Geographically Weighted Regression

Short Description	Geographically Weighted Regression is a spatial statistical technique used in urban health studies to examine the relationships between health outcomes and predictors while considering spatial variations.
Data
Suggested tools	GeodaStata
Category	Spatial Modelling
Variable	Multivariable

Overview

Geographically Weighted Regression (GWR) is a spatial statistical technique used in urban health studies to examine the relationships between health outcomes and predictors while considering spatial variations. It accounts for spatial heterogeneity by allowing the relationship between variables to vary across different locations within the study area.

Description

1. Data Collection: Gather data on health outcomes and predictor variables at the desired spatial resolution (e.g., neighborhood or census tract level) within the urban area.

2. Spatial Weight Matrix: Construct a spatial weight matrix to capture the spatial relationships between the data points. This matrix assigns weights to neighboring locations based on proximity.

2. Model Specification: Determine the form of the regression model to be used and select the appropriate variables as predictors. The model can be linear or non-linear, depending on the nature of the relationship between the variables.

Y_i = \beta_{0}(u_i, v_i) + \beta_{1}(u_i, v_i)X_{1i} + \beta_{2}(u_i, v_i)X_{2i} + \ldots + \beta_{p}(u_i, v_i)X_{pi} + \varepsilon_i

$Y_i$  represents the dependent variable for the $i$ th location.

$\beta_{0}(u_i, v_i)$ is the local coefficient for the intercept term at the $i$ th location.

$\beta_{1}(u_i, v_i)$ , $\beta_{2}(u_i, v_i)$ , ..., $\beta_{p}(u_i, v_i)$ are the local coefficients for the independent variables $X_{1i}$ , $X_{2i}$ , ..., $X_{pi}$ at the $i$ th location.

$\varepsilon_i$ represents the residual term for the $i$ th location.

Bandwidth Selection: Choose an appropriate bandwidth parameter that defines the spatial extent of the local analysis. The bandwidth determines the number of neighboring points used to estimate the regression coefficients at each location.

Local Model Estimation: Estimate the regression coefficients locally for each location within the study area using a subset of neighboring data points within the specified bandwidth.

Diagnostic Analysis: Evaluate the goodness-of-fit of the local models and assess the presence of spatial heterogeneity by examining diagnostic statistics such as local R-squared values and residual patterns.