Skip to main content
Log in

Dynamic survival models with spatial frailty

  • Published:
Lifetime Data Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Bastos LS (2003) Dynamic and static survival models with spatial frailty. Unpublished MSc Thesis, Universidade Federal do Rio de Janeiro, Brazil (In Portuguese)

  • Besag J, York J, Mollie A (1991) Bayesian image restoration, with two applications in spatial statistics (with discussion). Ann Inst Stat Math 43:1–59

    Article  MathSciNet  Google Scholar 

  • Carlin BP, Banerjee S (2002) Hierarchical multivariate CAR models for spatio-temporally correlated survival data. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 7. University Press, Oxford, pp 1–15

    Google Scholar 

  • Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151

    Article  MathSciNet  Google Scholar 

  • Cox DR (1972) Regression models and life tables. J Roy Stat Soc Ser B 34:187–220

    Google Scholar 

  • Cressie NAC (1993) Statistics for spatial data, revised edition. Wiley, New York

    Google Scholar 

  • Doornik JA (2002) Object-oriented matrix programming using ox, 3rd edn. Timberlake Consultants Press and Oxford, London, http://www.nuff.ox.ac.uk/Users/Doornik

  • Gamerman D (1991) Dynamic Bayesian models for survival data. Appl Stat 40:63–79

    Article  Google Scholar 

  • Gamerman D, Lopes HF (2006) Markov chain Monte Carlo: stochastic simulation for Bayesian inference, 2nd edn. Chapman & Hall, London

    MATH  Google Scholar 

  • Gamerman D, West M (1987) An application of dynamic survival models in unemployment studies. The Statistician 36:269–274

    Article  Google Scholar 

  • Gelfand AE, Kim HJ, Sirmans CF, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98:387–396

    Article  MathSciNet  Google Scholar 

  • Gelfand AE, Banerjee S, Gamerman D (2005a) Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics 16:465–479

    Article  MathSciNet  Google Scholar 

  • Gelfand AE, Kottas A, MacEachern SN (2005b) Bayesian nonparametric spatial modeling with Dirichlet process mixing. J Am Stat Assoc 100:1021–1035

    Article  MathSciNet  Google Scholar 

  • Hemming K, Shaw EH (2002) A parametric dynamical model applied to breast cancer survival times. Appl Stat 51:421–435

    MathSciNet  Google Scholar 

  • Henderson R, Shimakura S (2003) A serially correlated gamma frailty model for longitudinal count data. Biometrika 90:355–366

    Article  MathSciNet  Google Scholar 

  • Henderson R, Shimakura S, Grost D (2002) Modelling spatial variation in Leukaemia survival data. J Am Stat Assoc 97:965–972

    Article  Google Scholar 

  • Hougaard P (2000) Analysis of multivariate survival data. Springer

  • Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data. John Wiley and Sons, New York

    MATH  Google Scholar 

  • Kaplan E, Meyer P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481

    Article  Google Scholar 

  • Kauermann G (2005) Penalised spline fitting in multivariable survival models with varying coefficients. Comput Stat Data Anal 49:169–186

    Article  MathSciNet  Google Scholar 

  • Klein JP, Moeschberger ML (1997) Survival analysis: techniques for censured and truncated data. Springer

  • Li Y, Ryan L (2002) Modelling spatial survival data using semiparametric frailty models. Biometrics 58:287–297

    Article  MathSciNet  Google Scholar 

  • Paez MS, Gamerman D, de Oliveira V (2005) Interpolation performance of a spatio-temporal model with spatially varying coefficients: application to PM10 concentrations in Rio de Janeiro. Environ Ecol Stat 12:169–193

    Article  MathSciNet  Google Scholar 

  • Ripatti S, Palmgren J (2000) Estimation of multivariate frailty using penalized partial likelihood. Biometrics 56:1016–1022

    Article  MathSciNet  Google Scholar 

  • Sargent DJ (1997) A flexible approach to time-varying coefficients in the Cox regression setting. Lifetime Data Anal 1:13–25

    Article  Google Scholar 

  • Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measure of model complexity and fit (with discussion). J Roy Stat Soc Ser B 64(v4):583–641

    Google Scholar 

  • Vaida F, Xu R (2000) Proportional hazards model with random effects. Stat Med 19:3309–3324

    Article  Google Scholar 

  • Vaupel JW, Manton KG, Stallard E (1979) The impact of heterogeneity in individual frailty on dynamics of mortality. Demography 16:439–454

    Google Scholar 

  • West M, Harrison P (1997) Bayesian forecasting and dynamic models, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Xu R, Adak S (2003) Survival analysis with time-varying regression effects using a tree-based approach. Biometrics 56:305–315

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Leonardo Soares Bastos.

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.

$$\beta_{k.}^{\rm (prop)} \sim N_J \left( \beta_{k.}^{\rm (pre)}, V_{\beta_k} \Lambda_k^{-1} \right), \quad k=0,1,\ldots,p,$$
(24)

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

$$\alpha = \min \left\{ 1, \frac{p(\beta_{k.}^{\rm (prop)}|\cdots)}{p(\beta_{k.}^{\rm (pre)}|\ldots) } \right\}.$$
(25)

The full conditional of β k. is

$$\begin{array}{lll} p(\beta_{k.}|\cdots) & \propto & \exp\left\{ \sum\limits_{i=1}^{n} \left[ \left( \sum\limits_{j=1}^{J}\chi_{ij}X_{i}'\beta_{j} \right) + \delta_i(Z_i + W(s_i)) - B(t_i,X_{ki})\hbox{e}^{Z_i + W(s_i)} \right] \right\}\\ &\quad &\times \left\{ |\Lambda_k|^{1/2}\exp\left( -\frac{1}{2} (\beta_{k.} - b_{k.})^T\Lambda_k(\beta_{k.} - b_{k.}) \right) \right\}, \end{array}$$

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

$$\begin{array}{lll} p(\beta_{k.}|\cdots) & \propto & \exp\left\{ \sum\limits_{i=1}^{n} \left[ \left( \sum\limits_{j=1}^{J}\chi_{ij}X_{i}'\beta_{j} \right) + \delta_i(Z_i +W(s_i))- B(t_i,X_{ki})\hbox{e}^{Z_i + W(s_i)} \right] \right\} \\ &\quad& \times \exp\left\{-\frac{1}{2} \left[ (\beta_{1}-b)^T C^{-1} (\beta_{1}-b) + \sum\limits_{j=2}^{J}{\frac{ (\beta_{j}-\beta_{j-1})^T U^{-1}(\beta_{j}-\beta_{j-1}) }{a_j - a_{j-1}}} \right] \right\}. \end{array}$$
(26)

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

$$p(U)\propto |U|^{(r-(p+2))/2} \exp\left\{-\frac{1}{2}\hbox{Tr}(US) \right\},$$
(27)

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

$$\begin{array}{l} r^*=r + J - 1, \\ S^*=S+\sum_{j=2}^{J}{\frac{1}{a_j-a_{j-1}}(\beta_{j}-\beta_{j-1}) (\beta_{j}-\beta_{j-1})^T.} \end{array}$$

Unstructured frailties Z

These parameters do not possess full conditional distribution with easy sampling. Values for Z are generated from the random walk proposal

$$Z^{\rm (prop)} \sim N \left(Z^{\rm (pre)}, V_{Z} \sigma_Z^2 I_n \right),$$
(28)

where V Z is a tuning parameter used to control the acceptance rate in the 30%–80% range. The acceptance probability is

$$\alpha = \min \left\{ 1, \frac{p(Z^{\rm (prop)}|\cdots)}{p(Z^{\rm (pre)}|\cdots)} \right\}.$$
(29)

The full conditional of Z is

$$\begin{array}{lll} p(Z|\ldots) & \propto & \exp\left\{\sum\limits_{i=1}^{n}\left[\left( \sum\limits_{j=1}^{J}\chi_{ij}X_i'\beta_j \right) + \delta_i(Z_i + W(s_i)) - B(t_i,X_i)\hbox{e}^{Z_i + W(s_i)} \right] \right\}\\ &\quad & \times (\sigma_Z^2)^{-n/2}\exp\left(-\frac{1}{2 \sigma_Z^2}Z'Z \right). \end{array}$$
(30)

Spatial frailties W

These parameters do not possess full conditional distribution with easy sampling. Values for W are generated from the random walk proposal

$$W^{\rm (prop)} \sim N \left( W^{\rm (pre)}, V_{W} \sigma_W^2 \Omega \right),$$
(31)

where V W is a tuning parameter used to control the acceptance rate in the 30%–80% range. The acceptance probability is

$$\alpha = \min \left\{1, \frac{p(W^{\rm (prop)}|\cdots)}{p(W^{\rm (ant)}|\cdots) } \right\}.$$
(32)

The full conditional of W is

$$\begin{array}{lll} p(W|\cdots) & \propto & \exp\left\{ \sum\limits_{i=1}^{n} \left[ \left( \sum\limits_{j=1}^{J}\chi_{ij}X_i'\beta_j \right) + \delta_i(Z_i + W(s_i)) - B(t_i,X_i)\hbox{e}^{Z_i + W(s_i)} \right] \right\}\\ &\quad& \times (\sigma_W^2)^{-n/2}|\Omega|^{-1/2}\exp\left( -\frac{1}{2 \sigma_W^2}W' \Omega^{-1} W \right). \end{array}$$
(33)

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

$$\begin{array}{lll} a^*_{U_k} &=& a_{U_k}+J-1\, \hbox{ and}\\ b^*_{U_k} &=& b_{U_k}+\sum\limits_{j=2}^{J}\frac{(\beta_{kj} - \beta_{kj-1})^2}{a_{j}-a_{j-1}}, \end{array}$$

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

$$\begin{array}{lll} a^*_{Z} & = & a_{Z} + n \, , \\ b^*_{Z} & = & b_{Z} + Z'Z \, , \\ a^*_{W} & = & a_{W} + n \, \hbox{ and} \\ b^*_{W} & = & b_{W} + W'\Omega^{-1}W. \end{array}$$

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

$$\phi^{\rm (prop)} \sim N(\phi^{\rm (pre)},V_\phi) I( 0,\infty),$$
(34)

with density

$$p(\phi^{\rm (prop)}|\phi^{\rm (pre)},V_\phi) = ( 2 \pi V_\phi)^{-\frac{1}{2}} \exp\left\{ -\frac{1}{2}\frac{(\phi^{\rm (prop)}-\phi^{\rm (pre)})^2}{V_\phi} \right\} \frac{1}{1-\Phi\left(-\phi^{\rm (pre)}/\sqrt{V_\phi}\right)},$$
(35)

where V ϕ is a tuning parameter used to control the acceptance rate in the 30%–80% range.

The acceptance probability is

$$\alpha = \min \left\{1, \frac{p(\phi^{\rm (prop)}|\cdots)}{p(\phi^{\rm (pre)}|\cdots)} \frac{\left[1-\Phi\left(-\phi^{\rm (prop)}/\sqrt{V_\phi}\right)\right]} {\left[1-\Phi\left(-\phi^{\rm (pre)}/\sqrt{V_\phi}\right)\right]} \right\},$$
(36)

where Φ(·) is the distribution function of the standard normal distribution. The full conditional of ϕ is

$$p(\phi|\cdots) \propto \phi^{a_\phi-1} \exp\left( - b_\phi \phi \right) |\Omega|^{-1/2}\exp\left( -\frac{1}{2 \sigma_W^2}W' \Omega^{-1} W \right).$$
(37)

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10985-006-9020-2

Keywords

Navigation