Hierarchical Additive Modeling of Nonlinear Association with Spatial Correlations-An 
Application to Relate Alcohol Outlet Density and Neighborhood Assault Rates

 

Previous studies have suggested a link between alcohol outlets and assaults.  
In this research, we explore the effects of alcohol availability on assaults 
at the census tract level over time.  Also, we use a natural experiment to 
check whether a sudden lose of alcohol outlets is associated with deeper decreasing 
in assault violence.  Several features of the data raise statistical 
challenges: (1) the association between covariates (for example, the alcohol 
outlet density of each census tract) and the assault rates may be complex 
and therefore cannot be described using a linear model without covariates 
transformation; (2) the covariates may be highly correlated with each other;  
(3) there are a number of observations that have missing inputs; and (4) there 
is spatial association in assault rates at the census tract level.  We propose 
a hierarchical additive model, where the nonlinear correlations and the complex 
interaction effects are modeled using the multiple additive regression trees (MART) 
and the residual spatial association in the assault rates that cannot be explained 
in the model are smoothed using a Conditional Autoregressive (CAR) method. We develop 
a two-stage algorithm that connects the non-parametric trees with CAR to look for 
important covariates associated with the assault rates, while taking into account 
the spatial association of assault rates in adjacent census tracts. 
The proposed method is applied to the Los Angeles assault data (1990-1999).  
To assess the efficiency of the method, the results are compared with those obtained 
from a hierarchical linear model.