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# ANALYZING SPATIALLY VARYING RELATIONSHIPS

Several techniques for analyzing more complex instances of spatial inequality.  Often times, inequality results from a combination of factors and cannot be explained by mapping one variable alone.  In these cases, we can map how multiple factors overlap in space and determine places where relationships between factors are strong or weak.

## EXAMPLE: HOUSEHOLDS WITHOUT CARS AND POVERTY

In this example we are interested in a program that encourages alternatives to owning a car.  The program is more successful in some neighborhoods with better transit and walking opportunities but in other neighborhoods households’ lack of a car may be a function of poverty.  We consider car ownership in each neighborhood compared to the level of poverty in the area with the assumption that neighborhoods with more car-free households than poverty are more successful at creating alternatives to car ownership.

Using the GeoDa program, we begin with a standard deviation map of the percent of households without a car.  (See previous tutorial for more information on this step.)

There are some noticeable areas where the  “No Car Rate” is high, but we would like to know how much this is due to higher poverty as opposed to households having alternatives to driving.  We next create a standard deviation map of poverty by Census Tract.  It appears to have some similarity to the map above.

We can eyeball some similarities between the No-car rate and the poverty and a more precise way to consider this relationship is with a scatter plot.  In GeoDa we specify a scatter plot of the percent of households without car versus percent poverty.

The scatter plot suggests a linear relationship between poverty and the lack of a car, because most data points lie fairly close to a straight line.  The cluster of points well above the line are neighborhoods where the no-car rate is higher than the poverty rate.  We will investigate these neighborhoods further.  A more formal investigation of the relationship is done with a regression model (found under the “Methods” menu in GeoDa.

We specify the title and files name and check the option for the Moran’s I z-score (this option can increase the computation time). We set up a spatial lag model that considers poverty rate as an independent variable to explain car-less rate, the dependent variable.  The use of a spatial lag model takes spatial autocorrelation into account, and will help us pinpoint areas where the no-car rate is higher or lower than expected.  The spatial lag model requires that a weight file be selected to determine the spatial relationships.  (See the previous tutorial for more on how to create a spatial weights file). Once the model is set up as desired we hit “Run.”

Once the model has run, use the ‘Save’ button to save the predicted values, residuals and errors of the model.  These saved values are added to the attributes of the shapefile, allowing them to be mapped in later steps.

The regression result listing will appear after clicking “OK.”  This text contains the information for the overall model fit.

Although we selected a spatial age model, results for an ordinary least square (OLS) regression appear first. These results show that a model that only considers the poverty rate has a fairly good fit in explaining the no-car rate.  This model serves as a baseline for comparing the spatial lag presented below.

The spatial lag model includes an additional variable “W_NoCarRate” which is the spatial lag term.  The spatial lag term accounts for the effect of adjacent neighborhoods.  The fact that this term is statistically significant indicates that spatial factors are important in explaining the no-car rate.  We can then map the model results to better identify where spatial factors are important.

A standard deviation map of the model prediction error, “PRDERR,”  highlights where the poverty rate was least effective in predicting the no-car rate.  This term does not account for the spatial lag term, only the estimate based on the poverty rate.  The fact there is clustering of error values indicates that spatial factors are important in this model (which is confirmed in the model results previously discussed).  The negative values in southern neighborhoods indicate that the no-car rate is lower than expected given the poverty rate.  Red and orange clusters are areas where the no-car rate is higher than expected.

Lastly, we map the model residual, which accounts for the spatial lag term.  Spatial clusters are less apparent on this map because the influence of neighboring tracts has been incorporated into the model.  The residual, or remaining error, appears random indicating that the spatial lag model has largely accounted for spatial differences.

Returning to our original research questions, we can argue that spatial factors are important to whether households have alternatives to car ownership.  In the southern part of the city, the absence of a car seems to be due primarily to poverty.  In the central part of the city households are more likely to do without a car regardless of income.  It follows that neighborhood improvements may be needed in the southern parts of the city if more households are to be encouraged to do without a car.