Selecting the appropriate spatial unit of analysis, i.e. the operational definition for residential areas, is critical for minimizing aggregation errors 230. Aggregation error arises from the distribution of individuals around the centroid of spatial units 2. As spatial units vary in size from smaller areas, e.g. census blocks, to larger ones, e.g. census tracts, accessibility measured for smaller units is less subject to aggregation error than that measured for larger spatial units 2.

To evaluate the geographical accessibility of a service for a population living in a residential area, e.g. a census tract, three methods can be used 2; they are illustrated in Figure 1. The first method consists in computing the distance between the centroid of the census tract and the service (Figure 1.a). This method shows the inappropriateness of ignoring the spatial distribution of the population inside the census tract 2.

The second method consists of calculating the population-weighted mean centre of the census tracts (Equation 1) and then evaluating the distance between this new location and the service. Toward this end, smaller spatial units entirely contained by the census tracts can be used, such as dissemination areas, census blocks, or postal codes. This method accounts for the spatial distribution of the population inside the census tract in order to minimize aggregation error.

( x i ¯ , y i ¯ ) = ( ∑ b ∈ i w b x b ∑ b ∈ i w b , ∑ k ∈ i w b y b ∑ b ∈ i w b ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@63D3@

Where:

w_b = total population of spatial unit b completely within census tract i (i.e. dissemination area or census block or postal code).

x_b and y_b = X and Y coordinates of spatial unit b.

Finally, the third method consists of computing the distance between the services and each centroid of spatial units completely within census tracts, and then calculating the average of these distances weighted by the total population of each unit (Figure 1.b and 1c). In comparison with the previous methods, this one is more accurate because it more exactly accounts for the distribution of the population inside the census tract.

The five most commonly used measures of accessibility are: 1) the distance to the closest service, 2) the number of services within n metres or minutes, 3) the mean distance to all services, 4) the mean distance to n closest services, and 5) the gravity model. If the more accurate aggregation method detailed previously is selected, these accessibility measures can be written as:

Z i a = ∑ b ∈ i w b ( min ⁡ | d b s | ) ∑ b ∈ i w b , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdggaHbaakiabg2da9KqbaoaalaaabaWaaabuaeaacqWG3bWDdaWgaaqaaiabdkgaIbqabaGaeiikaGIagiyBa0MaeiyAaKMaeiOBa42aaqWaaeaacqWGKbazdaWgaaqaaiabdkgaIjabdohaZbqabaaacaGLhWUaayjcSdGaeiykaKcabaGaemOyaiMaeyicI4SaemyAaKgabeGaeyyeIuoaaeaadaaeqbqaaiabdEha3naaBaaabaGaemOyaigabeaaaeaacqWGIbGycqGHiiIZcqWGPbqAaeqacqGHris5aaaakiabcYcaSaaa@5225@

Where:

Zia MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdggaHbaaaaa@2FE3@ = mean distance between census tract i and closest service.

w_b = total population of spatial unit b completely within census tract i.

d_bs = distance between spatial unit b and service s.

Z i b = ∑ b ∈ i W b ∑ j ∈ S S j ∑ b ∈ i W b , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacbmGae8NwaO1aa0baaSqaaiabdMgaPbqaaiabdkgaIbaakiabg2da9maalaaabaWaaabuaeaacqWFxbWvdaWgaaWcbaGaemOyaigabeaaaeaacqWGIbGycqGHiiIZcqWGPbqAaeqaniabggHiLdGcdaaeqbqaaiab=nfatnaaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabgIGiolabdofatbqab0GaeyyeIuoaaOqaamaaqafabaGae83vaC1aaSbaaSqaaiabdkgaIbqabaaabaGaemOyaiMaeyicI4SaemyAaKgabeqdcqGHris5aaaakiabcYcaSaaa@4D15@

Where:

Zib MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdkgaIbaaaaa@2FE5@ = mean number of services within n metres or minutes of census tract i.

w_b = total population of spatial unit b completely within census tract i.

S = all services.

S_j = number of services within n metres or minutes of spatial unit centroid b with S_j = 1 where d_bs ≤ n and S_j = 0 where d_bs > n.

Z i c = ∑ b ∈ i w b d b s ∑ b ∈ i w b , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdogaJbaakiabg2da9KqbaoaalaaabaWaaabuaeaacqWG3bWDdaWgaaqaaiabdkgaIbqabaGaemizaq2aaSbaaeaacqWGIbGycqWGZbWCaeqaaaqaaiabdkgaIjabgIGiolabdMgaPbqabiabggHiLdaabaWaaabuaeaacqWG3bWDdaWgaaqaaiabdkgaIbqabaaabaGaemOyaiMaeyicI4SaemyAaKgabeGaeyyeIuoaaaGccqGGSaalaaa@4933@

Zic MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdogaJbaaaaa@2FE7@ = mean distance between census tract i population and all services.

w_b = total population of spatial unit b completely within census tract i.

d_bs = distance between spatial unit centroid b and service s.

Z i d = ∑ b ∈ i w b ∑ s d b s n ∑ b ∈ i w b , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdsgaKbaakiabg2da9KqbaoaalaaabaWaaabuaeaacqWG3bWDdaWgaaqaaiabdkgaIbqabaWaaabuaeaadaWcaaqaaiabdsgaKnaaBaaabaGaemOyaiMaem4CamhabeaaaeaacqWGUbGBaaaabaGaem4CamhabeGaeyyeIuoaaeaacqWGIbGycqGHiiIZcqWGPbqAaeqacqGHris5aaqaamaaqafabaGaem4DaC3aaSbaaeaacqWGIbGyaeqaaaqaaiabdkgaIjabgIGiolabdMgaPbqabiabggHiLdaaaOGaeiilaWcaaa@4E24@

Zid MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdsgaKbaaaaa@2FE9@ = mean distance between census tract i and n closest services.

w_b = total population of spatial unit b completely within census tract i.

d_bs = distance between spatial unit centroid b and service s, d_bs is sorted in ascending order.

n = number of closest services to be included in measure.

Z i e = ∑ b ∈ i w b ∑ S S w s d b s − α ∑ b ∈ i w b , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdwgaLbaakiabg2da9KqbaoaalaaabaWaaabuaeaacqWG3bWDdaWgaaqaaiabdkgaIbqabaWaaabuaeaacqWGtbWudaWgaaqaaiabdEha3jabdohaZbqabaGaemizaq2aa0baaeaacqWGIbGycqWGZbWCaeaacqGHsisliiGacqWFXoqyaaaabaGaem4uamfabeGaeyyeIuoaaeaacqWGIbGycqGHiiIZcqWGPbqAaeqacqGHris5aaqaamaaqafabaGaem4DaC3aaSbaaeaacqWGIbGyaeqaaaqaaiabdkgaIjabgIGiolabdMgaPbqabiabggHiLdaaaOGaeiilaWcaaa@533B@

Zie MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOwaO1aa0baaSqaaiabdMgaPbqaaiabdwgaLbaaaaa@2FEB@ = mean value of potential gravity.

w_b = total population of spatial unit b completely within census tract i.

S = number of services in study area.

d_bs = distance between spatial unit centroid b and service s.

α = friction parameter (usually 1, 1.5 or 2).

S_ws = weight given to the service s such as its size (for example, number of beds for a hospital).

Four types of distance are typically used for calculating accessibility measures: Euclidean distance (straight-line), Manhattan distance (distance along two sides of a right-angled triangle opposed to the hypotenuse), shortest network distance and shortest network time (Figure 2) 2831. Euclidean and Manhattan distances can easily be computed using geographic coordinates:

d i j = ( x i − x j ) 2 + ( y i − y j ) 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpdaGcaaqaaiabcIcaOiabdIha4naaBaaaleaacqWGPbqAaeqaaOGaeyOeI0IaemiEaG3aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaakiabgUcaRiabcIcaOiabdMha5naaBaaaleaacqWGPbqAaeqaaOGaeyOeI0IaemyEaK3aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkdaahaaWcbeqaaiabikdaYaaaaeqaaOGaeiilaWcaaa@46F8@

d_ij = |x_i - x_j| + |y_i - y_j|,

Where:

X_i and Y_i = X and Y coordinates of point i with a plane projection.

On the other hand, calculation of network distance and network time distance – which represent respectively the shortest and fastest paths between two points using a street network – is more complex. Indeed, the computation of these two distances necessitates geometric network files – with directions, speed limits, turning restrictions, and delays available for each street segment – integrated into GIS, and a GIS module or GIS software specialized in transportation analysis (ESRI Network Analyst Extension or TransCad software, for example).

Shortest network and network time distances represent two different objectives. Shortest network distance is useful for evaluating the path between two points as if taken on foot; consequently, it is frequently used in studies on the accessibility of "proximal" services and facilities 41820323334. Shortest time distance is more accurate for evaluating distances by car or public transportation.

Table 3 presents results for global correlation coefficients between the four types of distances computed for the entire sample of health services (n = 642). Three observations can be made. First, at the metropolitan scale, independently of the type of distance used, results are globally similar as indicated by high correlation coefficient values (greater than 0.95). Second, in comparison with Manhattan distance, Euclidean distance is most strongly correlated with the more accurate network path and time distances. Thus, if it is impossible to compute network distance in a study focussing on geographical accessibility in the Montréal CMA, Euclidean distance seems preferable to Manhattan distance. These results are in line with those of Apparicio et al. 28 for eight Canadian metropolitan areas (Toronto, Montréal, Vancouver, Ottawa-Hull, Calgary, Edmonton, Québec and Winnipeg), and with those of Fone et al. 12 for Caerphilly county borough in Wales. Finally, as expected, correlations between both network distances are almost perfect (0.992). This means that if directions and speed limits are unknown for computing the shortest network time, the shortest network distance is a very reliable alternative.

Although global correlations are high, they are not perfect (values differ from one). For this reason, local variations at the intra-metropolitan scale must exist and should be examined in detail. Figure 4 presents local Pearson coefficients between Euclidean distance and shortest network time, and between Euclidean and Manhattan distances for the geographical accessibility of the 642 health services computed from the centroids of census tracts, dissemination areas, and blocks.

Results show similar spatial patterns for the three spatial scales (census tract, dissemination area and block): with increasing distance from the central business district, correlations are reduced between Euclidean distance and shortest network time, and between Euclidean and Manhattan distances. For all spatial units in the centre of the Island of Montréal and those located on the south shore, correlations are higher. For those located on the periphery of the CMA, notably on the north shore, characterized by suburban areas, correlations are weaker, often lower than 0.9.

These results illustrate that for the Island of Montréal, integrating Euclidean distances at the census tract, dissemination area and block levels into statistical analysis, e.g. in regression or multilevel analysis, would yield similar results to those obtained if network distances were computed. However, if the focus is on the CMA as a whole, or on specific parts of the CMA, namely, those located in the northern suburbs, then results are likely to vary as a function of the distance type used to compute geographical accessibility.

The global analysis of aggregation errors is performed by means of Spearman correlations between the three methods of aggregation used to calculated 20 accessibility measures at the census tract level using the more accurate distances (network distances). Results are shown in Table 4 for hospitals only, although similar patterns of correlation were observed for other health services.

Correlations between the three aggregation methods are high (>0.9) for all measures of accessibility except for the number of services within 500, 1000 and 2000 metres. For example, correlation between the least exact aggregation method (census tract centroid) and the most exact based on blocks within census tracts is 0.588 for the number of hospitals within 500 metres, 0.776 for those within 1000 metres, and 0.898 for those within 2000 metres. This means that if we want to assess service provision in a close-proximity area around a census tract, it is preferable to use an aggregation method that precisely accounts for the distribution of population within it; if not, the risk of error might be considerable.

A second stage of comparison of aggregation methods consists in assessing the absolute difference between the geographical accessibility results obtained from the methods based on census tract and dissemination areas centroids in relation to the most accurate method based on blocks within census tracts. The descriptive statistics for local errors are reported in Table 5 for hospitals.

Not surprisingly, the local errors are on the whole quite small, though not insignificant: for example, compared with the most accurate method, the census tract centroid method misestimates the distance to the closest hospital by an average of 365 m, and the dissemination area method by an average of 134 m. Up to the third quartile (75%), the local errors are still quite small: for 75% of census tracts, the error associated with the census tract centroid approach is less than 365 m. However, in 10% of cases, the error is greater than 948 m, and in 5% of census tracts the error is greater than 1.5 km (Table 5). Despite the high correlations, significant errors in the measurement of geographical accessibility can occur in a small number of cases.

Absolute differences between aggregation methods for the closest hospital computed using shortest network distance and shortest distance time are further mapped in Figure 5. Again, stronger absolute aggregation errors are observed in suburban areas on the south and north shores of the CMA; errors remain smaller in central areas of the Island of Montréal.

For the purposes of statistical studies at a general level, the least precise aggregation method – based on census tract centroids – is adequate: it enables the broad identification of areas in Montréal that have the least access to health services. However, if one wishes to reach more precise conclusions for specific neighbourhoods, major errors appear for 5% to 10% of census tracts.