Advertisement

Applied Spatial Analysis and Policy

, Volume 11, Issue 4, pp 739–751 | Cite as

Spatial Interaction Healthcare Accessibility Model – an Application to Texas

  • Felipa de Mello-SampayoEmail author
Article
  • 69 Downloads

Abstract

A theoretical model was developed using the entropy approach to cope with the random component of the utility function to find that the spatial accessibility improves as the provider capacity increases or the opportunity cost of traveling to and from health provider decreases. The Kernel Density Estimation of the model show disparities in healthcare accessibility with extensive pockets of poor accessibility in rural and peripheral areas in Texas, when using hospitals’ location and number of hospital beds or counties’ centroid and data on Primary Care Physician. The model can be beneficially used to evaluate policies indicative of changes in the provision of health services, such as closures of rural hospitals or capacity increases, potentially have spatially very differentiated accessibility outcomes.

Keywords

Spatial interaction model Healthcare accessibility Kernel density estimation Texas 

Background

The main goal of most public healthcare systems is to improve or achieve a healthier population. Physical access to health services is one of the first steps in maintaining and improving population health. To ensure equitable distribution of healthcare services, we have to understand the existing geographical distribution of health services in relation to the populations they serve. Fundamental to addressing the issues of equity and equitable access to healthcare is the issue of geographical distribution (Oliver and Mossialos 2004). Accessibility to healthcare is a multidimensional concept and can be defined as the ability of a population to obtain healthcare services. It varies across space because neither health professionals nor residents are uniformly distributed (Luo and Wang 2003).

Spatial accessibility combines accessibility and availability, a term that is gaining some favor in the healthcare geography literature (Luo and Wang 2003; Luo 2004). Accessibility is travel impedance between patient and health service locations, and is usually measured in units of distance or travel time. Availability refers to the number of health service locations from which a patient can choose. The two dimensions should be combined. Spatial accessibility is a necessary, albeit not sufficient, prerequisite for quality healthcare services for all population segments of society, whether they reside in urban agglomerations or in peripheral rural areas. However, spatial barriers, most notably long travel distances to healthcare facilities, are important factors contributing to the exclusion from high-quality medical care. An important factor in obtaining quality care is physical access to healthcare, as a lack of spatial access can result in delayed treatment and poor health outcomes.

This paper focuses on the spatial dimension of healthcare accessibility and analyzes areas that are underserved or at risk of being underserved. We develop a theoretical model of spatial accessibility to healthcare professionals and facilities. The basis for modeling is the maximization of utility and then use the entropy approach to cope with the random component of the utility function to find that the spatial accessibility improves as the provider capacity increases or the opportunity cost of traveling to and from health provider decreases. Incorporating the theoretical foundations are important to policy researchers because they affect the data, specification, and econometric technique used to estimate the spatial model. Use of a theoretically grounded spatial model can lead to interpretations that are substantially different from those obtained via an intuitive formulation, and high-quality policy research and advice increasingly needs to be based on a rigorously established methodology.

The theoretical framework derived here departs from the interaction models based on random utility maximization (RUM) with the Multinomial Logit (MNL) model as an special case (Manski 1977; McFadden 1978, 1981; Boyce and Williams 2015) as we use the entropy approach to cope with the random component of the utility function. Our approach has the advantage of not having to assume some identically independent distribution of the error term. The specification of error term’s distribution invalidates the ordinal utility assumption and converts the utility specification to cardinal utility (Batley 2008). These assumptions also leave the model saddled with the “independence of irrelevant alternatives”(IIA) property (Fotheringham 1984). Since we do not assume a distribution for the error term, we use a non-parametric method, the kernel density estimation to estimate the theoretical model. Conversely, to the RUM, the focus is not on the choice,1 but rather on the share of the healthcare service.

We analyze healthcare accessibility across Texas using data on healthcare facilities and physicians, geo-referenced at a county spatial scale. According to US Census Bureau data, the population is unevenly distributed across 254 counties, with the largest county, Harris County, accounting for around 15% of the state’s 24.2 million residents in 2008. Texas has a relatively large share of rural dwellers with 12% of Texans living in rural counties. The healthcare policies evaluated thus focus on scenarios that potentially affect residents in rural areas, such as closures of rural healthcare facilities and consolidation of hospitals.

The remainder of this paper comprises the discussion of literature related to the application of the spatial interaction model to the analysis of healthcare. Then, we elaborate the theoretical model, and map the theoretical results into an empirical strategy. In the empirical application, we describe the data and present the empirical results. We discuss the empirical findings, and provide concluding comments in the closing section.

Related Literature

Spatial modeling has been widely applied to the issue of healthcare accessibility that is key to supporting policy makers in making rationing decisions that affect the spatial planning of health. Accessibility to healthcare varies across space because neither health professionals nor residents are uniformly distributed (Luo and Wang 2003; Luo 2004). Geographic information systems (GIS) have advanced the ability to both visualize and analyze these point data. Using GIS, point based maps can easily be aggregated to differing areal units and examined at varying resolutions (Guagliardo 2004; Guagliardo et al. 2004; Carlos et al. 2010). Hospital and primary care physicians’ localization and size lie at the heart of the equity/efficiency trade-off that characterizes national healthcare systems and is based on the principle that all inhabitants should have equal access to healthcare. The concentration of hospital services raises equity concerns because of its detrimental impact on accessibility. Efficiency gains can be obtained through economies of scale and scope.

Several studies adopt spatial modeling to investigate patient mobility. Recently, Lowe and Sen (1996), Congdon (2001), Levaggi and Zanola (2004), Cantarero (2006), Fabbri and Robone (2010), and de Mello-Sampayo (2015) adopt gravity modeling to investigate patient mobility. Lowe and Sen (1996) use the gravity model to study the flows for acute inpatient hospital care from the six-county metropolitan Chicago area to 92 hospitals in that same area in the year 1987. The model is used to forecast how potential changes in hospital financing policy can change patient flows. Congdon (2001) models patient flows to emergency units in 127 electoral wards in North East London and Essex and describes how such models may be adapted to allow for unit closures and expansion, or the opening up of other units. The estimation of the gravity model is based on simulation based Bayesian methods. Levaggi and Zanola (2004) study the net flows of people moving from one Italian region to another as determined by regional differences in the quality of healthcare and distance. The dataset they use is a sample of observations over the period 1994–1997. A similar analysis is developed by Cantarero (2006), working on patient mobility across Spanish regions during the period 1996–1999. Fabbri and Robone (2010) evaluate the extent to which the imbalances observed in the Italian geography of hospital admissions are due to scale effects or reflect the presence of other spatial factors in the distribution of healthcare resources. de Mello-Sampayo (2015) derives a competing-destinations gravity model and applies it empirically to patient mobility into State Mental Hospitals in Texas.

Neutens (2015) discusses recent accomplishments in terms of modeling accessibility and provides a systematic and comprehensive literature review of their application in empirical studies of healthcare delivery. In terms of the explanatory variables related to size and quality that have been associated with health service supply, studies consider the cardinality of the available set of services, but few have additionally considered other attributes such as staffing capacity (Spencer and Angeles 2007) and number of hospital beds (Mao and Nekorchuk 2013). On the demand side, scholars have relied on population size as a proxy for healthcare need without adjustment for socio-economic and other attributes.

As in many studies of spatial accessibility, our method concerns only potential spatial accessibility, not actual access or utilization of services. Only complex and expensive investigations can reveal the absolute significance of spatial accessibility for utilization, or the relative importance of spatial accessibility vis-a-vis the other components of healthcare access: affordability, acceptability and accommodation. Planners and policy makers continue to devote a great deal of attention to other components, but elimination of non-spatial barriers will not eliminate all barriers for the socially disadvantaged.

Material and Methods

Theoretical Framework

Consider a general utility function of health and wealth, U(h, w) where w denotes wealth (or alternatively, in a multi-period setting, consumption), and h denotes the health level. An important factor in obtaining quality care is spatial access to healthcare as a lack of spatial access can result in delayed treatment and poor health outcomes (McClellan et al. 1994). For simplicity, from this point we assume that h is proxied by spatial accessibility to healthcare services. Spatial accessibility, hrj, from population point r, r = 1, 2, ⋯, R may be a personal residence or the centroid of an area of interest such as a census tract, city or county to health suppliers location, j = 1, 2, ⋯, J. Let Hr = ∑jhrj be defined as the measure of total spatial accessibility in the economy, and we wish to model the spatial accessibility pattern in this region, i.e. hrj, the unknown amount of health service accessed by population point r at location j. The deterministic utility function, Ur is defined and calibrated over the population point r in terms of spatial accessibility to health at location j.
$$ {U}_r={\int}_j{h}_{rj}^{\mu }{w}_r^{1-\mu } dj, $$
(1)
where hrj stands for the health service accessed by residents point r at location j. An exogenous fraction μ of income is spent on health, and the remaining fraction 1 − μ on the consumption good, which is our numeraire. Assume that all patients have health insurance, the price of health service is given by d, i.e. distance-price or time-price (distance or time spent travelling has an opportunity cost – e.g. patients could have been working) that patients incur when traveling to and from health provider. Let yr be the populations’ income, which equals its expenditure level. Population point r optimal spatial health services’ accessibility is given by:
$$ {h}_{rj}=\frac{\mu {y}_r}{d}. $$
(2)
Equation (2) shows that hrj is increasing in income and decreasing in distance or time-price. Assume that the maximum optimum utility, \( {U}_r^{\ast } \), emerging from the utility maximization will exceed the observed utility Uro emerging from the observed hrjo by a random component . This error term is usually attributed to one or more of the following: the variation in tastes over the population; contribution of unobserved or unmeasured attributes (Manski 1977; McFadden 1978, 1981; Boyce and Williams 2015). To cope with this divergence, it is used the entropy approach (Roy 2004). Let E be the number of ways that the observed Hro = ∑jhrjo of health accessed at each location j may be allocated in groups hrj to location j times the number of ways total of distinct healthcare service accessed, Hr = ∑jhrj , made from patients at point r may be arbitrarily allocated to each of suppliers’ capacity Sj in location j.
$$ E=\frac{\prod_r{H}_{ro}!}{\prod_j{h}_{rj}!}\times {\prod}_j{{\mathrm{S}}_j}^{H_r}. $$
(3)
The natural logarithm of Equation (3) is taken, the Stirling approximation2 applied, and constant terms omitted. The entropy E then comes out as:
$$ E=-{\sum}_{rj}{h}_{rj}\left[\mathit{\ln}\left(\frac{h_{rj}}{S_j}\right)-1\right]. $$
(4)
The maximization of Eq. (4) is constrained by the model flows being induced to conform with certain aggregate base period quantities. If we have the total utility Uo based on the observed spatial accessibility in all population points r, the following behavioral constraint is applied:
$$ {\sum}_r{U}_r^{\ast }={U}_o. $$
(5)
Maximize Eq. (4) subject to Eq. (5) with multiplier α to obtain:
$$ {h}_{rj}={S}_j{e}^{\beta {(d)}^{1-\mu }}. $$
(6)

This gives the spatial accessibility as a function of providers’ capacity and distance or time-price, hrj improves as the provider capacity increases or the travel impedance decreases. The parameter (1 − μ) represents opportunity cost of travelling; β = α(1 − μ)(1 − μ)μμ is the travel friction coefficient and reflects the perception of travel as determinant of interactions by patients. We expect α to be negative because as the travel time or distance increases, the expected spatial accessibility decreases.

The variable Sj measures the service capacity of location j and is usually proxied by the provider-to-population ratio, defined as the ratio of health service capacity in the numerator (e.g., as the number of physicians or the number of hospital beds) and population size in the denominator. Provider-to-population ratios have frequently been used in studies linking the supply of healthcare services to health outcomes (Waldorf et al. 2007). For example, Shi and Starfield (2001) focused on the effects of primary care physician supply, measured by the population-to-provider ratio, on mortality among Blacks and Whites across U.S. metropolitan areas in 1990. Provider-to-population ratios also take on a pivotal role in the delineation of Health Professional Shortage Areas (HPSAs) and Medically Underserved Areas (MUA) (Guagliardo 2004; Guagliardo et al. 2004).

The estimation of standard form of the spatial accessibility pattern as presented in Eq. (6) has the problem of the delineation of the bounded areas, r and j, when measuring the provider-to-population ratio. First, the delineation of the bounded region strongly influences the results, as any change in the area definition will change both the numerator and the denominator of the supplier’s capacity. Second, Provider-to-population ratios are potentially misleading since they ignore internal variations of spatial accessibility, especially differences between rural and urban areas.

One way to address the above mention issues is to estimate the density pattern given by Eq. (6) from numerous points in a region. Kernel density estimation (KDE) is a non-parametric method of extrapolating point data over an area of interest without invoking the area unit problem or relying on fixed boundaries for aggregation (Waller and Gotway 2004). In KDE, a two-dimensional spatial point pattern is defined as a set of data points j (j = 1, ..., J) located in a two-dimensional study region R. Each data point j represents the location in R of one or more Sj e.g. hospital beds. In the analysis of spatial point patterns we are often interested in determining whether the distribution of the health providers’ capacity of interest within R exhibits some form of clustering, as opposed to being random. Areas with low values would correspond to areas of relatively poor access and the high point values would indicate areas of potential over-service.

The spatial point pattern of interest, hrj given by Eq. (6) by means of its intensity function is as follows:
$$ {\widehat{h}}_{rj_g}=\frac{c}{A_g}{\sum}_{j=1}^J{S}_jk\left({\delta}_{jg}/{\tau}_g\right). $$
(7)
where k(·) is the kernel function, δjg is the Euclidean distance between data point j and grid point g; τg is the kernel bandwidth, i.e. the radius of the kernel function at grid point g, Sj suppliers’ capacity at j; Ag is the area of the subregion over which the kernel function is evaluated at grid point g, possibly corrected for edge effects; and c is a constant of proportionality.

When the spatial weighting function is fixed or applied equally at each point, one assumes that the weight-distance relationship is globally applicable at all calibration points across space, which can be problematic for several reasons. Firstly, the global statement may not be true, for instance in situations where physical (built or natural) buffers such as highways or parks between two neighborhoods radically affect their impact on one another; secondly if data are sparse in parts of the larger area the local regressions may be based on too few data points. To account for these possibilities, a spatially adaptive weighting function was used instead. This allows for smaller bandwidths in which the data are dense and for larger bandwidths in which the data are sparse. Specifically, a bandwidth represents how far out from a focal neighborhood, j, the other neighborhoods will count in the calibration of parameters at point j.

Whereas the fixed bandwidth kernel density estimation model employs a bandwidth based on a geographic distance, the adaptive bandwidth method uses background data to calculate a kernel of varying size for each individual case. In deciding which method to choose, one needs to consider the research hypothesis. Using Counts of beds people use, if the primary concern is distance, a fixed bandwidth is preferable because it may be better to define the “neighborhood” or healthcare accessibility of each hospital based on distance. If the primary concern is the share of the service per person, or differences in healthcare accessibility, an adaptive bandwidth is preferred. Counts of beds people use are greater in higher population areas: there are more hospitals, clinics, doctors in places where more people live. As a result, the density of hospital beds should not be interpreted without knowledge of the underlying health providers’ distribution.

As shown in Equation (7) above, for each grid point g, kernel estimators compute a weighted sum of Sj making up the spatial point pattern of interest. The weight k(δjg/τg) applied to each object is a function of the ratio between the js distance from g (δjg) and the value of the kernel bandwidth τg. In turn, the way weights depend on δjg and τg is determined by the form of the kernel function k(·) used in the analysis. There are various choices for the function K; most will not significantly affect the outcome (Carlos et al. 2010). In order to approximate as close as possible to the form of Equation (6), it was used a truncated negative exponential for the kernel function of Equation (7), as follows:
$$ k\left({d}_{jg}/{\tau}_g\right)=\left\{\begin{array}{cc}{e}^{-3\frac{\delta_{jg}}{\tau_g}},& \mathrm{if}\ {\delta}_{jg}<{\tau}_g\\ {}0\kern7em ,& \mathrm{otherwise}\end{array}\right. $$
(8)
where τg is the radius of the smallest circle centered on g that circumscribes at least nth nearest neighbor from g. Instead of fixing the distance, the number of nearest neighbors was fixed allowing the kernel to go as far in space as needed in order to find that number of neighbours. When using a truncated negative exponential, the coefficient3 β is set to −3. and, \( d={\left({\delta}_{jg}/{\tau}_g\right)}^{\frac{1}{1-\mu }} \), i.e. distance-price or time-price increases with the Euclidean distance to the hospital but decreases with the options of health providers that can be accessed within the radius.

In less populous areas however, the bandwidth could expand beyond the distance where a case may influence healthcare accessibility. Therefore it was set a limit to the maximum distance of the bandwidth. The maximum distance parameter restricts the bandwidth from expanding further, even if the expected data points have not been reached.

Empirical Application

In this section the theoretical framework is estimated to analyze spatial accessibility across hospitals in Texas using data on hospitals during the year 2008. Data for this analysis, including patients, number of hospitals’ beds, health facilities’ locations, number of primary care physicians, and the population in each county, geographic coordinates of county centroids were collected from a number of different sources. Patients, health facilities’ locations and number of beds are reported in the Texas Inpatient Public Use Data File (PUDF) provided by the Texas Department of State Health Services (DSHS). All healthcare facilities were geo-coded using their ZIP code. The number of primary care physicians per county was also derived from DSHS. The population data were obtained from the US Census Bureau. Geographic data: county’s centroid and boundaries were created using the shapefile for Texas’ counties from TIGER 2008 from the US Census Bureau.

Texas is divided in 254 counties in 2008, as show in Fig. 1. According to America’s Health Rankings, Texas’ healthcare service provision ranks below the national average. Using the number of physicians per capita as an indicator, Texas ranked around 40th among the 50 states in 2008. Texas had only 95 physicians per 100,000 residents. The deficit of physicians affects potentially preventable hospitalizations (admissions to a hospital for certain acute illnesses, e.g., dehydration) or worsening chronic conditions (e.g., diabetes) that might not have required hospitalization had these conditions been managed successfully by primary care providers in outpatient settings. Texas is at the bottom of the national rank in 2008 with 87.6 per 1000 preventable hospitalizations. Texas is also at the bottom concerning the lack of insurance, with 24.9% of the population having no health insurance (private, employer, or government). Given these indicators, analyzing geographic areas that have poor healthcare accessibility is of primary concern.
Fig. 1

Texas counties

Figure 2 shows the spatial variation of healthcare accessibility as given by Equation (7) across the state, when Sj is proxied by number of beds per patient. Since the primary concern is to analyze differences in healthcare accessibility, we use an adaptive bandwidth up to a maximum distance. There are limitations to this spatial analysis, as the relatively arbitrary selection of bandwidth limits with both static and adaptive methods. Too large or small a bandwidth poses the risk of over or undersmoothing the original data, respectively. We followed (Carlos et al. 2010) and tested multiple parameters for bandwidth in a sensitivity analysis. We placed a 100 miles limit on the density calculation if the expected three hospitals data points’ threshold was not reached. We determined this maximum distance by testing a number of distances as well as choosing a limit based on behavior theory regarding healthcare accessibility (Grossman et al. 2017).
Fig. 2

Kernel density estimation of healthcare accessibility using beds per patient

The density map in Fig. 2 shows 7608 cells of one tenth square mile size, which comprise 540 hospital tracts. The hospitals’ location (data points) are shown in black dots. The area of highest density are the neighborhoods of the five big cities in Texas: Houston, San Antonio, Dallas, Austin, and Fort Worth. Areas of relatively poor accessibility (green areas) include a pocket in the west, next to El Paso, and a considerable number of counties in central Texas and from south of Texas up to the north. The area with the poorest access to health is the corridor from the southwest through the west of Texas and east of Panhandle (around Carson State). These blue areas are composed mainly of rural counties.

In a robustness test, Equation 7 was estimated using counties’ centroid and Sj proxied by Primary Care Physician (PCP) per capita as shown in Fig. 3. The method used was the adaptive KDE placing a 100 miles limit on the density calculation if the expected three data points’ threshold was not reached.
Fig. 3

Kernel density estimation of healthcare accessibility using PCP per capita

The density map in Fig. 3 shows 7608 cells of one tenth square mile size, which comprise 254 census tracts. The data points of counties’ centroid are shown in black dots. There are a few differences between Figs. 2 and 3. Similar to Fig. 1, the area of highest discharges to provider capacity density, represented by the red and orange areas, is the neighborhood of San Antonio, Dallas, Austin, and Fort Worth. Conversely to Fig. 1, the other higher density area is predominantly east of Panhandle (around Carson State). Areas of relatively poor accessibility (green areas) include a pocket in central Texas, and a considerable number of counties from South Texas up to Panhandle. The area with the poorest access to healthcare (blue areas) include a pocket in the west, around El Paso, and the corridor from the southwest up to north Texas. Again, these blue areas are composed mainly of rural counties.

The health accessibility disparities shown in Figs. 1 and 2 can be aggravated if geographic variation in transportation options is considered. More options are generally considered to improve accessibility. For a given travel distance, those who rely solely on public transportation may face more travel friction in the form of time delays and higher costs, thereby decreasing accessibility. Guagliardo et al. (2004) also recognized the importance of transportation mode alternatives in gravity modeling and accounting for percentage of car-owning households in their study. These authors conclude that fewer transportation mode options negatively affects spatial accessibility to healthcare.

Discussion

Our use of Kernel density approach to measuring spatial accessibility to health has a number of advantages. It is similar to classic spatial interaction models’ estimation in that it accounts for both availability and distance of multiple centroids, and the measure is gradually discounted as distance increases. But unlike classic spatial interaction models, our measure of spatial accessibility is in intuitive units that are comparable over many settings and areas of different size.

A potentially significant shortcoming of our method is that it accounts for only the supply side, with no adjustment for variation in demand for healthcare services. Carlos et al. (2010) used a grid format population database that allowed counts at different locations more spatially comparable. Due to lack of access to LandScan Global Population database or unavailability of street addresses, an adjusted demand side model was not estimated.

Straight-line or Euclidian distance, which is the basis for our density estimations, is by far the most popular proxy for travel friction in the health services literature, because it is easily computed from geo-coded locations. However, Euclidian distance may be an imperfect proxy for travel friction. Travel time and travel distance might be better, but these must be computed from transportation network data, a burdensome process that carries its own sources of error. Some authors use Euclidian distance and travel distance (Luo 2004; Guagliardo 2004; Guagliardo et al. 2004), while others favor time or travel distance over Euclidian distance (Luo and Wang 2003).

The previous discussion identifies several avenues for methodological improvement: the search for a satisfactory adjustment for demand on health service availability, an adjustment for travel mode options, assessment of the improvements achievable through use of travel time or travel distance, and collection and incorporation of data on mid-level and safety net providers.

It would also be worthwhile to conduct research to determine the most appropriate service radius to use in the creation of provider density layers. This would ideally be distance beyond which patients find it difficult to maintain a consistent relationship with the provider. A representative survey of city residents and an examination of medical records, indicating patient origin, adequacy of utilization, and consistency of patient-provider relationship, could reveal a better radius. Also, given that burden of transportation probably varies with socioeconomic status and neighborhood characteristics, it might be that the radius should vary with ecological circumstances.

Conclusion

This paper demonstrates the utility of spatial interaction models as a tool to assess spatial accessibility to healthcare services. The empirical application indicates that access to healthcare varies sharply across Texas counties, with pockets of deprived access. While population in centrally located and urban areas enjoy better access, rural counties in southern and western regions have the poorest access to healthcare.

The paper also demonstrates that the spatial interaction model can be used for evaluations of healthcare policies. The empirical application shows that changes in the provision of healthcare services, such as closures of rural hospitals or capacity increases, potentially have spatially very differentiated accessibility outcomes. Thus, the technique presented in this paper may not only help healthcare policy makers and planning authorities to identify and target areas and population groups with insufficient access to physician and hospital care, but also avoid policy-induced deepening of already existing accessibility disparities.

This research has several limitations that should be addressed in future research. First, the specification of the spatial measure ignores asymmetries in travel behavior. That is, it assumes equal probabilities that residents of a large city will travel to a hospital in a rural area and that a rural resident will travel to a similar hospital in an urban area. This symmetry implies that intervening opportunities that are typically much more numerous for urban compared to rural dwellers, are ignored. Finally, the measures can be refined by improving the geo-referencing of the population as well as the physicians, and by disaggregating by type of healthcare services and population characteristics. The differentiation by population attributes will be particularly important when integrating spatial accessibility with the social constraints of healthcare utilization (Wang and Luo 2005).

Footnotes

  1. 1.

    In the RUM context, accessibility can be interpreted as an economic evaluation measure of user benefit or consumer surplus (Williams 1977; Williams and Senior 1978).

  2. 2.

    The Stirling approximation is given by lnx !  = x(lnx − 1).

  3. 3.

    The value of the coefficient β is often unknown (Talen and Anselin 1998), particularly for health services. If we use β =  − 3, means that α varies around −3.57 when μ varies around 0.05. α approximates to the distance decay parameter estimate of 3.64 found by Griffith (2009) that analyzed Texas Journey to work Flows in Texas Counties. Given that out-of-pocket healthcare spending represents 5% of income in 2008 (Foster 2016), we will assume =0.05, i.e. the share of income spent on traveling to and from health provider is around 5%.

Notes

Funding

Funding from Fundação para a Ciência e Tecnologia, under UID/GES/00315/2013 grant is gratefully acknowledged. The funding body had no influence in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.

Compliance with Ethical Standards

Conflict of Interest

The author declares that she has no conflict of interest.

References

  1. Batley, R. (2008). On ordinal utility, cardinal utility, and random utility. Theory and Decision, 64(1), 37–63.CrossRefGoogle Scholar
  2. Boyce, D., & Williams, H. (2015). Forecasting Urban Travel: Past, Present and Future. Cheltenham, U.K. and Northampton, Massachusetts: Edward Elgar.CrossRefGoogle Scholar
  3. Cantarero, D. (2006). Health care and patients' migration across Spanish regions. European Journal of Health Economics, 7, 114–116.CrossRefGoogle Scholar
  4. Carlos, A., Shi, X., Sargent, J., Tanski, S., & Berke, E. M. (2010). Density estimation and adaptive bandwidths: A primer for public health practitioners. International Journal of Health Geographics, 9, 1–8.CrossRefGoogle Scholar
  5. Congdon, P. (2001). The development of gravity models for hospital patient flows under system change: A Bayesian modelling approach. Health Care Management Science, 4, 289–304.CrossRefGoogle Scholar
  6. de Mello-Sampayo, F. (2015). A spatial analysis of mental health care in Texas. Spatial Economic Analysis, 1–25.  https://doi.org/10.1080/17421772.2016.1102959.
  7. Fabbri, D., & Robone, S. (2010). The geography of hospital admission in a national health service with patient choice: Evidence from Italy. Health Economics, 19, 1029–1047.CrossRefGoogle Scholar
  8. Foster, A. C. (2016) Household healthcare spending in 2014. Beyond the Numbers: Prices and Spending, 5(13). https://www.bls.gov/opub/btn/volume-5/household-healthcare-spending-in-2014.htm. Accessed 1 July 2017.
  9. Fotheringham, A. S. (1984). Spatial flows and spatial patterns. Environment and Planning A, 16, 529–542.CrossRefGoogle Scholar
  10. Griffith, D. A., 2009, Spatial autocorrelation in spatial interaction, complexity-to-simplicity in journey-to-work flows. In Complexity and Spatial Networks, Ed. Aura Reggiani and Peter Nijkamp, pp 221–237 Springer.Google Scholar
  11. Grossman, D., White, K., Hopkins, K., & Potter, J. E. (2017). How greater travel distance due to clinic closures reduced access to abortion in Texas. PRC Research Brief, 2(2).Google Scholar
  12. Guagliardo, M. F. (2004). Spatial accessibility of primary care: concepts, methods and challenges. International Journal of Health Geographics, 3, 1–13.CrossRefGoogle Scholar
  13. Guagliardo, M. F., Ronzio, C. R., Cheung, I., Chacko, E., & Joseph, J. G. (2004). Physician accessibility: an urban case study of pediatric providers. Health and Place, 10, 273–283.CrossRefGoogle Scholar
  14. Levaggi, R., & Zanola, R. (2004). Patient's migration across regions: the case of Italy. Applied Economics, 36, 1751–1757.CrossRefGoogle Scholar
  15. Lowe, J. M., & Sen, A. (1996). Gravity model application in health planning: analysis of an urban hospital market. Journal of Regional Science, 36, 437–461.CrossRefGoogle Scholar
  16. Luo, W. (2004). Using a GIS-based floating catchment method to assess areas with shortage of physicians. Health and Place, 10, 01–11.CrossRefGoogle Scholar
  17. Luo, W., & Wang, F. (2003). Measures of spatial accessibility to health care in a GIS environment: synthesis and a case study in the Chicago region. Environment and Planning B: Planning and Design, 30, 865–884.CrossRefGoogle Scholar
  18. Manski, C. (1977). The structure of random utility models. Theory and Decision, 8(3), 229–254.CrossRefGoogle Scholar
  19. Mao, L., & Nekorchuk, D. (2013). Measuring spatial accessibility to healthcare for populations with multiple transportation modes. Health and Place, 24, 115–122.CrossRefGoogle Scholar
  20. McClellan, M., McNeil, B. J., & Newhouse, J. P. (1994). Does more intensive treatment of acute myocardial infarction in the elderly reduce mortality? Analysis using instrumental variables. JAMA, 272, 859–866.CrossRefGoogle Scholar
  21. McFadden, D. (1978). Modelling the choice of residential location. In A. Karlqvist, L. Lundqvist, F. Snickars, & J. Weibull (Eds.), Spatial Interaction Theory and Residential Location. Amsterdam: North-Holland.Google Scholar
  22. McFadden, D. (1981). Econometric models of probabilistic choice. In C. Manski & D. McFadden (Eds.), Structural Analysis of Discrete Data: With Econometric Applications. Cambridge: The MIT Press.Google Scholar
  23. Neutens, T. (2015). Accessibility, equity and health care: review and research directions for transport geographers. Journal of Transport Geography, 43, 14–27.CrossRefGoogle Scholar
  24. Oliver, A., & Mossialos, E. (2004). Equity of access to health care: outlining the foundations for action. Journal of Epidemiology and Community Health, 58, 655–658.CrossRefGoogle Scholar
  25. Roy, J. O. (2004). Spatial Interaction Modelling, A regional Science Context. Berlin Heidelberg New York: Springer.CrossRefGoogle Scholar
  26. Shi, L., & Starfield, B. (2001). The effect of primary care physician supply and income inequality on mortality among blacks and whites in US metropolitan areas. American Journal of Public Health, 91, 1246–1250.CrossRefGoogle Scholar
  27. Spencer, J., & Angeles, G. (2007). Kernel density estimation as a technique for assessing availability of health services in Nicaragua. Health Services and Outcomes Research Methodology, 7, 145–157.CrossRefGoogle Scholar
  28. Talen, E., & Anselin, L. (1998). Assessing spatial equity: an evaluation of measures of accessibility to public playgrounds. Environment and Planning A., 30, 595–613.CrossRefGoogle Scholar
  29. Waldorf, B., Chen, S., & Unal, E. (2007). A spatial analysis of health care accessibility and health outcomes in Indiana. Cambridge: Spatial Econometrics Association.Google Scholar
  30. Waller, L. A., & Gotway, C. A. (2004). Applied spatial statistics for public health data. Hoboken: Wiley.CrossRefGoogle Scholar
  31. Wang, F., & Luo, W. (2005). Assessing spatial and non-spatial factors for healthcare access: Towards an integrated approach to defining health professional shortage areas. Health and Place, 11, 131–146.CrossRefGoogle Scholar
  32. Williams, H. C. W. L. (1977). On the formation of travel demand models and economic evaluation measures of user benefit. Environment and Planning A, 9, 285–344.CrossRefGoogle Scholar
  33. Williams, H. C. W. L. and Senior, M. L. (1978) Accessibility, spatial interaction and the spatial benefit analysis of land use-transportation plans. In Karlqvist, A., Lunqvist, L., Snickars, F. and Weibull, J. W. (Eds.), Spatial Interaction Theory and Planning Models. North Holland, Amsterdam.Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of EconomicsInstituto Universitário de Lisboa (ISCTE-IUL)LisbonPortugal

Personalised recommendations