Lab 11 - Multivariate Regression, Diagnostics, and Regression in ArcGIS

This week we expanded our regression analysis from bivariate (or comparing two variables) to multivariate (or comparing more than two variables).  This type of analysis can be accomplished in ArcGIS by using the Ordinary Least Squares (OLS) script tool.

As suggested by ESRI staff, using the OLS tool is a must - even if your target is to run a Geographically Weighted analysis.  By using the OLS tool you can determine if your model is, in fact, the best fit to explain your data.  How one does this is by determining if the OLS results passes the "6 OLS checks", which are:

1.  Are the independent variables helping your model (are they statistically significant)?
2.  Are the relationships as expected (variables are either negatively or positively correlated)?
3.  Are any of the explanatory variables redundant?
4.  Is the model biased?
5.  Do you have all key explanatory variables?
6.  How well are you explaining your dependent variable?

Each of the above can be answered with the slew of stats generated by the OLS report.  For example, to check for model bias you review the Jarque-Bera test results.  This test assesses whether your residuals are normally distributed or not - if this test comes back as statistically significant then you have a problem with skewed (or biased) data.

To determine if you have all explanatory variables it necessary to run the Spatial Correlation (Global Moran's I) tool; the extremely helpful printout generated at the end not only shows your residual distribution, but also lets you know if any clustering or dispersion is statistically significant.  If you have problems here then you need to add more data.

To compare models one simply needs to know the Akaike's Information Criterion (AIC) score and the Adjusted R-squared residual... also helpfully provided within the OLS report.  And if there are issues with what an OLS generated statistic means, then there are plenty of ArcGIS Help files to help you out.  It's actually quite impressive what the ESRI folks have done to make regression analysis easier for the general user.