I have estimated a model using OLS that showed spatial autocorrelation (by Moran's I tool on the model residuals). So, I decided to estimate the model using GWR and compare the results. I am not very experienced with using GWR, but I am very confused as to where to find p values or signs of significance within the local models. I can compare the OLS coefficients with the median of the GWR local coefficients, but cannot really compare the models, with knowing the significance of individual predictors. Is this information somewhere in the output or do I need to specify certain options to get these results? I have seen that some studies have seemed to use Monte Carlo tests of spatial variability for this - can I calculate this test with ArcGis 10? Thanks so much for any guidance!
This is a great question, and one that we get quite a bit. With GWR, there is a local linear equation for each feature in the dataset. The equation is weighted so that nearby features have a larger influence on the prediction of yi than features that are farther away. I do know that our consultant�??s GWR software (Fotheringham, Charlton and Martin) does compute p-values for each coefficient in every one of the local linear equations. However, because doing so is really not appropriate (and we�??ve discussed this with our consultants and they agree with that assessment), we do not report the coefficient p-values for our ArcGIS GWR tool. I know our consultants are looking into other methods for computing those p-values that might be appropriate, but I don�??t believe they have come up with an ideal solution.
Because GWR does not have the strong diagnostics (like p-values, as just one example) that OLS does, we very strongly recommend finding a properly specified OLS model before moving to GWR. Unless you are only interested in predictions (not interested in the coefficients�?� in variable relationships), you cannot trust GWR results unless you can be sure you�??ve found all of the key explanatory variables to model your dependent variable (you can see this easily: run OLS or GWR with 1 important explanatory variable and examine the coefficients; add a second important explanatory variable and notice that the coefficient values change�?� sometimes dramatically; the coefficient values can change 180 degrees, in fact).
Having said that, we have just released a tool and documentation on the Geoprocessing Resource Center to help you find a properly specified OLS model: Exploratory Regression and �??What they don�??t tell you about regression analysis�?�. The Exploratory Regression tool is similar to Stepwise Regression except instead of just looking for high Adj R2 values, it looks for models that meet all of the assumptions of the OLS method (no variable redundancy, no spatial autocorrelation in regression residuals, no model bias, statistically significant coefficients�?�). To download this tool and the associated documentation, check out http://esriurl.com/spatialstats (look for Supplementary Spatial Statistics).
I took this text right out of this forum post, which also has some additional useful information about using condition numbers and local r-squared results of GWR to see how well you're doing, as well as some other information about moving to GWR from OLS.