Abstract
In many survival studies, covariates effects are time-varying and there is presence of spatial effects. Dynamic models can be used to cope with the variations of the effects and spatial components are introduced to handle spatial variation. This paper proposes a methodology to simultaneously introduce these components into the model. A number of specifications for the spatial components are considered. Estimation is performed via a Bayesian approach through Markov chain Monte Carlo methods. Models are compared to assess relevance of their components. Analysis of a real data set is performed, showing the relevance of both time-varying covariate effects and spatial components. Extensions to the methodology are proposed along with concluding remarks.
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Acknowledgments
The research of the second author was supported by a grant from CNPq-Brazil. This paper is based on the M.Sc. Dissertation of the first author under the supervision of the second author.
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Appendix
Appendix
MCMC sampling is performed in blocks. This Appendix describe the calculations involved for each of the blocks used to implement the algorithm in Table 1.
Time-varying effects β
This parameter does not possess full conditional distribution with easy sampling. Values are generated for each β k. separately from the random walk model proposal.
where (prop) indicates proposed value, (pre) indicates previous value and V β_k is a tuning parameter used to control the acceptance rate in the 30%–80% range. If the acceptance rate is high, then the autocorrelation of the chain is also high. On the other hand, if the acceptance rate is low then the variability of the chain will be underestimated. See Gamerman and Lopes (2006).
The acceptance probability of β k. is
The full conditional of β k. is
where \(B(t_i,X_{ki}) = \sum_{j=1}^{J}\hbox{e}^{X_{i}\beta_j} (f(t_{i},j)-a_{j-1})\), and f(·,·) is defined in (7).
If we do not assume prior independence among the regression coefficients, the prior distribution of β is given by (12), where U is a (p + 1) × (p + 1) evolution covariance matrix and C is a (p + 1) × (p + 1) hyperprior covariance matrix. Then, the full conditional of β k. is
The prior for this group can be completed with an inverse Wishart distribution with parameters r and S as the prior distribution for the variance matrix U with density
where r is a scalar and S is a (p + 1) × (p + 1) symmetric, positive-definite matrix. So, the full conditional distribution of the variance matrix U is an inverse Wishart with parameters r * and S * given by
Unstructured frailties Z
These parameters do not possess full conditional distribution with easy sampling. Values for Z are generated from the random walk proposal
where V Z is a tuning parameter used to control the acceptance rate in the 30%–80% range. The acceptance probability is
The full conditional of Z is
Spatial frailties W
These parameters do not possess full conditional distribution with easy sampling. Values for W are generated from the random walk proposal
where V W is a tuning parameter used to control the acceptance rate in the 30%–80% range. The acceptance probability is
The full conditional of W is
Evolution hyperparameters U 0,…,U p
Their full conditional distributions are inverse Gamma distributions with parameters \(a^*_{{U_{k}}}/2 \hbox{ e } b^*_{{U_{k}}}/2\) given by
for k = 0,1,…,p.
Variances of the frailties σ 2 Z and σ 2 W
Their full conditional distributions are inverse Gamma distributions with parameters (a * Z /2, b * Z /2) and (a * W /2, b * Z /2) respectively given by
Correlation parameter ϕ
This parameter does not possess full conditional distribution with easy sampling. Values are generated for ϕ are generated from a half normal distribution
with density
where V ϕ is a tuning parameter used to control the acceptance rate in the 30%–80% range.
The acceptance probability is
where Φ(·) is the distribution function of the standard normal distribution. The full conditional of ϕ is
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Bastos, L.S., Gamerman, D. Dynamic survival models with spatial frailty. Lifetime Data Anal 12, 441–460 (2006). https://doi.org/10.1007/s10985-006-9020-2
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DOI: https://doi.org/10.1007/s10985-006-9020-2