A Bayesian inference for the penalized
spline joint models of longitudinal and time-to-event data: A prior
sensitivity analysis

Huong Thi Thu Pham; Hoa Pham; Darfiana Nur

doi:10.1515/mcma-2020-2058

Published by De Gruyter February 22, 2020

A Bayesian inference for the penalized spline joint models of longitudinal and time-to-event data: A prior sensitivity analysis

Huong Thi Thu Pham , Hoa Pham and Darfiana Nur

From the journal Monte Carlo Methods and Applications

https://doi.org/10.1515/mcma-2020-2058

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Abstract

Bayesian approaches have been used in the literature to estimate the parameters for joint models of longitudinal and time-to-event data. The main aim of this paper is to analyze the impact of prior distributions on estimating parameters in a proposed fully Bayesian analysis setting for the penalized spline joint models. To achieve this aim, the joint posterior distribution of parameters in survival and longitudinal submodels is presented. The Markov chain Monte Carlo (MCMC) algorithm is then proposed, which consists of the Gibbs sampler (GS) and Metropolis Hastings (MH) algorithms to sample for the target conditional posterior distributions. The prior sensitivity analysis for the baseline hazard rate and association parameters is performed through simulation studies and a case study.

Keywords: Survival data; longitudinal data; joint models; Bayesian approach; random effects

MSC 2010: 62N02; 62F15; 62P10; 62P12

A The conditional posterior distribution for all parameters and the conjugate prior distribution of the variance matrix 𝐆

A.1

The conditional posterior distribution for parameters in the baseline hazard function:

p(𝜽h0|𝜽(-𝜽h0),𝐛,T,δ,𝐲)∝∏i=1n[h0(Ti)exp{𝜸T𝐰i+αmi(t)}]δi

×∏i=1nexp⁡(-∫0Tih0⁢(s)⁢exp⁡{𝜸T⁢𝐰i+α⁢mi⁢(s)⁢d⁢s})

(A.1)×|𝚺𝜽h0|-12⁢exp⁡{-12⁢(𝜽h0-μ𝜽h0)T⁢𝚺𝜽h0-1⁢(𝜽h0-μ𝜽h0)},

where the notation 𝜽(-θi) means all parameters in the joint model except for θi.

A.2

The conditional posterior distribution of the regression coefficients in the survival submodel:

p(𝜸,α|𝜽(-𝜸,-α),𝐛,T,δ,𝐲)∝∏i=1nexp{𝜸T𝐰i+αmi(t)}δi×exp(-∫0Tiexp{𝜸T𝐰i+αmi(s)ds})

×|𝚺𝜸|-12⁢exp⁡{-12⁢(𝜸-μ𝜸)T⁢𝚺λ-1⁢(𝜸-μ𝜸)}

(A.2)×(2⁢π⁢σα)-12⁢exp⁡{-12⁢σα2⁢(α-μα)2}.

A.3

The conditional posterior distribution of the regression coefficients 𝜷 in the linear mixed effects submodel:

p(𝜷|𝜽(-𝜷),𝐛,T,δ,𝐲)∝∏i=1nexp{𝜸T𝐰i+αmi(t)}δi×exp(-∫0Tiexp{𝜸T𝐰i+αmi(s)ds})

×∏i=1n∏j=1niexp⁡{-(yi⁢(ti⁢j)-𝐗iT⁢(ti⁢j)⁢𝜷-𝐗iT⁢(ti⁢j)⁢𝐯i-𝐙iT⁢(ti⁢j)⁢𝐮i)22⁢σε2}

(A.3)×|𝚺𝜷|-12⁢exp⁡{-12⁢(𝜷-μ𝜷)T⁢𝚺𝜷-1⁢(𝜷-μ𝜷)}.

A.4

The conditional posterior distribution of measurement error τ=1/σε2: The prior distribution of σε2 is an inverse gamma distribution with a scale of a0 and a shape of b0 as in (3.2), which is the conjugate prior distribution. The conditional posterior distribution of

τ=1σε2

is distributed as 𝒢⁢(α*,β*), where

(A.4){α*=a0+N2,β*=b0+12⁢∑i=1n∑j=1ni(yi⁢(ti⁢j)-𝐗iT⁢(ti⁢j)⁢𝜷-𝐗iT⁢(ti⁢j)⁢𝐯i-𝐙iT⁢(ti⁢j)⁢𝐮i)2.

A.5

The conditional posterior distribution for random effects 𝐛i in the linear mixed effects submodel:

p(𝐛i|𝜽(-𝐛i),T,δ,𝐲)∝exp{𝜸T𝐰i+αmi(t)}δiexp(-∫0Tiexp{𝜸T𝐰i+αmi(s)ds})

×∏j=1ni1(2⁢π⁢σε2)12⁢exp⁡{-(yi⁢(ti⁢j)-𝐗iT⁢(ti⁢j)⁢𝜷-[𝐗iT⁢(ti⁢j)𝐙iT⁢(ti⁢j)]⁢𝐛i)22⁢σε2}

(A.5)×|𝐆|-12⁢exp⁡{-12⁢(𝐛i)T⁢𝐆⁢(𝐛i)}.

A.6

There are two options for choosing the conjugate prior distribution of the variance matrix 𝐆. These are when 𝐆 is a diagonal matrix and when 𝐆 is a non-diagonal matrix. Therefore, we propose two conditional posterior distributions for the matrix 𝐆.

In the case when 𝐆 is a non-diagonal matrix, the conjugate prior distribution for 𝐆 is a Wishart distribution as in (3.4). The conditional posterior distribution for 𝐆-1 has the standard form

(A.6)𝐆-1|𝜽(-𝐆-1),𝐛,T,δ,𝐲∼𝒲[n+r,(𝐑+∑i=1n(bibiT))-1]q.

In the second case when 𝐆 is a diagonal matrix with elements σk2, k=1,…,q, the conjugate prior distribution for 𝐆 has the form

p⁢(σ12,…,σq2)=∏k=1qℐ⁢𝒢⁢(ak,bk).

Set τk=1/σk2 for k=1,…,q. The conditional posterior distribution for 𝐆-1 has the standard form

(A.7)τ1,…,τq|𝜽(-τ1,…,-τq),T,δ,𝐲∼∏k=1q𝒢(ak*,bk*),

where

ak*=ak+n2,

bk*=bk+12⁢∑i=1nbiT⁢bi.

B MCMC traces and posterior distribution plots for the parameters of the flat prior on λ and α

MCMC traces and posterior distribution plots for the parameters of the flat prior on λ and α (see Figures 1–3).

Figure 1

MCMC traces (left) and the marginal posterior distribution (right) plots for the parameters λ, γ and α. The thick lines indicate the positions of the true values.

$Figure 2 MCMC traces (left) and the marginal posterior distribution (right) plots for the parameters β0{\beta_{0}}, β1{\beta_{1}} and σ. The thick lines indicate the positions of the true values.$

Figure 2

MCMC traces (left) and the marginal posterior distribution (right) plots for the parameters β0, β1 and σ. The thick lines indicate the positions of the true values.

$Figure 3 MCMC traces (left) and the marginal posterior distribution (right) plots for the parameters D11{D_{11}}, D12{D_{12}} and D22{D_{22}}. The thick lines indicate the positions of the true values.$

Figure 3

MCMC traces (left) and the marginal posterior distribution (right) plots for the parameters D11, D12 and D22. The thick lines indicate the positions of the true values.

C The potential rate reduction factor plots of Gelman and Rubin diagnostic and the autocorrelation function (ACF) plots for the parameters of the flat prior on λ and α

The potential rate reduction factor plots of Gelman and Rubin diagnostic and the autocorrelation function (ACF) plots for the parameters of the flat prior on λ and α (see Figures 4 and 5).

Figure 4

The potential rate reduction factor plots of Gelman and Rubin diagnostic for all parameters in (5.1) and (5.2).

Figure 5

ACF plots for all parameters in (5.1) and (5.2).

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Received: 2019-12-07

Accepted: 2020-02-03

Published Online: 2020-02-22

Published in Print: 2020-03-01