmometry[44]. The model for a response variable y at time t of a rhythmfor a single subject is of the additive form:

y(t) = f (t, θ) + ε(t) (3.1)

where f (t, θ) is a deterministic function depending on parameters θ and ε(t) is thestochastic component. For short and sparse time series, ε(t) are assumed to be inde-pendent errors with zero mean. However, in general, the errors can be described usingautocorrelation structure models.

As described in Figure (1.1), the rhythmic nature can be characterized by f (t, θ)having a constant, a polynomial trend term and a cosine function with frequencies thatare harmonics of one basic frequency [3].

f (tj) = M + Acos(ωtj + φ) + αtj (3.2)

where tj represents the jth time of measurements for the ith individual, αtj is the polyno-mial trend which is assumed to be linear, M is the mean level- also known as Midline

19

Figure 3.2: Mean Ambulatory Systolic Blood Pressure Profiles for all 49 Subjects Ap-proximately 44 Hours Post-dialysis

Estimating Statistic of Rhythm (MESOR) of the cosine curve, A is the amplitude, ω is theangular frequency, and φ is the acrophase (horizontal shift) of the curve. We can rewrite3.2 as

f (tj) = M + A[cos(ωtj)cos(φ)− sin(ωtj)sin(φ)] + αtj (3.3)

Equivalently it can be expressed as:

f (tj) = M + βxj + γzj + αtj (3.4)

where the following substitutions are introduced to linearize the function:

β = Acosφ; γ = −Asinφ; xj = cosωtj and zj = sinωtj

For a circadian rhythm, generally, a period of 24-hour cycle is assumed i.e. ω =2πτ , where τ is the period length( e.g. 12 hours , 24 hours). The β and γ parameters are

used collectively to estimate the amplitude of the fitted curve. Amplitude, A is

A =√

β2 + γ2 (3.5)

The amplitude as noted in Figure (1.1) represents the half of the change in a cycle. Theconsinor model (3.1) at time tj is

y(tj) = M + βxj + γzj + αtj + εj (3.6)

20

A general model for i subjects having yij measurement at tj timepoint can be thought of:

yij = M + βxij + γzij + αtij + εij (3.7)

where j = 1, 2, ..., ni.In a matrix form:

Yi = XiB + εi (3.8)where

Xi =

1 xi1 zi1 ti11 xi2 zi2 ti2...

......

...1 xini zini tini

B =

Mβγα

εi =

εi1εi2...

εini

This is the familiar model discussed in Chapter 2. In the next sections, the differentmodels discussed in Chapter 2 are explored to find the best model that fits our data inhand.

3.4 Statistical Analysis

In this section, the statistical software R [54] is used to implement the classicalapproach hierarchical models as discussed in Section 2.3. Similarly, OpenBugs [59] andR package - BRugs [60] are used for Bayesian model parameter estimations.

3.4.1 Classical Approach

Single Cosinor Model Ignoring the inter-subjects variability, lets implement a singleregression model that has fixed parameters for all the subjects. For ith subject, the mea-surements collected over tij the model is

yij = M + βxij + γzij + αtij (3.9)

Using the variables in the data, equation(3.9) can be implemented in R with gls (gen-eralized least squares) function. It is a model fitting routine available in nlme library[52].

SYSTOLICi = M + β cos24ij + γ sin24ij + α tdhrij (3.10)

The estimates are obtained using the restricted maximum likelihood estimation (REML).The R-Output is in Listing (3.1).

Listing 3.1: Single Cosinor ModelGeneralized least squares fit by REML

Model: SYSTOLIC ~ cos24 + sin24 + tdhr

Data: bp1

AIC BIC logLik

31014.01 31044.66 -15502.01

21

Coefficients:

Value Std.Error t-value p-value

(Intercept) 149.91625 0.8080233 185.53455 0.0000

cos24 1.84869 0.5810015 3.18190 0.0015

sin24 -0.01930 0.5760345 -0.03351 0.9733

tdhr 0.36241 0.0323337 11.20835 0.0000

Correlation:

(Intr) cos24 sin24

cos24 0.106

sin24 0.280 0.086

tdhr -0.864 -0.055 -0.228

Standardized residuals:

Min Q1 Med Q3 Max

-3.49013035 -0.65608519 0.02652515 0.68397729 3.29894789

Residual standard error: 23.20945

Degrees of freedom: 3397 total; 3393 residual

The estimated overall regression model is:

SYSTOLICi = 149.92 + 1.85 cos24ij − 0.02 sin24ij + 0.36 tdhrij (3.11)

The amplitude of the fitted curve can be found using equation (3.5). Its value is 1.85. Apositive significant slope of 0.36 indicates the linear increment in systolic blood pressureover time. We make a note of Akaike information criterion(AIC), Bayesian informationcriterion (BIC) and log-likelihood ratio values. We compare these values with the othermodels to compare the fit of the model. Let us examine the fit of the model by examiningthe plot of predicted values using the coefficients from Listing 3.1.

As seen in the fit plot of the model, a single cosinor doesn’t seem to provide thebest fit to the data.

Another problem with this model is that it assumes the errors to be independent.But it is highly unlikely that eij is independent of eik, for j 6= k. As we can see in Figure(3.4), there is a significant autocorrelation between the measurements over time. Thedashed blue lines are approximate 95% confidence interval for the lack of autocorrela-tion. All the bars are above the dashed lines indicating high autocorrelation values. Asnoted in Listing (3.1), the correlation between time post dialyis and mean systolic bloodpressure is -0.864. We need a model that can reduce this autocorrelation.

Random Intercept model A model that addresses the errors(random effects) inducedby unmeasured subject factors (i.e. between-subject variability) can help us to refine ourmodel. As discussed in section 2.3, let us implement random intercept model.

Listing 3.2: R-output of Random Intercept ModelLinear mixed -effects model fit by REML

22

Figure 3.3: Single Cosinor Model Fit for 3 Subjects

Figure 3.4: Serial Autocorrleation of Residuals Using Single Cosinor Model

23

Data: bp1x

AIC BIC logLik

28791.21 28827.98 -14389.6

Random effects:

Formula: ~1 | subj

(Intercept) Residual

StdDev: 16.72537 16.21945

Fixed effects: SYSTOLIC ~ (cos24 + sin24 + tdhr)

Value Std.Error DF t-value p-value

(Intercept) 149.64543 2.4553966 3345 60.94552 0.0000

cos24 1.44396 0.4068750 3345 3.54890 0.0004

sin24 0.19422 0.4048365 3345 0.47975 0.6314

tdhr 0.38029 0.0226870 3345 16.76227 0.0000

Correlation:

(Intr) cos24 sin24

cos24 0.025

sin24 0.064 0.086

tdhr -0.199 -0.057 -0.225

Standardized Within -Group Residuals:

Min Q1 Med Q3 Max

-4.17220853 -0.61384603 0.01557292 0.61976496 3.96041948

Number of Observations: 3397

Number of Groups: 49

Figure 3.5: Random Intercept Model Fit

24

Figure 3.6: Serial Autocorrleation of Residuals Using Random Intercept Model

As noted from the R-output 3.2, the random effect of intercept i.e. between-subject variation is N ∼ (0, σu = 16.72). The standard deviation of residuals i.e. withinsubject variation is σe = 16.22. The proportion of the total variation that can be at-tributed to the between-subject variation can be found using equation (2.6) which is 0.51Almost the 51% variation in the data is due to the differences between the subjects.

The fixed effect estimate of systolic mean is 149.65 which is quite close to a modelestimated by single cosinor model (3.10). However, this estimate varies between thesubjects with standard deviation of 16.72. Hence, the intercepts are different for differentsubjects. The coefficients of covariates are fixed for all the subjects.

Random intercept and slope model The random intercept model can be extended tohave random slopes of the covariates in addition to random intercepts. Here, cosinecoefficient, sine coefficient and time trend coefficient are assumed to be normally dis-tributed with mean zero and σβ, σγ and σα respectively. Since the blood pressure mea-surements are unequally spaced in time, an unstructured covariance model is the mostappropriate model that we can use. This covariance model models the unexplainedvariances or the random effects of the data.

Listing 3.3: R-output of Random Intercept and Slope ModelLinear mixed -effects model fit by REML

Data: bp1x

AIC BIC logLik

28054.41 28146.35 -14012.2

Random effects:

Formula: ~cos24 + sin24 + tdhr | subj

25

Structure: General positive -definite , Log -Cholesky parametrization

StdDev Corr

(Intercept) 19.2664837 (Intr) cos24 sin24

cos24 7.4515672 0.162

sin24 6.9673527 -0.147 0.356

tdhr 0.4024575 -0.494 0.239 0.023

Residual 13.8320633

Fixed effects: SYSTOLIC ~ (cos24 + sin24 + tdhr)

Value Std.Error DF t-value p-value

(Intercept) 149.41059 2.7978020 3345 53.40285 0.0000

cos24 1.76832 1.1241798 3345 1.57298 0.1158

sin24 0.44841 1.0558422 3345 0.42470 0.6711

tdhr 0.38562 0.0610285 3345 6.31863 0.0000

Correlation:

(Intr) cos24 sin24

cos24 0.157

sin24 -0.119 0.327

tdhr -0.510 0.206 -0.006

Standardized Within -Group Residuals:

Min Q1 Med Q3 Max

-4.459404680 -0.613747333 -0.004609585 0.595065694 4.887675552

Number of Observations: 3397

Number of Groups: 49

In this model, the unexplained within-subjects variation of systolic blood pres-sure has an estimated standard deviation σ = 13.83; the subject intercepts have theestimated standard deviation of ˆσM = 19.26; the cosine coefficient’s estimated standarddeviation σβ = 7.45; the sine coefficent’s estimated standard deviation is σγ = 6.97and the estimated standard deviation of time post dialysis σα = 0.38. The estimatedcorrelation between intercept and time is −0.50. The estimated average coefficients arepresented as fixed effects. Since we are more interested in the individual models foreach subject the p-value of fixed effects parameter are not much of importance.

Figure (3.7) shows the individual intercept and slope model fitted to subject 1, 25and 49. The fit seems to be better than the previously discussed models. With additionalrandom effects of slopes, the individuals profiles seem to have much better fit. FromFigure (3.8), we can clearly see the decrease in autocorrelation with the lags.

Model Comparison The parameters obtained from the above discussed models aretabulated in Table (3.2).

Since we are comparing the nested models, the comparison of models can bedone using a likelihood-ratio test. The test statistic is:

2([−log(L1)]− [−log(L2)]]

)∼ χ2(d f = p1 − p2) (3.12)

26

Figure 3.7: Random Intercept and Slope Model Fit

Figure 3.8: Serial Autocorrleation of Residuals from Random Intercept and Slope Model

where, L1 is the likelihood of model 1, L2 is the likelihood of model 2, p1 and p2 arethe numbers of parameters in the two models. The likelihood of the function can becalculated as discussed in section 2.4.

The comparison of the single cosinor model vs the random intercept model yieldslog-likelihood ratio 2223.34 with a p-value of < .0001. Hence, we conclude randomintercept is better than single cosinor model. Similarly the log-likelihood ratio of randomintercept model vs random intercept and slopes model is 749.18 with a p-value of <

27

Table 3.2: Comparison of Model ParametersParameter Estimate Single-Cosinor

ModelRandom InterceptModel

Random Interceptand Slope Model

Fixed Effects:Intercept (M) 149.91(148.33,151.50) 149.65(144.88,154.41) 149.41(143.98,154.84)Cosine coefficient (β) 1.85(0.7,2.98) 1.44(0.65,2.24) 1.77(-.41,3.95)Sine coefficient (γ) -0.019(-1.15,1.11) 0.19(-0.60,0.99) 0.45(-1.6,2.50)Linear trend (α) 0.36(0.30,0.42) 0.38(0.34,0.42) 0.38(0.27,0.50)σ 23.20 16.21 13.83

Random Effects:σM 16.55 19.06σβ 7.36σγ 6.89σα 0.39

AIC 31010.4 28789.06 28057.88BIC 31041.06 28825.85 28149.84Log likelihood -15500.2 -14388.53 -14013.94Amplitude 1.85 1.45 1.52

0.001. Hence, random intercept and slopes model is significantly better than the othertwo models.

3.4.2 Bayesian Approach

Let us apply Bayesian perspespective to compare the models discussed in Sec-tions 2.5 and 3.5 by using OpenBUGS software [60] [59]. It is an open-source statisticalsoftware which can analyze a specified model based on Gibbs sampler [59]. As dis-cussed in section 2.5, we can fit the hierarchical models considered so far in Bayesianframework by adding prior distributional assumptions to the model parameters. Theassessment of convergence of the repeated samples to the target distribution is doneusing Gibbs sampler.

Single Cosinor Model The model described in section 3.4.1 can be written equiva-lently as follows:

yi ∼ N(µij, σ2)µij = M + β cos24ij + γ sin24ij + α tdhrij

(3.13)

where yi is the ambulatory systolic blood pressure of the ith subject. µij is the mean SBPof the target distribution. σ is the variation in measurements within subject.

The standard practice in Bayesian regression analysis is to have non-informativepriors for the location parameters (θ) such as intercepts and regression coefficients [21].

28

e.g.θ ∼ Uni f (−100, 100) or θ ∼ Normal(0, 10000)

Since OpenBUGS parameterizes the Normal distribution in terms of mean and preci-sion(τ = 1

σ2 ), precision parameter will be used instead of variances in the models dis-cussed here onwards. For the location parameters in model (3.13) let us assume non-informative normal priors as follows:

M ∼ N(0, 0.0001)β ∼ N(0, 0.0001)γ ∼ N(0, 0.0001)α ∼ N(0, 0.0001)

The prior for scale parameter i.e. σ2 or Jeffreys’ prior is the ’inverse’ prior [22].

p(σ2) ∝1σ2 ∝ Gamma(0, 0)

For the precision parameter (τ) a prior with small value

τ ∼ Gamma(1, 0.001)

is the conjugate prior and used as a proper prior for the precision of normal likelihood.A priori independence is assumed among the intercept, regression coefficients and τ andindependent normal priors on M, β, γ, α and an inverse gamma prior on the variance σ2

(or, a gamma prior on the precision).With the above assumed priors, the model (3.13) is run in OpenBUGS using

2 chains with each chain having different initial values. The model specification inOpenBUGS is available on Appendix C.1. Each chain was run for 1000 iterations firstand the trace plots of each parameter were analyzed for convergence. Each of thestochastic nodes M, β, γ, α and σ were monitored. Further 110000 iterations were doneto get sufficient simulations and strong convergence. The first 11000 pre-convergence’burn-in’ samples were discarded. The trace plots of convergence of two chains for allthe parameters are presented in Figure(3.9).

The posterior densities of the parameters are plotted in Figure (3.10). mean andsd are the approximations of the µ and σ of the posterior denisty respectively. The pos-terior standard deviation is analogous to the standard error in classical statistical infer-ence. MC error is the error in computation of the mean [59]. The summary statisticsof each parameter are generated with very low MC error. By default, the median andthe 2.5th and 97.5th percentiles are produced for each parameter statistic. Analogous to95% confidence interval but more meaningful 95% credible interval of the simulationscan be obtained using the 2.5th as the lower bound and the 97.5th percentile as the upperbound. start is the starting simulation and sample is the number of simulations used toobtain the posterior distribution. sample is not the sample size of the data N = 3397.

We can use the summary statistics to interpret the OpenBUGS results in Figure(3.10). The posterior distribution of M, the mean systolic ambulatory blood pressure,

29

Figure 3.9: Convergence History of Two MCMC Chains for the Parameters in SingleCosinor Model

Figure 3.10: Summary of Posterior Distribution Estimates of Single Cosinor Model Pa-rameters

is approximately normal with µ = 149.9 and σ = 0.8093. These values are compu-tationally accurate to about ±0.00666. Similarly, the distribution of β,γ, α and σ areapproximately normal (judging from the graphs) with mean 0.3626, 1.846,−0.0223 and23.21 respectively and standard deviation 0.03232, 0.5813, 0.5782 and 0.2823 respectively.All the parameters have very low MC error. Dbar, Dhat, DIC and pD are the deviance

30

parameters that can be used in the model comparisons. These are discussed in ModelComparison Section.

Random Intercept Model The model (3.13) can be re-written to include a random in-tercept as follows:

yi ∼ N(µij, σ2)µij = Mi + β cos24ij + γ sin24ij + α tdhrijMi ∼ N(µM, σ2

M)(3.14)

In the second level of model, the subject level systolic blood pressure means vary aboutan overall mean µM with variance σ2

M. Here, the subject level means are the randomeffects resulting from a normal population.

Mi ∼ N(µM, σ2M)

It is to be noted that µM is the overall population mean, a fixed effect. σ2M is the between-

subject variance. σ2 is the within-subject variance of the measurements. Various nonin-formative prior distributions have been suggested in the literature for the random effectsσM. As suggested in [21] , following priors are assumed for the hyperparameters.

µM ∼ N(0, 0.0001)

σ−2M ∼ Gamma(1, 0.001)

A vague but proper prior for the model is :

M, β, γ, α ∼ N(0, 0.0001)

σ−2 ∼ Gamma(1, 0.001)

The model was implemented in OpenBUGS using 2 chains with different initialvalues. The model specification in OpenBUGS is available on Appendix C.2. After dis-carding 11000 pre-convergence simulations, 100000 iterations were used for the analysis.The trace plots of the two chains of all the model parameters in Figure (3.11). The chainsmixed well to a stationary distribution. The posterior distributions of these parametersare plotted in Figure (3.12). The distributions are approximately normal with the meanand standard deviation values as summarized in the output.

The main interest in this model is the random effect i.e. σM. The posterior dis-tribution of σM is normal with mean 16.62 and standard deviation 1.721. The between-subject variability is 16.62 and the within-subject variability is 16.22. As from the clas-sical approach, the conclusion we can draw is that almost 50% of the variability in thedata is due to the between subject variation. The fixed effect of intercept, µM, is 149.6.

The individual means of the posterior distribution of intercepts of 49 subjects areplotted with their credible intervals. As seen in Figure (3.13), the individual interceptsvary from each other significantly. This justifies that the mean profiles of the subjects inthe study are different to each other. Hence, a single cosinor is not adequate to modelthe collected data.

31

Figure 3.11: Convergence History of Two MCMC Chains for the Parameters in RandomIntercept Model

Figure 3.12: Summary of Posterior Distribution Estimates of the Random InterceptModel Parameters

32

Figure 3.13: Caterpillar Plot of the Mean of the Posterior Distribution of Varying Inter-cept(M) of 49 Subjects

Random Intercept and Slopes Model The random intercept and slopes model can bewritten as 2 level hierarchical model as following:

yi ∼ N(µij, σ2)µij = Mi + βi cos24ij + γi sin24ij + αi tdhrijMi ∼ N(µM, σ2

M)βi ∼ N(µβ, σ2

β)

γi ∼ N(µγ, σ2γ)

αi ∼ N(µα, σ2α)

(3.15)

We can not ignore the correlation between intercepts and slopes in the model when theyvary between the subjects. Another way of writing equation(3.15) is in matrix form witha variance-covariance matrix to model it as a multivariate normal distribution.

yi ∼ N(XiBi, σ2)Bi ∼ N(µB, ΣB)

(3.16)

where

Xi =

1 cos24i1 sin24i1 tdhri11 cos24i2 sin24i2 tdhri2...

......

...1 cos24ini sin24ini tdhrini

Bi =

Miβiγiαi

µB =

µMµβ

µγ

µα

33

ΣB =

σ2

M ρMβσMσβ ρMγσMσγ ρMασMσα

ρMβσβσM σ2β ρβγσβσγ ρβασβσα

ρMγσγσM ργβσγσβ σ2γ ργασγσα

ρMασασM ραβσασβ ραγσασγ σ2α

A computationally convenient inverse-Wishart distribution is an appropriate prior forΣB [22] [21]. The inverse-Wishart distribution is a multivariate generalization of thescaled inverse-χ2 distribution. It is the conjugate distribution of the matrix ΣB [22].

ΣB ∼ Inv−Wishart(v, B)

where v is the degrees of freedom which is 1 more than dimension of B. B is the num-ber of coefficients which is 4 in this case. Doing this allows a uniform distribution oneach of the correlations as noted in ΣB matrix but this constraints the diagonal elements(variances) of ΣB. As suggested in [22], it is convenient to introduce a vector of scaleparameters ξ of length 4.

ΣB = Diag(ξ)QDiag(ξ)

, where Q is the unscaled covariance matrix. Now, we can assign a noninformative prioras

Q ∼ Inv−Wishart(I4×4)ξ ∼ Uni f (0, 100)

The variances and the correlations are of primary interest. The precision matrixΣ−1

B is assigned Wishart distribution, dwish(B[1 : 4, 1 : 4], 5), in OpenBUGS.

The above discussed model is run in OpenBUGS using 2 chains with different ini-tial values. The model specification in OpenBUGS is available on Appendix C.3. Eachchain was run for 21000 iterations first and the trace plots of each parameter were an-alyzed for convergence. Each of the stochastic nodes were monitored. Further 121000iterations were done to get sufficient simulations and mixture of the chains. The first21000 pre-convergence ’burn-in’ samples were discarded. The trace plots of conver-gence of two chains for all the parameters are presented in Figure(3.14).

The posterior distribution of fixed effect parameters, µM, µβ, µγ and µα look nor-mal with the mean and standard deviations as presented in Figure(3.15). The MC erroris very small for each parameter. The intercept is µM = 149.4 and the corresponding ran-dom effects of intercept varies with standard deviation σM = 19.72. Individual randomintercepts for all 49 subjects along with their credible intervals are presented in Figure(3.17). Clearly, the individual intercepts vary significantly between the subjects. As seenin Figure (3.16) the within-subject variability posterior distribution is normal with meanσ = 13.84 and standard deviation 0.1735. The variation between subjects is more thanwithin-subjects variability in the data.

The fixed effect parameter of β is µβ = 1.77 and corresponding random effectvariability of β is σβ = 7.677. As seen in Figure (3.17), the β parameter or the coefficientof the cosine part of the model varies quite a bit between the subjects.

The fixed effect parameter of γ is µγ = 0.452 and corresponding variability of γis σγ = 7.21. As seen in Figure (3.18), the γ parameter (the coefficient of the sine part ofthe model) varies quite a bit between the subjects.

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Figure 3.14: Convergence History of Two MCMC Chains for the Parameters in the Ran-dom Intercept and Slope Model

Figure 3.15: Summary of Posterior Distribution of the Random Effect Parameters

35

Figure 3.16: Posterior Distribution and Trace Plot of Convergence of Within-subject Vari-ation σ

Figure 3.17: Caterpillar Plots of Random Intercept M and Random Slope β

The fixed effect parameter of α is µα = 0.388 and corresponding variability of αis σα = 0.4115. As seen in Figure (3.18), the variablity between the subjects is small.It is the linear trend of the model. It gives us an idea that all the subjects more or less

36

Figure 3.18: Caterpillar Plots of Random Slopes : γ and α

exhibited similar linear trend in systolic blood pressure throughout the study session.Since there are significant between-subject and within-subject variations, the ran-

dom intercept and slope model seems to be the most appropriate to model the ambu-latory systolic blood pressure data. The comparison between the models are discussedin next section. Another benefit of considering random slopes is that we can modelthe correlations between the intercepts and slopes. After monitoring the parameters for100000 simulations, the correlation values are summarized and graphed in Figure (3.19).There is a weak correlation between intercept(M) and sine (β), ρMβ = 0.15 or cosine (γ)coefficients, ρMγ = 0.14. There is a negative correlation between linear trend α and M,ρMα = −0.45. The correlation between β and γ seem to be positive, ρβγ = 0.32. Thecorrelation between β and α, ραβ = 0.22. The correlation between γ and α, ραγ = 0.025,is negligible.

Model Comparison The literature suggests various approaches to compare Bayesianmodels [22] [58]. One of the popular approach similar to log likelihood ratio compar-isons for frequentist models is Deviance Information Criterion(DIC) [58]. The deviance isdefined as

D(θ) = −2 logp(yi|θ)It is calculated at each iteration given the parameters θ. Based on deviance, a Bayesianpredictive model comparison criterion has been proposed by Spiegelhalter et al [58].The idea here is to find a criterion that trades off between model fit and complexity. The

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Figure 3.19: Correlations Between Slopes and Intercept

38

posterior mean deviance isD = E [D(θ)|y]

The complexity can be described by the effective number of parameters, pD,

pD = D− D

where, D = D(E [θ|y]). This number can be negative also. DIC is defined by the for-mula:

DIC = D + pD

The DIC is easy to calculate in OpenBUGS. The rule of thumb for selecting the bettermodel is put in OpenBUGS documentation [59] as

It is difficult to say what would constitute an important difference in DIC.Very roughly, differences of more than 10 might definitely rule out the modelwith the higher DIC, differences between 5 and 10 are substantial, but if thedifference in DIC is, say, less than 5, and the models make very different in-ferences, then it could be misleading just to report the model with the lowestDIC.

Table 3.3 has D, DIC, D and pD parameters tabulated for all three models. DIC value ofRandom Intercept and Slope model is -135900 which is far lesser than DIC value of othertwo models. The negative value of pD parameter is due to large number of parameters inthe model. The D value of 27490 is also less than the other two models. Random Interceptmodel’s DIC value is 28570 which is lesser than Single-Cosinor model’s DIC value (31010)by 2440. This comparison gives us confidence to say that the random intercept andslopes model is suitable for the two level hierarchy of the data. The parameters obtainedfrom the above discussed models are tabulated in Table (3.3).

A comparison of results from Classical and Bayesian Approaches is described innext chapter.

3.5 Interpretation of Results

As the random intercept and slope model best fitted the ambulatory systolicblood pressure data, the estimates obtained can be interpreted to summarize the keyfindings of this study. By analyzing the fixed effects, we conclude that the mean systolicblood pressure of 49 subjects in the study was 149.4 mm Hg/h with a 95% credible inter-val of (143.7, 155). There was a linear increase in the blood pressure at 0.38 mm Hg perhour post dialysis during the inter-dialytic period of 44 hours. The average amplitudeof change in SBP was 1.52 mm Hg. Although the fixed effect sine and cosine coefficientshave credible intervals suggesting them to be not significantly different from 0, thesecoefficients are necessary to model the rhythmic nature of the blood pressure measure-ments over 24 hour period. Also, the credible intervals did not contain zero for each andevery subject.

There was 13.84 (13.51,14.19) mm Hg standard deviation in the within-subjectmeasurements of the SBP. The mean SBP varied greatly at 19.72 (15.94, 24.55) mm Hg

39

Table 3.3: Comparison of Bayesian Model ParametersParameter Estimate Single-Cosinor

ModelRandom InterceptModel

Random Interceptand Slope Model

Fixed Effects:Intercept (M) 149.9(148.3,151.5) 149.6(144.7,154.3) 149.4(143.7,155)Cosine coefficient (β) 1.846(0.69,2.98) 1.44(0.645,2.24) 1.77(-.50,4.07)Sine coefficient (γ) -0.023(-1.15,1.11) 0.19(-0.60,0.99) 0.45(-1.70,2.62)Linear trend (α) 0.36(0.30,0.43) 0.38(0.34,0.42) 0.38(0.26,0.52)σ 23.21(22.66,23.77) 16.22(15.84,16.61) 13.84(13.51,14.19)

Random Effects:σM 16.48(13.66,20.4) 19.72(15.94,24.55)σβ 7.67(6.12,9.67)σγ 7.21(5.73,9.08)σα 0.41(0.33,0.52)

Dbar 31010 28570 27490Dhat 31010 28520 190900DIC 31010 28620 -135900pD 5.007 52.37 -163400Amplitude 1.85 1.45 1.52

between the subjects. There was a variation of 0.41 (0.33,0.52) mm Hg in the linear trendof the SBP among the subjects. The cosine coefficient varied by 7.67 (6.12,9.67) mm Hgand sine coefficient varied by 7.21 (5.73,9.08) mm Hg between the subjects. Hence, wecannot ignore the sine and cosine coefficient terms in the model.

Since the hierarchical random intercept and slope model provides the separateestimates for each subject, it can be effectively used to model the individual subject pro-files. This model is flexible enough to give insight into the individual SBP profile of eachsubject. The ambulatory SBP has a large variability between subjects. A study of sepa-rate mean profiles for each subject would give more clinically significant conclusions.

40

CHAPTER 4

DISCUSSION

This thesis has demonstrated how biological rhythm data like ambulatory bloodpressure data can be analyzed using cosinor hierarchical models in frequentist andBayesian frameworks. In the following sections, the results obtained from both the clas-sical and Bayesian perspectives are discussed.

4.1 Comparison of Classical and Bayesian Approaches

A classical or a frequentist approach assumes that parameters are fixed and datais a repeatable random sample. It doesn’t make use of prior information to the modelspecification. A 95% confidence interval of a parameter is interpreted as ” a collection ofintervals with 95% of the intervals containing the true parameter”[6].

Whereas, Bayesian approach treats the parameters to be unknown with a proba-bility distribution and data is fixed. Bayesian approach accommodates the knowledgeof prior in the model building process. A 95% credible interval is interpreted as ”aninterval with 95% probability of containing the true parameter” [6].

Philosophically, the classical and Bayesian approach seem to differ. In Chapter3, we effectively used both the approaches for data analyses. They both are amazingstatistical tools which are useful for an applied statistician. In the next sections, theresults obtained from the models in Chapter 2 and 3 are compared.

4.1.1 Single Cosinor Model

The results obtained after fitting the single-cosinor model from both classical andBayesian frameworks are tabulated in Table 4.1. The parameter estimates are almostthe same for both the approaches. The reason behind this is because Bayesian modelswere fitted using the vague prior distributions. So, the information derived was fromthe likelihood. Bayesian model took around 5 minutes to run for 100000 iterations usingOpenBugs on Intel(R) Core 2 Duo CPU @2.83GHz. The classical model took less than aminute for convergence by using R on the same machine. Run-time wise, the classicalmodel performed better than the Bayesian model.

One difference we can see in the results is the estimation of within-subject error,σ. There is uncertainty associated with this parameter in Bayesian model but classicalmodel doesn’t take this into account for estimation.

41

Table 4.1: Comparison of Single Cosinor ModelParameter Estimate Classsical Model Bayesian Model

Intercept (M) 149.91 (148.33,151.50) 149.90 (148.30,151.50)Cosine coefficient (β) 1.85 (0.70,2.98) 1.85 (0.70,2.98)Sine coefficient (γ) -0.019 (-1.15,1.11) -0.023 (-1.15,1.10)Linear trend (α) 0.36 (0.30,0.42) 0.36 (0.30,0.43)σ 23.20 23.21 (22.66, 23.77)

4.1.2 Random Intercept Model

The results obtained from the random intercept model is tabulated in Table 4.2.Again, we can see that the parameter estimates are very similar from both the ap-proaches. Also, the lower and upper limit of 95% confidence or credible intervals seemto agree with each other. The Bayesian model took around 10 minutes to run for 100000iterations using OpenBugs on Intel(R) Core 2 Duo CPU @2.83GHz. The classical modeljust took less than a minute for convergence by using R on the same machine. Perfor-mance wise the classical models were faster in converging. As we can see, the Bayesianmodel lets us know the uncertainty in two random effect parameters, σ and σM whereasthe classical model doesn’t.

Table 4.2: Comparison of Random Intercept ModelParameter Estimate Classsical Model Bayesian ModelFixed effects:

Intercept (M) 149.65 (144.88,154.41) 149.60 (144.70,154.30)Cosine coefficient (β) 1.44 (0.65,2.24) 1.44 (0.65,2.24)Sine coefficient (γ) 0.19 (-0.60,0.99) -0.19 (-0.60,0.99)Linear trend (α) 0.38 (0.34,0.42) 0.38 (0.34,0.42)σ 16.21 16.22 (15.84,16.61)

Random effect:σM 16.55 16.48 (13.66,20.40)

4.1.3 Random Intercept and Slope Model

The results obtained from the random intercept model are tabulated in Table 4.3.We can see that the parameter estimates are very similar from both the approaches. TheBayesian model took around 145 minutes to run for 100000 iterations using OpenBugson Intel®Core 2 Duo CPU @2.83GHz. The classical model just took around a minutefor convergence by using R on the same machine. 95% credible intervals of the parame-ters are slightly wider than analogous 95% confidence intervals. The correlation matrix

42

estimates also look similar from both the models. A slightly significant correlation be-tween intercept and time trend has been confirmed by both the approaches. As seen inprevious models, one benefit of using Bayesian model over classical model is that it pro-vides uncertainties in standard deviation parameters too.Performance wise the classicalmodels are faster in converging.

Table 4.3: Comparison of Random Intercept and Slope ModelParameter Estimate Classsical Model Bayesian ModelFixed effects:

Intercept (M) 149.41 (144.88,154.41) 149.40 (143.70,155.00)Cosine coefficient (β) 1.77 (-0.41,3.95) 1.77 (-0.5,4.07)Sine coefficient (γ) 0.45 (-1.60,2.50) 0.45 (-1.70,2.62)Linear trend (α) 0.38 (0.27,0.50) 0.38 (0.26,0.51)σ 13.83 13.84 (13.51,14.19)σM 19.26 19.72 (15.94,24.55)σβ 7.45 7.67 (6.12,9.67)σγ 6.97 7.21 (5.73,9.08)σα 0.40 0.41 (0.33,0.52)

Random effects:Corrleation Matrix:

M β γβ 0.16γ −0.15 0.35α −0.49 0.24 0.023

M β γβ 0.15γ −0.14 0.32α −0.45 0.22 −0.025

From these comparisons, we can conclude that the results obtained from both theapproaches yield similar estimates when Bayesian models are run with non-informativepriors. Since the data used for the analysis was large enough, the use of informativeprior distributions would impact the computations little [20]. When the models are setup hierarchically in such a way that the parameters in each level share prior distribu-tions, then the prior distributions can be estimated from data [20]. Since the posteriordistributions of the parameters are proper (Normal)(as seen in the results in Section3.4.2), we can confirm that the use of non-informative priors of the parameters does agood job in analyzing the large data.

Computationally, classical models seem to be more efficient than Bayesian mod-els. Some literature suggest that an efficient way for implementing the hierarchical mod-els is to use the maximum likelihood methods for model exploration and selection dueto speed. Then, finally equivalent Bayesian model should be used to obtain accurateestimates by using MCMC alogrithm [25]. Although classical and Bayesian approachesseem to differ philosophically, both the approaches can be used in implementing thehierarchical models as per one’s convenience.

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4.2 Conclusions

With the advancement of computational capabilities and easy availability of opensource statistical packages like R and OpenBUGS, the use of hierarchical models is be-coming popular among statisticians and data analysts. This thesis has demonstratedhow repeated measures data can be modeled using hierarchical models. As an appli-cation, ambulatory blood pressure data was analyzed using the hierarchical model. Inliterature, classical approaches are popular for ambulatory blood pressure data anlaysis[31] [41]. In this thesis, it has been shown that more intuitive and flexible Bayesian mod-els can be equally effective as classical models. Thus, this thesis provides an option forapplied statisticians to model such data by using Bayesian methodologies.

The models that are discussed in this thesis can be extended in many ways. Forexample, treatment arms or various centers where the study was conducted can be in-cluded to increase the level of hierarchy. The data can be studied with other source ofvariations that can impact the mean blood profiles of the subjects. The cosinor modelwas used to model the rhythmic characteristic of the data. The alternatives to cosinormodels such as polynomial model [17] and cubic splines [33] can be explored and com-pared.

The non-hierarchical and hierarchical models were compared in classical andBayesian frameworks. A random effect model or a hierarchical model was found tobe superior to non-hierarchical model while analyzing a repeated measures dataset likeambulatory blood pressure data. We explored the within-subject variability and be-tween subject variability using hierarchical models. Since the ambulatory blood pres-sure readings were not taken in equal intervals and the number of measurements in ourdata was not same for all the subjects, the unstructured covariance model was used.There is a scope of future research to implement other covariance structures to modelthe within subject variability[36].

44

Appendices

45

APPENDIX A

SAS CODES

A.1 Individual Means with Standard Error Bars

/** Read the data ***/

PROC IMPORT OUT= WORK.bpx

DATAFILE= "F:\ PDbackup\sysbpdata.csv"

DBMS=CSV REPLACE;

GETNAMES=YES;

DATAROW =2;

RUN;

/** Calculate the means and standard errors per subject **/

Proc means data=bpx noprint;

by subj;

var SYSTOLIC;

output out=meansout mean=mean stderr=stderr;

Run;

Data reshape(keep=xvar yvar mean);

set meansout;

xvar=subj;

yvar=mean;

output;

yvar=mean - stderr;

output;

yvar=mean + stderr;

output;

run;

/** Graph it**/

goptions reset=all hsize =35cm vsize =20cm;

/* Define the title */

title1 ’Mean Systolic BP with Standard Error Bars ’;

/* Define the axis characteristics */

axis1 label=(’Subject ID ’) offset =(5,5) minor=none order =(1 to 50 by 1);

axis2 label=(’’) ;

/* Define the symbol characteristics */

symbol1 interpol=hiloct color=blue;

symbol2 interpol=none color=blue value=dot height =1;

46

/* Plot the error bars */

Proc gplot data=reshape;

plot yvar*xvar mean*xvar / overlay haxis=axis1 vaxis=axis2;

Run;

Quit;

47

APPENDIX B

R CODES

B.1 Individual Patient Profiles of 3 Subjects

setwd ("C:/Users/Prakash/OneDrive/Thesis/Data")

bp1 <-read.csv(" sysbpdata.csv")

head(bp1)

dataMat = bp1

yColName =" SYSTOLIC" ; yPlotLab = "Systolic BP (mmHg)"

xColName ="tdhr" ; xPlotLab ="Hours Post Dialysis"

subjColName ="subj" ; subjPlotLab ="Plot of Subject ID:"

fname = "ClassBP"

windows (25 ,6)

par( mar=c(4 ,4 ,1.75 ,1)+0.1 , mgp=c(2.5 ,0.8 ,0) )

layout(matrix(c(1:3), nrow=1,byrow=T))

xlims = c( min( dataMat[,xColName] ) , max( dataMat[,xColName] ) )

ylims = c( min( dataMat[,yColName] ) , max( dataMat[,yColName] ) )

sIdVec = unique( dataMat[,subjColName] )

# Plot data of individual subjects:

nSubjPlots = 3 # number of representative subject plots to make

subjIdxVec = round(seq(1,length(sIdVec),length=nSubjPlots ))

for ( sIdx in subjIdxVec) {

rVec = ( dataMat[,subjColName] == sIdVec[sIdx] )

plot( dataMat[rVec ,xColName] , dataMat[rVec ,yColName] , type="o" ,

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

}

B.2 Parameter Estimation using Maximum Likelihood

#setwd ("C:/Users/Prakash/OneDrive/Thesis/Data")

setwd ("C:/Users/pdhakal/OneDrive/Thesis/Data")

bp1 <-read.csv(" sysbpdata.csv")

head(bp1)

dataMat = bp1

library(nlme)

library(lme4)

#by(bp1$tdhr ,bp1$subj ,mean)

48

#bp1$tdhr <-bp1$tdhr -mean(bp1$tdhr)

bp1x <- groupedData(SYSTOLIC~tdhr|subj ,data=bp1)

yColName =" SYSTOLIC" ; yPlotLab = "Systolic BP (mmHg)"

xColName ="tdhr" ; xPlotLab ="Hours Post Dialysis"

subjColName ="subj" ; subjPlotLab ="Plot of Subject ID:"

fname = "ClassBP"

windows (25 ,6)

par( mar=c(4 ,4 ,1.75 ,1)+0.1 , mgp=c(2.5 ,0.8 ,0) )

layout(matrix(c(1:3), nrow=1,byrow=T))

xlims = c( min( dataMat[,xColName] ) , max( dataMat[,xColName] ) )

ylims = c( min( dataMat[,yColName] ) , max( dataMat[,yColName] ) )

sIdVec = unique( dataMat[,subjColName] )

# Plot data of individual subjects:

nSubjPlots = 3 # number of representative subject plots to make

subjIdxVec = round(seq(1,length(sIdVec),length=nSubjPlots ))

# Single Cosinor Model

mod1 = gls(SYSTOLIC ~cos24+sin24+tdhr ,data=bp1 ,method ="REML")

summary(mod1)

df<-data.frame(bp1$subj ,predict(mod1),bp1$SYSTOLIC ,bp1$tdhr)

for ( sIdx in subjIdxVec) {

rVec = ( dataMat[,subjColName] == sIdVec[sIdx] )

plot( dataMat[rVec ,xColName] , dataMat[rVec ,yColName] , type="o" ,

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

lines( df[rVec ,"bp1.tdhr"] , df[rVec ," predict.mod1 ."] ,

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,col=’red ’,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

}

windows ()

Autocorrelation =( residuals(mod1))

acf(Autocorrelation)

windows (25 ,6)

par( mar=c(4 ,4 ,1.75 ,1)+0.1 , mgp=c(2.5 ,0.8 ,0) )

layout(matrix(c(1:3), nrow=1,byrow=T))

##With Heirarchy . Random intercept model

lme1 <-lme(fixed=SYSTOLIC ~( cos24+sin24+tdhr),random =~1|subj ,

data=bp1x ,method ="REML")

summary(lme1)

df<-data.frame(bp1$subj ,predict(lme1),bp1$SYSTOLIC ,bp1$tdhr)

for ( sIdx in subjIdxVec) {

rVec = ( dataMat[,subjColName] == sIdVec[sIdx] )

plot( dataMat[rVec ,xColName] , dataMat[rVec ,yColName] , type="o" ,

49

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

lines( df[rVec ,"bp1.tdhr"] , df[rVec ," predict.lme1 ."] ,

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,col=’red ’,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

}

windows ()

Autocorrelation =( residuals(mod1))

acf(Autocorrelation)

## Random intercept and slope

lme2 <-lme(fixed=SYSTOLIC ~( cos24+sin24+tdhr),random =(~ cos24+sin24+tdhr),

data=bp1x ,method ="REML")

summary(lme2)

Autocorrelation=resid(lme2)

acf(Autocorrleation)

windows (25 ,6)

par( mar=c(4 ,4 ,1.75 ,1)+0.1 , mgp=c(2.5 ,0.8 ,0) )

layout(matrix(c(1:3), nrow=1,byrow=T))

df<-data.frame(bp1$subj ,predict(lme2),bp1$SYSTOLIC ,bp1$tdhr)

for ( sIdx in subjIdxVec) {

rVec = ( dataMat[,subjColName] == sIdVec[sIdx] )

plot( dataMat[rVec ,xColName] , dataMat[rVec ,yColName] , type="o" ,

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

lines( df[rVec ,"bp1.tdhr"] , df[rVec ," predict.lme2 ."] ,

ylim=ylims , ylab=yPlotLab , xlab=xPlotLab , cex.lab =1.5 ,col=’red ’,

pch=sIdx %%26 , lty=sIdx , main=bquote (.( subjPlotLab) *" "* .(sIdx)) ,

cex.main =1.75 )

}

50

APPENDIX C

OPENBUGS CODES

C.1 Single Cosinor Model

# BUGS model for Single Cosinor:

model {

#Likelihood

for( r in 1 : Ndata) {

y[r] ~ dnorm( mu[r], tau)

mu[r] <- M + beta*cos24[r] + gamma*sin24[r] + alpha*tdhr[r]

}

#Priors

M ~ dnorm (0 ,0.0001)

beta ~ dnorm( 0 ,0.0001 )

gamma ~ dnorm( 0 ,0.0001 )

alpha ~ dnorm( 0 ,0.0001 )

tau ~ dgamma (1 , 0.001 )

sigma <- sqrt (1/tau) #Normal errors

}

#Intial Values

#1st Chain

list( M=140,

beta=0, gamma =0.5, alpha =0.5, tau =.001

)

#2nd Chain

list( M=160,

beta=3, gamma =0.01, alpha =0.1, tau =.01

)

C.2 Random Intercept Model

# BUGS model for random intercepts:

model {

for( r in 1 : Ndata) {

y[r] ~ dnorm( mu[r] , tau)

mu[r] <- M[subj[r]] + beta*cos24[r]+ gamma*sin24[r] + alpha*(tdhr[r])

}

51

#Random Intercepts

for (s in 1: Nsubj)

{

M[s] ~ dnorm( muM , tauM )

}

#Priors

muM ~ dnorm (0 ,.0001)

tauM~ dgamma (1 ,.0001)

beta ~ dnorm( 0 ,0.0001 )

gamma~ dnorm( 0 ,0.0001 )

alpha~ dnorm( 0 ,0.0001 )

tau ~ dgamma(1, 0.001 )

sigma <- sqrt (1/ tau) #Normal errors

sigmaM <-sqrt (1/ tauM) ## Between subject errors

}

#initial values

#1st Chain

list( M=c(140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,

140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,

140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,140 ,

140 ,140 ,140 ,140) ,

beta =0.1, gamma =0.1, alpha =0.5,muM =149., tauM =0.002924 ,tau =.001)

#2nd Chain

#Auto Generated

C.3 Random Intercept and Slope Model

#Bugs Model for Random Intercept and Slope Model

model {

for( r in 1 : Ndata ) {

y[r] ~ dnorm( mu[r] , tau)

mu[r] <- M[subj[r]] + beta[ subj[r]]* cos24[r] +gamma[subj[r]]* sin24[r]+

alpha[subj[r]]* tdhr[r]

}

tau <-pow(sigma ,-2)

sigma ~ dunif (0 ,100)

#Subject level priors

for ( s in 1 : Nsubj ) {

M[s] <-xi.b0* B[s,1]

beta[s] <- xi.b1*B[s,2]

gamma[s] <- xi.b2*B[s,3]

alpha[s] <- xi.b3*B[s,4]

B[s, 1:4] ~ dmnorm( B.hat[s, ], Tau.B[ , ] )

52

B.hat[s,1] <- mu0Gr

B.hat[s,2] <- mu1Gr

B.hat[s,3] <- mu2Gr

B.hat[s,4] <- mu3Gr

}

mu0G <-xi.b0*mu0Gr

mu1G <-xi.b1*mu1Gr

mu2G <-xi.b2*mu2Gr

mu3G <-xi.b3*mu3Gr

mu0Gr ~ dnorm (0 ,.0001)

mu1Gr ~ dnorm (0 ,.0001)

mu2Gr ~ dnorm (0 ,.0001)

mu3Gr ~ dnorm (0 ,.0001)

xi.b0 ~ dunif (0 ,100)

xi.b1 ~ dunif (0 ,100)

xi.b2 ~ dunif (0 ,100)

xi.b3 ~ dunif (0 ,100)

#Wishart Distribution for Precision Matrix

Tau.B[1:4, 1:4] ~ dwish(W[,],df)

df <-5

Sigma.B[1:4 ,1:4] <- inverse(Tau.B[,])

sigma0 <-xi.b0*sqrt(Sigma.B[1,1])

sigma1 <-xi.b1*sqrt(Sigma.B[2,2])

sigma2 <-xi.b2*sqrt(Sigma.B[3,3])

sigma3 <-xi.b3*sqrt(Sigma.B[4,4])

W[1,1]<-1

W[2,2]<-1

W[3,3]<-1

W[4,4]<-1

W[1,2]<-0.00

W[1,3]<-0.00

W[1,4]<-0.00

W[2,1]<-0.00

W[2,3]<-0.00

W[2,4]<-0.00

W[3,1]<-0.00

W[3,2]<-0.00

W[3,4]<-0.00

W[4,1]<-0.00

W[4,2]<-0.00

W[4,3]<-0.00

#Correlations

rho01 <-Sigma.B[1,2]/ sqrt(Sigma.B[1,1]* Sigma.B[2,2])

rho02 <-Sigma.B[1,3]/ sqrt(Sigma.B[1,1]* Sigma.B[3,3])

rho03 <-Sigma.B[1,4]/ sqrt(Sigma.B[1,1]* Sigma.B[4,4])

53

rho12 <-Sigma.B[2,3]/ sqrt(Sigma.B[2,2]* Sigma.B[3,3])

rho13 <-Sigma.B[2,4]/ sqrt(Sigma.B[2,2]* Sigma.B[4,4])

rho23 <-Sigma.B[3,4]/ sqrt(Sigma.B[3,3]* Sigma.B[4,4])

}

#Initial values

# 1st chain

list(

B=structure(

.Data=c(140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,

140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5 ,140 ,22 ,17 ,0.5) ,. Dim=c(49,4)),

sigma =10,

mu0Gr =149.4117 ,

mu1Gr =1.759025 ,

mu2Gr =0.424634 ,

mu3Gr =0.38598 ,

Tau.B=structure( . Data=c(1,0,0,0,

0,1,0,0,

0,0,1,0,

0,0,0,1),.Dim=c(4,4)),

xi.b0=0.5,

xi.b1=0.5,

xi.b2=0.5,

xi.b3=0.5

)

#2nd Chain

#Auto generated

54

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Hierarchical Modeling of Biological Rhythm Data:Classical and Bayesian Approaches

by

Prakash Dhakal

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