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Department of Geography

 

A Bayesian approach to modelling binary data: the case of high-intensity crime areas

Research Team: Jane Law and Robert P. Haining

Abstract

This paper reports the fitting of a number of Bayesian logistic models with spatially structured or/and unstructured random effects to binary data with the purpose of explaining the distribution of high intensity crime areas (HIA) in the city of Sheffield, England. Bayesian approaches to spatial modelling are attracting considerable interest at the present time. This is because of the availability of rigorously tested software for fitting a certain class of spatial models. The paper considers issues associated with the specification, estimation and validation, including sensitivity analysis, of spatial models using the WinBUGS software. It pays particular attention to the visualization of results. We discuss a map decomposition strategy and an approach that examines properties of the full posterior distribution. The Bayesian spatial model reported provides some interesting insights into the different factors underlying the existence of the three police defined HIA in Sheffield.

Illustrations

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Figure 1: Model 1 Logistic Regression: Y(i) is the ith binary response variable, p(i) is the probability that the ith ED is an HIA, and is a set of covariates for the ith case.

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Figure 2: Model 2 Logistic Regression: Y(i) is the ith binary response variable, p(i) is the probability that the ith ED is an HIA, and is a set of covariates for the ith case. U(i) is the ith unstructured random effect, which is an independent normal random variable with mean zero and variance .

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Figure 3: Model 3 Logistic Regression: Y(i) is the ith binary response variable, p(i) is the probability that the ith ED is an HIA, and is a set of covariates for the ith case. U(i) is the ith unstructured random effect, which is an independent normal random variable with mean zero and variance . S(i) is the ith spatially structured normal random variable, and is the variance parameter that controls the amount of variability in {S(i)}.

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Figure 4: Model 4 Logistic Regression: Y(i) is the ith binary response variable, p(i) is the probability that the ith ED is an HIA, and is a set of covariates for the ith case. S(i) is the ith spatially structured normal random variable, is the variance parameter that controls the amount of variability in {S(i)}, and g is a spatial interaction or dependency parameter.

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Figure 5: Map decomposition of the odds into the component associated with the covariate and unstructured, and spatially structured random effects.

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Figure 6: Standard errors of posterior means of the spatially structured random effects, S.

Further reading

  • Law, J., and R. P. Haining (2004). “A Bayesian Approach to Modelling Binary Data: The case of High-Intensity Crime Areas.” Geographical Analysis 36:3 197-216.
  • Craglia, M., R. Haining, and P. Signoretta (2001). “Modelling High-Intensity Crime Areas in English Cities.” Urban Studies 38(11), 1921-1941.