Introduction to Bayesian spatial models
The small-area estimates reported in the Australian Cancer Atlas were generated by fitting statistical models, called Bayesian spatial models, to data for each type of cancer and sex. Bayesian models apply probabilities to the unknown terms in a model and allow these to be updated by new evidence. The term Bayesian comes from Bayes’ Theorem. The spatial aspect means that the models consider the small-area geographical structure of the data. Consistent with the first law of geography [Tobler 1970], we expect nearby areas to be more similar to each other than to areas further away. This assumption means we can smooth over nearby areas to increase the precision of the area-specific estimates, resulting in ‘spatial smoothing’.
Spatial smoothing is an important part of these models for two reasons. First, it reduces the risk of identifying individual people, and second, it improves the stability of the resulting estimates, providing more confidence that any observed differences are real and not just due to chance.
There are numerous ways to conduct spatial smoothing within Bayesian models, including through considering distance between areas, or adjacency. The general concept used in the models for the Australian Cancer Atlas involves defining a neighbourhood of adjacent areas for each of the small areas, such that the estimate for a given area is dependent on areas it shares a boundary with, making the estimate more similar to those of its neighbours. Areas which have small populations will be subjected to greater neighbourhood smoothing than areas with larger populations. Details of the neighbourhood structure and model formulation are given below.
For a specific area, the neighbours are defined as those SA2s having a shared boundary of any length and an estimated annual resident population of at least 5 people. There were additional refinements made to this default definition. The 12 island areas lacking neighbours were each assigned at least one neighbour, and often this was the closest mainland area. This neighbourhood structure was then represented in a matrix, called an ‘adjacency matrix’. Note that symmetry was required in the adjacency matrix, so that any area with a specific area assigned as a neighbour, is also assigned as the neighbour of that area. The adjacency matrix used can be provided on request. Please contact us for more details.
Tobler WR. A Computer Movie Simulating Urban Growth in the Detroit Region. Economic Geography. 1970; 46:234-240.