A multivariate spatio-temporal model of the opioid epidemic in Ohio: A factor model approach

Abstract

Opioid misuse is a significant public health issue and a national epidemic with a high prevalence of associated morbidity and mortality. The epidemic is particularly severe in Ohio which has some of the highest overdose rates in the country. It is important to understand spatial and temporal trends of the opioid epidemic to learn more about areas that are most affected and to inform potential community interventions and resource allocation. We propose a multivariate spatio-temporal model to leverage existing surveillance measures, opioid-associated deaths and treatment admissions, to learn about the underlying epidemic for counties in Ohio. We do this using a temporally varying spatial factor that synthesizes information from both counts to estimate common underlying risk which we interpret as the burden of the epidemic. We demonstrate the use of this model with county-level data from 2007-2018 in Ohio. Through our model estimates, we identify counties with above and below average burden and examine how those regions have shifted over time given overall statewide trends. Specifically, we highlight the sustained above average burden of the opioid epidemic on southern Ohio throughout the 12 years examined.

1. Introduction

The opioid epidemic in the United States is associated with unprecedented morbidity and mortality (Rudd et al., 2016a; Zibbell et al., 2015) and presents the most significant drug-related threat to the United States (Drug Enforcement Agency, 2016). From 2000 to 2014, drug overdose deaths increased in the United States by 200% (Rudd et al., 2016b). In 2017, drug overdose became the leading cause of injury-related death in the United States (Centers for Disease Control and Prevention, 2019). It was nationally estimated in 2018 that 10.3 million people aged 12 and older had misused opioids in the past year, which reflects 3.7% of the United States population (SAMHSA, 2019).

The opioid epidemic has hit the state of Ohio particularly hard. In 2017, Ohio had the second highest rate of overdose in the country, roughly double the national average (Division of Unintentional Injury Prevention, 2017; Scholl et al., 2019). In fact, the rate of overdose death in Ohio increased over 1,000% from 2000 to 2017 (Ohio Department of Health, Bureau of Vital Statistics, 2019). While the initial wave of the epidemic was largely tied to prescription pills, more recently overdose has been driven by the introduction of fentanyl (Daniulaityte et al., 2017).

Despite being a major public health problem, it is challenging to quantify the burden of the opioid epidemic on a local level. Here, we use local to refer to counties since that most closely reflects the system of local health districts in Ohio. Local estimates are critical for assisting policymakers with the allocation of resources and services to combat the epidemic. While surveys may provide the most direct information, they are rarely designed to generate estimates in small areas, often lag several years behind, and can be subject to reporting biases (Palamar et al., 2016). Alternatively, public health surveillance measures, in this case overdose deaths and treatment admissions, are routinely collected by health departments at the local level. While providing a finer spatial resolution, surveillance measures fail to directly capture facets of the epidemic like misuse and morbidity. In addition, individual surveillance measures reflect an incomplete and imperfect understanding of the full scope of the problem as they only count individuals who can be detected and reported by the system. Thus, we intend to leverage the joint information in overdose deaths and treatment admissions to gain a fuller understanding of the relative burden in counties across the state over the past 12 years.

To do so, we will extend the cross-sectional framework of Hepler et al. (2019) to a longitudinal setting. In Hepler et al. (2019), a spatial factor model (Wall and Wang, 2003) was used to estimate the county-level latent opioid burden at a single point in time given observed counts on overdose deaths and treatment admissions. This framework assumes a common, unobserved underlying process drives both surveillance outcomes. Since each outcome has different detection and reporting processes, they each contribute unique information about the true hidden process of interest. Here, we will make the same key assumption but will estimate temporally varying latent factors to assess the annual county-level burden for each of the 12 years studied. This will enable us to assess which counties exhibit above and below average burdens for each of the 12 years based on overdose deaths and treatment admissions. This information can be used to help gauge the relative impact of the epidemic on each county and serve as a guide for decision-makers as they target counties for interventions and resources. Thus, our main goal in this paper is estimation and inference related to the latent burden of the opioid epidemic.

The rest of the paper is organized as follows. We describe the methods used in Section 2. The main results from the analysis are presented in Section 3. We conclude in Section 4 with a discussion of the findings.

2. Methods

2.1. Data

Our goal is to jointly analyze two markers of county-level opioid misuse over the past 12 years in Ohio. Specifically, we will analyze annual counts of opioid-associated deaths and treatment admissions for opioid misuse from 2007-2018, which is the most recent available year. Both outcomes are routinely monitored surveillance measures collected for all 88 counties by agencies of the state of Ohio. We elect to use county-level data for two reasons. First and foremost, health districts in Ohio generally reflect county boundaries and are a policy relevant unit for considering the allocation of resources and delivery of services. Second and more practically, counties are a unit for which data are available with minimal suppression due to small counts.

Mortality data are publicly available in the Ohio Public Health Data Warehouse Ohio Resident Mortality Data (http://publicapps.odh.ohio.gov/EDW/DataCatalog), which is a resource compiled by the Ohio Department of Health. We define an opioid-associated death as any death of an Ohio resident where the death certificate mentions poisoning from any opioid. This is defined by International Classification of Disease (ICD)-10 multiple cause codes T40.0-T40.4 and T40.6. All deaths are indexed to the county of residence of the decedent. The observed death rates per 100,000 over this time period are shown in Supplemental Figure 1. Population estimates used to calculate rates were also obtained from the Ohio Department of Health Data Warehouse and were produced by the National Center for Health Statistics.

We obtained treatment admission counts through a data use agreement with the Ohio Department of Mental Health and Drug Addiction Services. Treatment admissions include any residential, intensive outpatient, or outpatient treatment for opioid misuse and were identified through the diagnostic codes listed in Supplemental Table 1. The treatment admissions count does not include patient visits to hospitals or medical facilities to receive treatment for overdose or complications from opioid misuse. As with death counts, treatment counts are indexed to the patient’s county of residence. Data were provided in two age groups: adolescent (under 20 years of age) and adult (20 years of age and older), but for this analysis we will only consider the total number of treatment admissions for all ages in each county. State policy requires that treatment counts between 1 and 9 be suppressed and so those counties will be treated as censored in the analysis. This primarily impacts the counts for the adolescents, which subsequently impacts the total. The observed treatment rates per 100,000 over this time period and censoring indicators are shown in Supplemental Figure 1. For exploratory purposes, counties with adolescent counts suppressed show the rate for the adults, and counties with both counts censored are gray. In addition, the scatterplot in Supplemental Figure 2 shows the correlation between the death and treatment rates in each county for each year.

2.2. Model

The motivation for our model is to reflect the general process through which the surveillance outcomes are generated and thus learn about that hidden underlying process that drives opioid-associated deaths and treatment admissions. To do so, we recognize that both opioid-associated deaths and treatment admissions are proxies for the underlying burden of opioid misuse in a county. While neither exactly reflects the underlying burden, each outcome provides indirect information about the burden of misuse. Using a spatial factor model (Wall and Wang, 2003), we can exploit this structure to estimate the hidden burden, quantified through the spatial factor, and make relative comparisons across counties. This work extends the spatial factor model of Hepler et al. (2019) to a longitudinal spatial factor model that enables estimation of the relative burden for each year.

The data model is based on a generalized common spatial factor model for bivariate Poisson outcomes. Let $Y_{it}^D$ be the count of deaths and $Y_{it}^T$ be the count of treatment admissions for county i in year t. Then for county i = 1, …, 88 and year t = 1, …, 12 for years 2007-2018, let:

$$Y_{it}^D\mid {\lambda }_{it}^D\stackrel{ind}{\sim }\text{Poisson}({\lambda }_{it}^D)$$ (1)

$$Y_{it}^T\mid {\lambda }_{it}^T\stackrel{ind}{\sim }\text{Poisson}({\lambda }_{it}^T),$$ (2)

where ${\lambda }_{it}^D$ is the mean death count and ${\lambda }_{it}^T$ is the mean treatment count for county i in year t. Note that we assume conditional independence of the death and treatment counts within a county and across time given ${\lambda }_{it}^D$ and ${\lambda }_{it}^T$ so that all dependence is captured through the mean structure.

Using the canonical log link function, we specify a multivariate generalized linear mixed effects model for the means in county i at time t. That is,

$$\log ({\lambda }_{it}^D)=O_{it}+{\mu }_t^D+{\alpha }_{it}{\nu }_{it}+{ϵ}_{it}^D$$ (3)

$$\log ({\lambda }_{it}^T)=O_{it}+{\mu }_t^T+{\nu }_{it}+{ϵ}_{it}^T,$$ (4)

where ${\mu }_t^D$ and ${\mu }_t^T$ are state-level intercepts at time t, ν_it is the latent spatial factor which is shared across outcomes, α_it is a spatially-varying factor loading, and ${ϵ}_{it}^D$ and ${ϵ}_{it}^T$ are independent error terms to account for outcome specific variation. We let O_it be an offset such that O_it = log(P_it) where P_it is the population of county i in year t. We assume ${ϵ}_{it}^D\stackrel{iid}{\sim }N(0,{\sigma }_D^2)$ and ${ϵ}_{it}^T\stackrel{iid}{\sim }N(0,{\sigma }_T^2)$.

As was mentioned in Section 2.1, the treatment counts are suppressed if they are between 1 and 9. However, we can incorporate the knowledge that suppressed counts are between 1 and 10 into our modeling by adapting the approach of Famoye and Wang (2004) for interval censoring. There are three cases of censoring that we encounter here, denoted by the indicator:

$$c_{it}=\{\begin{array}{cc}0\hfill & \text{total count observed}\hfill \\ \hfill & \hfill \\ 1\hfill & \text{adolescent count censored}\hfill \\ \hfill & \hfill \\ 2\hfill & \text{adolescent and adult counts censored}.\hfill \end{array}\phantom{\}}$$ (5)

For c_it = 0, we observe both the adult and adolescent counts and so the total treatment count is their sum. For c_it = 1, we observe the adult count, but the adolescent count is censored so we know that the total count must be between the adult count plus 1 and the adult count plus 9. Finally, for c_it = 2, both counts are censored so we know the total count must be between 2 and 18. Let $Y_{it0}^T$ be the adolescent treatment count and $Y_{it1}^T$ be the adult treatment count. Then, we have

$$Y_{it}^T=\{\begin{array}{cc}{Y}_{it0}^T+Y_{it1}^T\hfill & c_{it}=0\hfill \\ \hfill & \hfill \\ Y_{it1}^T\hfill & c_{it}=1\hfill \\ \hfill & \hfill \\ 0\hfill & c_{it}=2,\hfill \end{array}\phantom{\}}$$ (6)

and the likelihood from Equation 1 becomes

$$\begin{gathered} L({\lambda }^T,{\lambda }^D\mid {\mathbf{Y}}^T,{\mathbf{Y}}^D)=\left[\prod _{t=1}^{12}\prod _{i=1}^{88}f(Y_{it}^D\mid {\lambda }_{it}^D)\right]\times \\ \times [\prod _{t=1}^{12}\prod _{i=1}^{88}[f(Y_{it}^T\mid {\lambda }_{it}^T){]}^{I(c_{it}=0)}[F(Y_{it}^T+9\mid {\lambda }_{it}^T)-F(Y_{it}^T\mid {\lambda }_{it}^T){]}^{I(c_{it}=1)}\phantom{]} \\ \times \phantom{[}[F(18\mid {\lambda }_{it}^T)-F(1\mid {\lambda }_{it}^T){]}^{I(c_{it}=2)}] \end{gathered}$$ (7)

where $F(\cdot \mid {\lambda }_{it}^T)$ is the cumulative distribution function and $f(\cdot \mid {\lambda }_{it}^T)$ is the probability mass function of the Poisson distribution with mean ${\lambda }_{it}^T$ and $f(\cdot \mid {\lambda }_{it}^D)$ is the probability mass function of the Poisson distribution with mean ${\lambda }_{it}^D$.

Our main inferential interest lies in the latent spatial factors. The spatial factors represent a hidden, underlying process that is common to both deaths and treatment admissions. In this case, we believe the factor reflects the burden of opioid misuse since both outcomes are observable consequences of opioid misuse. Similar to Hepler et al. (2019), we will assume the spatial factor follows an intrinsic conditional autoregressive (ICAR) model (Banerjee et al., 2004; Besag, 1974). To incorporate temporal dependence, we introduce an autoregressive of order 1 (AR(1)) structure to reflect dependence with the factor of the previous year. That is, for t = 1 we conditionally let

$${\nu }_{it}\mid {\nu }_{-it}\sim N\left(\sum _j\frac{w_{ij}}{w_{i+}}{\nu }_{jt},\frac{{\tau }_v^2}{w_{i+}}\right)$$ (8)

and for t = 2, …, 12 we let

$${\nu }_{it}\mid {\nu }_{-it},{\nu }_{i(t-1)}\sim N\left(\eta {\nu }_{i(t-1)}+\sum _j\frac{w_{ij}}{w_{i+}}({\nu }_{jt}-\eta {\nu }_{j(t-1)}),\frac{{\tau }_v^2}{w_{i+}}\right)$$ (9)

where ν_−it is the vector of factors excluding county i at time t, and ν is an autoregressive parameter. We define counties to be neighbors if they are adjacent. Let w_ij be an indicator of if counties i and j are neighbors, and let w_i+ be the number of neighbors for county i.

The ICAR model is an improper probability distribution; however, it is a valid process model when appropriate centering constraints are enforced (Banerjee et al., 2004). Here, we enforce the constraint, ∑_i ν_it = 0, for each year, t, which centers the spatial factors within each year. Choosing this centering constraint also sets the interpretation of the latent factor. Here we interpret the factor as the burden within a county in a given year relative to the yearly state average for each outcome. Thus, conditional on the state average death and treatment rates, we are able to estimate counties who have relatively better or worse experiences for that year. This can allow us to identify counties where additional intervention may be needed and to examine shifts in regions that may have elevated burden. We note that this interpretation is limited to within a year and the factor estimates do not provide a meaningful longitudinal interpretation under these constraints. Thus, the temporal autocorrelation is incorporated to simply reflect the belief that if a county is above average in one year, it is likely to remain above average the following year.

As in Neeley et al. (2014), we let the factor loadings in the model vary spatially. Following similar notation as above, we assume an ICAR model for α_it. That is,

$${\alpha }_{it}\mid {\alpha }_{-it}\sim N\left(\alpha +\frac{1}{w_{i+}}\sum _{j\sim i}({\alpha }_{jt}-\alpha ),\frac{{\tau }_{\alpha }^2}{w_{i+}}\right)$$ (10)

where α is the mean loading across all years and counties. This implies that the loadings are centered on a common mean but exhibit spatial variation about that mean for each year. Note that we have implicitly assumed the factor loading for treatment in Equation (1) to be 1 for identifiability. Since we are using an ICAR model, we must also enforce a centering constraint within each year to ensure a proper posterior (Banerjee et al., 2004). For the loadings, we force ∑_i α_it = α. Mathematically, spatial loadings allow the covariance between the outcomes to vary across space and appropriately rescale the factor for the death outcome. This added flexibility is important because different regions may have different environments that affect the joint relationship between death and treatment rates. By subtracting α from α_it, we can interpret deviations from the average loading to learn about heterogeneity in the joint association, which highlight counties where death rates may be given more or less weight to the factor than average.

Since we are fitting the model within the Bayesian paradigm, we must specify prior distributions on all model parameters. For all mean parameters, ${\mu }_t^D$ and ${\mu }_t^T$, we assign uniform priors on the real line. For variance parameters, ${\sigma }_D^2$, ${\sigma }_T^2$, ${\tau }_v^2$, and ${\tau }_a^2$, we assign inverse gamma distributions with shape and scale parameters equal to 0.5. We let the autoregressive parameter η have a uniform distribution on the interval (0,1). For the mean factor loading, α, we assign a Gaussian prior with mean 1 and variance 2, which reflects a prior belief that each outcome contributes equally to the factor.

The model was fit using the R package NIMBLE (de Valpine et al., 2017), which implements an adaptive Metropolis-within-Gibbs Markov chain Monte Carlo (MCMC) algorithm. The MCMC sampler was run for 650,000 iterations, discarding the first 150,000 as burn-in and thinned by keeping every 50th iteration. Convergence was assessed visually using trace plots and code is provided in the Supplemental Material.

3. Results

In Figure 1, we show the posterior mean state intercepts for each year, ${\mu }_t^D$ and ${\mu }_t^T$. Note that these estimates are intercepts on the natural log scale. We notice that while scale of the treatment rates is larger than the death rates, both are increasing with time. In 2018, we observe a slight reduction in the state average death rate. While the treatment rate is consistently increasing over time, we observe an immediate upward shift in the general trend after 2014. This happens to correspond with Medicaid expansion in Ohio and likely reflects the general increased accessibility of treatment (Governor’s Cabinet Opiate Action Team, 2018).

Fig. 1.

Posterior mean state intercepts for death and treatment rates

Posterior mean estimates for the spatial factors for each year are shown in Figure 2. Recall that the factors are estimated conditional on the state rates for each outcome and so must be interpreted in light of the overall increasing trends as shown in Figure 1. In general, we see fairly consistent spatial patterns with southern and southwestern Ohio being consistently above average. These regions correspond to parts of the Appalachian region of Ohio and the Cincinnati area, respectively. During the later half of the period examined, we also see above average factors in the northeastern corner of the state. We also see parts of eastern and northwestern Ohio that are consistently below average. We further illustrate annual county comparisons with plots of the posterior means and 95% credible intervals in Figure 3. In Figure 3, we highlight estimates of the spatial factor for Franklin County (Columbus), Montgomery County (Dayton), and Ross County (rural) to illustrate changes in the relative burden over time. Full results for all counties and years are shown in the Supplemental Material ordered by the P(ν_it > 0) and colored by region. Through these plots, it is apparent that the relative burden is consistently highest in southeastern and southwestern Ohio counties. In particular, Scioto County has the highest estimated burden in each of the 12 years.

Fig. 2.

Posterior mean estimate of the factor and posterior probability of counties being above average (i.e., a factor greater than 0), 2007-2018.

Fig. 3.

Posterior mean factor and 95% credible intervals for selected counties and years.

In Figure 2, we also show the posterior probability that the factor estimate is greater than 0. For counties with probabilities close to 1, we have evidence that they have an above average burden, and for counties with probabilities close to 0, we have evidence they are below average. These probabilities follow the same general pattern as described for the factors with above average counties becoming more concentrated in later years on the eastern and southern borders of the state.

We can also detect informative patterns by looking at estimates of outcome specific variation in Figure 4. In general, we do not see many consistent patterns in the treatment error estimates, ${ϵ}_{it}^T$. We do see patterns emerging from the death error estimates, ${ϵ}_{it}^D$, reflecting variation not captured by the shared structure. Across the entire time period, we see higher death rates in both Cincinnati (southwest) and Cleveland (northeast). Particularly starting in 2012, we observe higher death rates in the southwestern and northeastern regions of the state. We also see elevated death rates in central Ohio, around Columbus, starting in 2014. This overdose death specific variation is likely attributable to the rise of fentanyl starting in 2013, particularly in and around the large cities of the state. This information highlights regions that could benefit from additional harm reduction interventions to specifically address overdose death.

Fig. 4.

Posterior mean estimates of error, ${ϵ}_{it}^D$ and ${ϵ}_{it}^T$, 2007-2018.

The other key parameter in our model is the factor loading, α_it. The posterior mean for the mean loading, α, is 0.61 with 95% credible interval (0.54, 0.69). This reflects that the amount of variation in the death rates is smaller than that of the treatment rates. In Supplemental Figure 17, we show the posterior mean of the county deviation from average, α_it – α, for each year. An estimate of 0 indicates that the contribution of death and treatment rates to the factor are roughly equal to the average for the state. Estimates less than 0 indicate areas where treatment rates are more influential than average, and estimates greater than 0 indicate areas where death rates are more influential than average. In general, we see a disproportionate influence of treatment rates in southern and to a lesser extent eastern Ohio. In southern Ohio, this is likely attributable to early interventions by the state to expand treatment resources in the region (Governor’s Cabinet Opiate Action Team, 2018). We also see a slightly stronger influence of death rates in western Ohio.

4. Discussion

In this paper, we extended the spatial factor modeling approach for quantifying the opioid epidemic of Hepler et al. (2019) to include longitudinal data. To do so, we used a generalized spatial factor model with multivariate Poisson generalized mixed effects models. We incorporated an ICAR model to account for spatial autocorrelation and an AR(1) structure to account for temporal autocorrelation. State-level mean trends were accounted for in overdose deaths and treatment admissions so the spatial factor was estimated residual to global state trends. This enables relative comparisons of counties within each year to assess areas with above and below average burden of the opioid epidemic.

Based on our model, we observed general increasing trends over time for rates of opioid associated overdose death and treatment admission. We estimated relatively stable spatial patterns of the latent burden with the highest levels estimated to be in southern and eastern Ohio. This is largely the Appalachian region of the state and is widely considered to be the epicenter of the epidemic. In addition, we noted additional risk of overdose death in southwestern and northeastern Ohio, which suggests potential need for additional intervention. We also observed additional overdose death risk around Ohio’s major cities, particularly since the proliferation of fentanyl around 2013.

While the factor estimates cannot be directly interpreted longitudinally, they provide annual assessments of counties with above and below average burden relative to the common trends in the state. This information can be useful to public health officials as they work to identify areas where the relative need is higher so that interventions and resources can be allocated to try to mitigate the epidemic. In addition, by looking at the snapshots of relative burden over time, we can assess how areas of need may or may not have shifted over time. While we observe some changes in the above and below average counties across time, we also see that the burden is consistently above average in the southeastern and southwestern parts of the state, reflecting the ongoing urgency of the opioid misuse in that region.

There are several limitations to this work. We are using surveillance data which we assume to be correctly reported. We know that this may not always be the case, particulary with the classification of overdose deaths (Slavova et al., 2015). However, by leveraging multiple surveillance outcomes to estimate the burden, we hope to mitigate some of the error that may be specific to either individual outcome. We could further mitigate this by extending the model to include additional data sources like hospitalizations. We have also constructed our model to estimate factors that must be interpreted within a given year so assessments of county-level change from one year to the next are not meaningful. Our analysis was also conducted entirely at the county-level and so all interpretations must be made at the county-level to avoid the ecological fallacy (Pianntadosi et al., 1988). Our current work is of limited utility to local health departments for whom finer spatial resolution would be desired. Future work should explore ways to incorporate data sources that may be collected at finer spatial scales.

In conclusion, we have extended the use of methodology for spatial factor models to longitudinally assess the annual relative county-level burden of the opioid epidemic in Ohio. By incorporating multiple surveillance outcomes, we can more fully characterize the impact of the epidemic on counties. These estimates can provide useful information to policymakers as they work to effectively allocate interventions and resources to counties exhibiting the most relative need.