Investigating spatio-temporal similarities in the epidemiology of childhood leukaemia and diabetes

Abstract

Childhood acute lymphoblastic leukaemia (ALL) and Type 1 diabetes (T1D) share some common epidemiological features, including rising incidence rates and links with an infectious aetiology. Previous work has shown a significant positive correlation in incidence between the two conditions both at the international and small-area level. The aim was to extend the methodology by including shared spatial and temporal trends using a more extensive dataset among individuals diagnosed with ALL and T1D in Yorkshire (UK) aged 0–14 years from 1978–2003. Cases with ALL and T1D were ascertained from 2 high quality population-based disease registers covering the Yorkshire region of the UK and linked to an electoral ward from the 1991 UK census. A Bayesian model was fitted where similarities and differences in risk profiles of the two diseases were captured by the shared and disease-specific components using a shared-component model, with space-time interactions. The extended model revealed a positive correlation of at least 0.70 between diseases across all time periods, and an increasing risk across time for both diseases, which was more evident for T1D. Furthermore, both diseases exhibited lower rates in the more urban county of West Yorkshire and higher rates in the more rural northern and eastern part of the region. A differential effect of T1D over ALL was found in the south-eastern part of the region, which had a more pronounced association with population mixing than with population density or deprivation. Our approach has demonstrated the utility in modelling temporally and spatially varying disease incidence patterns across small geographical areas. The findings suggest searching for environmental factors that exhibit similar geographical-temporal variation in prevalence may help in the development and testing of plausible aetiological hypotheses. Furthermore, identifying environmental exposures specific to the south-eastern part of the region, especially locally varying risk factors which may differentially affect the development of T1D and ALL, may also be fruitful.

Appendix

The two diseases spatial temporal model

In the presence of longitudinal data for two diseases, an extension of the BYM model is given by:

$$ \begin{gathered} \log (R_{it1} ) = \alpha_{0t1} + \beta_{t1}^{T} x_{it} + u_{it1} + v_{it1} \hfill \\ \log (R_{it2} ) = \alpha_{0t2} + \beta_{t2}^{T'} x_{it} + u_{it2} + v_{it2} \hfill \\ \end{gathered} $$

(1)

where α _0tj is an intercept term on the log relative risk scale for disease j at time t; x _it is a covariate risk vector with the corresponding disease-specific coefficient parameter vector β _tj at time t; and u _itj and v _itj represent unstructured and spatially structured random effects, respectively, in area i at time t for disease j. The unstructured random effects are assumed to follow a normal distribution with time varying variance such that $u_{itj} \sim N\left( {0,\sigma_{tj}^{2} } \right)$ for j = 1, 2. The spatially structured effects are modelled by the intrinsic conditional autoregressive normal (CAR Normal) prior [14, 26], which specifies a conditional distribution for a specific spatially structured effect v _itj as:

$$ f(v_{itj} |v_{ktj} ,k \ne i) = N\left( {{\frac{{\sum\nolimits_{{k \in \Uptheta_{i} }} {W_{iktj} v_{ktj} } }}{{\sum\nolimits_{{k \in \Uptheta_{i} }} {W_{iktj} } }}},{\frac{{\lambda_{tj}^{2} }}{{\sum\nolimits_{{k \in \Uptheta_{i} }} {W_{iktj} } }}}} \right) $$

(2)

at time t for disease j, where Θ _i is the set of areas adjacent to area i; W _iktj is the weight reflecting spatial dependence between area i and k at time t for disease j and $\lambda_{tj}^{2}$ is the (conditional) structured variance. For simplicity, we assume that the weighting factor W _iktj is constant over time and disease, which seems reasonable as we do not expect the influence of neighbouring areas to change over time and to be different between the two diseases, i.e. W _iktj  = W _ik . The most common and simplest adjacent specification is to set W _ik  = 1 if areas i and k are neighbours that share a common boundary and W _ik  = 0 otherwise. Thus, for disease j at time t, a CAR Normal prior specifies the conditional distribution of each area-specific effect v _itj , given all the other v’s, to be a normal distribution with mean equal to the average of the v’s of its neighbours, and variance inversely proportional to the number of neighbours; the more neighbours an area has, the greater the precision for that area effect. This scenario is reasonable for population density, as urban areas are likely to have more neighbours that sparsely populated rural areas. For brevity, following [14, 26], a conditional autoregressive normal prior on the spatially structured effects will be denoted by

$$ v_{itj} \sim {\text{CAR Normal}}\left( {W,\lambda_{tj}^{2} } \right) $$

where W is simply the neighbourhood adjacency weight matrix.

A common and specific spatial-temporal model

The spatio-temporal model we consider to estimate relative risks of the two diseases in space and time is

$$ \begin{gathered} \log (R_{it1} ) = \alpha_{01} + \beta_{t1}^{T} x_{it} + \phi_{i} \kappa + \psi_{t} \omega + \zeta_{it} + \varepsilon_{it1} \hfill \\ \log (R_{it2} ) = \alpha_{02} + \beta_{t2}^{T} x_{it} + {\frac{{\phi_{i} }}{\kappa }} + {\frac{{\psi_{t} }}{\omega }} + \zeta_{it} + \theta_{i} + \gamma_{t} + \varepsilon_{it2} \hfill \\ \end{gathered} $$

(3)

where now α _0j is the overall risk for disease j (j = 1, 2), φ _i and ψ _t are the main shared spatial and temporal effects, respectively; θ _i and γ _t represent disease 2 (T1D) specific spatial and temporal components, which capture the differential effect between the two diseases; ζ _it are the shared space-time interaction effects, capturing departure from space and time main effects, which may highlight space-time clusters of risk, and ɛ _itj are the disease-specific heterogeneous effects, capturing possible extra-Poisson variation not explained by terms included above. Thus, the shared-temporal factor model (3) partitions the risk profile for the two diseases into disease-specific components and shared spatial and time components. Parameters κ and ω are included to allow for a differential gradient on the main shared spatial and temporal components, respectively. The ratio κ ² compares the risk of disease 1 (ALL) to the risk of disease 2 (T1D) associated with the main shared spatial component and the ratio ω ² compares the risk associated with the main temporal effect.

Ideally, in model (3) we could have split the main shared spatial components into unstructured and structured effects or the main shared temporal component into unstructured and structured time trends or similarly the diseases-specific components. This could have resulted into 8 × 5 = 40 extra random effects, as well as the space-time interaction terms. This seems overly complex and could lead to identifiability problems, so we decided to minimise the number of random effects. We could have also included specific components for disease 1, but such a symmetric formulation can result in identifiability problems [9].

Prior specification

There are various choices of prior distributions for the space-time interaction effects, ζ _it . They can simply be taken to be independent for every period and time with a constant variance over time, as in [9], or, depending on the length of the time periods, can evolve smoothly over time with a polynomial temporal trend as in [27, 28]. Alternatively, they can be allowed to vary independently or be spatially structured for every period, with variances that are either independently and identically distributed or temporally structured as proposed in Nobre et al. [10]. In the present study, we define only five periods, too few to show any reliable space-time jumps in risk. Thus, we assume a spatially structured prior distribution at every period for the shared interaction term, $\zeta_{it} \sim {\text{CAR Normal}}\left( {W,\sigma_{\zeta t}^{2} } \right)$ (i.e. a time-independent prior but linked in space) with variances that are allowed to change smoothly over time,$\log \sigma_{\varsigma t}^{2} = \log \sigma_{\varsigma (t - 1)}^{2} + e_{t}$, where $e_{t} \sim N\left( {0,\sigma_{e}^{2} } \right)$. The pair of disease-specific heterogeneity terms $(\varepsilon_{it1} ,\varepsilon_{it2} )^{T}$are also assumed to evolve smoothly over time according to a first-order autoregressive (AR(1)) multivariate normal prior distribution with covariance matrix Σ to allow for correlations between the ALL and T1D risks in each space-time unit. Thus, $(\varepsilon_{i11} ,\varepsilon_{i12} )^{T} \sim {\text{MVN}}(0,\Upsigma )$ and $(\varepsilon_{it1} ,\varepsilon_{it2} )^{T} \sim {\text{MVN}}\left( {(\varepsilon_{i(t - 1)1} ,\varepsilon_{i(t - 1)2} ),\Upsigma } \right);t = 2, \ldots ,5$. Since we are using the CAR Normal prior, with sum-to-zero constraints on the random effect terms, we assign a flat prior on the overall disease risk terms, α _0j .