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### Regression in geoDA

### Geographically Weighted Regression

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

Running Regression in geoDA: I

1

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

Running Regression in geoDA: II

Select one dependent variable

One or more independent variables

Selecttype 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

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 Savebefore clicking OK

Click Reset to run a different regression

The results

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

Regression for Provinces: n = 35

Briggs Henan University 2010

- 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!)

Display table: Table >PromotionPlot using: Explore >ScatterPlot

Results for simple regressionNote: 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

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

Y

Y

(Y-Ỹ)

Y

Ỹ

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

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 = 1368.89/33=41.4816

SE of regression=standard error of the estimate=√41.4816=6.44062

Identical in simple regression

Results for each regression coefficient

Y= 11.3146 - 6.578X

Briggs Henan University 2010

Simple Regression Results from GeoDA:spatial

n = 35

Moran’s I for regression residuals

--not statistically significant (p=.09)

Space > Univariate Moran

for variable: OLS_Resid

Same results!

Briggs Henan University 2010

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

Multiple Regression Results n = 29Illiteracy with % Pop Urban and Urban Income

Overall Results

Results for each variable

significant

Not significant

Spatial Results

Not significant

Briggs Henan University 2010

Residual Analysis:Illiteracy v. Urban Pop % and UrbanIncomePerCapita

Moran’s I = .0226

p = 0.5520

Not statistically significant

No Spatial autocorrelation in residuals

Briggs Henan University 2010

Spatial Error Model Resultsillustrative only: not needed

Spatial error not significant

Briggs Henan University 2010

Spatial Lag Model Resultsillustrative only: not needed

Spatial lag not significant

Briggs Henan University 2010

Regression Results Summary

Briggs Henan University 2010

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

Table >> Add Column Table >> Field Calculator

Improving the modelRelationship is Non-linearUse log of IlliteracyBriggs Henan University 2010

The same plots using ExcelRelationship is Non-linear

Illiteracy

Log of Illiteracy

Urban pop %

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

Log of Illiteracy:makes relationship linear

Urban Income now significant, and % urban not significant.

--these two variables are highly intercorrelated

--see next slide

Briggs Henan University 2010

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

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

Other software options for multiple regression

Briggs Henan University 2010

- 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
- http://cran.r-project.org/web/views/Spatial.html
- CrimeStat III has some support for spatial regression http://www.icpsr.umich.edu/NACJD/crimestat.html
- For a good list of spatial software sources, go to: http://en.wikipedia.org/wiki/List_of_spatial_analysis_software

What have we learned today?

Briggs Henan University 2010

- 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

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

Problems with Geographically Weighted Regression

Xi

Briggs Henan University 2010

- 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

GWR in ARCGIS

Briggs Henan University 2010

- 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 -1.79e+308
- Use source_ID to join with FID of original data to identify observations

Usage Tips from ArcGIS Help

Briggs Henan University 2010

- 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

Running GWR in ArcGIS

Briggs Henan University 2010

Execution Dialog for GWR in ArcGIS

Results presumable for global regression?????

--R2 value does not agree with results from geoDA?

Briggs Henan University 2010

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

Join with the original shapefile using FID and Source_Id in order to identify provinces

Briggs Henan University 2010

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

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

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