In this paper we consider the analysis of LBW counts at county level within the counties of Georgia and South Carolina for the years 1997–2006. The county level counts of LBW consist of 205 units and so have a considerable spatial variation. The temporal period represents a considerable time span which could be considered to yield evidence for spatial-temporal latent structure within the LBW risk. We consider two basic models for this risk in these data. First, we have considered a conventional space-time random effect model with separate space-time (ST) components and a ST interaction term (

Knorr-Held, 2000). This allows for a parsimonious model representation of space-time variation (

Lawson, 2009), and is commonly applied to describe ST variation of disease risk. We call this the SREST model (Standard Random Effect Space-Time model). Let Y

_{ij} denote the number of low-birth weight babies for county

*i* (

*i* = 1, …,

*I*) at time point

*j* (

*j* = 1, …,

*J*) and E

_{ij} denote the expected count. The count is assumed to follow a standard Poisson distribution as Y

_{ij}~POis (E

_{ij}_{ij}), where

_{ij} is the relative risk.

In the SREST model, the log relative risk can be specified as

where

**x**_{ij} = (1,

**X**_{ij1}, …,

**X**_{ijP})

^{T} is the vector of intercept and p predictors and

**β** = (β

_{0},β

_{1} … β

_{P})

^{T} is the corresponding coefficient vector. The parameters u

_{i} and v

_{i} are the uncorrelated spatial random effect and the correlated spatial random effect, respectively. Similarly, the parameters η

_{j} and γ

_{j} are the uncorrelated temporal random effect and the correlated temporal effect, respectively. The parameter ε

_{ij} is the space-time interaction term.

The prior distribution of the correlated spatial component v

_{i} is assigned to be a conditional autoregressive (CAR) distribution (

Besag et al., 1991),

where

is the overall variance parameter,δ

_{1} is the set of labels of the neighbors of county

*i,* and n

_{i} is the number of neighbors (adjacent counties) of county

*i*. The correlated temporal component γ

_{j} is assigned to be a random walk Gaussian distribution,

. The prior distributions of the other random components u

_{i}, η

_{j}, and ε

_{ij} are specified as

, and all the standard deviance parameters in the model are assumed to have a uniform distribution, Uniform(0,5). The coefficient vector

**β** has an independent non-informative Gaussian prior distribution with large variance.

In contrast to this model we also examine the latent structure model of

Lawson et al. (2010) which allows there to be a disaggregation of temporal profiles in risk for LBW. The approach is based on the idea that regions have a set of underlying risks that they can support and these risks are temporally varying. The number of these latent risk profiles is usually unknown in advance and must be estimated in the analysis. This model is called the STLS model (Space-time Latent Structure model) and is given by

where L is the number of components, χ

_{lj} is the temporal component that explains the underlying temporal pattern in relative risk, w

_{il} is the corresponding weight that depends on space, and ψ

_{1} is the entry parameter that controls the selection of the temporal components. The temporal component χ

_{lj} can be assumed to have various temporal dependency structures, and in this paper we assume a random walk Gaussian distribution,

. The weight parameter w

_{il} describes the proportion of

*l*th temporal component contribution for

*i*th county, and it has two conditions: w

_{il} ≥ 0 and

. Thus, the weight w

_{il} is modeled as

, where

≥ 0 is the unnormalized weight and is assumed to have a log-normal distribution with the space-dependent mean ζ

_{il} and the variance

The mean parameter ξ

_{il} has a CAR distribution to take into account the spatial dependency structure of the weights. The number of temporal components (L) is assumed to be a large value apriori in order to find the true temporal components. The entry parameter is used to allow components to enter or be removed from the model during updating. The entry parameter has a value 0 or 1 so the

*l*th temporal component is included in the model if ψ

_{1} and is not included if ψ

_{1} = 0. We assume that the entry parameter has a Bernoulli prior distribution, ψ

_{1} ~Bern(0.5), where 0.5 is a non-informative value. Using entry parameters in the model, it is not necessary to find the number of components in advance and this model can allow for the estimation of the number of components included in the model.

To identify the spatial clusters each of which has a homogeneous temporal trend in relative risk, we consider a post hoc method using the estimated weight values. We define the spatial cluster indicator Z

_{i}(=1, …, L) as

The indicator Z_{i} means the index of the temporal component with the largest weight in county *i*, which becomes the primary temporal pattern of the county in relative risk. Using the indicator Z_{i}, we can allocate a fixed component to a given region.

Since a component identifiability problem can arise in Bayesian STLS modeling due to the invariance of the likelihood with respect to the permutation of the component labels (

Stephens, 2000), we make the assumption in the model that the latent components have time-dependent structures while the corresponding weights have only space-dependent structures. Also, in the STLS model, it is possible for components to switch labels during Markov Chain Monte Carlo (MCMC) simulation if multiple chains are used. A single chain run can avoid this problem. Thus, in this paper, we use a single chain and obtain the converged estimates of the parameters from the posterior sampling from the joint models via two packages

R and

WinBUGS. We use the posterior mean values for estimates of all the parameters except the cluster indicator Z

_{i} and use the posterior mode values for the estimation of Z

_{i} as Z

_{i} is the nominal value.