# Representing spatial dependence and spatial discontinuity in ecological epidemiology: a scale mixture approach

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## Abstract

Variation in disease risk underlying observed disease counts is increasingly a focus for Bayesian spatial modelling, including applications in spatial data mining. Bayesian analysis of spatial data, whether for disease or other types of event, often employs a conditionally autoregressive prior, which can express spatial dependence commonly present in underlying risks or rates. Such conditionally autoregressive priors typically assume a normal density and uniform local smoothing for underlying risks. However, normality assumptions may be affected or distorted by heteroscedasticity or spatial outliers. It is also desirable that spatial disease models represent variation that is not attributable to spatial dependence. A spatial prior representing spatial heteroscedasticity within a model accommodating both spatial and non-spatial variation is therefore proposed. Illustrative applications are to human TB incidence. A simulation example is based on mainland US states, while a real data application considers TB incidence in 326 English local authorities.

## Keywords

Spatial Bayesian Conditional autoregressive Heteroscedasticity Scale mixture Tuberculosis## 1 Introduction

Modelling variation in disease or other events underlying observed totals for geographic areas is important for detecting elevated rates (Beale et al. 2008). In disease mapping, the observations often consist of incidence totals for chronic or infectious disease. Such data are subject to stochastic variations, and the underlying area specific incidence risks are often the focus in data mining studies. In such studies, the objects include extraction of underlying spatial and spatiotemporal patterns, including detection of elevated risk (hotspots) and spatial outliers (Shekhar et al. 2015). The particular focus of this paper is on ecological epidemiology, in the sense of focusing on population aggregates (Morgenstern 1995), namely geographic areas, and on environmental and socio-economic risk factors for infectious disease (Ploubidis et al. 2012). The applications are to human infectious disease, namely TB incidence.

Different forms of spatial correlation analysis or model have been proposed in disease applications (human and veterinary), environmental science, ecology, crime and other settings. For example, Wikle (2003) reviews hierarchical spatial models applied in environmental science, including irregular lattice data (such as geographic areas) and regular lattice data (such as air pollution grids). Beale et al. (2010) consider how regression findings for spatial ecology data are affected by the method used (if at all) to reflect spatial dependence. To exemplify hierarchical models for veterinary data, Pioz et al. (2012) apply simultaneous autoregressive (SAR) models to investigate bluetongue spread in French municipalities, while Farnsworth and Ward (2009) apply Bayesian conditional autoregressive (CAR) models to avian influenza H5N1 outbreak data. In such applications, identifying elevated risk in particular areas, detecting elevated risk clusters, or assessing significant predictors of risk, are emphasized, in methods recognizing the explicitly spatial structure of the data. However, the underlying assumptions of such techniques should be assessed, and subject to modification when indicated.

Hierarchical models involving spatial random effects, both CAR and SAR forms, can be estimated by classical methods (Alam et al. 2015; Horabik and Nahorski 2010) or Bayesian methods (Waller and Carlin 2010; Lesage 1997). CAR spatial priors imply local smoothing of outcome rates, that is smoothing towards the local rather than global average (Gelman 1996; Waller and Carlin 2010). Such local discontinuity is demonstrated in the England TB application considered below. Marked variability in risks has been detected in other area studies of infectious disease (Duarte-Cunha 2015; Varga et al. 2015; Ploubidis et al. 2012), whereas spatial variability in relative risks of chronic diseases (cancer, diabetes, etc.) is generally less pronounced. When there are spatial discontinuities in risk, it is preferable to allow differing strengths of association between neighbouring areas, as opposed to uniform local smoothing under CAR priors (Gelman 1996; Smith et al. 2015).

Bayesian applications in disease mapping and ecological epidemiology commonly employ a CAR prior (Lee 2011) to express spatial clustering in underlying risks (Besag et al. 1991; Best 1999), including human TB incidence (Nunes 2007; Maciel et al. 2010). Most applications of CAR priors assume a normal density for the underlying risks combined with uniform local smoothing. However, normality assumptions may be vitiated by heteroscedasticity linked to spatial outliers or to marked discrepancies in risk between neighbouring areas. It is also desirable that spatial disease models represent variation in area disease risks that is not attributable to spatial dependence (i.e. heterogeneity as against clustering). Some spatial priors may represent this feature by using more than one set of random effects, but at the cost of identifiability.

This paper considers modification of the local smoothing principle when there are spatial discontinuities, namely discrepant levels of outcome rates (e.g. disease or crime incidence) between neighbouring areas. In particular, we consider modifications of the normality assumption for area random effects based on a scale mixture version of the Leroux et al. (1999) model, allowing for heterogeneity and clustering in a single set of random effects, but with the scale mixture providing adaptivity to local discontinuity and spatial outliers. The relevance of such an approach is illustrated with simulated data on TB incidence in 49 mainland US states, and an application to observed TB incidence in 326 English local authorities.

## 2 Defining conditional spatial priors

## 3 Conditional autoregressive spatial priors

_{i}denotes an area specific relative risk, and \( \upphi \) is a variance term for iid unstructured effects \( {\text{h}}_{\text{i}} \). A drawback with this scheme is that identifiability may be impeded by the presence of two sets for random effects representing one underlying aspect of the data, namely variation in area illness risks.

## 4 The Leroux et al. spatial prior

## 5 Adaptiveness to non-normality and spatial discontinuities

From (9) it can be seen that small \( \upkappa_{j} \) values indicate areas discrepant in risk from their neighbours (i.e. they indicate outliers in spatial terms), and reduce the amount of spatial borrowing of strength. Equivalently stated, a clustering of small \( \upkappa_{j} \) values can be taken as indicators of spatial volatility, namely discrepant illness risks in a set of adjacent areas. In regression applications, small \( \upkappa_{j} \) values will also indicate where the regression predictions in the neighbourhood of area \( \text{i} \), and their implied neighbourhood relative risk \( \sum\limits_{{{\text{j}} \in {\text{N}}_{{\text{i}}} }} {\upmu _{{\text{j}}} } /\sum\limits_{{{\text{j}} \in {\text{N}}_{{\text{i}}} }} {{\text{E}}_{{\text{j}}} } \), are discrepant from the modelled relative risk in area \( \text{i} \) itself \( \upmu_{\text{i}} /\text{E}_{\text{i}} \).

Identification of random effects in spatial disease models is often problematic (e.g. MacNab 2014; Nathoo and Ghosh 2013), especially for models including multiple random effects, or when disease counts are relatively small. In the case of the model just discussed, identification of outliers (e.g. in terms of significantly low \( \upkappa_{i} \)), as well as identification of elevated risks \( \text{s}_{\text{i}} , \) will be improved for larger disease counts and/or longer observation periods. Identification of hyperparameters may also be problematic, especially with small samples. For example, in student t binary regression with data augmentation, Gelman et al. (2004, p 447) recommend a robust analysis with \( \nu \) not estimated but preset at 4.

## 6 Simulation example

A simulation example of the heteroscedastic LLB prior involves TB incidence with a spatial framework provided by the \( \text{n} = 49 \) mainland states (including the District of Columbia). Expected TB incidence counts \( \text{E}_{\text{i}} \) are obtained by applying actual US-wide age specific rates for TB in 2013 to state population estimates for 2013, taken from the US National Cancer Institute SEER site (http://seer.cancer.gov/popdata/). TB incidence rates are from the CDC National Tuberculosis Surveillance System, with just over 9500 incident cases in 2013, and an all ages rate of 3 per 100,000. Highest rates (over 6 per 100 thousand) are for the 75-84 and 85 + age groups.

We simulate TB incidence counts using these expected counts as offsets. The LLB hyperparameters (guide values) are set as \( \uplambda = 0.7, \) \( \uptau = 3 \), and with \( \nu \) taking values 3,10, and 25. Although the student t is defined for degrees of freedom of 2 or less, it has infinite variance, and Gelman et al. (2004) mention that “t’s with one or two degrees of freedom have infinite variance and are not usually realistic in the far tails”. One hundred sets of random effects are generated from the multivariate normal \( {\text{s}}_{{ 1\;\; :\;\;{\text{n}}}} \sim {\text{N(0,}}\,{\text{Q}}^{ - 1} ) \). Simulated TB incidence counts are then obtained via a Poisson simulation \( {\text{y}}_{\text{i}} \sim {\text{Po(E}}_{\text{i}} \uprho_{\text{i}} ) \), with \( \log (\uprho_{\text{i}} ) = \upbeta_{0} + \text{s}_{\text{i}} , \) where \( \upbeta_{0} = - 0.1 \), and \( \uprho_{i} \) is the simulated disease relative risk in state \( \text{i} \) (relative to that expected on the basis of US wide incidence levels). The R code used is set out in “Appendix”. Note that each of the 100 simulations involves a separate sample of \( \upkappa_{\text{i}} \sim Ga(0.5\nu ,\,0.5\nu ) \).

Analyses to estimate the parameters from the 100 sets of simulated data \( \{ y,\,E\} \) (with \( \text{E} \) as in the simulations) are carried out using the WINBUGS package (Lunn et al. 2009). An exponential prior with mean 10 is adopted for \( \nu \,\; \)(Fernandez and Steel 1998; Geweke 1993), a gamma prior with shape 1 and index 0.01 assumed for the inverse variance parameter \( \uptau \), a normal prior with mean zero and precision 0.001 assumed for the fixed effect \( \upbeta_{0} \), and a uniform \( U(0,\,1) \) prior assumed on \( \uplambda \). Estimates are based on the last 5000 iterations from two chain runs of 10,000 iterations, with convergence assessed using Brooks–Gelman–Rubin diagnostics (Brooks and Gelman 1998).

Recovered parameter estimates from 100 simulated datasets

Parameter | Percentiles of posterior means | Samples with 95 % credible interval containing guide value | ||
---|---|---|---|---|

20th | 50th | 80th | Percent | |

(a) ν set to 3 | ||||

λ | 0.44 | 0.59 | 0.67 | 99 |

ν | 2.4 | 4.5 | 8.4 | 95 |

log(ν) | 0.76 | 1.29 | 1.88 | 95 |

β | −0.26 | −0.12 | 0.04 | 88 |

(b) ν set to 10 | ||||

λ | 0.52 | 0.65 | 0.72 | 99 |

ν | 7.8 | 10.4 | 12.5 | 100 |

log (ν) | 1.68 | 2.09 | 2.36 | 97 |

β | −0.26 | −0.14 | 0.01 | 91 |

(c) ν set to 25 | ||||

λ | 0.53 | 0.65 | 0.73 | 100 |

ν | 16.1 | 23.4 | 31.0 | 99 |

log(ν) | 2.22 | 2.65 | 3.03 | 99 |

β | −0.28 | −0.15 | −0.02 | 84 |

## 7 Application: TB incidence for England local authorities

Model assessment and parameter summaries, models without and including predictors

Model fit and checks | LLB constant scale | Scale mixture LLB |
---|---|---|

Model without predictors | ||

Fit measures | ||

PPL (k = 0.5) | 645.8 | 630.6 |

PPL (k = 5) | 677.4 | 661.9 |

Predictive checks | ||

Total observations overpredicted, with Pr(y | 5 | 0 |

Total observations underpredicted, with Pr(y | 0 | 0 |

Total observations overpredicted, with Pr(y | 30 | 25 |

Total observations underpredicted, with Pr(y | 38 | 33 |

Model with predictors | ||

Fit Measures | ||

PPL (k = 0.5) | 622.2 | 616.3 |

PPL (k = 5) | 666.6 | 660.2 |

Predictive checks | ||

Total observations overpredicted, with Pr(y | 5 | 3 |

Total observations underpredicted, with Pr(y | 0 | 0 |

Total observations overpredicted, with Pr(y | 31 | 27 |

Total observations underpredicted, with Pr(y | 31 | 25 |

Parameter Summaries (posterior mean, 95 % credible intervals) | LLB constant scale | Scale mixture LLB |
---|---|---|

Model without predictors | ||

λ Spatial dependence | 0.59 (0.37, 0.89) | 0.57 (0.37, 0.86) |

ν Scale mixing parameter | 8.1 (4.2, 15.9) | |

Model with predictors | ||

λ Spatial dependence | 0.76 (0.49, 0.98) | 0.75 (0.51, 0.98) |

ν Scale mixing parameter | 10.4 (4.2, 27.3) | |

β | 4.98 (4.01, 6.11) | 4.96 (3.71, 6.23) |

β | 1.34 (0.98, 1.7) | 1.25 (0.75, 1.81) |

Also presented are predictive checks based on replicate observations. Posterior predictive probabilities \( \Pr (\text{y}_{\text{i},\,\text{rep}} > \text{y}_{\text{i}} |\text{y}) \) in extreme tails (e.g. values under 0.1 or over 0.9) indicate poorly fitted cases. The mixed predictive scheme (Marshall and Spiegelhalter 2003), providing checks that are close to leave-one-out cross validation (Green et al. 2009), was also applied. This involves sampling new random effects \( \text{s}_{\text{i},\,\text{rep}} \), and then sampling replicate data \( \text{y}_{\text{i},\,\text{rep},\,\text{mixed}} \) conditional on these new effects.

Areas ranked by outlier status, no predictors

Number | Name | Events | Relative risk (MLE) | Spatial lag relative risk (MLE) | Heteroscedastic LLB | Constant scale LLB | |||||
---|---|---|---|---|---|---|---|---|---|---|---|

κ | μ | Pr(y | Pr(y | μ | Pr(y | Pr(y | |||||

1 | Peterborough | 170 | 2.12 | 0.33 | 0.48 | 167.6 | 0.48 | 0.03 | 166.3 | 0.40 | 0.01 |

2 | Leicester | 528 | 3.57 | 0.44 | 0.48 | 526.2 | 0.49 | 0.03 | 524.7 | 0.46 | 0.00 |

3 | Blackburn-Darwen | 157 | 2.55 | 0.79 | 0.49 | 154.1 | 0.39 | 0.03 | 152.7 | 0.39 | 0.00 |

4 | Preston | 106 | 1.66 | 0.30 | 0.55 | 103.9 | 0.40 | 0.05 | 103.4 | 0.41 | 0.02 |

5 | Tandridge | 3 | 0.09 | 1.12 | 0.56 | 5.7 | 0.78 | 0.98 | 6.6 | 0.87 | 1.00 |

6 | Bromsgrove | 2 | 0.05 | 1.59 | 0.57 | 5.0 | 0.79 | 0.98 | 5.9 | 0.91 | 1.00 |

7 | Rushmoor | 81 | 1.86 | 0.35 | 0.58 | 78.9 | 0.42 | 0.04 | 78.2 | 0.40 | 0.02 |

8 | Luton | 257 | 2.74 | 0.29 | 0.60 | 255.3 | 0.44 | 0.05 | 255.0 | 0.45 | 0.02 |

9 | Crawley | 75 | 1.47 | 0.40 | 0.60 | 72.9 | 0.42 | 0.06 | 72.2 | 0.39 | 0.03 |

10 | Reading | 159 | 2.02 | 0.37 | 0.63 | 156.9 | 0.44 | 0.06 | 156.7 | 0.44 | 0.02 |

11 | Slough | 248 | 3.91 | 1.71 | 0.64 | 246.2 | 0.44 | 0.07 | 245.6 | 0.45 | 0.02 |

12 | Sheffield | 279 | 1.04 | 0.32 | 0.64 | 276.6 | 0.45 | 0.07 | 276.4 | 0.45 | 0.04 |

13 | Birmingham | 1238 | 2.54 | 1.20 | 0.64 | 1234.0 | 0.43 | 0.04 | 1233.0 | 0.45 | 0.00 |

14 | Southampton | 130 | 1.07 | 0.17 | 0.65 | 128.3 | 0.42 | 0.06 | 127.7 | 0.43 | 0.04 |

15 | South Staffordshire | 4 | 0.09 | 0.85 | 0.65 | 7.1 | 0.78 | 0.97 | 7.4 | 0.83 | 0.98 |

16 | Newcastle upon Tyne | 122 | 0.84 | 0.19 | 0.66 | 119.9 | 0.44 | 0.07 | 119.8 | 0.43 | 0.04 |

17 | Redditch | 46 | 1.31 | 0.24 | 0.69 | 44.2 | 0.40 | 0.07 | 44.0 | 0.39 | 0.04 |

18 | Woking | 57 | 1.34 | 0.37 | 0.70 | 55.2 | 0.43 | 0.08 | 55.1 | 0.41 | 0.06 |

19 | Rossendale | 6 | 0.20 | 1.13 | 0.71 | 8.5 | 0.71 | 0.95 | 8.8 | 0.74 | 0.98 |

20 | Halton | 2 | 0.04 | 0.36 | 0.71 | 4.2 | 0.77 | 0.96 | 4.5 | 0.82 | 0.98 |

Areas ranked by outlier status, regression with predictors

Name | Events | Scale mixture LLB | Constant scale LLB | |||
---|---|---|---|---|---|---|

κ | μ | Model RR (μ | Model neighbourhood RR | μ | ||

Brent | 896 | 0.57 | 892.9 | 7.22 | 2.41 | 889.7 |

Peterborough | 170 | 0.60 | 166.3 | 2.06 | 0.35 | 165.1 |

Barnsley | 20 | 0.60 | 23.7 | 0.24 | 0.85 | 24.8 |

Swale | 8 | 0.62 | 11.1 | 0.20 | 0.50 | 11.7 |

Woking | 57 | 0.67 | 54.2 | 1.26 | 0.39 | 53.7 |

North Lincolnshire | 47 | 0.68 | 43.5 | 0.61 | 0.27 | 43.0 |

Kirklees | 287 | 0.69 | 283.1 | 1.53 | 1.00 | 281.9 |

Newham | 1072 | 0.71 | 1068.0 | 9.34 | 2.99 | 1068.0 |

Tandridge | 3 | 0.71 | 6.8 | 0.21 | 1.10 | 7.5 |

Rushmoor | 81 | 0.72 | 78.0 | 1.79 | 0.36 | 77.6 |

## 8 Conclusion

Different forms of spatial correlation analysis or modelling have been proposed in disease applications, ecological epidemiology, environmental science and other settings. Both Bayesian and frequentist estimation have been used. Common themes include identifying elevated risk areas or clusters of areas, and finding predictors of risk, while recognizing the explicitly spatial structure of the observations. For example, in a review of regression findings from spatial species abundance data, Dorfmann (2007) shows that ignoring spatial dependence (e.g. in regression residuals) leads to possible bias in parameter estimates and optimistic standard errors. However, while it is important to incorporate spatial dependence in models for area data, the assumptions of such techniques should be assessed, and subject to modification when the data so indicate. In particular, spatial discontinuities suggest a modification to the principle of uniform local smoothing.

In particular, Bayesian analyses of spatially arranged data often employ a conditionally autoregressive prior, which can express spatial clustering commonly present in the underlying risks, but typically assume a normal density for risks and uniform conditional association. However, a more sensitive parameterisation with utility in detecting outliers and locally irregular risk patterns may be obtained by allowing for non-normality. Commonly applied conditionally autoregressive priors, such as the proper CAR prior and the convolution prior, also have potential deficits when the observations contain a mixture of spatial dependence and unstructured heterogeneity. The present paper has proposed a scale mixture version of the Leroux et al. (1999) spatial prior, combining the benefit of adaptability when risks are discrepant in adjacent areas, and also a less problematic approach to representing a mixture of clustering and heterogeneity.

The analyses here show improved fit to infectious disease data, which may often show pronounced risk variability between areas. In England, high risk areas are often major urban centres, whereas the neighbouring suburban or rural hinterlands of such centres may be low risk. In such situations some modification of the uniform local borrowing of strength principle may be beneficial.

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