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date: 19 August 2018

# The Spatial Dimension of Health Systems

## Summary and Keywords

The spatial dimension of supply and demand factors is a very important feature of healthcare systems. Differences in health and behavior across individuals are due not only to personal characteristics but also to external forces, such as contextual factors, social interaction processes, and global health shocks. These factors are responsible for various forms of spatial patterns and correlation often observed in the data, which are desirable to include in health econometrics models.

This article describes a set of exploratory techniques and econometric methods to visualize, summarize, test, and model spatial patterns of health economics phenomena, showing their scientific and policy power when addressing health economics issues characterized by a strong spatial dimension. Exploring and modeling the spatial dimension of the two-sided healthcare provision may help reduce inequalities in access to healthcare services and support policymakers in the design of financially sustainable healthcare systems.

# Introduction

Differences across individuals, for example in their health status, are not only due to personal characteristics, which constitute an intrinsic natural endowment, but also to collective forces external to individuals, such as contextual factors, social interaction processes, and global shocks.

In the last decades, there has been a growing empirical literature on the role of contextual effects in determining health connected behaviors and outcomes (Hogan & Kitagawa, 1985; Macintyre, 1997; Harding, 2003; Jokela, 2014). Contextual and compositional factors, such as the natural and socioeconomic environment of the place where people live, have a direct impact on their health, as well as an indirect effect, by shaping individual preferences and behavior (Macintyre, 1997). In health economics, this perspective associates such external forces to the notion of neighborhood, which is an aggregation of individuals on the basis of a set of socioeconomic characteristics, usually coinciding with a geographical area (Dietz, 2002). Contextual factors affect the health needs of a population across a wide geographical area, causing geographical concentration of needs and risk factors that is often detected in the data. Such observation has encouraged public health research to propose disease mapping approaches to explore spatiotemporal pattern in disease risk and identify high-risk subregions for focused action and surveillance of disease outbreaks (Aregay, Lawson, Faes, & Kirby, 2017).

Traditional models in health economics usually view agents (e.g., individuals, patients, GPs, and hospital managers) as isolated entities, ignoring the existence of social relations and interdependence. However, it is clear that agents are rooted into networks of relations that provide opportunities and constraints, such as information flows, and the provision and enforcement of norms (Manski, 1993; Brock & Durlauf, 2001). The set of interactions between individuals or groups of individuals determines a social structure (social network) that has profound effects on individuals regarding a variety of decisions concerning their health, such as the diet to follow, smoke or alcohol consumption, which healthcare provider to use, etc. (Moscone, Tosetti, & Vittadini, 2011). Social cohesion within the neighborhoods may shape the strength of interaction and social structure of the neighborhood, ultimately affecting disease prevalence and needs (e.g., Sampson, Raudenbush, & Earls, 1997; Macintyre, 1997; Cohen, Finch, Bower, & Sastry, 2006). Networks may also exist among healthcare providers that allow exchange of information and ideas through professional contacts, for example leading to diffusion of medical technologies, standards of practice, and organization of services (Coleman, Katz, & Menzel, 1966; Moscone, Knapp, & Tosetti, 2007). Interaction processes have also been recently hypothesized for several other phenomena in health economics, such as local competition among hospitals in quality and efficiency (Gravelle, Moscelli, Santos, & Siciliani, 2016).1 The geographical concentration of risk factors and needs and network effects, as well as mechanisms of social interaction among individuals, can lead to an emergent collective behavior that empirically translates into a structure of correlation in the data, also known as spatial dependence (Anselin, 2010).

Interdependence between economic agents not only occurs through local, contextual factors and social interaction, but may also arise when agents react in a similar manner to external forces and unanticipated events, such as technological advances, health shocks, the implementation of new health policies, and sociological structural changes (Andrews, 2005). For example, innovations in medical devices and therapies such as a new vaccine might render treatable health conditions that were incurable in the past. Epidemics or diseases whose incidence suddenly increases regionally or worldwide might generate a risk-adverse behavior that translates into the accumulation of drugs and increase of pharmaceutical expenditure. The implementation of new health policies, such as campaigns through the media on highway regulations, may reduce avoidable accidents, ultimately impacting on the costs of the health system. These shocks are often unobservable to the econometrician and perturb the health system as a whole, simultaneously affecting the behavior of agents (recipients, providers, etc.), ultimately impacting on health costs. Correlation arises because the responses to such common perturbations is similar—though not identical—across individuals (Moscone & Tosetti, 2010). This combination of common unanticipated events impacts micro-level population units and is often responsible for observable co-movements of a large number of health economics time series, such as mortality and morbidity.

This article provides an overview of the importance of spatial dependence and global shocks2 in health economics and health policy–related research. It is directed towards health economists, graduate students, scholars, policymakers, and more generally researchers at any level who are interested in understanding the spatial dimension of healthcare systems. It provides a platform to introduce the reader to a set of exploratory techniques and econometric methods to visualize, summarize, test, and model spatial patterns of health economics phenomena, and show their scientific and policy power when addressing health economics issues characterized by a strong spatial dimension. The article first reviews methods and applications that assume the structure of the network in interaction processes is known a priori by the investigator, usually exploiting information on geographical distance or group membership. Hence it introduces methods from the statistical and computer science literature that allow one to estimate network structures. Widely adopted in areas such as medicine and biology, these methods could be exploited to improve existing health econometrics models, offering new insights to existing policy debates. Exploring and modeling the spatial dimension of the two-sided healthcare provision may help reducing inequalities in the access to healthcare services, and support policymakers in the design of financially sustainable healthcare systems.

Section “Exploratory Spatial Analysis of Health Data” introduces the reader to a set of exploratory spatial data techniques for the analysis of the geographical distribution of health data and identification of spatial patterns. Section “Incorporating Spatial Dependence in Health Econometrics Models” describes a set of spatial econometrics techniques to estimate and test for spatial dependence in a regression framework and summarizes existing literature in health economics that have adopted such methods to analyze a number of health economics issues. Section “Unknown Network Structure and the Graphical Modeling Approach” briefly reviews a set of techniques that allow one to estimate the network when the structure is not known. Section “Global Shocks in Healthcare Systems” considers econometric techniques that allow one to control for correlation between agents that arises from global, unobserved shocks, and related applications in health economics. The last section concludes the article summarizing the importance of the spatial dimension in healthcare analysis and the challenges for future research.

# Exploratory Spatial Analysis of Health Data

Over the last decades, the exploitation of geographic variation for providing directions on how to improve health system quality and efficiency has drawn increasing attention from healthcare managers and policy makers, in situations where countries have been facing rising in both healthcare expenditure and disparities in quality and access. Methods from geography and spatial statistics can help policy makers to visualize in a simple way the geographic variation of healthcare, trying to link it to differences in population health and needs (Haining, 2003).

Geographical mapping is an important tool for visualizing and describing the spatial distribution of a variable of interest. Maps are part of a wide collection of techniques, known as Exploratory Spatial Data Analysis (ESDA), aimed at describing and visualizing spatial distributions, detecting atypical localizations or spatial outliers, discover clusters, patterns of spatial association, and identify heterogeneity across space (Anselin, 1999). Several institutions across the world produce a set of maps, or health atlas, to inform different stakeholders (policymakers, the general audience, and scientists) on the geographical variation of patients’ outcomes and the availability and use of medical resources. Some important examples around the world are the Dartmouth Health Atlas for the United States, NHS RightCare for the United Kingdom, and Eurostat’s Statistical Atlas for Europe. Health atlases are not only important tools for presenting information, but can also be effectively exploited to support the development of healthcare policies. One example is the introduction of a musculoskeletal triage service by some English health authorities, after having observed through the NHS RightCare atlas their unusually high rates of musculoskeletal patients being referred directly to secondary care services when compared to nearby health authorities (Davies et al., 2016).

The geographical unit used for mapping is usually an administrative area, such as the zip code, health authority, municipality, or country, although also non-administrative areas can be used. For example, the Dartmouth Health Atlas uses hospital service area and hospital referral regions, the NHS RightCare in England focuses on health authorities (Primary Care Trusts in 2010 and Clinical Commissioning Groups in 2015), and Eurostat adopts the Nomenclature of Territorial Units for Statistics (NUTS). Health economists should take considerable care when choosing the geographical unit for mapping. For example, Figure 1 reports the map of 2015 indirect standardized rate of preventable emergency admissions3 for England at Clinical Commissioning Group (CCG) level. The areas in red, belonging to the highest decile, are the CCGs with more preventable emergency admissions than expected given the population age and gender, while the green areas, belonging to the lowest decile, are CCGs with less preventable emergency admissions than expected. Looking at the map, it is interesting to observe that CCGs located in the North of England show the highest concentration of indirect standardized rates of preventable emergency admissions.

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Figure 1. Map of indirect standardized rate of preventable emergency admissions at CCG level.

However, noting that population varies considerably across CCGs (ranging from 65,434 to 882,800), large variation in the rate of preventable emergency admissions is likely to occur within CCGs. Figure 2 reports the map of indirect standardized rate at neighborhood level, where neighborhoods are defined by the Lower Super Output Area (LSOA).4 The map shows that only a few neighborhoods in the North of England, mostly located in urban areas near the coast, belong to the highest decile of indirect standardized rate of preventable emergency admissions. Hence an intervention to reduce preventable emergency admissions in CCGs of the North of England should only target their local communities in the costal, urban areas.

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Figure 2. Map of indirect standardized rate of preventable emergency admissions at neighborhood level.

An alternative approach is to produce maps using point data rather than regional averages, exploiting information on the location of agents. This is useful when the variable of interest is at provider-level, such as hospital quality (e.g., Cookson, Laudicella, & Donni, 2013; Gravelle, Santos, & Siciliani, 2014), health expenditure, or health needs obtained from disease registers such as the primary care clinic. When using point data, interpolation methods are usually employed to predict the value in the non-sampled space between points. Triangulated Irregular Networks (TIN) and Inverse Distance Weighting (IDW) are the most commonly used interpolation techniques (Longley, Goodchild, Maguire, & Rhind, 2015). TIN interpolation creates a surface formed by triangles of nearest neighbor points, by creating circumcircles around the nearest points which intersections are connected to a network of non-overlapping and compact triangles. One disadvantage of the TIN interpolation is that the surface is not smooth and may give a uneven appearance, usually due to the discontinuity in the slopes at the triangle edges and sparse data points. IDW is a deterministic, nonlinear interpolation technique that creates surface layers from data points. The non-sampled locations are estimated by calculating a weighted average of the values recorded in nearby sampled locations/points. The main disadvantages of the IDW are that the function cannot estimate above maximum or below minimum data point values and the frequent presence of “bull’s eye” effect, namely higher values near observed locations, as well as edgy appearance. The IDW technique is more adequate to fit data at the healthcare provider level, given that providers’ point density might be low in rural areas. Under this case, the TIN interpolation would show discontinuities, while the IDW will generate a map with the “bull’s eye” effect around areas with higher density of providers with similar outcome.

Figure 3 shows the 2015 indirect standardized rate of preventable emergency admissions at General Practitioner (GP) practice level, using the IDW approach. The preventable emergency admissions, being admissions for conditions that could be avoided by appropriate management in primary care, are linked by definition to primary care. In 2015, there were over 7,000 GP practices covering more than 8,000 surgeries, with some practices having more than one surgery. This map shows some similarities with Figure 2, although it also indicates some variation within CCGs. In particular, focusing on the North England CCGs, the map shows a concentration of high rates in the costal line also detected in Figure 2, but also across the GP practices in the border of CCGs located in the center-North, namely Hambleton, Richmondshire and Whitby, Durham Dales, Easington and Sedgefield, Cumbria, and Northumberland, highlighted in blue in the figure.

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Figure 3. Map of indirect standardized rate of preventable emergency admissions at GP practice level.

Central to ESDA is the notion of spatial correlation, namely the identification of similarity in values in similar locations. The figure above shows concentration of high values of indirect standardized rates of preventable emergency admissions in the North, as opposed to a concentration of low values in the South East and the East Midlands regions. However, to conclude that there exists geographical concentration in this variable, one should carry formal testing of the hypothesis that the distribution of indirect standardized rate of preventable emergency admissions is at random across territory against the alternative of spatial correlation. If either high or low indirect standardized rates of preventable emergency admissions are found in a healthcare provider and nearby providers, it would be an instance of positive spatial correlation. In contrast, negative spatial correlation would occur if a healthcare provider with a high indirect standardized rate of preventable emergency admissions is surrounded by providers with low values of indirect standardized rates of preventable emergency admissions, or a provider with a low indirect standardized rate of preventable emergency admissions is surrounded by providers with high indirect standardized rates. In order to measure and test for spatial correlation, a metric of distance between geographical units needs to be introduced, which is usually expressed by the means of a non-negative matrix, known as a spatial weights matrix.

## Spatial Weights Matrix

In a spatial weights matrix, often indicated by $W$, the rows and columns correspond to the cross section observations (e.g., healthcare providers, regions, or countries), and the generic element, $wij$, can be interpreted as the strength of potential interaction between units $i$ and $j$ (regions or points), with $wij≠0$ implying that units $i$ and $j$ are neighbors. By convention, the diagonal elements of the weights matrix are set to zero, implying that an observation is not a neighbor to itself. Further, the weights matrix is typically row-standardized so that the sum of the weights for each row is one.

When using regional data, the most common spatial weights matrix is the contiguity matrix, also known as queen matrix by analogy to how the queen moves in a chess game. Under the contiguity criterion, region $i$ is connected equally to all the other regions with which it shares a boundary.

Spatial weights matrices can also be defined using the geographical or travel distance between units, by assuming $wij=f(dij)$ if $i≠j$. The most common example of distance function in a spatial weights matrix is the inverse weight distance $f(dij)=1dij$, which assigns smaller weights to units that are further away, since these are assumed have a weaker relationship with the unit $i$. The distance, $dij$, is defined using information on the distance between the geographical centroids (represented, for example, by the blue circle in the Figure 4 (b)) or between the population weighted centroids, which usually coincide with the city or town in the region with the highest population density. Under this specification, a common assumption is to restrict weights by imposing a threshold, so that only the areas with a centroid (or points) within the predefined distance will have a nonzero weight (represented, for example, by the brown areas in Figure 4 (b)). The use of a threshold is common when it is reasonable to assume that agents, such as patients, or healthcare providers, cannot interact if they are located beyond a certain distance from each other.

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Figure 4. Neighboring regions according to (a) contiguity, (b) inverse of distance, and (c) 10-nearest-neighbors criteria.

Noting that the distance between units might be influenced by urbanicity of the areas, with urban areas having a high concentration of smaller areas or health providers within a given radius, an alternative, distance-based, specification is to set as neighbors a given number of nearest units, leading to the $k$-nearest neighbors spatial weights matrix. This implies that all spatial units will have the same number of neighbors, and the cutoff/threshold distance will vary across units. Figure 4 displays, for a given region $i$ (in red), all regions sharing a boundary with region $i$ (in brown), for which $wij$ is nonzero according to three alternative criteria: contiguity (Figure (a)), inverse of distance (Figure (b)), and 10-nearest-neighbors (Figure (c)) criterion. Compared to figure (b), the 10-nearest-neighbors in Figure (c) sets $wij$ to zero for any two regions that are located within the radius but do not belong to the group of the 10 nearest neighbors. The opposite will happen in the presence of wider regions, since larger regions/areas will have less neighbors within a cutoff distance.

In the case of point/location data, while it is still possible to use the distance-based weights matrices, weights matrices that exploit boundaries between regions/areas, like the contiguity matrix, cannot be employed. In addition, there are often challenges due to the fact that a healthcare provider, a hospital or family doctor clinic, can have more than one site. In this case, the strength of the relationship with other healthcare providers can be set by taking the distance between all pairs of sites or by using the minimum distance between the sites of healthcare provider $i$ and $j$. In the example in Figure 5, healthcare providers A and E have two sites, namely A1 and A2 and E1 and E2, respectively, having different sets of neighbors within a given radius. In this case, the (unique) spatial proximity between providers A and E, $wAE$, can be set as a function of the minimum distance between provider A and E sites, i.e., $wAE=F(min{dA1E1,dA1E2,dA2E1,dA2E2})=F(dA1E2)$.

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Figure 5. Set of neighbors for two healthcare providers, for a given radius.

The choice of the radius in distance-based weights matrices is a key factor that can greatly affect the results obtained, and should ideally be based on some theoretical or practical justification. One interesting example is that provided by Gravelle, Santos, and Siciliani (2014), who estimated spatial models to analyze the competition process between healthcare providers in the United Kingdom, to establish whether the healthcare quality of a hospital is influenced by that of its neighboring hospitals, or rivals. To determine the set of neighbors for each hospital, the authors tried various threshold distances by considering providers that are distant within a car drive of 30, 60, and 98 minutes. Results appeared to be sensitive to the choice of the radius when using quality measures other than overall mortality, with larger values for the radius leading to nonsignificant effect of rivals’ quality on hospital’s healthcare quality. The authors observed that, while there is no theoretical reason or regulation stating that healthcare providers cannot compete with providers as distant as a 98-minute car drive, descriptive statistics on patients’ hospital choice show that it is very likely for patients to choose a nearer provider, thus supporting the choice of a 30-minute car drive threshold.

## Moran’s I Global and Local Statistic

The first general measure for spatial autocorrelation was proposed by Moran (1950) and then further developed by Cliff and Ord (1973). Let $xit$ be a variable of interest for spatial unit $i$ at time $t$, with $i=1,2,…,N$, having mean $μt$. The Moran’s I is defined as

$Display mathematics$
(1)

where $xit$ is the variable of interest, for example a health outcome or medical resource for unit $i$ in year $t$; $μt$ is the mean of $xit$ in year $t$; $n$ is the number of units; $wij$ is an element of the spatial weights matrix; and $S0=∑i=1N∑j=1Nwij$. The above statistic was originally developed to test for the presence of spatial correlation in regression residuals, although it has also been used to test the randomness of the spatial pattern of variables of interest (see, among others, Le Gallo & Ertur, 2003). Moran’s I is proportional to the covariance between $xit$, and $wijxjt$, and when using a row-standardized spatial weights matrix, noting that $nS0=1$, it can be interpreted as the correlation coefficient between $x$ in unit $i$ and its spatial lag. Cliff and Ord (1973) and Cliff Ord (1981) developed Moran’s I statistics moments (mean and variance) under the assumption that the variable of interest is normally distributed, or adopting a random permutation approach. While moments under the normal assumption can be exploited to test if the regression residuals are spatially correlated, the random permutation approach can be adopted to test if the spatial pattern of a variable of interest is significantly spatially clustered. Kelejian and Prucha (2001) have further studied the asymptotic distribution of the Moran’s I for widely used spatial regression models (see section “Incorporating Spatial Dependence in Health Econometrics Models”), as well as in the context of limited dependent variable models.

Table 1 displays results for the Moran’s I to measure and test whether the spatial patterns in preventable emergency admissions are distributed at random or not, using various levels of aggregation of data and different types of spatial weights matrices, and adopting the random permutation approach. The positive and significant values for the Moran’s I indicate that there is positive global spatial correlation, with nearby CCGs and neighborhoods presenting similar values, although the choice of the spatial weights matrix has an impact on the strength of spatial correlation detected. From these results one can conclude that the spatial pattern in preventable emergency admissions is not random, with certain areas (and GP practices) in England showing a high concentration of high/low preventable emergency admissions rates.

Table 1. Moran’s I Statistics for Preventable Emergency Admissions at CCG Level (Top Panel) and Neighborhood Level (Bottom Panel)

Row-Standardized Weights Matrix

Moran’s I

p-value

CCG level

Contiguity

0.403

<0.0001

Six nearest neighbors

0.281

<0.0001

Distance threshold

0.044

0.0131

Neighborhood Level

Contiguity

0.452

<0.0001

Six nearest neighbors

0.443

<0.0001

Distance Threshold

0.133

<0.0001

GP practice level (point data)

Distance Threshold(*)

0.0056

<0.0001

Notes: (*): For calculation of this index, a row-standardized 5-nearest-neighbors (as shown in Figure 3) has been adopted.

The Moran’s I test is a global spatial statistic, that is, it tests whether the spatial pattern is random or spatially clustered, but it doesn’t identify the local spatial clusters. Anselin (1995) proposed a Moran’s I Local Indicator of Spatial Association (LISA). The author defined as LISA any statistic that satisfies the following two conditions: (a) for each observation it gives an indication of the extent of significant spatial clustering of similar values around that observation; (b) its sum across all observations is proportional to a global indicator of spatial association:

$Display mathematics$
(2)

where the notation is as above. When using a row-standardized spatial weights matrix, the mean of all Moran’s I LISA equals the global Moran’s I statistics.

As an illustration example, the Moran’s I LISA local statistics have been calculated for preventable emergency admissions at CCG, neighborhood, and GP practice level. When adopting a row-standardized contiguity spatial weights matrix, the spatial pattern of preventable emergency admissions at CCG level does not show significant local clusters. This was expected, since most of the variation is observed within the CCGs. Figure 6 and 7 show local spatial clusters identified by the Moran’s I LISA as statistically significant, using moments for Moran’s I LISA under the randomization assumption (Anselin, 1995).

As for the neighborhood-level analysis, Figure 6 reports local clusters of high values of preventable emergency admissions in red, labeled HH (High-High), and low values, in blue, having the label LL (Low-Low) when using a row-standardized contiguity weights matrix. The map indicates the presence of a cluster covering several neighborhoods having a high rate of preventable emergency admissions (in red) in the North of England and West Midlands, as well as large significant clusters characterized by low rates of preventable emergency admissions (in blue) in the South East of England. From the 9,895 neighborhoods classified into a HH cluster, only 2,896 were statistically significant, and of the LL clusters with more than 14,000 neighborhoods only 1,921 neighborhoods formed significant spatial clusters.

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Figure 6. LISA significant clusters—2013–2015 Indirect Standardized Preventable emergency admissions at LSOA level.

When the analysis is carried at the GP practice level, using a row-standardized distance weights matrix, only clusters with high values of indirect standardized rate of preventable emergency admissions are detected. Of these, only 92 out of more than 950 GP practice classified into a HH spatial cluster belong to a significant spatial clusters. These are reported in Figure 7. Contrary to the previous neighborhood-level analysis, the significant spatial clusters are now located not just in the North and West Midlands but also in the South of England. This suggests that there are some small clusters of GP practices having high rates of preventable emergency admissions that, given the size of their catchment area, are not detected when carrying the analysis at neighborhood level.

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Figure 7. LISA—Indirect Standardized rate of preventable emergency admissions in 2015 at GP practice level.

Clearly, results of a spatial analysis of health data can change dramatically depending on the scale of the analysis, highlighting the importance of understanding the relevant geographical scale of the study. In the example provided above, observed variations at healthcare-provider level and at neighborhood level can be explained by contextual factors, such as demographic and socioeconomic characteristics, the population’s (un)healthy lifestyle, and accessibility to an appointment with the provider.

# Incorporating Spatial Dependence in Health Econometrics Models

The data-driven approach outlined above can support the decision to incorporate spatial dependence arising from contextual effects and/or social interaction in health econometrics models. Multilevel models are often adopted to account for the fact that individuals are part of a larger group, for example a neighborhood, a hospital, or some other unit of aggregation. A consolidated literature in health economics exists on linear and nonlinear multilevel models representing the impact of individual and higher-level or contextual factors on key outcome variables; see Rice and Jones (1997) and Snijders (2011) for a review of hierarchical methods and their empirical application in the area of health economics. Although multilevel models can integrate the spatial component at the group level, they are not explicitly concerned with identifying interdependence among statistical units. Social interaction models have been proposed in health economics to link the health behaviors and outcomes of one individual to those of socially connected individuals. One common assumption of this literature is that each individual belongs to a specific social group, for example classmates, employees in teams, or those living in the same neighborhood, where individuals are assumed to interact with each other; see Fletcher (2014) for a review of social interaction, or peer effects, models in health economics and Patuelli and Arbia (2016) for a review of estimation issues emerging in this literature. An alternative econometric approach to represent and measure contextual effects and social interactions is offered by the spatial econometrics literature (see Anselin & Bera, 1998; Arbia, 2014; LeSage & Pace, 2009; and Arbia, Espa, & Giuliani, 2018). Consider $N$ agents (e.g., patients, hospitals, neighborhoods) observed over $T$ time periods, and let $yit$ be a key outcome variable for $i$th agent at time $t$, with $i=1,…,N$ and $t=1,…,T$. In its most general formulation, we can consider the following linear panel augmented with spatial lags of the dependent variable, of regressors, and with spatially correlated errors:

$Display mathematics$
(3)

$Display mathematics$
(4)

where $xit$ is a $k×1$ vector of observed individual-specific regressors on the $i$th cross-section unit at time $t$, $εit$ are ID random errors, and $wij$ are elements of the spatial weights matrix, summarizing the network structure among agents. The spatial coefficient $ρ$ measures how the outcome of one agent (for example, individual’s health, or quality of hospitals) is affected by outcomes in the network, conditional on observed regressors. Including a spatial lag of the dependent variable in the model is consistent with a social interaction process, and is often adopted to represent peer effects in behavior or diffusion mechanisms in contagious diseases. The coefficient, $γ$, attached to the spatial lag of regressors, $∑j=1Nwijxjt$, represents the impact of contextual effects on the dependent variable. The spatial structure of the error term in (4), often referred to as correlated effects under Manski (1993)’s categorization, is adopted to represent the spatial concentration arising from unmeasured individual- and/or contextual-specific covariates. This could be due to the existence of risk factors that are difficult to measure and geographically concentrated, such as lifestyle factors or antisocial behavior, leading regression disturbances to be spatially correlated. Spatial error dependence could also be caused by measurement problems that health economists often encounter in their applied work. For instance, there may be a mismatch between the observed spatial unit, such as the county, and the spatial scale of the phenomenon under study, such as the neighborhood, or there could be locational uncertainty arising when individuals are arbitrarily associated to area centroids or when their locational is intentionally masked due to confidentiality issues (Arbia, Espa, & Giuliani, 2016). Finally, spatial error dependence can be induced by the sampling scheme used by the investigator to select statistical units (Anselin, 1988). For example, under a clustered sampling scheme, potential correlation may arise between units sampled from the same cluster (Pepper, 2002). Indeed, respondents sharing observable characteristics such as location or income may also share unobservable characteristics that would cause the regression disturbances to be correlated (Moulton, 1990).

In applied research, model (3)–(4) is often adopted, assuming that only one of the spatial coefficients is nonzero, leading to spatial lag models ($γ=0,λ=0$), spatial Durbin ($ρ=0,λ=0$), and spatial error models ($ρ=0,γ=0$). Choosing whether to employ a spatial lag, spatial Durbin, or spatial error model as the proper specification should be driven by theory, and also supported by a set of statistical diagnostic tests (Baltagi & Yang, 2013).

Ignoring spatial dependence in a regression model might have serious drawbacks on estimation and inference of regression coefficients. Estimation of spatial models is not trivial, as it faces a number of methodological problems due to violations of the assumptions of traditional econometric approaches. Maximum Likelihood, Instrumental Variables, and Generalized Method of Moments approaches have been proposed; the reader is referred to Moscone and Tosetti (2014) for a discussion on methods for estimation of spatial models. Various approaches have been proposed in the literature to estimate spatial nonlinear models, where $yit$ takes discrete random values. Many phenomena in health economics involve a binary choice as an outcome variable (e.g., choosing to smoke), or a multinomial choice such as selecting among a set of hospital. Expectation Maximization algorithm-based procedures (McMillen, 1992; LeSage, 2000), maximum likelihood estimator based on recursive importance sampling approach (Beron & Vijverberg, 2004), partial maximum likelihood approach (Wang, Iglesias, & Wooldridge, 2013), and generalized method of moments (Pinkse & Slade, 1998; Klier & McMillen, 2008) have been proposed. However, most of these approaches pose substantial computational challenges when the number of cross-section units, $N$, is of even moderate size.

Following on the example on preventable emergency admissions in section “Exploratory Spatial Analysis of Health Data,” Table 2 reports estimation of a spatial lag model with fixed effects augmented with spatial lags of regressors (model (3)), using data at the GP practice level over the years 2006 to 2012. Estimation of the spatial model is carried under a social interaction game framework, where all GP practices make quality decisions simultaneously. For each GP practice, its neighbors are specified as all practices within the health authority. The elements of the spatial weights matrix are weighted by the GP practice size, since larger GP practices might have more influence than others. Each spatial regression controls for a set of patient list demographic, socioeconomic characteristics, and morbidity as well as GP practice characteristics. Looking at the results in Table 2, it is interesting to observe a significant positive spatial lag coefficient in all the four quality variables, indicating the presence of peer effects, namely that good practices will spillover across peers. From a policy perspective, this result is quite important, as it suggests that health authorities may use peer pressure to spread guidelines and best quality practices. See Santos, Gravelle, and Rice (Forthcoming) for further details and results on this topic.

Table 2. Summary Results of the Spatial Durbin Model (see Santos et al., Forthcoming)

Quality Outcome Framework Variables

Patient Experience

Total QOF Points

Population Achievement

Proportion of Preventable Emergency Admissions (in 1,000s Patients)

Able to Book Urgent Appointment

Spatial

0.395**

0.454**

0.686**

0.431**

0.367**

lag coef.

(0.015)

(0.011)

(0.007)

(0.013)

(0.014)

Notes: Standard errors in parentheses.

* p <0.05,

** p <0.01,

*** p <0.001.

Each spatial panel regressions controls for patient list demographic, socioeconomic characteristics, morbidity, and GP practice characteristics.

Spatial econometric models have been widely adopted to study health economics phenomena. There exists a growing literature adopting spatial econometric methods to model geographical clustering of various health conditions such as mortality (e.g., Lorant, Thomas, Deliege, & Tonglet, 2001; Sridharan, Tunstall, Lawder, & Mitchell, 2007; Congdon, 2011; Ferrandiz, Lopez, Liopis, Morales, & Tejerizo 1995), obesity (Congdon, 2011), and diseases, both communicable, like poliomyelitis, influenza, HIV, and cancer (Cliff & Haggett, 1993; Trevelyan, Smallman-Raynor, & Cliff, 2005; Carroll et al., 2017), and noncommunicable, like diabetes, cardiovascular diseases, and mental health disorders (Congdon, 2002, 2009). These studies find a significant geographical concentration of the health outcomes considered and a consistent improvement in the explanatory and prediction power of their models when accounting for spatial dependence. Works in health economics and in the medical literature indicate that one important element explaining variations in health expenditure is represented by the spillover effect, that is, expenditure on health services in one locality can have beneficial or harmful effects across a wider geographical area (Revelli, 2006; Moscone, Knapp, & Tosetti, 2007; Moscone, Tosetti, & Knapp, 2007; Moscone & Knapp, 2005; Atella, Belotti, Depalo, & Piano Mortari, 2014).

Spatial lag models have found a recent interesting application in the estimation of hospital reaction functions, useful to study the impact of competition on hospital prices and quality. Mobley (2003) adopted a spatial lag model to estimate the slope of a price reaction function, using data for a set of Californian hospitals between 1993 and 1999, and found that the slope of the reaction function is positive and significant, and does not change significantly over time. Using the same data, Mobley, Frech, and Anselin (2009) explored the impact of market structure on hospital pricing, while accounting for spatial effects arising from local competition among hospitals. The authors estimated a market factor effect that is much higher than that estimated when spatial competition was not taken into account, having implications for the US definition of hospital markets for antitrust purposes. Gravelle, Santos, and Siciliani (2014) presented a theoretical model with regulated prices, specifying conditions on demand and cost functions that determine whether a hospital will have higher quality when its rivals have higher quality. Accordingly, the authors estimated a spatial lag model using data on 16 quality indicators for 99 English hospitals in 2009/10. The authors found evidence of a positive and significant spatial lag quality coefficient for 6 of the 16 quality indicators, although results are sensitive to the choice of the threshold in the distanced-based spatial weights matrix. A similar framework is adopted by Longo et al. (2017), who estimated a spatial lag model on data on quality and efficiency indicators for 140 English trusts over the years from 2010/11 to 2013/14. Using a distance-based spatial weights matrix with 30 km straight-line threshold distance, the authors found a significant spatial lag coefficient for only 3 of the 12 indicators, indicating that hospitals generally do not respond to rivals’ quality and efficiency.

From the above, it emerges that the use of spatial econometrics methods in health economics is justified not only by the importance of the spatial distribution of health, healthcare services, and amenities, but also by the economic theory that underlies certain health economics models. The application of spatial econometrics models to studies on the behavior of healthcare providers offers new findings and new insights that are of extreme importance to decision-makers.

# Unknown Network Structure and the Graphical: The Modeling Approach

One common feature of empirical studies reviewed in this article is that the structure of the network within which economic agents are embedded, summarized by the spatial weights matrix $W$, is assumed to be known, usually based on geographical distance between agents. However, in several applications, assuming that interaction only occurs locally between geographically closed agents is not realistic. Connections between individuals could be related to other types of metrics, such as economic, policy, or social proximity, often only partially observable. We observe that if the elements of the spatial weights matrix involve socioeconomic variables, endogeneity issues may frequently arise; see Kelejian and Piras (2014) and Qu and Lee (2015) for a review of this case. Connections may be unknown or too complex to be represented by a simple rule such as geographical distance or contiguity. For instance, in the hospital competition setting described in section “Incorporating Spatial Dependence in Health Econometrics Models,” hospitals may have an incentive to compete with healthcare providers located outside their catchment area, leading to network structures that are only partially observable. The good reputation of a provider for its championing of particular services, for example providing better process control, or more energetically developing procedures for user involvement in decision-making, may encourage mimicking behavior by other providers, regardless of their geographical location. Under these cases, rather than assuming a priori the nature of potential interaction between agents, the researcher should estimate the spatial weights matrix, $W$.

One important statistical technique for estimating network structures is the graphical modeling approach. Introduced by the statistical and computer science literature, graphical modeling studies the relationships among random variables via their joint distribution. Under this approach, the normality distribution is usually assumed, to allow the conditional dependence structure (i.e., the network) to be completely determined by the inverse of the covariance matrix (precision matrix). Specifically, let $yit$ be the observation on the $i$th cross section unit at time $t$, with $i=1,2,…,N$ and $t=1,2,…,T$, and assume that $yt=(y1t,y2t,.,,,.yNt)′∼N(μt,Σ)$, where $Σ$ is a $N×N$ symmetric and positive definite matrix, having inverse $Θ=Σ−1$. In most applications, the mean of the joint distribution is assumed to be a linear function of observable exogenous variables, namely $μt=Xtβ$. One key result in the Gaussian graphical modeling literature is that estimating parameters and identifying zeros in the precision matrix are equivalent to parameter estimation and model selection in the corresponding Gaussian graphical model:

$Display mathematics$
(5)

$Display mathematics$
(6)

where $wij$ belongs to the spatial weights matrix $W$. The above model is also known in the spatial econometrics literature as the Conditional Autoregressive model, a widely used alternative to the spatial lag model (3). In (5), $W$ is often written as $δW∗$, where $W∗$ is a matrix prespecified by the user, namely the network structure, while $δ$ is an unknown parameter that needs to be estimated, measuring the amount of spatial dependence in the data. It is possible to show that the ($i,j$)th element of the precision matrix, $θij$, is zero if and only if $wij=0$. To estimate $Θ$, or, equivalently, $W$, the literature has proposed the use of regularization methods where the likelihood function is appropriately penalized to achieve sparsity as well as efficient and stable inference, leading to the well-known LASSO and Graphical LASSO approaches (Banerjee et al., 2008; Friedman et al., 2008). These methods, initially developed for the case of a continuous, normally distributed dependent variable, have been extended to the case where $yit$ is a discrete random variable. One important model used for estimating the graph associated with a binary random variable is the Ising model, originally introduced in statistical physics for describing magnetic interactions and more recently adopted for modeling phenomena such as the outcomes of political elections (Banerjee et al., 2008), the adoption of new technologies (Laciana & Rovere, 2011), and genes association (Cheng, Levina, Wang, & Zhu, 2014).

These methods have been widely employed in biology and medicine to infer the interactions between biological entities, such as proteins and metabolites, in a biological system under specific conditions (such as disease and time) (Abegaz & Wit, 2013; Vinciotti et al., 2016) or for the inference of brain networks from time-series fMRI data (Cribben & Yu, 2017). In the field of health and health economics, only a few recent studies have applied graphical modeling techniques. Anker et al. (2017) adopted a Graphical LASSO estimation approach of the network of comorbidity conditions for a sample of individuals affected by alcohol-use disorders, in order to identify the key factor that should be targeted in interventions for reducing such disorders. Similar network analysis has been proposed by Levinson et al. (2017) to assess the core symptoms of bulimia nervosa. Using Italian data, Lisi et al. (2017) adopted graphical modeling techniques to identify for each hospital the set of rivals with which it competes, showing that hospitals do not compete only locally but also outside their catchment area.

While the adoption of graphical modeling techniques in the area of health economics is still at early stage, its application could provide new insights to existing debates and offer improvements of existing health economics models. One important area of application is in modeling the diffusion of new medical technologies across hospitals, when studying differences in productivity (Skinner & Staiger, 2005). Another interesting potential application of graphical models is the analysis of healthcare needs. Graphical models could also be employed to enhance the power of existing public health and epidemiologic model for analyzing disease incidence, to enhance prediction power of the spread of a disease and estimation of local disease risk (Lee, Rushworth, & Sahu, 2014; Lawson & Lee, 2017).

# Global Shocks in Healthcare Systems

Interdependence between economic agents not only occurs through social interaction processes, but may also occur when agents react in a similar manner to unanticipated events or global shocks (Andrews, 2005). These shocks are often unobservable to the econometrician and perturb the health system as a whole, simultaneously affecting the behavior of agents (recipients, providers, etc.), ultimately impacting on health costs. An important feature of these shocks is that they induce a correlation between pairs of statistical units that does not depend on how close they are in geographical space. Accordingly, we will refer to this type of correlation as long-range or global interdependence.

When data contain cross-section dependence, conventional estimators such as ordinary least squares are inefficient, and the estimated standard errors are biased (Andrews, 2005). Phillips and Sul (2003) proved that when cross-section dependence occurs but is not incorporated in the panel regression, the pooled ordinary least squares (OLS) estimator provides little gain in precision compared with a single OLS equation. Further, least squares may be biased if regressors are correlated with the source of interdependence.

One popular approach to incorporating global shocks in the regression equation is to adopt a common factor representation. Under this framework, the error term is a linear combination of a few common time-specific effects with heterogeneous factor loadings plus an idiosyncratic (individual-specific) error term:

$Display mathematics$
(7)

where $f1t,f2t,…,fmt$ are $m$ unobserved common effects and $εit$ are the independently distributed error terms, with zero mean and variance $σiε2$. Equation (7) expresses the error term as a linear combination of $m$ common, or pervasive, factors, plus an orthogonal idiosyncratic residual. The coefficients $γi$ are called factor loadings, and represent the sensitivity of statistical units to movements in the factors. If $m, this formulation allows for contemporaneous correlation between shocks, represented as a simpler, lower-dimensional matrix than the unconstrained. Notice that in multifactor models interdependence arises from common correlated reaction of units to some external events. According to this representation, correlation between any pair of units does not depend on how far apart these observations are, and violates the distance decay effect that underlies spatial interaction theory.

Common factor models were originally developed in the psychometric literature to measure human intelligence on the basis of the observation of a large number of variables related to human ability (Spearman, 1904; Garnett, 1920). The aim of these studies was to explain the correlations among the observed variables in terms of a smaller number of unobservable or hypothetical variables, also called factors, using statistical techniques such as principal component (Hotelling, 1933) and factor analysis. In economics, unobserved factor models have gained popularity in the fields of macroeconomics, finance, and international finance, where long runs of data are available for several countries. A number of estimators of regression models with a common factor structure for the error term have been proposed (see, among others, Bai, 2009; Pesaran, 2006). One popular approach is the Common Correlated Effects (CCE) approach proposed by Pesaran (2006), which consists of approximating the latent factors with the cross-section averages of the dependent variable, so that standard panel data methods can be used.

In health economics, unobserved factor models have been adopted to study the income elasticity of healthcare, a widely debated topic in the literature. Works by Baltagi and Moscone (2010), Moscone and Tosetti (2010) and Baltagi et al. (2016) have analyzed the non-stationarity of healthcare expenditures and income and cointegration between healthcare expenditures and income, allowing for cross-sectional dependence. After controlling for country-specific characteristics and correlation arising from local dependence and unobserved common shocks, these studies find an income elasticity below one, indicating that healthcare is a necessity good and supporting the idea of public involvement in healthcare. However, results are quite heterogeneous according to the geographical area taken into consideration. Specifically, Baltagi et al. (2016) carried an analysis using data on 167 countries around the world, over the period 1995–2012. Results showed that when moving from wealthier to poorer countries, the income elasticity rises to around unity. Various factors can explain this result, such as the consistent variability in healthcare productivity across countries, differences in what is regarded as “essential” in healthcare, as well as differences in the degree of complexity of the various healthcare systems, with wealthier countries having highly regulated and complex healthcare systems that reduce their capacity to adjust their health spending to variations in income.

# Concluding Remarks

The use of health atlases and the set of techniques offered by ESDA are powerful tools to analyze geographical variation and detect spatial concentration in risk factors, health outcomes, and health costs and identify mismatches between allocated resources and the health needs of a population. A spatial exploratory analysis approach may help health economists to gain a better understanding of the spatial structure of the phenomenon, which can be used to improve the explanatory and prediction power of their health econometrics models. For example, it might be interesting to explicitly take into account the spatial nature of certain risk and supply factors in the formula allocation model when distributing health resources across territory, to close the well-known gap between the formula base allocation and the actual level of health needs across territory.

Spatial models can be used to understand the relative importance of individual characteristics, contextual factors, and peer effects. Part of the controversy over the individual versus the contextual effect has been exacerbated by the policy implications of the empirical studies, namely whether resources (broadly defined) should be directed to single individuals or neighborhoods. For example, in the case of mental health, if the conclusion is in favor of individual characteristics, then the focus of policy efforts to alleviate certain common mental health disorders should be people and their households rather than neighborhoods. For instance, a local authority can offer direct payments to people who are eligible. Direct payments provide greater independence and flexibility in support arrangements, and for those who experience mental health problems can facilitate social inclusion, giving support to access mainstream activities. Conversely, if one recognizes a significant impact of place on health, then interventions should target the neighborhoods rather than single individuals, for example improving the recreational services for the elderly and children, the quality of the places where people live, reducing juvenile delinquency, crime, diffusion, or incidence of diseases. Estimation of significant peer effects may also induce policymakers to design specific policies to reduce inequalities due to asymmetric information. For example, peer effects in patients’ decisions of being admitted in a particular hospital may mislead patients, who end up in low-quality institutions. In this case, policymakers may decide to diffuse guidelines and hospital rankings to inform patients about true hospital quality.

The range of spatial techniques adopted until now in health economics is rather limited, often using simple cross-section analysis, under strong assumptions of homogeneity, and mostly using continuous dependent variables. This is due also to the fact that existing methods for nonlinear spatial models with network effects are computationally very expensive and cannot be adopted when dealing with large data sets. Hence, the future of this field of research depends on the development of computationally efficient methods that incorporate the spatial effects in nonlinear models.

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## Notes:

(1.) Throughout the chapter, the terms social interaction, network or spillover effects, and peer or social influences are used as synonymous to indicate the process through which the behaviors and outcomes of one individual vary with those of connected individuals (Manski, 1993).

(2.) While spatial dependence is known as a form of weak dependence, global shocks, or common factors, are assumed, carrying a strong form of dependence.

(3.) Preventable emergency admissions and Ambulatory Care Sensitive Conditions are internationally recognized as emergency admissions for conditions that could be avoided by appropriate management in primary care (Purdy, Griffin, Salisbury, & Sharp, 2009). In this chapter, preventable emergency admissions are defined as in the NHS Outcome Framework (Department of Health, 2013), which was set to evaluate the effectiveness of primary and community care.

(4.) The indirect standardized rates for LSOA were calculated using data from 2013 to 2015.