Skip to main content
Biostatistics (Oxford, England) logoLink to Biostatistics (Oxford, England)
. 2018 Sep 25;21(3):369–383. doi: 10.1093/biostatistics/kxy038

A Bayesian mixture modeling approach for public health surveillance

Areti Boulieri 1,, James E Bennett 1, Marta Blangiardo 1
PMCID: PMC7307974  PMID: 30252021

Summary

Spatial monitoring of trends in health data plays an important part of public health surveillance. Most commonly, it is used to understand the etiology of a public health issue, to assess the impact of an intervention, or to provide detection of unusual behavior. In this article, we present a Bayesian mixture model for public health surveillance, which is able to provide estimates of the disease risk in space and time, and also to detect areas with unusual behavior. The model is designed to deal with a range of spatial and temporal patterns in the data, and with time series of different lengths. We carry out a simulation study to assess the performance of the model under different scenarios, and we compare it against a recently proposed Bayesian model for short time series. Finally, the proposed model is used for surveillance of road traffic accidents data in England over the years 2005–2015.

Keywords: Bayesian hierarchical analysis, Mixture modeling, Public health surveillance, Road traffic accidents, Small-area detection, Spatio-temporal modeling

1. Introduction

In public health surveillance, it is typically of interest to describe spatio-temporal trends of a disease, as well as to detect areas where the disease risk follows a temporal pattern that deviates from the expected one. This can lead to important findings such as the identification of a localized risk factor, and can therefore, be a useful tool for public health researchers and practitioners. The situation considered here is the small area context, where the study region is divided into small bounded areas that typically represent administrative regions, each corresponding to an aggregated observed number of cases and an associated expected number of cases or population count.

There is a large literature on surveillance systems which can be distinguished into test statistics and model-based methods (Lawson and Kleinman, 2005; Robertson and others, 2010). The first class seeks to determine whether a defined subset (temporal, spatial, or spatio-temporal) is unusual compared to the incidence in the study region as a whole. Test statistics have a long history; see for instance the standard tests by Knox and Bartlett (1964) and Mantel (1967) which determine whether pairs of cases are close in space and time. These were followed by the scan statistics (Kulldorff, 1997) which normally use a circular window that passes over the study region and determines the significance of the number of cases within the window based on likelihood ratio statistics. SaTScan (Kulldorff and others, 2005) is perhaps the most popular scan statistic which has been extensively used for health care applications and is available through a user-friendly software (http://www.satscan.org/).

Model-based methods, on the other hand, are used to describe the underlying distribution of the data. Compared with the test statistics, they allow for the inclusion of random effects and explanatory variables to adjust the disease risk, thus providing a much more flexible framework. Bayesian hierarchical models have received considerable attention over the last two decades, due to advances in computer power allowing the implementation of sophisticated algorithms. These have been extensively applied in disease mapping studies aiming at providing estimates for the incidence of a disease (see for instance Waller and others, 1997; Sun and others, 2000; Knorr-Held and Richardson, 2003; Best and others, 2005; Lawson, 2013; Rushworth and others, 2014), while recently they have been used for the detection of change (Abellan and others, 2008; Zhou and Lawson, 2008; Li and others, 2012; Wakefield and Kim, 2013; Lee and Lawson, 2014; Napier and others, 2016). Li and others (2012) proposed BaySTDetect, a promising method specifically designed to detect areas with unusual temporal patterns for non-communicable diseases, which has been applied to several studies (Li and others, 2014; Duncan and others, 2016; Boulieri and others, 2016).

Briefly, BaySTDetect is a mixture of two model components: the first describes the background effect of the disease accounting for spatial and temporal correlations and estimates one time trend for the whole study region, while the second estimates a time trend for each area. A mixing parameter Inline graphic allocates each area Inline graphic to one of the two components, classifying these as common and unusual. A Bayesian estimate of the false discovery rate (FDR) is also used in order to control for the multiple testing problem. The method has shown its potential for detecting changes in time trends under various realistic scenarios, and performed better when compared to SaTScan, however, it suffers from important limitations. For instance, the mixing parameter Inline graphic is constant over time, making the method perhaps too restrictive for long time series, as it assumes that the unusual trend applies to the entire time period. In addition, the prior specification of Inline graphic does not account for spatial correlation in the unusual areas, which might produce biased results when unusual areas in the form of clusters are present.

In this article, we describe a modeling approach for epidemiological surveillance of public health data. By using BaySTDetect as the baseline model, we adopt its mixture specification and further extend it to address its limitations. Our proposed model is able to detect temporal changes in the spatial setting for scenarios where the baseline model lacks performance. The remainder of the article is structured as follows: in Section 2, we describe the baseline model, its limitations, and our proposed approach to overcome these. In Section 3, we present a simulation study to assess the performance of our model against the baseline model. A case study on road traffic accidents data in England is carried out to illustrate the method in Section 4, and we raise discussion points in Section 5.

2. Methods

2.1. Baseline model

Following the epidemiological structure on disease mapping, let Inline graphic and Inline graphic denote disease counts and expected counts respectively in area Inline graphic and time point Inline graphic. In the first level of the hierarchy, a Poisson lognormal model is assumed for the disease counts,

graphic file with name M9.gif (2.1)

where Inline graphic represents the relative risk of the disease in area Inline graphic at time point Inline graphic. This parameter is modelled as a mixture of two alternative models, a Common model and an Area-Specific model. The Common model assumes one global time trend for all areas, while the Area-Specific model estimates a time trend for each area independently. This aims to distinguish between areas that follow the expected time trend and those that exhibit unusual behavior. More specifically, the second level of the hierarchy models Inline graphic as follows:

graphic file with name M14.gif (2.2)
graphic file with name M15.gif (2.3)
graphic file with name M16.gif (2.4)

and Inline graphic is the mixing parameter that selects between estimates Inline graphic from the Common model and estimates Inline graphic from the Area-Specific model.

The Common model (Eq. 2.3) follows a standard disease mapping approach where the relative risk Inline graphic consists of a global spatial component Inline graphic and a global temporal component Inline graphic. The spatial component Inline graphic is assigned a convolution prior, widely known as BYM, which was proposed by Besag and others (1991). This is a Gaussian prior, Inline graphic where Inline graphic is a spatially structured term following an intrinsic conditional autoregressive prior (ICAR)

graphic file with name M26.gif (2.5)

The parameter Inline graphic represents the entries of the adjacency matrix Inline graphic of size Inline graphic specifying the spatial neighborhood structure, such that Inline graphic if areas Inline graphic and Inline graphic share borders and Inline graphic otherwise. Inline graphic is the number of neighbors of area Inline graphic.

Similarly, the temporal component Inline graphic follows a Gaussian random walk prior of order 1 Inline graphic, which is implemented as the temporal analogue of the ICAR prior in Eq. (2.5), Inline graphic, where the entries of the adjacency matrix Inline graphic are given by Inline graphic if Inline graphic and Inline graphic otherwise, with Inline graphic and Inline graphic indexing units of time.

An overall intercept which follows a flat prior Inline graphic is included due to the sum-to-zero constraints of the ICAR priors Inline graphic and Inline graphic such that the model is identifiable (Thomas and others, 2004).

The Area-Specific model (Eq. 2.4) consists of an area-specific intercept that follows a weakly informative prior Inline graphic and an area-specific temporal component Inline graphic where the adjacency matrix Inline graphic is defined as above.

The hyperparameters Inline graphic, Inline graphic, and Inline graphic are assigned a weakly informative half Normal prior Inline graphic, while Inline graphic, where Inline graphic, bounded below by 0, and Inline graphic (for details see Li and others, 2012).

The mixing parameter Inline graphic follows a Bernoulli prior assuming that around 5% of the areas are unusual:

graphic file with name M59.gif (2.6)

The model is fitted to the data set through Markov chain Monte Carlo (MCMC) sampling, and, at each iteration, the mixing parameter Inline graphic takes values 1 or 0 taking into account which model (Common or Area-specific) is more likely under the data. At the end of the MCMC sampling, the posterior mean of Inline graphic, namely Inline graphic = P(Inline graphic = 1Inline graphicdata), reflects the posterior probability that area Inline graphic is common or unusual over the whole time period. In order to account for the multiple testing problem, a detection rule is employed based on a Bayesian estimate of the FDR (Newton and others, 2004; Ventrucci and others, 2010). According to the rule, the areas with Inline graphic below threshold Inline graphic are classified as unusual; Inline graphic is the maximum integer such that Inline graphic, with Inline graphic denoting the Inline graphic ordered posterior probability and Inline graphic a preset level usually taken equal to 0.05 (for details see Li and others, 2012).

2.2. Limitations of the baseline model

The model described above has shown its potential for detection of unusual behavior within the small area context, however, it suffers from drawbacks that limit its applicability. More specifically:

  1. The mixing parameter Inline graphic (Eq. 2.6) assumes that the probability that area Inline graphic follows the common or unusual trend remains the same across the whole time period. This assumption is perhaps too restrictive and makes the model appropriate only for short time series, (i.e. Li and others (2012)) tested the model on simulated data with 8 time points and recommend that it can be used for a maximum length of 10.

  2. The posterior estimates of Inline graphic give information regarding the location of the unusual behavior but not the corresponding time point/points when this occurs.

  3. The prior on Inline graphic (Eq. 2.6) assumes independence across unusual areas, however, in epidemiological studies these can be spatially correlated, and not accounting for this can result in unreliable estimates.

  4. In addition, Inline graphic assumes that a priori 5% of areas are unusual (Eq. 2.6), which might induce bias when this does not agree with the data, for instance when no outbreaks exist or a much larger proportion of areas are unusual. This is particularly likely to happen when very sparse data are considered, i.e. short time series and/or low counts, where the prior can have a strong effect on the posterior. The impact of this prior specification on the detection power of the model has been described in Li and others (2011).

  5. Because of the nature of Inline graphic, multiple independent tests are performed and hence the multiple testing problem arises. The Bayesian FDR specification is incorporated in the baseline model to address this, however, as can be seen in the original paper (Figure 3 in Li and others, 2012), the proportion of false positives is not adequately low for all scenarios considered in the simulation study.

2.3. Proposed model

The proposed model follows the Poisson lognormal specification of the baseline model (Section 2.1), with Equations (2.1) to (2.4) remaining the same. We address the limitations of the baseline model in the rest of this section.

To make the approach able to deal with long time series, we allow the mixing parameter Inline graphic in Eq. 2.2 to become Inline graphic; the probability of whether area Inline graphic is common or unusual therefore changes across time (Limitation 1). In addition, the posterior mean Inline graphic provides information on both where and when unusual behavior is observed (Limitation 2).

We allow the parameter Inline graphic to follow a hierarchical model:

graphic file with name M84.gif (2.7)
graphic file with name M85.gif (2.8)
graphic file with name M86.gif (2.9)

where Inline graphic). The spatial component Inline graphic is assigned an ICAR Inline graphic and the temporal component Inline graphic is assigned the temporal analogue of ICAR, Inline graphic, where Inline graphic and Inline graphic are the spatial and temporal adjacency matrices respectively as specified in Section 2.1. Through this specification, we account for dependences of the unusual observations in space and time (Limitation 3); it is realistic to assume that the probability that area Inline graphic is unusual at time point Inline graphic depends on the corresponding probability of a neighboring area at time point Inline graphic, and the corresponding probability of area Inline graphic at time points Inline graphic and Inline graphic.

The hyperparameters Inline graphic and Inline graphic are assigned a weakly informative half Normal prior Inline graphic. All remaining parameters and hyperparameters of Equations (2.2) to (2.4) are specified as in Section 2.1, except for the area-specific variance Inline graphic which now follows a less restrictive half Normal prior Inline graphic. Given the hierarchical structure of Inline graphic, and the fact that the proposed model is appropriate for longer time series, more information is now available, and therefore, it is no longer needed to specify an informative prior on the area-specific variances, but rather allow for more flexibility in the detection mechanism of the model.

We also include in the model the term Inline graphic; this shifts the expectation of Inline graphic from 0.5 to parameter Inline graphic which is now assigned a prior distribution. The rationale behind the choice for the prior of Inline graphic is that it needs to be relatively high so that a small proportion of observations follows the Area-Specific model (roughly 5–10%), thus characterized as unusual, while at the same time it allows for flexibility to deal with scenarios with no unusual observations. We, therefore, specify a uniform prior with a range between 0.9 and 1, assuming that the proportion of unusual observations varies from 0 (when no unusual behavior is evident) to 10% (Limitation 4).

An additional positive impact of modeling Inline graphic hierarchically is that we achieve low proportions of false positives (Limitation 5), while we no longer need to incorporate the Bayesian FDR to adjust for the multiple testing problem. Within the Bayesian modeling framework, the interpretation of the posterior Inline graphic is straightforward; as long as the model is properly specified, accounting for known sources of variability, Inline graphic = P(Inline graphic=1Inline graphicdata) represents the posterior probability that area Inline graphic follows the common trend at time point Inline graphic (Gelman and others, 2012). We select areas with probability Inline graphic less than the standard statistical significance level of 0.05, and we class them as unusual at the corresponding time point/points Inline graphic.

3. Simulation study

We carried out a simulation study to evaluate the performance of our proposed model under various realistic scenarios, as well as to compare it against the baseline model. For comparison purposes, we followed closely the simulation design used in Li and others (2012).

3.1. Data generation and study design

We generated count data for 211 areas in England over 15 time points under the Common model (Eq. 2.3). These areas represent Clinical Commissioning Groups, an authority responsible for decisions on health related matters at the local level. We used data on asthma hospital admissions in the same areas to (i) calculate the expected number of cases after adjusting for age and sex characteristics using indirect standardization and (ii) obtain posterior parameter estimates from the Common model, which were then used to simulate the outcome variable. Summary statistics of the asthma hospitalization data set can be found in Table 3 in the supplementary material available at Biostatistics online.

Table 3.

Performance for scenario S10

  Baseline model Proposed model
Simulations that detect at least one aberration (%) 26 8
Mean number of aberrations detected 1.31 1.25

The simulated counts represent the behavior of the disease under normal conditions. In order to evaluate whether the model is able to pick up unusual behavior, we selected a number of areas to follow a temporal trend that differs from the national one. For the selection of areas, we considered two different spatial scenarios: (i) one where unusual areas are isolated (they do not share geographical borders) and (ii) one where they form clusters (they share geographical borders).

Accounting for the fact that the spatial risk and the magnitude of the expected cases will affect the ability of the model to detect a particular area, we selected 15 unusual areas in total (for details see Section 1 in the supplementary material available at Biostatistics online). For each of the two spatial scenarios, we considered three temporal scenarios: (i) one where unusual time points are isolated, (ii) one where they are consecutive and characterized by a high degree of variability, and (iii) one where they are consecutive and show a stable pattern, i.e. they do not show high variability (Figure 1a–c in the supplementary material available at Biostatistics online). This will allow us to evaluate whether the model is able to detect unusual behavior when spatial and temporal dependences are evident.

Fig. 1.

Fig. 1.

Map of posterior accident rates in England.

In addition to the above six scenarios, we also include the following: (i) one scenario with 30 time points in order to evaluate the performance of the model on much longer time series (Figure 1d in the supplementary material available at Biostatistics online), (ii) one scenario with reduced expected cases, and (iii) one scenario with increased expected cases, in order to assess the impact of the magnitude of the expected counts (see Table 3 in the supplementary material available at Biostatistics online). Finally, we consider (iv) one scenario with no aberrations in order to assess whether the model produces positives when there is no signal. All 10 scenarios are summarized in Table 1. Details about the construction of all unusual temporal trends can be found in Section 2 in the supplementary material available at Biostatistics online.

Table 1.

Scenarios for the simulated data

Scenario Spatial pattern Temporal pattern Expected cases
S1 Isolated Isolated Normal
S2 Isolated Consecutive-variable Normal
S3 Isolated Consecutive-stable Normal
S4 Clustered Isolated Normal
S5 Clustered Consecutive variable Normal
S6 Clustered Consecutive stable Normal
S7 Isolated Long time series Normal
S8 Isolated Consecutive variable Increased
S9 Isolated Consecutive variable Reduced
S10 No aberrations

In order to ensure that the proposed model performs well also in the case of short time series, where the baseline model has shown to be appropriate, we carried out a simulation study with eight time points. This is presented in Section 3 in the supplementary material available at Biostatistics online.

Fifty simulated data were generated in total for each scenario.

3.2. Results

For scenarios Inline graphicInline graphic (Table 1), we are interested to detect as many aberrations as possible while keeping a low proportion of false positives.

In the case of our proposed model, an aberration is a combination of area Inline graphic and time point Inline graphic for which Inline graphic is below 0.05. In the case of the baseline model, an aberration is an area Inline graphic for which Inline graphic is below threshold Inline graphic, which satisfies the rule described in Section 2.1 with the preset level Inline graphic set to 0.05. Therefore, given the simulated data set of 15 time points and 211 areas, Inline graphic for the proposed model, while Inline graphic for the baseline model. Model performance was assessed through sensitivity Inline graphic, specificity Inline graphic, proportion of false positives Inline graphic, and global error Inline graphic(FP+FN)/(TN+TP+FP+FN)Inline graphic, where TP, FP, TN, and FN are the numbers of true positives, false positives, true negatives, and false negatives, respectively. For the proportion of false positives, when there are no declared unusual observations (Inline graphic), this is set to 0. In order to assess the performance of the scenario with no aberrations (Inline graphic), we take the proportion of simulations (of the total 50) with at least one aberration declared, and the corresponding average number of these.

Table 2 presents the averaged values of the performance criteria across 50 simulations for scenarios Inline graphic to Inline graphic. As can be seen, the mean proportion of false positives from the baseline model is above 0.2 for all scenarios, which says that at least 20% of the declared unusual observations are not truly unusual. The range of values is detailed by the scenario characteristics: the minimum, equal to 0.2, corresponds to the case where reduced expected cases are assumed, whereas the highest values, 0.32 and 0.33, correspond to the scenarios with a longer time series (Inline graphic) and increased expected cases (Inline graphic), respectively. From this we can conclude that when more information is added, i.e. the data are less sparse, the performance of the baseline model in terms of false positives becomes worse. Although sensitivity is very high for the same model, and specificity and global error have adequate values, the high proportion of false positives makes the model inappropriate to use for scenarios with long time series, i.e. 15 time points and above. In Table 1 in the supplementary material available at Biostatistics online, we can see that for short time series (eight time points), the baseline model is powerful and gives relatively low proportions of false positives, as has been previously shown.

Table 2.

Performance for scenarios S1–S9; parentheses indicate 95% credible intervals (the 2.5 and 97.5 percentiles of the distribution from 50 simulations)

  FP proportion Sensitivity Specificity Global error
Baseline model
S1 0.250 (0.211, 0.286) 0.965 (0.933, 1.000) 0.976 (0.969, 0.980) 0.025 (0.019, 0.028)
S2 0.264 (0.211, 0.296) 1.000 (0.985, 1.000) 0.972 (0.969, 0.980) 0.027 (0.019, 0.032)
S3 0.271 (0.222, 0.318) 0.967 (0.933, 1.000) 0.972 (0.964, 0.980) 0.029 (0.024, 0.033)
S4 0.268 (0.250, 0.318) 1.000 (0.999, 1.000) 0.971 (0.964, 0.974) 0.027 (0.024, 0.033)
S5 0.286 (0.286, 0.318) 1.000 (1.000, 1.000) 0.969 (0.964, 0.969) 0.029 (0.028, 0.033)
S6 0.295 (0.250, 0.318) 1.000 (1.000, 1.000) 0.968 (0.964, 0.974) 0.030 (0.024, 0.033)
S7 0.322 (0.286, 0.306) 1.000 (0.933, 1.000) 0.963 (0.959, 0.969) 0.035 (0.028, 0.041)
S8 0.327 (0.286, 0.348) 1.000 (1.000, 1.000) 0.962 (0.959, 0.969) 0.035 (0.028, 0.038)
S9 0.200 (0.143, 0.222) 0.857 (0.800, 0.933) 0.984 (0.980, 0.990) 0.025 (0.019, 0.032)
Proposed model
S1 0.022 (0.000, 0.036) 0.710 (0.671, 0.750) 1.000 (0.999, 1.000) 0.006 (0.005, 0.007)
S2 0.019 (0.015, 0.031) 0.796 (0.763, 0.827) 1.000 (0.999, 1.000) 0.005 (0.004, 0.006)
S3 0.150 (0.118, 0.187) 0.799 (0.762, 0.814) 0.998 (0.997, 0.999) 0.005 (0.004, 0.006)
S4 0.030 (0.000, 0.052) 0.881 (0.838, 0.917) 0.999 (0.999, 1.000) 0.003 (0.002, 0.003)
S5 0.029 (0.014, 0.041) 0.928 (0.907, 0.947) 0.999 (0.999, 1.000) 0.002 (0.002, 0.003)
S6 0.225 (0.190, 0.266) 0.932 (0.911, 0.956) 0.996 (0.995, 0.997) 0.005 (0.004, 0.006)
S7 0.027 (0.022, 0.034) 0.750 (0.710, 0.770) 0.973 (0.966, 0.978) 0.008 (0.007, 0.008)
S8 0.057 (0.040, 0.075) 0.969 (0.960, 0.987) 0.999 (0.998, 0.999) 0.002 (0.002, 0.003)
S9 0.012 (0.000, 0.027) 0.480 (0.427, 0.507) 1.000 (1.000, 1.000) 0.013 (0.012, 0.014)

Our proposed method, on the other hand, gives adequately high sensitivity values and low proportion of false positives for most scenarios (Table 2). More specifically, we observe that for scenarios Inline graphic, Inline graphic, Inline graphic, Inline graphic, Inline graphic, and Inline graphic, our proposed model outperforms the baseline model: sensitivity ranges from 0.71 to 0.97, whereas the proportion of false positives is always below 0.05. When the temporal pattern is stable (scenarios Inline graphic and Inline graphic), the proportion of false positives increases to 0.15 and 0.23, respectively, still lower than the corresponding values of the baseline model, i.e. 0.26 and 0.29. In the case of reduced expected cases (Inline graphic), our model has a mean proportion of false positives equal to only 0.01, however, sensitivity is 0.5, indicating that it is not able to pick up unusual behavior sufficiently. In the case of short time series, it seems that the baseline model is slightly better than our proposed model overall, however, the associated mean proportion of false positives is higher than 0.05 for all scenarios.

In the extreme case of no aberrations (Inline graphic), a proportion of 8% simulations (of the total 50) produces positives, while these give an average number of declared aberrations equal to 1.25, contrary to a corresponding 26% with an average 1.31 produced by the baseline model (Table 3). This suggests that the proposed model seldom produces false positives, and when this happens, the corresponding number tends to be very small.

In order to show that the model parameters are estimated well, the simulated temporal trend under the Common model is plotted against the corresponding estimated trends for scenarios Inline graphicInline graphic (Figure 3 in the supplementary material available at Biostatistics online). In addition, the simulated temporal trend under the Area-Specific model is plotted against the estimated trends of the 15 unusual areas for scenarios Inline graphicInline graphic (Figure 4 in the supplementary material available at Biostatistics online).

4. Case study: road traffic accidents in England

We applied the proposed method to a set of road traffic accidents data at small area level in England over a period of 11 years. Road traffic accidents are considered to be a major public health issue, and they are typically analyzed by methods developed for non-communicable diseases (Miaou and others, 2003; MacNab, 2003; Song and others, 2006; Aguero-Valverde and Jovanis, 2008; Boulieri and others, 2017). We seek to describe the overall risk across space and time, while we focus on the detection of areas with unusual temporal trends.

4.1. Data

Road traffic accidents data in Great Britain are recorded on a STATS19 report from police officers and are held by the Department for Transport. The data include information on the postcode, date, and severity level (slight, severe, or fatal) of each accident. We analyze severe and fatal accidents data together for 326 districts in England over the years 2005–2015.

The annual average daily flow (AADF) data for the major road network (motorways and A-roads) in England were obtained from the Department for Transport. These are the traffic counts for each junction-to-junction link on the major roads containing spatial information in the form of a postcode. The AADF data set was joined to the major road network shapefile, also obtained from the Department for Transport. Each AADF point was associated to a road segment based on unique road names, or, if not available, based on distance. For the road segments with no corresponding AADF count, this was estimated as the average AADF of the bordering road segments, and in the few occasions when this was not possible the middle value of AADF was used.

The resulting shapefile, containing information on all road segments in England with their corresponding AADF values was intersected with the districts shapefile. The traffic volume of each intersected road segment was then calculated by multiplying the length of the road segment by its AADF value. Finally, the district level traffic volume was estimated by summing the traffic volumes of all road segments lying within each district:

graphic file with name M155.gif (4.1)
graphic file with name M156.gif (4.2)

where TV is the traffic volume, D represents the district level and rs represents the road segment.

The traffic volume was calculated for all years considered in the study, i.e. 2005–2015, and the correlation was found to be 0.99. We, therefore, decided to use the traffic volume for the middle year 2010 for our analysis. The summary statistics for the accidents data and the accident rates (number of accidents divided by traffic volume) are presented in Tables 4 and 5 in the supplementary material available at Biostatistics online.

4.2. Results

The model was implemented in R using the R2OpenBUGS package. We run two chains for 80 000 MCMC iterations storing every 2nd iteration, from which 20 000 were discarded as a burn-in. The model took roughly 5 h on an Intel Core processor at 3.40 GHz with 16 Gbytes of RAM. We carried out convergence diagnostics through visual check of the trace plots, Brooks–Gelman–Rubin statistic, autocorrelation plots, and assessing the Monte Carlo error (see Figure 5 in the supplementary material available at Biostatistics online).

Figure 1 plots the posterior estimate of the spatial component Inline graphic (Eq. 2.3) on the exponential scale showing the overall pattern of accident risk across England; in the disease mapping context this is usually referred to as a relative risk, however, in our example, where the offset is the traffic volume and not an expected count of accidents, we interpret it as an accident risk. The map suggests higher accident risk in London and other big cities such as Manchester and Leeds, while low accident risk is observed along certain motorways such as M1 and M5. The posterior estimate of the temporal component Inline graphic (Eq. 2.3) on the exponential scale, shown in Figure 2, shows the pattern of accident risk over the period 2005–2015, which can be thought of as the national trend. There is a nearly linear downward trend over time up until 2010, while afterwards the trend becomes flatter.

Fig. 2.

Fig. 2.

The national trend of England against the time trends of three selected unusual areas. (a) District of Eden. (b) District of Southampton. (c) District of Elmbridge.

Under the model, 96 observations were detected as unusual out of which 45 correspond to unique areas (see map in Figure 6 in the supplementary material available at Biostatistics online). These are in the form of isolated areas as well as clusters. We present the area-specific temporal trends of three selected areas plotted against the national trend in Figure 2. These correspond to the parameter Inline graphic (Eq. 2.4) on the exponential scale, where Inline graphic is the selected unusual area.

The district of Eden in Cumbria shows a temporal trend with a much higher accident rate than the national average at the beginning of the time period, with an aberration detected by the model in year 2005. According to published reports, the rates of deaths and serious injuries in Cumbria are significantly higher than the national average, with the district of Eden having the highest rate in the county (Brown, 2015; Critchley and Whitfield, 2015).

The district of Southampton follows a nearly opposite temporal pattern to the national one, with aberrations detected at several time points over the study period and the highest accident risk observed in 2011. This might be related to the increased popularity of cycling in Southampton potentially leading to high accident rates. In fact, the time trend of the accident risk in Southampton somewhat resembles the trend corresponding to cyclist numbers in the area (Southampton City Council, 2016).

Similarly, the district of Elmbridge in Surrey exhibits an overall upward trend with an aberration detected in year 2015, a finding which is in line with recent reports ranking Surrey among the areas with the sharpest increase in the numbers of people killed or seriously injured on the roads (Surrey Police, 2015). In specific, road accidents involving vans and lorries increased by a third in Surrey in 2015 compared to the previous year, while the number of pedestrians who were injured on the roads of Elmbridge rose by 54% for the same year (Scott, 2016a, 2016b).

5. Discussion

In this article, we have described a Bayesian spatio-temporal model for public health surveillance that can be used for detection of areas with unusual behavior over time. We use the BaySTDetect method as a baseline, and we overcome its limitations by extending it to a more generic framework. Allowing the mixing parameter to change across time results in a model which is appropriate for long time series, a case where the baseline model showed lack of performance. Information on the time points when unusual behavior occurs is also provided to facilitate interpretation and etiological investigation. The prior of the mixing parameter changes to a less informative hierarchical prior that accounts for spatial and temporal dependences making the model more flexible for detection of several realistic unusual patterns.

Surveillance methods are typically optimized for one sole criterion depending on the problem in hand. In epidemiological surveillance, where the financial resources are usually limited and the priority is to spend expenditure wisely, we need to stress the importance of controlling the proportion of false positives. The baseline model, for instance, which is able to correctly detect nearly all unusual observations as such, at the cost of excess numbers of false detections, is not appropriate in this context. Our proposed model, on the other hand, has the great advantage of keeping the proportion of false positives below 5% for most scenarios. This suggests that we can be confident that the observations that are detected as unusual under the model are indeed unusual. In addition, unusual behavior is detected only when this is evident, as shown from the scenario with no aberrations (Inline graphic), thus confirming our hypothesis that the strong prior specification on Inline graphic of the baseline model (Eq. 2.8) induces bias when this does not agree with the data. At the same time, the corresponding sensitivity and specificity values are satisfactory.

An additional strength of the proposed model is that there is no need to adjust for the multiple testing problem as in the case of the baseline model. This arises when multiple tests are performed, and false detections are likely to occur due to pure chance. Although the importance of controlling for multiple testing is clear in classical inference, the relevant problem in the Bayesian setting is not always considered valid. Gelman and others (2012) argue that when the data are properly modelled through a Bayesian hierarchical framework, there should be no concern about false detections; the estimates are naturally smoothed towards the mean and they become more conservative and therefore less prone to false positives. On the other hand, it is likely that the estimates suffer from a higher proportion of false negatives which may lead to under-detection of important risks.

We note that the model is designed specifically for detection of a small proportion of unusual observations when these exist, ranging a priori between 0 and 10%. The upper bound could be chosen based on each specific case; however, a much larger proportion of unusual trends, say 20% and more, would not be seen as unusual but rather as a different common trend. The objective would then be clustering of multiple time trends, and a model following non-parametric Bayesian approaches would be more appropriate (see for instance Dass and others, 2015).

From a policy making perspective, the proposed model could be especially valuable for healthcare professionals who can use it for spatial monitoring at the population level, in order to detect areas with unusual behavior over time. This might indicate the impact of a policy/intervention, or the presence of a localized risk factor. Detecting areas with increased demand is particularly important for implementation of targeted intervention strategies, however, further investigation would be normally required to better understand unusual patterns, and generate hypotheses regarding associated risk factors.

While our model has shown its potential in detecting unusual patterns under various temporal and spatial scenarios, its performance was less satisfactory when the unusual time trend was stable (i.e. consecutive-stable patterns), as it gave a relatively high proportion of false positives. On the other hand, when the unusual trend was more variable (i.e. isolated and consecutive-variable patterns), the proportion of false positives was very low, at the cost of a slight decrease in power. In order to test whether this is due to the space-time separable structure of the probability model in Eq. 2.10, we carried out two simulations using a modified version of the model, where an additional latent variable is included to allow for space-time interactions. The results show that the additional parameter does not provide any important benefit on the performance of the model, suggesting that the conservative performance of our model is not due to the lack of flexibility on the parameter Inline graphic in Eq. (2.10). Instead, it is more related to the overall hierarchical structure of the parameter Inline graphic, which being more flexible, it is more appropriate for longer time series, while addressing the issues of identifiability and multiple testing. This results in a low proportion of false positives, which comes at the cost of losing slightly in power. Details about the additional simulations and the corresponding results can be found in Section 4 in the supplementary material available at Biostatistics online.

In addition, the model was not able to adequately pick up signal when expected cases were reduced to half. This might be due to the area-specific variances, which are defined on a specific scale of data. Potentially a way to address this is to assume that these are relative to the variance of the Common model.

It was also shown that for short time series (eight time points) there is no additional benefit of using our proposed model rather than the baseline model, suggesting that it is probably too complex for such short time series. We, therefore, recommend that the baseline model should be used for times series of length less than 10, as was also recommended by Li and others (2012), while the proposed model should be preferred otherwise. In the context of epidemiological surveillance, nevertheless, the availability of data has recently shifted the public interest from yearly to monthly or weekly time resolution, hence resulting in longer time series where the proposed approach would be beneficial.

An example of such analyses is the syndromic surveillance which makes use of data on pre-diagnostic syndromes rather than confirmed cases of specific diseases, such as ambulance calls or drug prescriptions, with the aim to provide early warning detection (Zou and others, 2011; Corberán-Vallet and Lawson, 2014). A natural extension of the proposed model would be the prospective setting, such that syndromic surveillance can be performed.

The mixture modeling approach that we propose is specifically designed for detection of unusual behavior allowing the implementation of two separate components. Alternatively, it is common in the disease mapping context to assume a single component model where an interaction term is used to highlight spatio-temporal deviations. A comparison among the two approaches through a simulation study would be useful to identify their respective benefits and drawbacks.

Furthermore, we are currently developing a web application that integrates the proposed model and allows for easy implementation and the creation of interactive data visualizations. This could be particularly useful to public health researchers and practitioners who may lack adequate programming skills and advanced statistical knowledge to carry out complex analyses.

6. Software

The code is provided on Github at https://github.com/aretib/bayes_mixture.git.

Supplementary Material

kxy038_Supplementary_Materials

Acknowledgments

We would like to thank both reviewers and the Associate Editor for their valuable comments and suggestions. Conflict of Interest: None declared.

Funding

This research is funded by an MRC DTP scholarship.

References

  1. Abellan J. J., Richardson S. and Best N. (2008). Use of space–time models to investigate the stability of patterns of disease. Environmental Health Perspectives 116, 1111. [DOI] [PMC free article] [PubMed] [Google Scholar]
  2. Aguero-Valverde J. and Jovanis P. (2008). Analysis of road crash frequency with spatial models. Transportation Research Record: Journal of the Transportation Research Board 1, 55–63. [Google Scholar]
  3. Besag J., York J. and Mollié A. (1991). Bayesian image restoration, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics 43, 1–20. [Google Scholar]
  4. Best N., Richardson S. and Thomson A. (2005). A comparison of Bayesian spatial models for disease mapping. Statistical Methods in Medical Research 14, 35–59. [DOI] [PubMed] [Google Scholar]
  5. Boulieri A., Hansell A. and Blangiardo M. (2016). Investigating trends in asthma and COPD through multiple data sources: a small area study. Spatial and Spatiotemporal Epidemiology 19, 28–36. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Boulieri A., Liverani S., Hoogh K. and Blangiardo M. (2017). A space–time multivariate Bayesian model to analyse road traffic accidents by severity. Journal of the Royal Statistical Society: Series A (Statistics in Society) 180, 119–139. [Google Scholar]
  7. Brown R. E. (2015). Avoidable Mortality in Cumbria: A Review of 73 Fatal Road Traffic Collissions. http://www.cph.org.uk/wp-content/uploads/2015/04/A-Review-of-73-Fatal-Road-Traffic-Collisions-in-Cumbria_RBrown-2.pdf. Accessed 11 September 2017. [Google Scholar]
  8. Corberán-Vallet A. and Lawson A. B. (2014). Prospective analysis of infectious disease surveillance data using syndromic information. Statistical Methods in Medical Research 23, 572–590. [DOI] [PubMed] [Google Scholar]
  9. Critchley K. and Whitfield M. (2015). TIIG Cumbria: Themed Report. http://www.cph.org.uk/wp-content/uploads/2015/02/TIIG-Cumbria-Road-traffic-collisions-across-Cumbria-2011-12-to-2013-14.pdf. Accessed 05 September 2017. [Google Scholar]
  10. Dass S. C., Lim C. Y., Maiti T. and Zhang Z. (2015). Clustering curves based on change point analysis: a nonparametric Bayesian approach. Statistica Sinica 1, 677–708. [Google Scholar]
  11. Duncan E. W., White N. M. and Mengersen K. (2016). Bayesian spatiotemporal modelling for identifying unusual and unstable trends in mammography utilisation. BMJ Open 6, e010253. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. Gelman A., Hill J. and Yajima M. (2012). Why we (usually) don’t have to worry about multiple comparisons. Journal of Research on Educational Effectiveness 5, 189–211. [Google Scholar]
  13. Knorr-Held L. and Richardson S. (2003). A hierarchical model for space–time surveillance data on meningococcal disease incidence. Journal of the Royal Statistical Society: Series C (Applied Statistics) 52, 169–183. [Google Scholar]
  14. Knox E. G. and Bartlett M. S. (1964). The detection of space-time interactions. Journal of the Royal Statistical Society. Series C (Applied Statistics) 13, 25–30. [Google Scholar]
  15. Kulldorff M. (1997). A spatial scan statistic. Communications in Statistics-Theory and Methods 26, 1481–1496. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Kulldorff M., Heffernan R., Hartman J., Assunção R. and Mostashari F. (2005). A space–time permutation scan statistic for disease outbreak detection. PLoS Medicine 2, e59. [DOI] [PMC free article] [PubMed] [Google Scholar]
  17. Lawson A. B. (2013). Bayesian Disease Mapping: Hierarchical Modeling in Spatial Epidemiology. United States; CRC Press. [Google Scholar]
  18. Lawson A. B. and Kleinman K. (2005). Spatial and Syndromic Surveillance for Public Health. United Kingdom: John Wiley & Sons. [Google Scholar]
  19. Lee D. and Lawson A. (2014). Cluster detection and risk estimation for spatio-temporal health data. arXiv preprint arXiv:1408.1191. https://arxiv.org/abs/1408.1191.
  20. Li G., Best N., Hansell A. L., Ahmed I. and Richardson S. (2012). BaySTDetect: detecting unusual temporal patterns in small area data via Bayesian model choice. Biostatistics 13, 695–710. [DOI] [PubMed] [Google Scholar]
  21. Li G., Haining R., Richardson S. and Best N. (2014). Space–time variability in burglary risk: a Bayesian spatio-temporal modelling approach. Spatial Statistics 9, 180–191. [Google Scholar]
  22. Li G., Richardson S., Fortunato L., Ahmed I., Hansell A., Toledano M. and Best N. (2011). Data mining cancer registries: retrospective surveillance of small area time trends in cancer incidence using BaySTDetect In: 2011 IEEE 11th International Conference on Data Mining Workshops. vancouver, BC: IEEE. pp. 885–890. [Google Scholar]
  23. MacNab Y. C. (2003). A Bayesian hierarchical model for accident and injury surveillance. Accident Analysis and Prevention 35, 91–102. [DOI] [PubMed] [Google Scholar]
  24. Mantel N. (1967). The detection of disease clustering and a generalized regression approach. Cancer Research 27, 209–220. [PubMed] [Google Scholar]
  25. Miaou S.-P., Song J. J. and Mallick B. K. (2003). Roadway traffic crash mapping: a space-time modeling approach. Journal of Transportation and Statistics 6, 33–58. [Google Scholar]
  26. Napier G., Lee D., Robertson C., Lawson A. and Pollock K. G. (2016). A model to estimate the impact of changes in MMR vaccine uptake on inequalities in measles susceptibility in Scotland. Statistical Methods in Medical Research 25, 1185–1200. [DOI] [PubMed] [Google Scholar]
  27. Newton M. A., Noueiry A., Sarkar D. and Ahlquist P. (2004). Detecting differential gene expression with a semiparametric hierarchical mixture method. Biostatistics 5, 155–176. [DOI] [PubMed] [Google Scholar]
  28. Robertson C., Nelson T. A., MacNab Y. C. and Lawson A. B. (2010). Review of methods for space–time disease surveillance. Spatial and Spatiotemporal Epidemiology 1, 105–116. [DOI] [PMC free article] [PubMed] [Google Scholar]
  29. Rushworth A., Lee D. and Mitchell R. (2014). A spatio-temporal model for estimating the long-term effects of air pollution on respiratory hospital admissions in Greater London. Spatial and Spatiotemporal Epidemiology 10, 29–38. [DOI] [PubMed] [Google Scholar]
  30. Scott P. E. (2016a). More than 400 pedestrians hurt in road accidents in Surrey last year. http://www.getsurrey.co.uk/news/surrey-news/more-400-pedestrians-hurt-road-11706152. Accessed 11 September 2017.
  31. Scott P. E. (2016b). Worrying increase in lorry and van crashes according to new figures. http://www.getsurrey.co.uk/news/surrey-news/worrying-increase-lorry-van-crashes-11819808. Accessed 11 September 2017. [Google Scholar]
  32. Song J. J., Ghosh M., Miaou S. and Mallick B. (2006). Bayesian multivariate spatial models for roadway traffic crash mapping. Journal of Multivariate Analysis 97, 246–273. [Google Scholar]
  33. Southampton City Council. (2016). Cycling Southampton: A Plan for Our City 2016-2026. https://www.southampton.gov.uk/Images/Cycling-Southampton-2016-2026-DRAFT-Nov%202016_tcm63-389640.pdf. Accessed 11 September 2017. [Google Scholar]
  34. Sun D., Tsutakawa R. K., Kim H., He Z. (2000). Spatio-temporal interaction with disease mapping. Statistics in Medicine 19, 2015–2035. [DOI] [PubMed] [Google Scholar]
  35. Surrey Police. (2015). The Increase in Fatal Road Traffic Collissions. http://www.surrey-pcc.gov.uk/wp-content/uploads/2015/01/03_The-Increase-in-Fatal-RTCs-final.pdf. Accessed 5 September 2017. [Google Scholar]
  36. Thomas A., Best N., Lunn D., Arnold R. and Spiegelhalter D. (2004). GeoBugs User Manual. Cambridge, MA: Medical Research Council Biostatistics Unit. [Google Scholar]
  37. Ventrucci M., Scott E. M. and Cocchi D. (2010). Multiple testing on standardized mortality ratios: a Bayesian hierarchical model for FDR estimation. Biostatistics 12, 51–67. [DOI] [PubMed] [Google Scholar]
  38. Wakefield J. and Kim A. (2013). A Bayesian model for cluster detection. Biostatistics 14, 752–765. [DOI] [PMC free article] [PubMed] [Google Scholar]
  39. Waller L. A., Carlin B. P., Xia H. and Gelfand A. E. (1997). Hierarchical spatio-temporal mapping of disease rates. Journal of the American Statistical Association 92, 607–617. [Google Scholar]
  40. Zhou H. and Lawson A. B. (2008). EWMA smoothing and Bayesian spatial modeling for health surveillance. Statistics in Medicine 27, 5907–5928. [DOI] [PubMed] [Google Scholar]
  41. Zou J., Karr A. F., Banks D., Heaton M., Datta G., Lynch J. and Vera F. (2011). Bayesian methodology for spatio-temporal syndromic surveillance. Statistical Analysis and Data Mining. [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

kxy038_Supplementary_Materials

Articles from Biostatistics (Oxford, England) are provided here courtesy of Oxford University Press

RESOURCES