Exploratory Mapping

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Exploratory Analysis of the Sacramento metropolitan area:

Prior to calculating spatial autocorrelation, let’s first map the variable “eviction rate” to see if it looks like it clusters across space. In fact, it does look like eviction rates cluster. In particular, there appears to be a concentration of high eviction rate neighborhoods in the downtown and northeast portions of the metropolitan area.

Spatial weights

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Before modelling dependency, let’s check how neighborhoods are spatially connected to one another by defining 1. Neighbor connectivity (who is you neighbor?), and 2. Neighbor weights (how much does your neighbor matter?). Once Queen connectivity is specified, the next step is to construct weights for the 20-mile distance based neighbor objects and plot the centroids.

Plot I

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Moran Scatterplot

The first map showed that neighborhood eviction rates appear to be clustered in Sacramento. To explore this further, a Moran scatterplot can be used. Here, standardized eviction rates on the x-axis are plotted against the standardized average eviction rate of one’s neighbors (also known as the spatial lag) on the y-axis.

The scatterplot shows a fairly strong positive association: the higher your neighbors’ eviction rate, the higher your eviction rate.

Global spatial autocorrelation

The most popular test of spatial autocorrelation is the Global Moran’s I test. A rule of thumb is a spatial autocorrelation higher than 0.3 and lower than -0.3 is meaningful.


    Moran I test under randomisation

data:  sac.tracts$evrate  
weights: sacw    

Moran I statistic standard deviate = 21.952, p-value
< 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic       Expectation          Variance 
     0.5677628263     -0.0020618557      0.0006737946

A correlation of 0.567 is fairly high indicating strong positive clustering. Moreover, this correlation is statistically significant (p-value basically at 0).

Plot II

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Local spatial autocorrelation

The Moran’s I tells us whether clustering exists in the area. It does not tell us, however, where clusters are located. Local Indicators of Spatial Association (LISAs) have the primary goal of providing a local measure of similarity between each unit’s value (in this case, eviction rates) and those of nearby cases (i.e. hot or cold spots). A popular local measure of spatial autocorrelation is Getis-Ord (Gi* value):

Plot III

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Local spatial autocorrelation

Designation of hot and cold spots as tracts with different Z-scores (significance level)

Plot IV

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Local spatial autocorrelation (interactive)

Designation of hot and cold spots as tracts with Z-scores above 1.96 and below -1.96 (5% significance level)

Plot V

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Local spatial autocorrelation (interactive)

Local Moran’s I can capture spatial outliers. Positive values (similar values are being clustered) indicate similarity between neighbors; negative values (dissimilar (high and low) values are clustered) indicate dissimilarity between neighbors.