1. Briggs Henan University 2010
Regression in geoDA
Example regression analyses for Illiteracy Rate ( ILLITERACY)
ChinaData.shp (n=35)
1. Simple regression with URBAN_POP_
ChinaData_29 (n=29)
2. Simple regression with URBAN_POP
3. Multiple regression with URBAN_POP
and RMB_PC_UR_
4. Spatial lag and error multiple regression
5. Multiple regression with log of Illiteracy
Briggs Henan University 2010

2. Running Regression in geoDA: I1
File>Open Shape File
ChinaData
Tools>Weights>
Open or Create
Need weights to test for spatial autocorrelation.
Generally, always use a weights file.
You can begin with Method>Regress if
--very large number of observations (over 1,000)
--no spatial weights
--data only in a .dbf file 2
Methods>Regress
Place as below
If you have a large number of observations, do not
Need this for Moran’ s I for residuals

3. Running Regression in geoDA: II
Select one dependent variable
One or more independent variables
Select type of regression:
Classic or Lag or Error
Warning-bug!
Use Suggested name.
The names are
reversed here!
Click OK to save these.
Saves values for Predicted Y and Residuals in the table
--use Table>>Promotion to see them in table.
--you can map them or draw graphs
--use Table >> Save to Shapefile if you want to keep them permanently
Click RUN, then Click SAVE

4. Running Regression in geoDA: III
Results are saved in this text file.
It is saved in the same folder as the shapefile.
You can rename it and change location.
Click OK to see the results.
(You can also open the file later with a program such as Notepad)
--scroll to end of file since results are added to end if file already exists
Warning: if you want the residuals (see previous slide) you must click Save before clicking OK
Click Reset to run a different regression
The results

5. Summary: Running Regression in geoDA
Warning-bug!
Use Suggested name.
The names are reversed here!
Select variables as below.
Select type of regression:
Classic Lag Error
File>Open Shape File
ChinaData
Tools>Weights>
Open or Create
(need weights to test for spatial autocorrelation in residuals)
Methods>Regress
Place as below
Click OK to save these.
Use Table>Promotion to see them in table.
Click OK in Regression window to see results
--scroll to end of file since results are added to end if file exists already
Click RUN, then Click SAVE

6. Regression for Provinces: n = 35
Next slide shows results from running a simple regression with ChinaData.shp
Y = Illiteracy rate (ILLITERACY)
X = % of population urban (URBAN_POP_)
All provinces included
Note problems with
Extreme value for Xizang/Tibet
Zeros (0) for missing data on X variable
(Taiwan, Macau, Hong Kong, P’eng-hu)
Solution: Reduced data set to 29 using ArcGIS
(do not know how to do this in geoDA!)
Briggs Henan University 2010

7. Results for simple regression
Display table: Table >Promotion Plot using: Explore >ScatterPlot
Results for simple regression
Note: mean of residuals is always zero
Residual Variation
OLS_Resid v. Urban Pop%
Total Variation
Illiteracy v. Urban Pop%
Predicted by Regression
OLS_Predict v. Urban Pop%
Extreme
value identified by linking:
Xizang/Tibet
Briggs Henan University 2010

8. Partitioning the Variance on Y
Residual Variation
OLS_Resid v. Urban Pop%
Total Variation
Illiteracy v. Urban Pop%
Predicted by Regression
OLS_Predict v. Urban Pop%
(Y-Ỹ) Y Ỹ Y
SS Residual
or Error Sum of Squares
SS Total
or Total Sum of Squares
SS Regression
or Explained Sum of Squares
Briggs Henan University 2010

9. Simple Regression Results from GeoDA: general
Statistics for dependent variable
n = 35
Not statistically significant
Results for overall regression
explains only 4.6% of variance in Y
Sigma-square= Variance of the estimate = /33=
SE of regression=standard error of the estimate=√ =
Identical in simple regression
Results for each regression coefficient
Y= X
Briggs Henan University 2010

10. Simple Regression Results from GeoDA: spatial
Moran’s I for regression residuals
--not statistically significant (p=.09)
Space > Univariate Moran
for variable: OLS_Resid
Same results!
Briggs Henan University 2010

11. Results with omitted observations: much better!
Now explains 33.41%
But probably non-linear
Statistically significant
Spatial autocorrelation not a problem
Data for China Provinces 29:
excludes Xizang/Tibet, Macao, Hong Kong, Hainan, Taiwan, P'eng-hu
Briggs Henan University 2010

12. Briggs Henan University 2010
Multiple Regression Results n = 29 Illiteracy with % Pop Urban and Urban Income
Overall Results
Results for each variable
significant
Not significant
Spatial Results
Not significant
Briggs Henan University 2010

13. Residual Analysis: Illiteracy v. Urban Pop % and UrbanIncomePerCapita
Moran’s I = .0226
p =
Not statistically significant
No Spatial autocorrelation in residuals
Briggs Henan University 2010

14. Spatial Error Model Results illustrative only: not needed
Spatial error not significant
Briggs Henan University 2010

15. Spatial Lag Model Results illustrative only: not needed
Spatial lag not significant
Briggs Henan University 2010

16. Regression Results Summary
Overall
Urban Pop
Urban Income
*Spatial Term R2
Adj2
Akaike F
F-prob
coeff
Test Stat
prob
coef
Simple-35
0.046
0.017
231.65
1.60
0.215
-6.58
-1.263
0.1636
1.678
0.0934
Simple-29
0.334
0.309
155.42
13.55
0.001
-16.15
-3.681
0.0272
0.578
0.5631
Multiple
0.384
0.337
155.16
8.11
0.002
-26.80
-3.151
0.000
1.452
0.159
0.383
0.7015
Spatial Error
0.385
155.13
-27.02
-3.411
1.572
0.116
-0.162
0.8716
Spatial Lag
0.387
157.05
-26.00
-3.128
0.006
1.486
0.137
0.0720
0.340
0.7339
*Spatial Term
OLS: for Moran's I
For Multiple Regression 29
Lag: for W_Illiteracy
Robust LM (lag)
1.312
0.2520
Error: for Lambda
Robust LM (error)
1.220
0.2693
Briggs Henan University 2010

17. Note on: Variables Saved for Spatial Models
Again, labels are reversed. Use suggested variable names.
ERR_ indicates use of Spatial Error model.
LAG_indicates use of Spatial Lag Model
OLS_ indicates use of classic model
For the spatial lag model, there is a distinction between the residual and the prediction error. The latter is the difference between the observed value and the predicted value that uses only exogenous variables, rather than treating the spatial lag Wy as observed. (Documentation for 905i, page 53)
Prediction error (xxx_PRDERR): calculated without including spatial term.
Residual error (xxx_RESIDU): calculated including spatial term
Briggs Henan University 2010

18. Improving the model Relationship is Non-linear Use log of Illiteracy
Table >> Add Column Table >> Field Calculator
Improving the model Relationship is Non-linear Use log of Illiteracy
Briggs Henan University 2010

19. The same plots using Excel Relationship is Non-linear
Illiteracy
Log of Illiteracy
Urban pop %
Briggs Henan University 2010

20. Briggs Henan University 2010
Y = Log of Illiteracy
R2 increases from
38% to 83% !
Urban Income now significant and Urban Population is not!
Briggs Henan University 2010

21. Log of Illiteracy: makes relationship linear
Overall
Urban Pop
Urban Income
*Spatial Term R2
Adj2
Akaike F
F-prob
coeff
Test Stat
prob
coef
Simple-35
0.046
0.017
231.65
1.60
0.215
-6.58
-1.263
0.1636
1.678
0.0934
Simple-29
0.334
0.309
155.42
13.55
0.001
-16.15
-3.681
0.0272
0.578
0.5631
Multiple
0.384
0.337
155.16
8.11
0.002
-26.80
-3.151
0.000
1.452
0.159
0.383
0.7015
Multiple Log Y
0.837
0.824
560.07
66.69
-1.800
0.083
-2.975
0.006
-0.548
0.5839
*Spatial Term
OLS: for Moran's I
Urban Income now significant, and % urban not significant.
--these two variables are highly intercorrelated
--see next slide
Briggs Henan University 2010

22. Inter-Correlation between Urban Population and Urban Income
R2 for Urban Pop versus Urban Income 0.84
R is .92
N=29
Urban Population
Urban Income
Briggs Henan University 2010

23. Briggs Henan University 2010
Table >> Add Column then use Table >> Field Calculator
Creating a better model
Transforming dependent and/or independent variables can often improve the predictive capability of regression models
geoDA has several capabilities to support this.
Briggs Henan University 2010

24. Other software options for multiple regression
Multiple regression of the type discussed here is not available in ArcGIS
Only geographically weighted
regression available
(there is a multiple regression for raster data
but it is only in ArcInfo Workstation—difficult to use)
Use geoDA to create spatial lag variables, then use standard statistical packages such as SAS, SPSS or STATA
Use R
Free open source software, but difficult to use
CrimeStat III has some support for spatial regression
For a good list of spatial software sources, go to:
Briggs Henan University 2010

25. What have we learned today?
How to use geoDA to run
classic regression models
Spatial Lag models
Spatial Error Models
Importance of examining data for “problems”
Can have a very large affect on results
Missing data and zeros
Extreme values can dominate results
Using transformations to create a better model
Briggs Henan University 2010

26. Briggs Henan University 2010

27. Geographically Weighted Regression
Briggs Henan University 2010

28. Geographically Weighted Regression
The idea of Local Indicators can also be applied to regression
Its called geographically weighted regression
It calculates a separate regression
for each polygon and its neighbors,
then maps the parameters from the model, such as the regression coefficient (b) and/or its significance value
Mathematically, this is done by applying the spatial weights matrix (Wij) to the standard formulae for regression
See Fotheringham, Brunsdon and Charlton Geographically Weighted Regression Wiley, 2002 Xi
Briggs Henan University 2010

29. Problems with Geographically Weighted Regression
Each regression is based on few observations
the estimates of the regression
parameters (b) are unreliable
Need to use more observations than just those with shared border, but
how far out do we go?
How far out is the “local effect”?
Need strong theory to explain why the regression parameters are different at different places
Serious questions about validity of statistical inference tests since observations not independent Xi
Briggs Henan University 2010

30. Briggs Henan University 2010
GWR in ARCGIS
Requires ArcInfo, Spatial Analyst or Geostat. Analyst license
Shapefile is created:
Open its table to see results
for each polygon there are standard regression results
Condition variable: indicates when the results are unstable due to local multicollinearity
Results not good if condition > 30, Null, or e+308
Use source_ID to join with FID of original data to identify observations
Briggs Henan University 2010

31. Usage Tips from ArcGIS Help
Use projected data
Observations included in each regression depend on kernal type, bandwidth method and bandwidth distance parameters set by user
Max of 1,000 observations in any one local regression
Multicollinearity can be a problem
if variables cluster spatially
if use binary/nominal/categorical variables
Never use dummy variables (1/0) to index spatial regions
(Multicollinearity: intercorrelation between independent variables)
Not appropriate for small data sets: need several hundred observations
Shapefiles cannot store “nul l” values: treated as zero. Be sure there is no missing data
Briggs Henan University 2010

32. Briggs Henan University 2010
Running GWR in ArcGIS
Briggs Henan University 2010

33. Execution Dialog for GWR in ArcGIS
Results presumable for global regression?????
--R2 value does not agree with results from geoDA?
Briggs Henan University 2010

34. Mapping Results from GWR in ArcGIS
(Default) standardized residuals
--the bigger the absolute value the poorer the prediction?
Regression coefficient for % Urban Pop
--larger impact of urban pop in south east China.
Briggs Henan University 2010

35. Briggs Henan University 2010
Join with the original shapefile using FID and Source_Id in order to identify provinces
Briggs Henan University 2010

36. GWR output: R2 and Y values
Output table (part)
(Columns reordered.
Highlighted columns obtained from join with original data.)
Observed: values on the dependent variable Y
Predicted values and residuals are based upon each local regression and are not the same as those for a global regression.
Briggs Henan University 2010

37. GWR output: regression coefficients and standard errors
Standard error of the estimate
Regression coefficients (b)
Standard error of the coefficients
No statistical significance results provided
--statistical significance tests in GWR have been severely criticized.
Briggs Henan University 2010