Alternative Distance Metrics Alternative Distance Metrics for Enhanced Reliability of for Enhanced Reliability of

Spatial Regression Analysis of Spatial Regression Analysis of Health DataHealth Data

Stefania Bertazzon & Scott Stefania Bertazzon & Scott Olson, 2008Olson, 2008

Here, a spatial regression model is calibrated to increase its reliability by specifying a spatial weighting matrix that best captures neighbourhood connectivity, hence the spatial dependence in the observed variables.

IntroductionIntroduction

The method we proposed to achieve this goal involves altering the method used for calculating the distance metrics inherent to the foundation of the spatial weighting matrix in spatial autoregressive models. This alternative approach can reflect overall spatial connectivity more accurately than the traditionally utilized distance metrics.

IntroductionIntroduction

Locational foundation of spatial regression.

Spatial RegressionSpatial Regression

Y = Xβ + ρWY + ε

Locational foundation of spatial regression.

Spatial RegressionSpatial Regression

Y = Xβ + ρWY + ε

Y = Xβ + ρ Y + ε

Locational foundation of spatial regression.

Contiguity MatrixContiguity Matrix

point point ‘j’‘j’

Distance MetricsDistance Metrics

Euclidean Distance:dij = [(xi – xj)2 + (yi –

yj)2]1/2

Manhattan Distancedij = | xi – xj | + | yi –

yj |

point point ‘i’‘i’

Combination of all historical street development patterns:

Evolution of street patterns since 1900 showing gradual adaptation to the car.

From: Southworth M. (1997). Streets and the Shaping of Towns and Cities. New York: McGraw-Hill.

Distance MetricsDistance Metrics

source: Jacobs, Allan. (1993). Great Streets. Cambridge, Mass.: MIT Press.

Calgary, Canada

The purpose is not to mimic the city road network but to select a distance metric that best represents neighbourhood connectivity, which is consequently defined by the interplay of road network and urban design.

point point ‘j’‘j’

Alternative Distance Alternative Distance MetricMetric

Euclidean distance: dij = [(xi – xj)2 + (yi – yj)2]1/2

Manhattan distance dij = | xi – xj | + | yi – yj |

Minkowski distance dij = [(xi – xj)p + (yi – yj)p]1/p

[ 1 1 ] 1/1

point point ‘i’‘i’

Case StudyCase Study

Where:Y = number of catheterization cases;X1 = number of 2 parent families with children at home***; X2 = number of persons with a post-secondary, non-university degree**;X3 = family median income**;X4 = number of persons with grade 13 or lower education***.

Negative relationship with YPositive relationship with Y

Y = βo+ ρWY + β1X1 + β2X2 + β3X3 + β4X4 + ε

Case StudyCase Study

Manhattan distance

p=1

Euclidean distance

p=2

Minkowski p=1.6

Study area: Calgary, Canada using data at the Census Tract scale

FindingsFindings

FindingsFindings

FindingsFindings

• Decision making

• When utilizing spatial regression models, alternative distance metrics should not be neglected.

ImportanceImportance

• An investigation of specific urban design and connectivity will aid the interpretation of optimal p value;

• We envisage the extension of this work to include local analyses;

• A procedure to semi-automate the optimal distance metric selection.

Next StepsNext Steps

Thank You;Grazie