Objectives

• Identify clusters with the Local Moran cluster map and significance map

• Identify clusters with the Local Geary cluster map and significance map

• Identify clusters with the Getis-Ord Gi and Gi* statistics

• Identify clusters with the Local Join Count statistic

• Interpret the spatial footprint of spatial clusters

• Assess potential interaction effects by means of conditional cluster maps

• Assess the significance by means of a randomization approach

• Assess the sensitivity of different significance cut-off values

• Interpret significance by means of Bonferroni bounds and the False Discovery Rate (FDR)

Getting started

With GeoDa launched and all previous projects closed, we load the Guerry sample data set from the Connect to Data Source interface. As before, we save this file in a working directory, so that we can easily add spatial weights files. We continue to use the ESRI Shape file format, so that we now have a file, say guerry_85 to load. This brings up the familiar themeless base map, showing the 85 French departments, as in Figure 1.

To carry out the spatial autocorrelation analysis, we will need a spatial weights file, either created from scratch, or loaded from a previous analysis (ideally, contained in a project file). The Weights Manager should have at least one spatial weights file included, e.g., guerry_85_q for first order queen contiguity, as shown in Figure 2.

For this univariate analysis, we will focus on the variable Donatns (charitable donations per capita). This variable displays an interesting spatial distribution, as illustrated in a natural breaks map (using 6 categories), in Figure 3. The global Moran’s I is 0.353, using queen contiguity, and highly significant at p<0.001 (not shown here).

Principle

The Local Moran statistic was suggested in Anselin (1995) as a way to identify local clusters and local spatial outliers. Most global spatial autocorrelation statistics can be expressed as a double sum over the i and j indices, such as \(\sum_i \sum_j g_{ij}\). The local form of such a statistic would then be, for each observation (location) \(i\), the sum of the relevant expression over the \(j\) index, \(\sum_j g_{ij}\).

Specifically, the Local Moran statistic takes the form \(c. z_i \sum_j w_{ij} z_j\), with \(z\) in deviations from the mean. The scalar \(c\) is the same for all locations and therefore does not play a role in the assessment of significance. The latter is obtained by means of a conditional permutation method, where, in turn, each \(z_i\) is held fixed, and the remaining z-values are randomly permuted to yield a reference distribution for the statistic. This operates in the same fashion as for the global Moran’s I, except that the permutation is carried out for each observation in turn. The result is a pseudo p-value for each location, which can then be used to assess significance. Note that this notion of significance is not the standard one, and should not be interpreted that way (see the discussion of multiple comparisons below).

Assessing significance in and of itself is not that useful for the Local Moran. However, when an indication of significance is combined with the location of each observation in the Moran Scatterplot, a very powerful interpretation becomes possible. The combined information allows for a classification of the significant locations as high-high and low-low spatial clusters, and high-low and low-high spatial outliers. It is important to keep in mind that the reference to high and low is relative to the mean of the variable, and should not be interpreted in an absolute sense.

Implementation

The Univariate Local Moran’s I is started from the Cluster Maps toolbar icon, shown in Figure 4.

It is included as the top level option in the resulting drop down list in Figure 5.

Alternatively, this option can be selected from the main menu, as Space > Univariate Local Moran’s I.

Either approach brings up the familiar Variable Settings dialog in Figure 6, which lists the available variables as well as the default weights file, at the bottom (guerry_85_q). We select Donatns as the variable name.

Clicking OK brings up a dialog to select the number and types of graphs to be created. The default is to provide the Cluster Map only, which is typically the most informative. This is shown in Figure 7. However, in addition, a Significance Map and a Moran Scatter Plot can be brought up as well. To continue, we select the Significance Map as well.

The results, for the default setting of 999 permutations and a p-value of 0.05 (and with both significance map and cluster map options checked) are illustrated in Figures 8 and 9.

The significance map shows the locations with a significant local statistic, with the degree of significance reflected in increasingly darker shades of green. The maps starts with p < 0.05 and shows all the categories of significance that are meaningful for the given number of permutations. In our example, since there were 999 permutations, the smallest pseudo p-value is 0.001, with four such locations (the darkest shade of green).

The cluster map augments the significant locations with an indication of the type of spatial association, based on the location of the value and its spatial lag in the Moran scatter plot (see also the discussion below). In this example, all four categories are represented, with dark red for the high-high clusters (eight in our example), dark blue for the low-low clusters (seventeen locations), light blue for the low-high spatial outliers (two locations), and light red for the high-low spatial outliers (two locations).

We now consider the various options and interpretation more closely.

Randomization options

The Randomization option is the first item in the options menu for both the significance map and the cluster map. It operates in the same fashion as for the Moran scatter plot. As shown in Figure 10, up to 99999 permutations are possible (for each observation in turn), preferably using a specified random seed to allow replication.

The effect of the number of permutations is typically marginal relative to the default of 999. In our example, selecting 99999 results in two minor changes, but the total number of significant locations remains the same. As shown in Figure 11, there are now 21 locations significant at p < 0.05, four at p < 0.01, three at p < 0.001, and one at p < 0.00001, as illustrated by different shades of green in the map.

The cluster map is affected in the same two locations, but now we can assess the effect on the four classifications. As illustrated in Figure 12, one of the high-low spatial outliers disappears (adjoining the large low-low cluster in the south of the country, it was significant at p < 0.05 for 999 permutations), and one new high-high cluster is added (to the west of the existing high-high cluster in the Brittany region, also significant at p < 0.05). In general, it is good practice to assess the sensitivity of the significant locations to the number of permutations, although this typically only affects locations with a p-value of 0.05 (see below, for further discussion of the interpretation of significance).

Significance

An important methodological issue associated with the local spatial autocorrelation statistics is the selection of the p-value cut-off to properly reflect the desired Type I error. Not only are the pseudo p-values not analytical, since they are the result of a computational permutation process, but they also suffer from the problem of multiple comparisons (for a detailed discussion, see de Castro and Singer 2006). The bottom line is that a traditional choice of 0.05 is likely to lead to many false positives, i.e., rejections of the null when in fact it holds.

There is no completely satisfactory solution to this problem, but GeoDa offers a number of strategies through the Significance Filter item in the options menu, shown in Figure 19.

A straightforward option is to select one of the pre-selected p-values from the list provided (i.e., 0.05, 0.01, 0.001, or 0.0001). For example, choosing 0.01 immediately changes the locations that are displayed in the significance and cluster maps, as shown in Figure 20. Now, there are only eight significant locations, compared to 29 with a pseudo p-value cut-off of 0.05.

Note that we can obtain exactly the same locations by selecting the three categories of p-values 0.01, 0.001 and 0.0001 in the original significance map of Figure 11 (click on the rectangle next to p=0.01, followed by a shift click on the rectangles next to p = 0.001 and p = 0.0001). The status bar in Figure 21 confirms that the selection consists of eight values (4 for p = 0.01, 3 for p = 0.001, and 1 for p = 0.0001).

A more refined approach to select a proper p-value is available through the Custom Inference item of the significance filter, shown in Figure 22. The associated interface provides a number of options to deal with the multiple comparison problem.

The point of departure is to set a target \(\alpha\) value for an overall Type I error rate. In a multiple comparison context, this is sometimes referred to as the Family Wide Error Rate (FWER). The target rate is selected in the input box next to Input significance. Without any other options, this is the cut-off p-value used to select the observations. For example, this could be set to 0.1, a value suggested by Efron and Hastie (2016) for big data analysis. In our example, since the 85 observations hardly constitute big data, we keep the value at 0.01.

We now consider the two more refined options, i.e., the Bonferroni bound and the False Discovery Rate.

Interpretation of clusters

Strictly speaking, the locations shown as significant on the significance and cluster maps are not the actual clusters, but the cores of a cluster. In contrast, in the case of spatial outliers, they are the actual locations of interest.

In order to get a better sense of the spatial extent of the cluster, there are a number of ways to highlight cores, their neighbors, or both in the Select All… option in Figure 26.

The first option selects the cores, i.e., all the locations shown as non-white in the map. This is not so much relevant for the cluster or significance map, but rather for any maps that are linked. The selection of the Cores will select the corresponding observations in any other map or graph window.

The next option does not select the cores themselves, but their neighbors. Again, this is most relevant when used in combination with linked maps or graphs.

The third option selects both the cores and their neighbors (as defined by the spatial weights). This is most useful to assess the spatial range of the areas identified as clusters. For example, with the p-value set at 0.01, selection of the cores and neighbors yields the regions in Figure 27, with the non-significant neighbors shown in grey.

An interesting application of this feature is to superimpose the cores and neighbors selected for a given p-value, say 0.01, onto the cluster cores identified with a different p-value, say 0.05. In the example in Figure 28, the cores and clusters for 0.01 mostly match the locations identified as cluster cores at p < 0.05. In our example, there are 23 locations selected, compared to 26 significant high-high and low-low locations for p < 0.05. The major difference is that the high-high region in the center of the country is totally missed, but the high-high cluster in Brittany and the low-low cluster in the south of the country is almost exactly matched.

Similarly, checking the cores and neighbors selected in the significance map, clearly illustrates how some of the neighbors at p < 0.01 are identified as significant cores for p < 0.05, as shown in Figure 29.

In sum, this drives home the message that a mechanical application of p-values is to be avoided. Instead, a careful sensitivity analysis should be carried out, comparing cores of clusters identified for different p-values, including the Bonferroni and FDR criteria, as well as the associated neighbors to suggest interesting locations that may suggest new hypotheses or discover the unexpected.

Conditional local cluster maps

A final option for the Local Moran statistic is that the cluster maps can be incorporated in a conditional map view, similar to the conditional maps we covered earlier. This is accomplished by selecting the Show As Conditional Map option in Figure 30.

The resulting dialog is the same as for the standard conditional map we reviewed in an earlier chapter. In our example, we take literacy (Litercy) as the conditioning variable for the x-axis, and Clergy as the conditioning variable for the y-axis. In addition, we change the default 3 by 3 micromaps to a 2 by 2 setup, selecting the median as the cut point for each variable (select quantile > 2).

The result is as shown in Figure 31, depicting four micromaps. The maps on the left show the location of clusters and outliers (using p < 0.05 in this example) for those departments with literacy below the median, whereas the maps on the right show the corresponding departments above the median. The main difference seems to be between the maps on the lower end of clergy versus the upper end. The upper end seems to have more high-high cluster cores, with the lower end with more low-low cores.

However, this example is purely illustrative of the functionality available through the conditional cluster map feature, rather than as a substantive interpretation. As always, the main focus is on whether the micromaps suggest different patterns, which would imply an interaction effect with the conditioning variables.

Principle

The Local Geary statistic, first outlined in Anselin (1995), and further elaborated upon in Anselin (2018), is a Local Indicator of Spatial Association (LISA) that uses a different measure of attribute similarity. As in its global counterpart, the focus is on squared differences, or, rather, dissimilarity. In other words, small values of the statistics suggest positive spatial autocorrelation, whereas large values suggest negative spatial autocorrelation.

Formally, the Local Geary statistic is \[LG_i = \sum_j w_{ij}(x_i - x_j)^2,\] in the usual notation.

Inference is again based on a conditional permutation procedure and is interpreted in the same way as for the Local Moran statistic. However, the interpretation of significant locations in terms of the type of association is not as straightforward. In essence, this is because the attribute similarity is not a cross-product and thus has no direct correspondence with the slope in a scatter plot. Nevertheless, we can use the linking capability within GeoDa to make an incomplete classification.

Those locations identified as significant and with the Local Geary statistic smaller than its mean, suggest positive spatial autocorrelation (small differences imply similarity). For those observations that can be classified in the upper-right or lower-left quadrants of a matching Moran scatter plot, we can identify the association as high-high or low-low. However, given that the squared difference can cross the mean, there may be observations for which such a classification is not possible. We will refer to those as other positive spatial autocorrelation.

For negative spatial autocorrelation (large values imply dissimilarity), it is not possible to assess whether the association is between high-low or low-high outliers, since the squaring of the differences removes the sign.

Implementation

In the same way as for the Local Moran, the Local Geary can be invoked from the Cluster Maps toolbar icon, as Univariate Local Geary in the drop down menu, shown in Figure 32.

Alternatively, it can be started from the main menu, as Space > Univariate Local Geary.

The subsequent step is the same as before, bringing up the Variable Settings dialog that contains the names of the available variables as well as the spatial weights. Everything operates in the same way for all local statistics, so we will not dwell on those aspects here. We again select Donatns as the variable, with guerry_85_q as the queen contiguity weights.

The following dialog offering different window options is slightly different, in that there is no Moran scatter plot option. The only options are for the Significance Map and the Cluster Map. The default is that only the latter is checked, as in Figure 33.

After selecting the Significance Map option as well, the OK button generates two maps, using a default p-value of 0.05 and 999 permutations, as shown in Figures 34 and 35. In our example, there are 28 significant locations, highlighted in Figure 34.

As discussed above, some of the locations with a positive spatial autocorrelation can be distinguished between the high-high and low-low cases. As shown in Figure 35, there are 9 such high-high locations and 17 low-low locations. There are no locations with positive spatial autocorrelation classified as other in this case. There are two observations with negative spatial autocorrelation, although, as discussed, it is not possible to characterized the type of spatial outliers they correspond with.

All the options operate the same for all local statistics, including the randomization setting, the selection of significance levels, the selection of cores and neighbors, the conditional map option, as well as the standard operations of setting the selection shape and saving the image.

Below, we only discuss the interpretation and how to save the results, which differ slightly in each case.

Interpretation and significance

Before proceeding further, we change the randomization option to 99999 permutations. This results in minor changes in the cluster map, with two of the marginal (i.e., only significant at p < 0.05) low-low locations removed. As a result, there are now 26 significant locations.

To illustrate the rationale behind the classification of the local clusters, we link the locations identified as high-high with a matching Moran scatter plot. As usual, we select the observations in question by clicking on the red rectangle in the legend next to High-High. This highlights the corresponding locations in the cluster map (the other locations become more transparent) and simultaneously selects the matching points in the Moran scatter plot. As illustrated in Figure 36, the type of association is between locations above the mean and a spatial lag that is also above the mean, which we have characterized as high-high.

Similarly, selecting the low-low cluster cores (click the orange rectangle in the legend) shows the corresponding points in the lower-left quadrant of the Moran scatter plot in Figure 37.

For negative spatial autocorrelation, there is no unambiguous classification, since the squared differences eliminate the sign of the dissimilarity between an observation and its neighbors. The corresponding points in the Moran scatter plot are not informative, as shown in Figure 38.

Principle

A third class of statistics for local spatial autocorrelation was suggested by Getis and Ord (1992), and further elaborated upon in Ord and Getis (1995). It is derived from a point pattern analysis logic. In its earliest formulation the statistic consisted of a ratio of the number of observations within a given range of a point to the total count of points. In a more general form, the statistic is applied to the values at neighboring locations (as defined by the spatial weights). There are two versions of the statistic. They differ in that one takes the value at the given location into account, and the other does not.

The \(G_i\) statistic consist of a ratio of the weighted average of the values in the neighboring locations, to the sum of all values, not including the value at the location (\(x_i\)). \[G_i = \frac{\sum_{j \neq i} w_{ij} x_j}{\sum_{j \neq i} x_j}\]

In contrast, the \(G_i^*\) statistic includes the value \(x_i\) in both numerator and denominator: \[G_i^* = \frac{\sum_j w_{ij} x_j}{\sum_j x_j}.\] Note that in this case, the denominator is constant across all observations and simply consists of the total sum of all values in the data set.

The interpretation of the Getis-Ord statistics is very straightforward: a value larger than the mean (or, a positive value for a standardized z-value) suggests a high-high cluster or hot spot, a value smaller than the mean (or, negative for a z-value) indicates a low-low cluster or cold spot. In contrast to the Local Moran and Local Geary statistics, the Getis-Ord approach does not consider spatial outliers.³

Inference is based on conditional permutation, using an identical procedure as for the other statistics.

Implementation

The implementation of the Getis-Ord statistics is largely identical to that of the other local statistics. Each statistic an be selected from the drop down menu generated by the Cluster Maps toolbar icon, either as Local G, or as Local G* in Figure 43.

Alternatively, the same two options are also available from the main menu, as Space > Local G or Space > Local G*.

The next step brings up the Variable Settings dialog. Again, we select Donatns as the variable, with guerry_85_q as the queen contiguity weights. This is followed by a choice of windows to be opened. The latter is again slightly different from the previous cases. The default, shown in Figure 44, is to use row-standardized weights and to generate only the Cluster Map. The Significance Map option needs to be invoked explicitly by checking the corresponding box.

The Getis-Ord statistics also allow the use of binary weights (i.e., not row-standardized), by having the row-standardized weights box unchecked. In practice, the results rarely differ much.

Using the default settings of 999 permutations, with p at 0.05, yields the significance map for the \(G_i\) statistic shown in Figure 45.

The corresponding cluster map, illustrated in Figure 46, shows 10 high-high cluster cores or hot spots (in red on the map), and 19 low-low cluster cores or cold spots (in blue on the map). Note that these are the exact same locations as identified for the Local Moran, except that the spatial outliers are now classified as part of the clusters (one in the high-high group and one in the low-low group).

In this particular example, the cluster map for the \(G_i^*\) statistic, shown in Figure 47, gives the identical results. This is often the case, but not always, so there is a point in computing both statistics.

For the Getis-Ord statistics, all the same options are available as for the Local Moran and the Local Geary statistics, and we refer to those discussions for details.

Interpretation and significance

In the same way as for the other statistics, changing the number of permutations to 99999 provides a more detailed insight into the importance of the different locations, as indicated in the significance map. While 21 locations are deemed to be significant for 0.05, there are only four such locations for 0.01, three for 0.001 and one for 0.00001. This is the exact same result as for the Local Moran, illustrated in Figure 48 for the \(G_i\) statistic (the results for the \(G_i^*\) statistic are the same).

Using the Significance Filter, we can assess the effect of a change of critical p-value to 0.01. In Figure 49, only one high-high cluster core remains, whereas the low-low cluster is reduced to seven observations.

As shown in Figure 50, the FDR criterion further reduces the number of significant locations to three in the South of the country. These are the same three locations also identified by the Local Moran statistic.

The result for the Bonferroni bound is again the same as for the Local Moran, with only one significant location (see Figure 23).

Principle

Recently, Anselin and Li (2019) showed how a constrained version of the \(G_i^*\) statistic yields a local version of the well-known join count statistic for spatial autocorrelation of binary variables, popularized by Cliff and Ord (1973). Expressed as a LISA statistic, a local version of the so-called BB join count statistic is \[BB_i = x_i \sum_j w_{ij}x_j,\] where \(x_{i,j}\) can only take on the values of 1 and 0, and \(w_{ij}\) are the elements of a binary spatial weights matrix (i.e., not row-standardized). For the most meaningful results, the value of 1 should be chosen for the case with the fewest observations (of course, the definition of what is 1 and 0 can easily be switched).

The statistic is only meaningful for those observations where \(x_i = 1\), since for \(x_i = 0\) the result will always equal zero. A pseudo p-value is obtained by means of a conditional permutation approach, in the same way as for the other local spatial autocorrelation statistics, but only for those observations with \(x_i = 1\). The same caveats as before should be kept in mind when interpreting the results, which are subject to multiple comparisons and the sensitivity of the pseudo p-value to the actual simulation experiment (random seed, number of permutations). Technical details are provided in Anselin and Li (2019).

Implementation