I have derived an OD Cost Matrix for all of the New Mexico census tracts using the Origin Destination Cost Matrix Tool. The DGR gravity model can now be applied with road based distances. For now, I will use the 2002 primary care physician data that was previously available during the earlier work performed by DGR for the New Mexico Health Policy Commission (NM HPC) from 1988 through 2002. Although this previous work was based on ZIP Codes, recent adjustments ( spatial join) have been performed (see the Story Map) to use census tracts and 2002 population estimates from UNM GPS. These results are very informative (see 2002 Gravity Model - Road Distance Map). The census tract test case results can be further modified to disregard county boundaries using the ArcGIS Inverse Distancw Weighted Tool (see IDW - Road Distance Map), but this smoother statistical surface should be interpreted with more caution. A Web Map and Web Map Application). are also available with these results. If the recent physician data for census tracts (499) or ZIP Codes (426) are eventually made available, the DGR gravity model results can provide a higher resolution and more useful picture of geographic accessibility.

Begun to focus on the primary research goal, to statistically compare various geographic access and gravity models (DGR, 2SFCA, and E2SFCA) at various levels of geography including those with higher resolution than New Mexico counties (by census tract, ZIP Code, and perhaps census block group). Also to compare the results for each model using both Euclidean and road based distances. Preliminary results have been informative and somewhat suprising. There are pronounced differences ( measured using both relative difference and correlation coefficent) between the results when comparing Euclidean and road distances for the DGR gravity model at the county (r = 0.8088695) and census tract (r = 0.6584075) levels of geography (see 2015 County Relative Difference Map or 2002 Census Tract Relative Difference Map or Web Map or Web Map Application). Obviously accessibility depends on where you are located with respect to the network of roads. Assumming equal accessibility as measured by Euclidean distance (isotrophic surface) is not very realistic. The geography of New Mexico supports this, although I did not think the differences woud be as large as they are in some places.

I have started using the Two-Step Floating Catchment Area (2SFCA) methods with the ArcGIS Desktop (ArcPy) script tools provided by Dr. Fahui Wang ( Wang, 2017). A Web Map showing results for the generalized 2SFCA method has been prepared using the 2002 primary care physician data that was previously available during the earlier work performed by DGR for the New Mexico Health Policy Commission (NM HPC) from 1988 through 2002. This class provided an excellent review and more in-depth background for various Python modules including Python - GeoPandas. I have re-programmed the generalized two-step floating catchment area methods using both SAS and Python3 - GeoPandas within Jupyter Notebook (see class PowerPoint Presentation). However, I did not make as much progress as I would have liked on the statistical comparison portion of my research project and more work will continue this summer and during the fall semester.

Exploratory Data Analysis: Some very preliminary results and data visualizations are now available that have been prepared using the SAS University Edition. These results (see summary statistics PDF) suggest that there is a significant difference in the results obtained from the 2SFCA methods compared with the 1SHGM method. However, these results based on a standard ANOVA should be interpreted with caution as several of the basic ANOVA assumptions are not met, primarily the independence of the data that are spatially autocorrelated and also not normally distributed plus non-common variances. More work is currently underway to research and apply more appropriate statistical methods. These additional statistical methods within R will be used to compare results from various models (one-step vs. two-step with various distance decay functions) and will be available as progress is made (see R Results). In addition, both absolute and relative differences will be calculated to investigate how results vary at each census tract location (see Preliminary Results (1SHGM and 2SFCA) Web Mapping Application - Note: a more efficient version is being prepared using the ArcGIS API for JavaScript).

Hybrid-Zonal Model Results with Various Distance Decay Functions: The boxplot (below) shows the distribution of Physicians per Population (PHYS_PER_POP) from the various models (Acc Methods - Accessibility Methods), all using a hybrid-zonal approach with various distance decay functions (1SED - 1SHGM with DGR type power distance decay; 2SEE - 2SFCA with exponential distance decay; 2SEG - 2SFCA with gaussian distance decay; 2SEP - 2SFCA with power distance decay). Note: the means and medians are close for the 2SFCA and only slightly different than the 1SHGM. It may be that the ANOVA results (can reject the null hypothesis that the means of the different methods are the same p < 0.0001 and accept the alternative hypothesis that at least one mean is not equal to the others) have been greatly influenced by the different variances and presence of extreme outliers.

Power Based Distance Decay Results: This boxplot (below) and histograms and boxplot PDF. shows the distribution of Physicians per Population (PHYS_PER_POP) from various models (Acc Methods - Accessibility Methods), using a hybrid-zonal approach with power based distance decay functions and also the distribution using the county based service area approach (COSVAR). This visual comparison indicates that the one-step methods (1SED and 1SEP) tend to produce more similar results to the widely used county based service area method (COSVAR) than a reasonably comparable two-step method (2SEP). An interactive web map has been prepared showing these results and the relative difference between the one-step and two-step methods at each census tract. There are also some special web mapping applications (see 1SEP and 2SEP, 1SED and 2SEP, and 1SED and 1SEP) that are more useful for comparing these results side-by-side. Note: The county based service area method (COSVAR) is only a population-to-physician ratio, it's the service capacity standard or "rational service area" used by the U.S. Department of Health and Human Services (DHHS) for defining physician shortage areas.

Exponential Distance Decay Results: This boxplot (below) and histograms and boxplot PDF. shows the distribution of Physicians per Population (PHYS_PER_POP) from various models (Acc Methods - Accessibility Methods), using a hybrid-zonal approach with exponential based distance decay functions and also the distribution using the county based service area approach (COSVAR). This visual comparison indicates that the one-step method (1SEE) tends to produce more similar results to the widely used county based service area method (COSVAR) than a reasonably comparable two-step method (2SEE). An interactive web map has been prepared showing these results and the relative difference between the one-step and two-step methods at each census tract. There is also a web mapping application that is more useful for comparing these results side-by-side. Note: The county based service area method (COSVAR) is only a population-to-physician ratio, it's the service capacity standard or "rational service area" used by the U.S. Department of Health and Human Services (DHHS) for defining physician shortage areas.

Gaussian Distance Decay Results: This boxplot (below) and histograms and boxplot PDF. shows the distribution of Physicians per Population (PHYS_PER_POP) from various models (Acc Methods - Accessibility Methods), using a hybrid-zonal approach with Gaussian based distance decay functions and also the distribution using the county based service area approach (COSVAR). This visual comparison indicates that the one-step method (1SEG) tends to produce more similar results to the widely used county based service area method (COSVAR) than a reasonably comparable two-step method (2SEG). An interactive web map has been prepared showing these results and the relative difference between the one-step and two-step methods at each census tract. There is also a web mapping application that is more useful for comparing these results side-by-side. Note: The county based service area method (COSVAR) is only a population-to-physician ratio, it's the service capacity standard or "rational service area" used by the U.S. Department of Health and Human Services (DHHS) for defining physician shortage areas.

Summary of Results: My initial goal for this research project was to compare the one-step and two-step models given an assumption or hypothesis that there would not be any significant differences as both are essentially gravity models. However, I have been surprised by the results. There is clear evidence of a significant difference between the one-step and two-step models that have similar constructs (hybrid-zonal) and employ the same distance decay methods (power, exponential, or Gaussian).

It is important to note that there is no indication that either model is a better measure of reality than the other. This can perhaps only be determined when actual observational provider data records are reviewed, or statistical patient-based surveys are conducted . As these models are only approximations of reality, a given model can only be evaluated as somewhat more appropriate or suitable than the other given the availability and quality of input data, plus how well it matches the desired usage or purpose.

This project has been an exceptional learning exercise that has enabled me to gain a better understanding of statistical, spatial, and GIS software and techniques. I have focused on presenting the results as objectively as possible. However, I do have some personal observations that are worth presenting for discussion purposes. Hopefully other researchers will review these results and make future suggestions that will aid in my interpretations.

The one-step models are easier to operationalize but conceptually provide lower resolution than the two-step models. They can require using less computational resources and GIS facilities. However, it is necessary to calculate road-based distances and use a GIS to display results on a map. Also standard statistical packages and programming environments can be used to perform most of the calculations. The results seem to be more similar to the traditional county-based service capacity standards used by the federal government for measuring physician shortage areas. The mean values are similar to the statewide means and have smaller variances. Although they appear to spread out the potential accessibility measures more evenly, they can also over-estimate in actual shortage areas. Regardless, the one-step model may be a good initial first step beyond the traditional county-based methods. They are easier to explain to decision makers such as state legislators as they are a simple extension of the standard service capacity ratio measure. A one-step model may be more suitable at the state level of geography than a similarly constructed two-step model. As such, it may be a more appropriate choice for healthcare planners seeking necessary increased funding.

The two-step models require more effort to operationalize but conceptually provides higher resolution than the one-step models. More computational resources and GIS facilities are required. A more involved programming effort within the GIS and related scripting environments is necessary as the calculations are more complicated. The resulting mean values are noticeably larger and variances greater than the traditional county-based service capacity standards used by the federal government for measuring physician shortage areas. They seem to not spread out potential accessibility measures as evenly as the one-step models and may perhaps over-estimate and severely under-estimate in some areas. Regardless, they represent a definite technological advancement that should be applied when appropriate. However, they are more difficult to explain to a non-professional audience. A two-step model may be more suitable at the urban and regional levels of geography than a similarly constructed one-step model. Several varieties of two-step models are continuing to be developed by academic researchers and their parctical applications have been demonstrated.

There is more research work to be undertaken in evaluating the similarities and differences (comparing and contrasting) between the one-step and two-step models. This study used the ZIP-Code (post office locations or centroids) to locate physicians because this older data was all that was available. More recent physician data, if available, could be used to more accurately geocode addresses. From a more technical perspective, the effects of both the modifiable areal unit problem (MAUP) and mathematical aspects of the calculations need more evaluation. It is also apparent that healthcare planners could benefit if better scripting applications were developed. These developments could make it easier to review results and allow for better selection of an appropriate model to employ.

I spent more time than expected this summer viewing selected technical sessions from the recent Esri Developer Summit and becoming familiar with the ArcGIS API for Python and the new Spatially Enabled DataFrame (SEDF). Replacing GeoPandas with the SEDF was easy and the calculations of a test 2SEE model were successful. However, the map visualization of results within Jupyter Notebook was more problematic. Using an ArcGIS Online WebMap with the Jupyter Notebook Mapping Widget (arcgis.widgets.MapView) was the best alternative as it allows for a map legend to also be displayed (see below). But I have had problems after adding a results layer directly to the webmap. I can't display the polygon symnology without saving it to ArcGIS Online for rendering with smart mapping and then loading it again in another jupyter notebook cell to display. I hope to learn more about the proper way to do this and resolve my problems shortly.

The additional statistical analyses of results has clearly shown that significant differences are apparent in the comparisons of the one-step (1SHGM) and two-step (G2SFCA) models. A series of scatterplots were prepared and correlation coefficients were calculated that support the results obtained from the previous histograms and T-Tests. The table (see below) summarizes these results. The most similar results when comparing the one-step and two-step models was for those that used a power based distance decay method (1SEP and 2SEP). The one-step (1SEP) model also seemed to produce the most realistic results with a mean value close to the statewide mean (0.6225) and a more even numerical distribution of census tracts below (244) and at or above (255) the national proviter-to-population benchmark (0.79). However, there was not a very even geographical distribution (see map below) as most of the census tracts below the benchmark seem to be in the larger rural census tracts with some exceptions in the southern and southeastern urban areas close to New Mexico's borders with Arizona, Texas and Mexico. Additional research using spatial statistical regression models is planned that will evaluate how results from these geographic access models are related to various measures of health inequality and health disparities.