Nonparametric Bayesian estimation of a bivariate density with interval censored data

A multivariate Polya tree model for meta-analysis with event-time distributions

We develop a nonparametric Bayesian prior for a family of random probability measures by extending the Polya tree ($\mbox{PT}$) prior to a joint prior for a set of probability measures $G_1,\dots ,G_n$, suitable for meta-analysis with event-time outcomes. In the application to meta-analysis, $G_i$ is the event-time distribution specific to study $i$. The proposed model defines a regression on study-specific covariates by introducing increased correlation for any pair of studies with similar characteristics. The desired multivariate $\mbox{PT}$ model is constructed by introducing a hierarchical prior on the conditional splitting probabilities in the $\mbox{PT}$ construction for each of the $G_i$. The hierarchical prior replaces the independent beta priors for the splitting probability in the PT construction with a Gaussian process prior for corresponding (logit) splitting probabilities across all studies. The Gaussian process is indexed by study-specific covariates, introducing the desired dependence with increased correlation for similar studies. The main feature of the proposed construction is (conditionally) conjugate posterior updating with commonly reported inference summaries for event-time data. The construction is motivated by a meta-analysis over cancer immunotherapy studies.

Density estimation on multivariate censored data with optional Pólya tree

Analyzing the failure times of multiple events is of interest in many fields. Estimating the joint distribution of the failure times in a non-parametric way is not straightforward because some failure times are often right-censored and only known to be greater than observed follow-up times. Although it has been studied, there is no universally optimal solution for this problem. It is still challenging and important to provide alternatives that may be more suitable than existing ones in specific settings. Related problems of the existing methods are not only limited to infeasible computations, but also include the lack of optimality and possible non-monotonicity of the estimated survival function. In this paper, we proposed a non-parametric Bayesian approach for directly estimating the density function of multivariate survival times, where the prior is constructed based on the optional Pólya tree. We investigated several theoretical aspects of the procedure and derived an efficient iterative algorithm for implementing the Bayesian procedure. The empirical performance of the method was examined via extensive simulation studies. Finally, we presented a detailed analysis using the proposed method on the relationship among organ recovery times in severely injured patients. From the analysis, we suggested interesting medical information that can be further pursued in clinics.

Smooth estimation of survival functions and hazard ratios from interval-censored data using Bayesian penalized B-splines

We discuss the use of Bayesian P-spline and of the composite link model to estimate survival functions and hazard ratios from interval-censored data. If one further assumes proportionality of the hazards, the proposed strategy provides a smoothed estimate of the baseline hazard along with estimates of global covariate effects. The frequentist properties of our Bayesian estimators are assessed by an extensive simulation study. We further illustrate the methodology by two examples showing that the proportionality of the hazards might also be found inappropriate from interval-censored data.