Joint Modeling Analysis of Multivariate Skewed-longitudinal and Time-to-event Data with Application to Primary Biliary Cirrhosis Study

Abstract

Background: Many clinical and public health researches collect data including multiple longitudinal measures and time-to-event outcomes, where characteristics of the pattern of exposure change and the association between features of longitudinal biomakers and the primary survival endpoint are of interest. Methods: Many existing statistical models for longitudinal-survival data might not provide robust inference when more than one longitudinal exposures which were significantly correlated and longitudinal measurements exhibit skewness and/or heavy tails; ignoring these data features may lead to biased estimation. In this article, we offered a multivariate joint model with the skew-normal (SN) distribution with application to the Mayo clinic primary biliary cirrhosis (PBC) study to assess simultaneous effects. Results: With the multivariate joint modeling associated with the skew-normal (SN) distribution, the subject-specific baseline (HR=2.390 with 95% CI: (1.429, 4.112)) and change rate (HR=2.588 with 95% CI:(1.845, 3.967)) of Bilirubin in natural log scale were positively associated with the risk of death; the higher the subject-specific change rate (HR=0.191 with 95% CI: (0.037, 0.915)) of Albumin in natural log scale was associated with a decrease in mortality rate; the subject-specific of SGOT levels in natural log scale did not affect the risk of death for PBC patients significantly. The results of the skewness parameters of natural log-transformed Bilirubin (δ 1 =0.42), Albumin (δ 2 =−0.03) and SGOT (δ 3 =0.095) were estimated to be significant, indicating the skewness of three biomarkers existed. Conclusions: Our results revealed the Bilirubin and Albumin levels may be involved in predicting risk of death for PBC patients, except for SGOT. The multivariate joint modeling associated with SN distribution provides better fit to the data, gives less biased parameter estimates for those longitudinal biomarkers in comparison with its counterpart where the normal distribution is assumed (data not shown here). The introduced modeling approach is generally applicable to other situations where longitudinal measurements and time-to-event outcomes are available.