Bayesian penalized spline models for the analysis of spatio-temporal count data

Abstract

In recent years, the availability of infectious disease counts in time and space has increased, and consequently, there has been renewed interest in model formulation for such data. In this paper, we describe a model that was motivated by the need to analyze hand, foot, and mouth disease surveillance data in China. The data are aggregated by geographical areas and by week, with the aims of the analysis being to gain insight into the space-time dynamics and to make short-term predictions, which will aid in the implementation of public health campaigns in those areas with a large predicted disease burden. The model we develop decomposes disease-risk into marginal spatial and temporal components and a space-time interaction piece. The latter is the crucial element, and we use a tensor product spline model with a Markov random field prior on the coefficients of the basis functions. The model can be formulated as a Gaussian Markov random field and so fast computation can be carried out using the integrated nested Laplace approximation approach. A simulation study shows that the model can pick up complex space-time structure and our analysis of hand, foot, and mouth disease data in the central north region of China provides new insights into the dynamics of the disease.

Keywords: Bayesian spatio-temporal analysis; Gaussian Markov random field; INLA; infectious diseases; penalized splines; surveillance count data.

Figure 1

Summaries of the central north region of China HFMD data from 2009–2010: (a) weekly numbers of cases, (b) weekly expected numbers, (c) marginal (across time) SMR, (d) centroids of prefectures (red dots) and location of spline bases (blue crosses), after scaling the study region.

Figure 2

Simulated Data: (a) true values of the basis functions b_kt versus time; (b) locations of the basis coefficients, color coded as in (a); the grey dots are the locations of the observed counts; (c) number of cases aggregated over time (with the size of the circles being proportional to the number of cases); (d) number of cases, aggregated over space, plotted by week.

Figure 3

Estimated basis coefficients with four different priors in the simulation study: true values are colored as in Figure 2, while the grey lines are the estimates.

Figure 4

Comparison of the log relative risk in four selected areas.

Figure 5

Estimated temporal component γ and ϕ, and spatial component u and υ from the Type IV interaction model with the China central north HFMD data.