Statistical Methods for Evaluating Continuous and Functional Diagnostic Markers 公开
Jang, Jeong Hoon (Summer 2019)
Published
Abstract
The proposed statistical research in this dissertation is motivated by a renal study conducted at Emory University. The study consists of kidneys with suspected obstruction, whose initial diagnoses were provided by nuclear medicine experts and residents. This study also includes functional markers (renogram curves) that have been collected as a noninvasive mean of interpreting kidney obstruction. The overarching scientific goal of this study is two-fold: (1) to understand the reliability of experts' and residents' interpretations of kidney obstruction; and (2) to evaluate the diagnostic utility of renogram curves for detecting kidney obstruction.
First research topic aims at developing new agreement indices based on root mean square of pairwise differences (RMSPD) that can be used to quantify agreement among multiple heterogeneous raters. The advantages of the proposed indices are: (1) interpretations are tied to the measurement scale; (2) satisfactory agreement is conveniently determined via pre-specified tolerable RMSPD. The proposed indices are applied to the Emory renal study to quantify the reliability in interpretations of kidney obstruction.
Quantitative features of functional markers (maximum, time to minimum, average velocity, etc.) are increasingly being used to diagnose diseases. Second research topic aims to study their alignment according to an ordinal reference test. I propose a class of summary functionals, which flexibly represent various quantitative features, and study its alignment via broad sense agreement (BSA, Peng et al., 2011). Asymptotic properties of the proposed BSA estimator are established. This work is applied to the Emory renal study to unveil quantitative features of renogram curves that closely replicate experts' interpretations.
Third research topic aims to assess the diagnostic accuracy of quantitative features based on area under the receiver operating characteristic curve (AUC). I propose a non-parametric AUC estimator that addresses discreteness and measurement error in functional data and establish its asymptotic properties. To describe the heterogeneity of AUC in different subpopulations, I propose a sensible adaptation of a semi-parametric regression model, whose parameters can be estimated by the proposed estimated estimating equations. This work is applied to the Emory renal study to identify quantitative features with high AUCs, and to investigate their relationship with patients' characteristics.
Table of Contents
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . 2
1.2 Literature Review . . . . . . . . . . . . . . 3
1.2.1 Statistical methods for assessing agreement . . . . . . . . . . . 3
1.2.2 Statistical methods for evaluating diagnostic accuracy of markers. . . . . . . . . . . . . . . . . . . .9
1.2.3 Latent class models for evaluating diagnostic accuracy of markers under no gold standard . . . . . . 12
1.3 Motivating Data . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 18
1.4 Statistical Problems and Contributions . . . . . . . . . . . . . . . . . 22
2 Overall Indices for Assessing Agreement Among Multiple Raters 25
2.1 Introduction . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 26
2.2 Methods . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 29
2.2.1 Existing unscaled and summary agreement indices for two raters . . . . . 29
2.2.2 Overall agreement indices for multiple raters . . . . . . . . . . 30
2.2.3 Estimation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 37
2.2.4 Inference . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 38
2.3 Simulations . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 42
2.4 Renal Study . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47
2.5 Discussion . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 52
3 Assessing Alignment Between Functional Markers and Ordinal Outcomes Based on Broad Sense Agreement 56
3.1 Introduction . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 57
3.2 Methods . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 61
3.2.1 Review of broad sense agreement. . . . . . . . . . . . . . . 61
3.2.2 General formulation of the summary functional . . . . . . . . 63
3.2.3 Proposed BSA framework . . . . . .. . . . . . . . . . . . . . 63
3.2.4 Nonparametric estimation . . . . .. . . . . . . . . . . . . . . 64
3.2.5 Asymptotic properties . . . . . .. . . . . . . . . . . . . . . . 65
3.2.6 Estimation of standard error and confidence interval . . . . . . 66
3.3 Illustration of the Proposed BSA Framework . . . . . . . . . . . . . . 67
3.3.1 Three special cases of summary functionals . . . . . . . . . . . 67
3.3.2 Nonparametric estimation of the special-case summary functionals . . . . . . . . . . . . 69
3.4 Statistical Test for Selecting a Summary Functional . . . . . . . . . . 71
3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Renal Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7 Discussion . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 82
4 Evaluating Quantitative Features of Functional Markers Based on Area Under the Receiver Operating Characteristic Curve 84
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Representing Quantitative Features via Summary Functionals . . . . 91
4.2.1 General formulation of a summary functional . . . . . . . . . 91
4.2.2 Three special (widely-used) cases of summary functionals . . . 91
4.2.3 Estimation of summary functionals . . . . . . . . . . . . . . . 93
4.3 AUC Analysis of Quantitative Features . . . . . . . . . . . . . . . . . 95
4.3.1 Formulation and estimation . . . . . . . . . . . . . . . . . . . 95
4.3.2 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . 97
4.3.3 Statistical inference . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 Covariate-specific AUC Analysis of Quantitative Features . . . . . . . 99
4.4.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.2 Estimated estimating equations . . . . . . . . . . . . . . . . . 101
4.4.3 Estimation with continuous covariate . . . . . . . . . . . . . . 102
4.4.4 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6 Application to Renal Study . . . . . . . . . . . . . . . . . . . . . . . 111
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 A Novel Statistical Approach to Evaluate Functional Markers Without a Gold Standard 117
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 A FPCA Approach for Evaluating Functional Markers Without a Gold Standard . . . . . . . . . . .. . . 124
5.2.1 FPCA for univariate functional markers . . . . . . . . . . . . 124
5.2.2 FPCA for multivariate functional markers (MFPCA) . . . . . 125
5.2.3 Estimated FPCA scores: a lower dimensional representation of a functional marker . .. . . . . . . 128
5.2.4 FPCA-based ROC analysis without gold standard . . . . . . . 130
5.2.5 Estimation and inference of the ROC model . . . . . . . . . . 132
5.2.6 FPCA-based approach to predict disease status of future observations . . . . . . . . . . . . . 133
5.3 A FPLS Approach to Incorporate Imperfect Reference Test . . . . . . 135
5.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.5 Application to Renal Study . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6 Future Research 154
Appendix A 158
A.1 Derivation of quadratic form (2.5) . . . . . . . . . . . . . . . . . . . . 158
A.2 Steps for two-sample hypothesis testing . . . . . . . . . . . . . . . . . 159
Appendix B 161
B.1 Proof of Theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B.2 Specification of Kernel Function. . . . . . . . . . . . . . . . . . . . . 164
B.3 Consistency of the estimators for the three special-case summary functionals. . . . . . . . . . . . . . . . . . 165
B.3.1 AUC-type functionals . . . . . . . . . . . . . . . . . . . . . . . 166
B.3.2 Magnitude-specific functionals . . . . . . . . . . . . . . . . . . 167
B.3.3 Time-specific functionals . . . . . . . . . . . . . . . . . . . . . 168
B.4 Additional Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.4.1 Evaluation of the proposed hypothesis testing procedure . . . 169
B.4.2 Finite-sample performance at the first derivative level of the summary functionals . . . . . . . . . . . 171
Appendix C 174
C.1 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C.2 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Appendix D 186
D.1 The EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
D.2 Standard Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . 189
D.3 Estimation and Prediction for FPLS . . . . . . . . . . . . . . . . . . 192
D.4 Parameter Setup for Simulation Settings . . . . . . . . . . . . . . . . 194
Bibliography 197
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