A New Class of Bivariate Weibull Distribution to accommodate the Concordance Correlation coefficient for Left-censored Data

Open Access
Author:
Domthong, Uthumporn
Graduate Program:
Public Health Sciences
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
October 06, 2014
Committee Members:
  • Vernon Michael Chinchilli, Dissertation Advisor
  • Lan Kong, Committee Member
  • Tonya Sharp King, Committee Member
  • W. Brian Reeves, Committee Member
  • David Spencer Phelps, Committee Member
Keywords:
  • concordance correlation coefficient (CCC)
  • lower limit of detection (LLD)
  • Weibull distribution
  • The maximum likelihood method
Abstract:
In many clinical studies, Lin’s concordance correlation coefficient (CCC) is a common tool to assess the level of agreement of a continuous response measured under two different conditions. However, the complicating feature is that the assay for measuring a specific biomarker typically cannot provide accurate numerical values below the lower limit of detection (LLD), which results in left-censored data. In addition, the CCC is based on a squared distance function, and it can be very sensitive to the effects of the outliers. In this work, we propose a new class of bivariate survival functions based on functions of univariate survival functions and univariate cumulative hazard functions. We focus on using the univariate Weibull distribution to obtain a bivariate Weibull survival function. The likelihood function can be determined via this new class of bivariate Weibull survival functions.Then, we take a parametric approach to derive the estimates of the means, variances, and covariance to construct the CCC. This new class of bivariate survival functions can be extended to the situation with p > 2 random variables. The maximum likelihood method based on three distributions, (1) bivariate Weibull distributions (2) bivariate Farlie-Gumbel-Morgenstern distributions and (3) bivariate lognormal distributions, were evaluated via simulation studies. The simulation results confirmed that overall in terms of accuracy, that is small relative bias, the estimator of CCC based on FGM-Weibull works relatively well in general cases when the correlation is not too strong even with the high percentage of censoring. For a skewed underlying distribution with moderate or weaker correlation between two variables, the CCC estimated by a FGM-Weibull model is more robust. However, when the data are generated from the bivariate lognormal, the ML approach based on the bivariate lognormality assumption still performs best. Finally, we use data from an ancillary study of the Assessment, Serial Evaluation, and Subsequent Sequelae of Acute Kidney Injury (ASSESS-AKI) Consortium, and an asthma clinical trial for demonstration.