1 Introduction
I recently attended a class on spatial regression with Prof Bob Haining. He described the issue of spatially correlated errors and the problems this poses in spatial regression.
The key issue is that spatial data often violates the assumption in regression models that the errors are independent.
A simple regression model applied to spatial data based on ZONES:
\(Y_{i} = \beta_{0} + ZONE_{i} + \beta_{1} X_{1i} + e_{i}\)But with spatial data it is likely that the errors are spatially correlated. This is likely to mean the point estimate of beta 1 is OK but the Standard Error is wrong.
This might be due to the scale of the study units, which may not capture the variation of exposure and outcome adequately. Or there might be unmeasured explanatory variables that have not been accounted for.
I am mostly concerned with EXPLANATORY modelling in which a particular exposure of interest is to be assessed. Examples include a weather variable (temperature), an air pollutant (PM10) or some measure of socio-economic deprivation in an area (SEIFA scores in Australian census data). In these models I tend to include a number of 'nuisance' parameters to control for confounding; or interaction terms to account for effect modification. In this type of model the performance of the model over-all is not that important, I just want to control for the most important confounders so that my estimate of the exposure of interest is as rigorous as possible.
Therefore the problem that spatially correlated errors pose for these models is slightly different to that which affects models aimed at PREDICTION: I am not concerned so much with the model's fit to the data, rather the confidence around the point-estimate of the parameter for the exposure of interest.
Simplistically I took away the following messages:
2 The Spatial Error Model
So we could model allowing for correlated errors:
\(Y_{i} = \beta_{0} + ZONE_{i} + \beta_{1} X_{1i} + \eta_{i}\)Where:
\(\eta_{i}\) = Spatially autocorrelated errors.
3 The Spatial Lag Model
Or we could include a term for the neighbours, thus absorbing the correlated errors:
\(Y_{i} = \beta_{0} + ZONE_{i} + \beta_{1} X_{1i} + \rho(Neighbours Y_{ij}) + e_{i}\)Where:
\(\rho_(Neighbours Y_{ij})\) = is an additional explanatory variable which is the value of the dependent variable in neighbouring areas.
4 Spatially Lagged Independent Variable(s)
This is almost a variation of the spatial lag model, except that we include a term for the exposure variable in the neighbours, and therefore 'smooth' the effect of the exposure from what was observed in any area to make it relevant to it's neighbours as well:
\(Y_{i} = \beta_{0} + ZONE_{i} + \beta_{1} X_{1i} + \beta_{2L} X_{2ij} + e_{i}\)Where:
\(\beta_{2L} X_{2ij}\) = is the independent variable X2 that is spatially lagged.
5 Discussion
5.1 How to decide which model to fit?
So the burning question is how to choose between the various spatial models? Prof Haining had some suggestions, but he noted that sometimes two could be equally appropriate. He suggested that the spatial lag model makes the strong assumption that there is a relationship between the outcome in a neighbouring area with the index zone. This suggests some kind of contagion or dispersion effect. He was not keen to fit this model in circumstances where the causal mechanism did not support such a relationship, suggesting the spatially weighted error model was more suited, but that "in practice they often give the same result".
In my situation where I am not concerned with the actual autocorrelation but with tightening up the standard error on my exposure of interest, I think I might plead forgiveness and try fitting the spatial lag model as it seems easier.
6 Conclusion
Stay tuned.
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