1 Inference and Prediction Using Functional Principal Components Analysis Application to Diabetic Kidney Disease Progression in the Chronic Renal Insufficiency Cohort CRIC Study

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Inference and Prediction Using Functional Principal Components Analysis: Application to

Diabetic Kidney Disease Progression in the Chronic Renal Insufficiency Cohort (CRIC)

Study

Brian Kwan1,2,

, Wei Yang3, Daniel Montemayor4,5, Jing Zhang2, Tobias Fuhrer6,

, Amanda H.

Anderson7, Cheryl A.M. Anderson8, Jing Chen9, Ana C. Ricardo10, Sylvia E. Rosas11, Loki

Natarajan1,2, and the CRIC Study Investigators*

1Division of Biostatistics and Bioinformatics, Herbert Wertheim School of Public Health, University of California,

San Diego, La Jolla, CA, USA;

2Moores Cancer Center, University of California, San Diego, La Jolla, CA, USA;

3Department of Biostatistics, Epidemiology and Informatics, Center for Clinical Epidemiology and Biostatistics,

Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, USA;

4Division of Nephrology, Department of Medicine, University of Texas Health San Antonio, San Antonio, TX,

USA;

5Center for Renal Precision Medicine, University of Texas Health San Antonio, San Antonio, TX, USA;

6Institute of Molecular Systems Biology, ETH Zurich, Zurich, Switzerland;

7Department of Epidemiology, Tulane School of Public Health and Tropical Medicine, New Orleans, LA;

8Herbert Wertheim School of Public Health, University of California, San Diego, La Jolla, CA, USA;

9Department of Medicine, Tulane School of Medicine, New Orleans, LA;

10Department of Medicine, University of Illinois, Chicago, IL, USA;

11Joslin Diabetes Center and Harvard Medical School, Boston, MA, USA;

*A list of the CRIC Study Investigators appears in the Acknowledgements.

Correspondence: Loki Natarajan, Division of Biostatistics, Herbert Wertheim School of Public Health, University

of California San Diego, 3855 Health Sciences Dr #0901, La Jolla, CA 92093, USA. Email: lnatarajan@ucsd.edu.

Moved new affiliations:

a Department of Biostatistics, Fielding School of Public Health, University of California, Los Angeles, Los Angeles,

CA, USA

Biostarks, Geneva, Switzerland

Abstract

Repeated longitudinal measurements are commonly used to model long-term disease

progression, and timing and number of assessments per patient may vary, leading to irregularly

spaced and sparse data. Longitudinal trajectories may exhibit curvilinear patterns, in which

mixed linear regression methods may fail to capture true trends in the data. We applied

functional principal components analysis to model kidney disease progression via estimated

glomerular filtration rate (eGFR) trajectories. In a cohort of 2641 participants with diabetes and

up to 15 years of annual follow-up from the Chronic Renal Insufficiency Cohort (CRIC) study,

we detected novel dominant modes of variation and patterns of diabetic kidney disease (DKD)

progression among subgroups defined by the presence of albuminuria. We conducted inferential

permutation tests to assess differences in longitudinal eGFR patterns between groups. To

determine whether fitting a full cohort model or separate group-specific models is more optimal

for modeling long-term trajectories, we evaluated model fit, using our goodness-of-fit procedure,

and future prediction accuracy. Our findings indicated advantages for both modeling approaches

in accomplishing different objectives. Beyond DKD, the methods described are applicable to

other settings with longitudinally assessed biomarkers as indicators of disease progression.

Supplementary materials for this article are available online.

Keywords: functional data analysis, longitudinal study, model fit, permutation tests, renal

disease, sparse data

1 Introduction

Diabetes mellitus is the leading cause of chronic kidney disease (CKD) (Bailey et al.

2014; Centers for Disease Control and Prevention 2017; Koro et al. 2009; Koye et al. 2018;

United States Renal Data System 2018). Patients with CKD are at increased risk of kidney

failure, potentially requiring treatment by kidney transplant or dialysis. Yet, there is substantial

heterogeneity in the development of kidney disease for patients with diabetes (Gheith et al.

2016).

Diabetic kidney disease (DKD) progression is typically characterized as the estimated

glomerular filtration rate (eGFR) trajectory over time (de Boer et al. 2009; Robinson-Cohen et al.

2014). Linear mixed effect models are often used to estimate change in eGFR; however,

characterizing the trajectory is complicated by observing sparse or irregularly spaced time series

data which may exhibit nonlinear trends as depicted in Supplementary Figure 1 within the

Supplementary Materials. Thus, implementing flexible statistical methods that characterize

eGFR trajectories in key subgroups could offer insights into DKD progression and treatment.

We propose the functional principal components analysis (FPCA) approach to model

long-term trajectories while accounting for complexity in curve estimation, i.e., nonlinearity,

sparsity, and irregularity. The subject of functional data analysis and the development of FPCA

in particular is well studied (Ramsay and Silverman 2005; Wang et al. 2016) with applications of

FPCA to sparse functional data dating back several decades (James et al. 2000; Paul and Peng

2009; Peng and Paul 2009; Rice and Wu 2001; Shi et al. 1996; Staniswalis and Lee 1998; Yao et

al. 2005; Yao and Lee 2006). FPCA has also been applied to study various disease progression

studies, e.g., lung, HIV (Szczesniak et al. 2017; Xie et al. 2017; Yan et al. 2017).

A salient feature of FPCA is that this approach can project infinite-dimensional curves

into finite-dimensional vector scores and detect major modes of curve variation which would

elucidate the leading patterns in disease (i.e., DKD) progression. Important for our context,

expanding the FPCA framework to characterizing and predicting DKD progression by clinically

meaningful subgroups will further elucidate the heterogeneous trends in the development of

kidney disease across patients with diabetes. In this work, we will consider subgroups based on

albuminuria, an established clinical biomarker for kidney disease based on the excess amount of

albumin in the urine often used in conjunction with eGFR to classify patient CKD stage (Kidney

Disease: Improving Global Outcomes (KDIGO) CKD Work Group 2013), although our

approach could readily be applied to any other subgroups of interest. A key question in model

development is whether a single full cohort FPCA model, trained using data from all patients

with diabetes irrespective of albuminuria groups, is sufficient for characterizing and accurately

predicting the long-term eGFR trajectory within specific albuminuria groups. If not, we consider

multiple albuminuria group-specific models, each fitted using data from only patients of a

specific albuminuria group, to prospectively predict eGFR trajectories for new subjects of the

same group. To decide whether this group-level approach is needed, we examine differences in

longitudinal eGFR patterns, i.e., mean and correlation functions, between albuminuria subgroups

as well as model fit, via our proposed goodness-of-fit procedure, and future prediction accuracy.

As a whole, our work applies and extends FPCA methodology to improving inference and

prediction of trajectories of kidney function decline in heterogeneous subpopulations of patients

with diabetes, with the methods generalizable to research applications beyond DKD.

The structure of this paper is as follows. Section 2 presents our methods. This describes

the FPCA approach, inferential permutation tests to test for differences in longitudinal eGFR

patterns between groups, comparison of model fits and accuracy, our CRIC study cohort,

assignment of albuminuria groups, and computational tools. Section 3 describes our statistical

analysis results in detail. Lastly, Section 4 discusses the findings, limitations, and future

directions of our work.

2 Methods

2.1 Functional Principal Components Analysis (FPCA)

Let    be the observed outcome at time , where  is the

measurement-error free outcome for subject  at time  and  are measurement errors assumed

to be identically and independently distributed normal with mean zero and variance  such that

  and   .

We model the individual trajectories as a smooth random function  with unknown

mean function  and covariance function , where   and  is a bounded and

closed time interval. Let be the outcome trajectory for the th individual and  be years of

follow-up. Under the Karhunen-Loeve expansion (Karhunen 1946; Loève 1946), the th

individual’s trajectory can be expressed as 





where  is the th functional principal component (FPC) and  is the associated th FPC

score for the th individual. The individual scores  are uncorrelated random variables with

mean zero and variance , where    and  

. The covariance function

 can be defined    



 

We briefly recapitulate the workflow and software implementation of the PACE algorithm by

Yao et al. (2005) to estimate these model components in the Supplementary Materials. Since the

outcome trajectory is often well approximated by the top  FPCs and their associated scores, we

select  as the number of FPCs that explained at least 95% of the total variance in the outcome

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1InferenceandPredictionUsingFunctionalPrincipalComponentsAnalysis:ApplicationtoDiabeticKidneyDiseaseProgressionintheChronicRenalInsufficiencyCohort(CRIC)StudyBrianKwan1,2,a,WeiYang3,DanielMontemayor4,5,JingZhang2,TobiasFuhrer6,b,AmandaH.Anderson7,CherylA.M.Anderson8,JingChen9,AnaC.Ricardo10,SylviaE....

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1 Inference and Prediction Using Functional Principal Components Analysis Application to Diabetic Kidney Disease Progression in the Chronic Renal Insufficiency Cohort CRIC Study

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