Hierarchical Bayes models have been used in disease mapping to examine small scale geographic variation. State level geographic variation for less common causes of mortality outcomes have been reported however county level variation is rarely examined. Due to concerns about statistical reliability and confidentiality, county-level mortality rates based on fewer than 20 deaths are suppressed based on Division of Vital Statistics, National Center for Health Statistics (NCHS) statistical reliability criteria, precluding an examination of spatio-temporal variation in less common causes of mortality outcomes such as suicide rates (SRs) at the county level using direct estimates. Existing Bayesian spatio-temporal modeling strategies can be applied via Integrated Nested Laplace Approximation (INLA) in R to a large number of rare causes of mortality outcomes to enable examination of spatio-temporal variations on smaller geographic scales such as counties. This method allows examination of spatiotemporal variation across the entire U.S., even where the data are sparse. We used mortality data from 2005–2015 to explore spatiotemporal variation in SRs, as one particular application of the Bayesian spatio-temporal modeling strategy in R-INLA to predict year and county-specific SRs. Specifically, hierarchical Bayesian spatio-temporal models were implemented with spatially structured and unstructured random effects, correlated time effects, time varying confounders and space-time interaction terms in the software R-INLA, borrowing strength across both counties and years to produce smoothed county level SRs. Model-based estimates of SRs were mapped to explore geographic variation.

The use of Bayesian methods in the areas of disease mapping, epidemiology, and small area health applications is well established. The Bayesian inference combines the prior distribution on model parameters and the data likelihood to derive the posterior distribution which summarizes the behavior of the parameters in light of the observed data. (

Traditionally, Markov Chain Monte Carlo (MCMC methods) have been used to approximate the posterior marginals in Bayesian Hierarchical models and are computationally intensive and time consuming. Two basic methods, namely, Gibbs sampling and Metropolis-Hastings are designed in Winbugs software (

The INLA method has been introduced as an alternative to MCMC to approximate the posterior marginals of latent Gaussian models and significantly reduces the computation time. (

In this study, we examine existing Bayesian spatio-temporal models in the software R-INLA for the purposes of mapping less common causes of mortality outcomes on small geographic scales and consider suicide rates (SRs) at the county level as one particular application. Mapping county level estimates provides greater understanding of the trends and variability in spatio-temporal patterns of less common causes of mortality outcomes not possible by examination of direct national and state estimates (

Specifically, in a Bayesian spatio-temporal model, the spatially structured and unstructured random effects are used to model the inherent spatial autocorrelation in the data, the correlated and uncorrelated time effects model the time dependent structure of the data, time varying covariates model the extra uncertainty in the data due to measured confounders, and the space-time interaction effects model the residual spatio-temporal variation that are unaccounted for by the county and time random effects to produce reliable model based yearly county level estimates.

The posterior distributions for the parameters in Bayesian Hierarchical spatio-temporal models in this study are simulated in the software R-INLA, to reduce the computation time often incurred when analyzing large spatial datasets. A variety of prior distributions for model parameters and random effects can be specified in R-INLA. The Bayesian spatio-temporal modeling approach borrows strength across both counties and years to produce smoothed yearly county level estimates and allows examination of spatial and temporal variability in less common causes of mortality outcomes over time. This method can be applied to a large number of rare causes of mortality outcomes to examine small-scale geographic variation and temporal variability with model based smoothed and robust small area estimates. The accuracy of the INLA estimates compared to MCMC estimates have been examined in large number of study areas. (

The spatiotemporal models in R-INLA smooth the time trends by borrowing strength from adjacent times. Since 2015 is the most recent year of data that is available from the National Vital Statistics System (NVSS) files, this study incorporates the years 2005–2015 to examine the county level spatio-temporal variation in SRs using 11 years of NVSS data. The smoothed model based county level SRs are mapped and compared for the years 2005 and 2015 to examine the geographic variations and the broad scale trend in spatio-temporal patterns. The absolute difference in the increase in SRs over time is also mapped to examine the overall increase in SRs from the start of the analyses year (2005) to the end of the analyses year (2015).

Section 2 describes the general space-time model for analyzing rare causes of mortality outcomes on small geographic scales, such as counties. Section 3 describes one particular application of the proposed existing methodology to examine county level spatio-temporal variation in SRs. Specifically, Section 3.1 contains information on SRs data and the respective sources of covariates used in this study. Section 3.2 describes model and prior distribution assumptions for modeling county level SRs, and Section 3.3 discusses model selection criteria for selecting the best model for county level SRs. In Section 3.4 we discuss model accuracy. Section 3.5 discusses results with respect to model covariates, and Section 3.6 outlines the broad scale trend and variability in geographic patterns and the usefulness of the Bayesian spatio-temporal technique in mapping small area outcomes such as SRs by using the proposed existing Bayesian spatio-temporal technique in R-INLA. Sections 4 summarizes the discussion.

The hierarchical Bayes statistical models employ multiple levels of modeling specified in a hierarchical order to estimate the posterior distributions of the model parameters using the Bayes method. The observed data is combined with the multiple sub-level model specifications (prior distributions) and possible covariates to estimate the posterior distribution via Bayes theorem. The hierarchical Bayes models can be used to model to model grouped data: temporally (repeated in time) or spatially structured (exhibiting spatial autocorrelation).

The small-scale geography (e.g. county level) data for a less common cause of mortality outcome, in general, often exhibits strong spatial autocorrelation. (_{it}_{it}_{it}_{it}_{it}_{it}_{it}

logit (_{it}_{0} + _{i}_{t}_{it}_{it}

Grand intercept _{0}.

Spatial component accounting for existent spatial autocorrelation _{i}

Time component accounting for fixed and random time effects _{t}

Space-time interaction term accounting for residual spatial variation not accounted for by the main time and space effects _{it.}

Covariates which can be time varying or time-invariant _{it}_{it}

The posterior distributions of the parameters in the hierarchical Bayesian model can be estimated via Integrated Nested Laplace Approximation (INLA) in R, borrowing strength across both counties and years to produce smoothed yearly county level estimates even where the data are sparse. Depending on the nature of the data, a variety of latent models such as random walk-1, random walk-2, besag, convolution etc. can be implemented via R-INLA software package to model the small area outcome and produce reliable smoothed estimates. (

Data were obtained from the 2005–2015 National Vital Statistics System (NVSS) Multiple Cause of Death Files (restricted-use geography files). (

Data on time-varying county-level characteristics were obtained from several sources, including Area Health Resource Files, (

Geographic boundaries for some counties changed during the study period. To provide constancy in the total number of counties during the study period (2005–2015), several counties in Alaska were aggregated and Bedford City, VA was merged with Bedford County, VA, resulting in a combined national file that included 3,140 counties (

The total numbers of crude counts and percentages of the numbers of suicides equal to zero, less than 10, and less than 20 that were extracted from the NVSS files are shown in

logit (_{it}_{0} + _{i}_{i}_{1}_{t}_{2}_{t}_{it}_{it}

logit link function log (_{it}_{it}_{it}

an overall intercept term _{0}. The intercept, _{0} was assigned a flat prior: P(_{0}) ∝ constant, (where, P indicates probability).

_{it}_{it}_{it}

the spatial effects, _{i}

where, _{u}_{i}_{δi} is the number of neighbours,
_{ij}_{u}

non-spatial random effects _{i}_{v}_{v}

correlated random time effects, _{1}_{t}_{1}_{t}_{φ}_{1}; where
_{φ}_{1} ~ Gamma (1, 0.001) prior.

an uncorrelated time dependent random effect _{2}_{t}_{φ}_{2};
_{φ}_{2} ~ Gamma (1, 0.001) prior.

the space time interaction term, _{it}_{ψ}

(The precisions for the intercept, fixed effects and the random effects are assigned priors that are default in R-INLA. INLA assigns log (precisions) ~log-gamma (1, 0.001) priors) (

A set of models following the above general space time modeling approach were explored to determine the contribution of different components, namely, the correlated and uncorrelated random time effects, spatially structured and unstructured random effects, space time interaction term and the different covariates to examine spatio-temporal variation in county level SRs. Alternative models such as proper CAR, Besag proper, and (ZIP) were also explored but did not provide any improvement in model fit, as assessed using the Deviance Information Criterion (DIC). (

Model fit was evaluated using the Deviance Information Criterion (DIC) with lower values indicating better fit. For context, a DIC difference of 3–5 is considered significant. (

The county level SR estimates from the best model and more parsimonious models (no covariates or only a subset of statistically significant covariates) were highly correlated (R2 from 0.88–0.99, see

The estimated marginals of the coefficients of the fixed effects and the estimated marginals of the precisions of the prior variances for all the random components from the best model were also checked for convergence. _{i}_{i}_{it}_{1}_{t}_{1}_{t}

Residual analysis was conducted to compare the state level direct estimates with the aggregated state level model-based estimates to check the model accuracy and performance. The county-level model-based posterior predictions for each year were summed by state, weighted by county population size as a proportion of state population size, to calculate the state-level model-based estimates. The comparison of the state-level directly estimated SRs and the aggregated model-based state-level SRs for the best model for different years is shown in

As an additional model check, the shrinkage between the direct state-level SRs and the aggregated model-based state-level SRs is plotted and can be seen in

Sensitivity analysis was conducted to compare the Delaunay triangulation and sphere of influence spatial weighting schemes. The estimated county level SRs and the associated posterior standard deviations from the two spatial weighting schemes were highly correlated (R2 =0.99) as seen in

The inclusion of covariates can enhance the predictive power of small area estimation models. (

The actual number of deaths due to suicide at the county level for the year 2015 (number of deaths less than 20 are suppressed) are shown in

In 2005 and 2015, counties with the highest model-based SRs were predominantly located across the western US while the lowest rates were observed across southern California, western Texas, along the Mississippi river, and in areas along the East Coast. These patterns were largely consistent over time. Multiple studies have described state-level variation in suicide rates (SRs), with higher rates noted in Western states lending credibility to the model based county level SRs. (

The absolute differences in the model based county level SRs in the U.S. from 2005–2015 are shown in

County-level direct estimates of less common mortality outcomes are often highly unstable. Many prior studies on county-level variations in less common causes of mortality outcomes have relied on estimates aggregated over time or larger geographic areas. However, this type of aggregation precludes the examination of detailed temporal and spatial trends. To overcome these limitations, this study uses hierarchical Bayesian methods to generate robust model-based estimates of yearly county-level SRs to examine the spatio-temporal variations across a span of 11 years.

This study contributed to the existing literature by applying an existing methodology, namely hierarchical Bayesian spatio-temporal models in R-INLA to estimate county-level SRs in order to examine spatiotemporal variation in SRs. Although there are a variety of alternative models with different assumptions that we did not explicitly explore, this study is the first to incorporate spatiotemporal random effects along with time varying confounders to estimate annual county level estimates for SRs for the years 2005–2015. There was substantial geographic variation in SRs. The majority of the counties across the U.S. demonstrated an increase in suicide death rates over this time period and no counties exhibited a decline. The existing Bayesian spatio-temporal modeling techniques in R-INLA can potentially be applied to a large number of rare causes of mortality-related outcomes from vital statistics data to examine geographic and temporal variation.

The use of R-INLA method resulted in substantially reduced computation time for this study, an average of twenty four for the best model with a full set of covariates and six to twenty four hours for models with no or few covariates, as opposed to weeks of time required for simulations based Markov Chain Monte Carlo (MCMC) via WINBUGS with large spatial datasets. (

The model fit was examined via DIC comparisons. Amongst the models fitted, the contribution of different space and time components was examined by the subsequent reduction in DIC values and the effective number of parameters to estimate. The best-fitting model captured the spatial autocorrelation and the time dependence structure of the data and was further improved by using time varying covariates accounting for the extra variability that was not captured by the main time and county effects. The best fitting model was found to have the lowest DIC with small number of effective parameters to estimate as compared to the model without time varying covariates. However, the temporal random effect was found to be an autoregressive process of order 1 which dampens out after a certain period of time. This suggests that future analyses might not require too long of a stretch of data in time in order to compute stable county level SRs. The comparison of state-level directly estimated SRs and the aggregated model-based state-level SRs for the different years showed that the majority of estimates fell on the line of equality indicating a close correspondence between the model-based state SRs and the direct state SRs at larger geographic scales.

The limitations of this study are as follows. Although a large number of models were implemented, alternative models incorporating different covariates from other data sources and space and time components might have improved the predictions. Secondly, R-INLA software can implement a variety of traditional models that are built-in, however there are a class of models such as latent mixture models that still need to be implemented. Moreover, the prior specifications that are not built-in in R-INLA need to be programmed. Thirdly, this study incorporated a large number of covariates to account for measured covariates, however suicides rates vary by gender and age groups and future studies can look at suicide rates by these mechanisms. Lastly, there is underreporting in suicides numbers and the actual number of suicides are always larger than the reported. Underreporting and measurement errors in suicides cannot be understated and have been studied in the literature (

Future research exploring spatial clustering of less common causes of mortality outcomes over time, including at sub-county levels, would provide further understanding of how the small-scale geographic variation may be spatially patterned across the U.S. Lastly, the R-INLA package has provided a new, flexible and substantially faster alternative to MCMC methods.

Area Health Resource Files (2015). US Department of Health and Human Services, Health Resources and Services Administration, Bureau of Health Workforce. Area Health Resources Files (Rockville, MD).

Comparison of state-level direct estimates (y-axis) and model-based estimates (x-axis), by year.

Shrinkage of suicide rates for each state, by population size for 2015. Crude death rates are plotted at the start of the arrows, and model-based death rates are located at the end of the arrows. Shrinkage is greater in states with smaller populations (left side of the chart) and more extreme suicide rates.

Crude county level deaths due to suicides for the year 2015. Number of deaths less than 20 are suppressed.

Predicted county-level suicide death rates in 2005 (top) and 2015 (bottom).

Absolute differences in model based county level suicide rates in the U.S. from 2005–2015. (The legend corresponds to the increase in suicide number of deaths per 100,000)

Spatially structured random effect representing correlated heterogeneity in suicide rates across U.S. counties.

Spatially unstructured random effect representing uncorrelated heterogeneity in suicide rates across U.S. counties.

Counts and percentages for numbers of suicides extracted from the NVSS data files reported to be equal to 0, less than 10, and less than 20 for years 2005, 2009, and 2015 respectively.

Equal to 0 | Less than 10 | Less than 20 | ||||
---|---|---|---|---|---|---|

| ||||||

Year | Count | Percent | Count | Percent | Count | Percent |

2005 | 475 | 15.12 | 2405 | 76.59 | 2775 | 88.37 |

2009 | 427 | 13.6 | 2349 | 74.8 | 2716 | 86.5 |

2015 | 360 | 11.5 | 2186 | 69.6 | 2646 | 84.3 |

Alternative Model Specification and Fit Statistics

Terms: _{0} represents the intercept or grand mean; _{i}_{i} | |||
---|---|---|---|

Model | Components | DIC | n.eff |

1. Simple random effects, _{i} | _{0} + _{i} | 150371.4 | 2316 |

2. Spatial _{i}_{i} | _{0} + _{i}_{i} | 149966.2 | 2316 |

3. Correlated time effects, _{1}_{t} | _{0} + _{i}_{i}_{1}_{t} | 148008.6 | 1884 |

4. Uncorrelated time effects, _{2}_{t} | _{0} + _{i}_{i}_{2}_{t} | 148010.3 | 1886 |

5. Space time interaction term, _{it} | _{0} + _{i}_{i}_{2}_{t}_{it} | 147821.9 | 2766 |

6. All components and covariates | _{0} + _{i}_{i}_{1}_{t}_{it}_{it} | 147181.1 | 1896 |

_{1}_{t}_{2}_{t}_{it}_{it}

Model hyperparameters: posterior mean, posterior standard deviation, 95% Bayesian credible intervals and posterior mode for the estimated marginals of the precisions of the prior variances of non-spatial effects, _{v}_{u}_{φ}_{1} and iid space time interaction effect, _{ψ}

Precisions | Posterior mean | Posterior standard deviation | 0.025 quantile | 0.975 quantile | Posterior mode |
---|---|---|---|---|---|

_{v} | 22380.75 | 2009.316 | 2523.08 | 75191.57 | 7201.39 |

_{u} | 21.27 | 1.439 | 18.85 | 24.46 | 20.68 |

_{φ}_{1} | 5935.05 | 3180.924 | 2062.85 | 14120.89 | 4046.36 |

_{ψ} | 461.13 | 78.305 | 335.38 | 641.15 | 428.46 |