In this paper we introduce a Markov gamma random field prior for modelling relative
risks in disease mapping data. This prior process allows for a different dependence
effect with different neighbours. We describe the properties of the prior process
and derive posterior distributions. The model is extended to cope with covariates
and a data set of respiratory infections of children in Mexico is used as an
illustration.
Annual Statistics Book
(2002) Produced by the Secretary of Health
under the direction of information and evaluation of performance.
Mexico
.
2.
Banerjee S
,
Wall MM
and
Carlin BP
(2003)
Frailty modeling for spatially correlated survival data, with application
to infant mortality in Minnesota
. Biostatistics, 4,
123–42
.
3.
Besag J
(1974)
Spatial interaction and the statistical analysis of lattice systems
(with discussion)
. Journal of the Royal Statistical Society, Series B,
36,
192–236
.
4.
Besag J
,
York J
and
Mollié A
(1991)
Bayesian image restoration, with two applications in spatial statistics
(with discussion)
. Annals of the Institute of Statistical Mathematics,
43,
1–59
.
5.
Breslow NE
and
Clayton DG
(1993)
Approximate inference in generalized linear mixed models
. Journal of the American Statistical Association,
88,
9–25
.
6.
Clayton DG
and
Kaldor J
(1987)
Empirical Bayes estimates of age-standardized relative risks for use in
disease mapping
. Biometrics, 43,
359–70
.
7.
Cox DR
(1972)
Regression models and life-tables (with discussion)
. Journal of the Royal Statistical Society, Series B,
34,
187–220
.
8.
Denison DGT
and
Holmes CC
(2001)
Bayesian partitioning for estimating disease risk
. Biometrics, 57,
143–49
.
9.
Green P
and
Richardson S
(2002)
Hidden Markov models and disease mapping
. Journal of the American Statistical Association,
97,
1055–70
.
10.
Ickstadt K
and
Wolpert RL
(1999) Spatial regression for marked
point processes. In
JM Bernardo
,
JO Berger
,
AP Dawid
and
AFM Smith
eds, Bayesian statistics, 6, pp.
323–41.
11.
Marshall RJ
(1991)
A review of methods for the statistical analysis of spatial patterns of
disease
. Journal of the Royal Statistical Society, Series A,
154,
421–41
.
12.
Nieto-Barajas LE
and
Walker SG
(2002)
Markov beta and gamma processes for modelling hazard rates
. Scandinavian Journal of Statistics, 29,
413–24
.
13.
Richardson S
(1992) Statistical methods for
geographical correlation studies. In
P Elliott
,
J Cuzick
,
D English
and
R Stern
eds, Geographical and environmental epidemiology, pp.
181–204.
Oxford: Oxford University Press
.
14.
Ripley BD
(1981) (ed.) Spatial
Statistics.
New York: Wiley
.
15.
Smith AFM
and
Roberts GO
(1993)
Bayesian computations via the Gibbs sampler and related Markov chain Monte
Carlo methods
. Journal of the Royal Statistical Society, Series B,
55,
3–23
.
16.
Tierney L
(1994)
Markov chains for exploring posterior distributions
. Annals of Statistics, 22,
1701–22
.
17.
Wakefield J
and
Morris S
(1999) Spatial dependence and
errors-in-variables in environmental epidemiology. In
JM Bernardo
,
JO Berger
,
AP Dawid
and
AFM Smith
eds, Bayesian Statistics, 6, pp.
657–84.
18.
Wolpert RL
and
Ickstadt K
(1998)
Poisson/gamma random field models for spatial statistics
. Biometrika, 85,
251–67
.
