Global Moran's I

[RichardMassatti]

Hey everyone! I???m a beginner/intermediate arc user, and I have a question about testing for spatial; auto-correlation with Moran???s I. I???m running some test data through ARCMAP to prepare for my dissertation analyses. Since I am analyzing all Ohio census tracts, I???m wondering about the best way to conceptualize spatial relationships. From what I???ve read, I???ve narrowed the best methods down to inverse distance weighting, distance band, and zone of indifference, but I???m still not sure which one to choose. (I didn???t choose the polygon contiguity options since the tracts are very different sizes and shapes.) When I use the Global Moran???s I test, I get statistically significant results for each of these methods; results are most similar for inverse distance weighting and distance bad with z-scores being lower for zone of indifference. A distance band was applied to each of the analyses. Do you have any opinions on the best method?

If it helps, the data with the polygon is continuous.

[EricKrause]

Any of the three can be justified, but I prefer fixed distance band among those three.  This is because fixed distance band is the conceptualization where it is most clear that the test statistic will converge to a normal distribution with increasing sample size.

If you're getting the results you want/expect from fixed distance band, I would stick with that.

[JeffreyEvans]

None of these methods make sense to me with polygon input. The spatial weights matrix (Wij) for lattice data (polygons) should be based on Nth order contingency. Distance based weights will be biased due to variability in polygon size. This is quite well established in the literature.

You may want to look at the distribution of your data. If it is non-normal, consider applying a statistical transformation. Moran's-I relies on the mean and if the data is skewed the statistic will be biased. Normally, you can account for this through a permutation test. However, since the ArcGIS implementation of Moran's-I does not perform a permutation significance test then this is not corrected for.

[RichardMassatti]

Thanks everyone for the replies. Jeffery, should I use a weighting matrix based on contiguity if I go your route? (The continuous data itself is normally distributed.) Also, would GeoDa be a good program to use for the Moran's I permutation test? Thanks!

Also, I'll be using GWR with my data, so should I create the spatial weights matrix in GeoDa and then import it into ArcMap?

[JeffreyEvans]

I would highly recommend GeoDA. It has a nice GUI, is easy to use and has a very robust permutation test. It also provides a local autocorrelation test (LISA), which is a requirement for local regressions such as GWR. You should not apply a GWR approach without quantifying the presence of nonstationarity based on a local test. I should note that the LISA statistic is also available in ArcGIS. The Moran's-I is a first order statistic and does not represent 2nd order spatial structure that would justify specification of a GWR. Global autocorrelation can easily be accounted for with a autoregression term in an OLS, mixed effects models or spatial regression (available in GeoDa). 

The GWR method has come under some serious criticism and is currently not considered a viable method in the spatial statistics community. Since this is in support of your graduate work, I would recommend reading Páez et al., (2011) before making the decision to implement GWR. It would be a shame to put forth considerable effort for unpublishable results.  

Páez A, Farber S, Wheeler D, (2011) A simulation-based study of geographically weighted regression as a method for investigating spatially varying relationships. Environment and Planning 43(12):2992�??3010. http://www.envplan.com/abstract.cgi?id=a44111

Wheeler & Calder (2007) propose a Bayesian approach that is a viable alternative to GWR.

Wheeler, D., C.A. Calder (2007) An assessment of coefficient accuracy in linear regression models with spatially varying coefficients. Journal of Geographical Systems. 9(2):145-166. http://link.springer.com/article/10.1007/s10109-006-0040-y?no-access=true