## 3 Data and Methods  
### Study Area and Data  
Shenzhen was chosen for the empirical analysis. It was selected because existing studies have revealed hierarchical features in healthcare facilities and in accessibility to these facilities in Shenzhen [32]. Shenzhen is one of the special economic zones and megacities in China. It is located in the Pearl River Delta region in China. Shenzhen has undergone rapid socio-economic development in the last four decades since China's reform and opening in 1978. By 2018, Shenzhen has 11 million permanent residents and 1997 square kilometers of land area. It is composed of 10 administrative districts, 55 sub-districts (or _jiedao_ in Chinese), and 771 communities (or _Shequ_ in Chinese) (see Figure 1). In Shenzhen, like in other Chinese cities, public healthcare facilities play a predominant role in the provision of healthcare services to the residents. According to the \"Medical Regulations of Shenzhen Special Economic Zone\" issued by Shenzhen's municipal government in 2016, healthcare facilities in the city are organized as a three-level system. They are the tertiary, secondary, and primary healthcare facilities from the top to the bottom. The primary facilities consist of the community health service centers (CHSCs) serving the city's 771 communities. The secondary and tertiary facilities serve the 10 districts and the city, respectively. The current hierarchical healthcare system in Shenzhen is not well established, wherein the referral system between various levels has not been formulated, and patients are free to choose healthcare facilities at various levels [16]. The data used in this study comprised three types:  
1. community-level population counts;
2. point-level healthcare facilities with attribute information such as names, hierarchy, number of physicians, and addresses;
3. the travel time between community centroids to healthcare facilities. The population counts are from the sixth population census of China, which was conducted in 2010 and is the most up to date of its kind available to the public. Centroids of communities and their geospatial information such as longitude and latitude coordinates are obtained via the geocoding API of Baidu Maps. The average population size of each community is 13,500. Each community is treated as a demand node in our ensuing analyses. Detailed information concerning all the three levels' facilities is available at the official website of Shenzhen Municipal Health Commission [45]. As of December 2018, there were 19 tertiary healthcare facilities, 35 secondary healthcare facilities, and 612 primary healthcare facilities in Shenzhen. The average numbers of physicians in these facilities were 422, 182, and 6 physicians, respectively (Table 1). The primary facilities are the largest in quantity and the smallest in the average number of physicians. They are widely distributed in all districts in Shenzhen (Figure 1). The primary facilities provide incredibly wide-ranging functions and service qualities that they usually become the first choice of patients residing in proximity. Therefore, the maximal equality model may be not applicable to these facilities. Furthermore, existing studies have found that the spatial accessibility to primary hospitals is relatively equal in Shenzhen [32]. Given this, only the tertiary and secondary facilities were considered in this study. In brief, tertiary and secondary healthcare facilities are different mainly in three aspects. First, tertiary facilities usually provide more complicated and higher-level services than secondary facilities. Second, tertiary facilities usually have larger coverage areas than secondary facilities, which can be reflected by the different catchment sizes in the model. Third, the average size (number of physicians) of tertiary facilities is significantly larger than that of secondary facilities. The measurement of spatial accessibility relies on the travel times between demand nodes and different facility locations.