A class of mixtures of dependent tail-free processes

Abstract

SUMMARYWe propose a class of dependent processes in which density shape is regressed on one or more predictors through conditional tail-free probabilities by using transformed Gaussian processes. A particular linear version of the process is developed in detail. The resulting process is flexible and easy to fit using standard algorithms for generalized linear models. The method is applied to growth curve analysis, evolving univariate random effects distributions in generalized linear mixed models, and median survival modelling with censored data and covariate-dependent errors.

Modeling county level breast cancer survival data using a covariate-adjusted frailty proportional hazards model

“…That is, $$e_{i}false|G_{{\mathrm{bold-italicx}}_i}\stackrel{ind.}{~}{G}_{{\mathrm{bold-italicx}}_i},$$ where for every x ∈ 𝒳, G x is a probability measure defined on ℝ; this specifies a probability model for the entire collection of probability measures 𝒢 𝒳 = { G x : x ∈ 𝒳}, such that its elements are allowed to smoothly vary with the cluster-level covariates x . Specifically, we consider a mixture of linear dependent tailfree processes (LDTFP) prior (Jara and Hanson, 2011) for 𝒢 𝒳 , denoted as $${𝒢}^{𝒳}false|J,h,\theta ,c,\rho ~\mathrm{normalL}\mathrm{normalD}\mathrm{normalT}\mathrm{normalF}\mathrm{normalP}false(h,{\mathrm{\Pi }}^{J,\theta },{𝒜}^{J,c,\rho }false),$$ and $$cfalse|Q~Q,$$ where J ∈ ℕ is the level of specification of the process, c ∈ ℝ + is a prior precision parameter controlling the prior variability of the process, $hfalse(·false)=\frac{\text{exp}false\{·false\}}{1+\text{exp}false\{·false\}}$, Π J ,θ is a J -level sequence of binary partitions of ℝ, depending on the scale parameter θ ∈ ℝ + , 𝒜 J,c ,ρ = {2 n / c ρ(1), …, 2 n / c ρ( J )} is a collection of positive numbers depending on J, c and ρ, ρ : ℕ → ℝ + is an increasing function, and Q is a probability measure defined on ℝ + .…”

“…As shown by Jara and Hanson (2011), dependent tailfree processes have appealing theoretical properties such as continuity as a function of the predictors, large support on the space of conditional density functions, straightforward posterior computation relying on algorithms for fitting generalized linear models, and the process closely matches conventional Polya tree priors (see, e.g., Hanson, 2006a) at each value of the predictor, which justify its choice here. Polya trees have been extensively studied in the literature and have desirable properties in terms of support and posterior consistency.…”

A Bayesian Nonparametric Test for Minimal Repair

“…The N (0, 2/cρ(j )) prior on λ k ( 0) mimics a beta(cρ(j ), cρ(j )) prior for Polya tree conditional probabilities {π k ( 0)} (Jara and Hanson 2011). A common choice that we adopt is ρ(j ) = j 2 .…”

“…Priors on the space of cumulative hazard functions include gamma processes, weighted gamma processes, beta processes; see Lo (1992), Kuo and Ghosh (1997), and Ishwaran and James (2004). Often, the random CDF F (s) is centered at a parametric distribution G θ in the sense that E{F (s)} = G θ (s) for all s > 0, that is, G θ is the "prior mean" of F. Our proposed framework uses tailfree priors (Fabius 1964;Ferguson 1974;Jara and Hanson 2011) to model F centered at the Weibull family, E{F (s)} = 1 − e −(s/γ ) α given (α, γ ), but allows for substantial data-driven deviations from Weibull. Our approach naturally tests whether Weibull is adequate, as well as incorporating maintenances where no failure has actually occurred (i.e., censored failures).…”

Generalized accelerated failure time spatial frailty model for arbitrarily censored data

“…This model extends the linear dependent tailfree process of Jara and Hanson (2011) to the interval-censored data setting, and also incorporates exchangeable or spatially varying areal-level frailties. In contrast to the aforementioned index models (Pang et al, 2015), the entire shape of the residual density changes smoothly with strata covariates.…”