Some methods for measuring and correcting for spatial autocorrelation

Abstract

Two new approaches to spatial autocorrelation (SAC) were examined for their ability to measure SAC: correlation of distance scores and k nearest spatial neighbor regression. Furthermore, the methods were compared to a more traditional measure of SAC, Moran’s I. All methods were found to give similar results as measured by their intercorrelations.

Three approaches, correlation of distance scores, k nearest spatial neighbor regression and spatial local regression, were used on simulated datasets to see if they could successfully distinguish between a true and a spurious cause. The correlation of distances approach was found to be lacking, while the other two methods clearly distinguished between the spurious and true causes and gave similar results.

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Published 2015/10/20.
Some methods for measuring and
correcting for spatial autocorrelation
Emil O. W. Kirkegaard
Abstract
Two new approaches to spatial autocorrelation (SAC) were examined for their ability to measure SAC:
correlation of distance scores and k nearest spatial neighbor regression. Furthermore, the methods were
compared to a more traditional measure of SAC, Moran's I. All methods were found to give similar
results as measured by their intercorrelations.
Three approaches, correlation of distance scores, k nearest spatial neighbor regression and spatial local
regression, were used on simulated datasets to see if they could successfully distinguish between a true
and a spurious cause. The correlation of distances approach was found to be lacking, while the other
two methods clearly distinguished between the spurious and true causes and gave similar results.
Key words: spatial autocorrelation, Moran's I, correlation of distances, k nearest spatial neighbor
regression, KNSNR, spatial local regression, SLR
1. Introduction
Much research analyzes data from countries, states, weather stations or other units have that a location.
Spatial autocorrelation (SAC) is when there are spatial patterns in the dataset (Hassall & Sherratt, 2011;
Radil, 2011). This can be both positive (nearby cases are similar), neutral (neighbor cases have no
particular relationship, absence of SAC), or negative (nearby cases are dissimilar). Figure 1 illustrates
this.
The presence of SAC in a dataset means that the cases are not independent which means that the
degrees of freedom are overestimated. There are methods for correcting the degrees of freedom, but
these are not the focus of this paper (Gelade, 2008; Hassall & Sherratt, 2011). Instead, the purpose of
this article is to explore ways to measure and correct for SAC with respect to effect sizes.
2. Correlation of distances method
A conceptually simple method for examining SAC is the correlation of distance scores method (CD).
CD was used by Davide Piffer (Piffer, 2015) but is so simple that it would be surprising if it had not
been thought of before. The first step is to calculate distance scores for all cases for all variables of
interest. Note that the kind of distance score depends on the type of variable. Piffer analyzed, among
other things, phylogenetic distances (genetic distance between two organisms or classes of organisms)
and used Fst values. However, if one's data concern e.g. countries, one would use spherical geometric
distances. If one is analyzing data from flatland (of any hyperspace with flat planes), one could use
euclidean distances. For single metric type variables, one could use the absolute difference.
Alternatively, one could combine several non-spatial variables and employ various well-researched
distance measures such as complete linkage (James, Witten, Hastie, & Tibshirani, 2013, p. 401).
The distance scores results in a new dataset which has N(N-1)/2 cases (choose 2 of N, order irrelevant),
where N is the number of cases in the original dataset.
The idea behind the method is that given two variables that are linearly related, pairs of cases that are
far from each other on one variable should also be far from each other on another variable. To better
understand this, it is worth looking at some example data. Figure 2 shows the spatial distribution of
some datapoints in flatland. An outcome variable also exists and is shown by color coding.
There are clear spatial patterns in the data: 1) we see 5 fairly distinct clusters, 2) neighbors tend to have
similar outcomes, and 3) points with higher x and y values tend to have higher outcome values.
Using the distance scores, one can examine (2) with a standard Pearson correlation, as shown in Figure
3.
As expected we see a positive correlation, i.e. cases closer to each other do have more similar
outcomes. We also see substantial non-linearity in the plot. This is because the cases are not uniformly
spatially distributed and that distances between all pairs of cases are used.
If one wants to control for SAC, one can use standard statistical methods such as multiple regression or
semi-partial correlations. Suppose one has a predictor that is spatially distributed as shown in Figure 4.
We see that there is some similarity to Figure 2, but it is not immediately obvious whether the
relationship owes to a pattern at the cluster-level or whether the pattern is general. In the original data
the correlation between predictor and outcome is .66 and using CD it is .36.
The semi-partial correlation between predictor and outcome is .13, a fair bit lower than .36, which
gives us some evidence that the relationship owes much of its size to the cluster pattern. If one instead
uses multiple regression the standardized betas are .16 and .36 for predictor and spatial distance,
respectively. In fact, the true correlation of predictor and outcome once SAC is taken into account is
near 0. The relationship between them is purely due to their common association with the clusters we
see (common cause scenario).
2.1. A second dataset
What about a similar situation where the relationship between predictor and outcome is not spurious?
Consider the data in Figures 5 and 6.
We see the same overall pattern as before, but it is subtly different. The correlation in the original
dataset between predictor and outcome is nearly the same, .68 vs. .66 before. However, using the
distance scores, the correlation between outcome and spatial distance is only .17 vs. .45 before. The
distance-based correlation between predictor and outcome is .43 vs. .36 before.
The semi-partial correlation for predictor and outcome correcting the latter for spatial distance is .39 vs.
.13 before. A fairly large difference. Using multiple regression the betas are .42 and .04 for predictor
and spatial distance, respectively.
Clearly, the distance scores approach shows that the association between predictor and outcome is
different in the two datasets.
2.2. The relationship between correlations of distances and correlations in the
original data
Before we go further, it is worth examining the relationship between correlations of distance data and
correlations of the original data. From my experimentation, correlations of distance scores seem to be
an r2-type statistic (Hunter & Schmidt, 2004, p. 189). This can be seen if one simulates normally
distributed data with a known correlation and then correlate their distance scores. The results are shown
in Figure 7.
As can be seen, the correlation of distance scores is almost exactly the same as the square of the
correlation, or conversely, the square root of the correlation of distance scores is almost the same as the
original correlation score.
Going back to the results in the previous section, this means that we need to take the square root of the
semi-partial correlations and standardized betas to get an r-type measure. These values are shown in
Table 1.
Relationship Dataset 1 Dataset 2
r (predictor x outcome), original data .66 .68
sqrt. r (predictor x outcome), distance data .60 .66
sqrt. r (spatial x outcome), distance data .67 .41
sqrt. rsp (predictor x outcome, spatial), distance
data
.36 .62
sqrt. beta (predictor x outcome), distance data .40 .65
sqrt. beta (spatial x outcome), distance data .60 .20
Table 1: Table of numbers from datasets 1 and 2.
There are several things worth noting. 1) The square root conversion seems to work because the
distance-based correlations are similar to the ones based on the original data. 2) Distance-based
correlations were higher for the first dataset as expected. 3) Semi-partial correlations of distance-data
showed that the predictor had a substantial size (.36) in the first dataset despite this relationship being
entirely spurious. However, it did correctly indicate a stronger effect in the second dataset (.62). 4)
Similarly, multiple regression showed that the predictor had substantial validity in the first dataset and
reversely that spatial distance had some validity in the second dataset. Neither should be the case. On
the positive side, in both cases did the true causes have larger effect sizes (.60>.40; .65>.20).
3. k nearest spatial neighbors regression
Given the problems of the CD method, I developed a new method. It is a variant of the general
approach of using the neighboring points to predict values, called k nearest neighbors (KNN) (James et
al., 2013). The implementations of KNN I could find for R did not support spatial data, so I wrote my
own implementation which I dub k nearest spatial neighbors regression (KNSNR) The approach is
fairly simple:
1. For each case, calculate the distance (spherical, euclidean or otherwise) to each every other
case.
2. For each case, find the k nearest neighbor cases.
3. For each case, calculate the mean value of the neighbors' scores and use that as the prediction.
One can alter k in (2) to tune the method. This value should probably be found thru cross-validation
(James et al., 2013).
(3) can be altered to make more advanced versions such as taking into account the relative distances
among the nearest neighbors (closer neighbors are more important) or the case weights (if using
aggregated data or uncertain data).
The function I wrote will output one of four things: 1) predicted values, 2) correlation of actual values
with predicted values, 3) residuals, and 4) the correlation of a predictor with the residuals of the
outcome. Each has its use.
The correlation of actual values with predicted values is a measure of the SAC in a given variable. It
corresponds to the correlation of a given variable and spatial distance when using distance data. Which
k is the right to use? It depends. What k does here is change the zoom level. The value at each k tells
you how much SAC there is in the dataset given a particular level of zooming in on every case. Figure
8 shows the amount of SAC in the outcome variable according to the first 100 values of k for multiple
datasets.
In general, k should be larger than 1. k=3 often works well. k reached plateaus for datasets 1-2 around
k=10. For these datasets k=50 makes more sense because there are 50 cases in each cluster. At all
levels, however, do we see that SAC is stronger for the odd numbered datasets, just as was found using
CD.
The predicted scores from KNSNR can be used in a multiple regression model where they can compete
with other variables. The residuals can be correlated with a predictor to easily perform semi-partial
correlation with SAC controlled for in the chosen variable. The semi-partial correlation between
predictor and outcome controlling outcome for SAC in the first dataset is .01 while it is .42 in the
second dataset (using k=50). In other words, the method tells us that the without SAC in the outcome
the predictor doesn't work in the first dataset, but it does in the second. The semi-partial correlations of
spatial prediction x outcome controlled for predictor are .34 and .01, for the first and second datasets
respectively. In other words, spatial proximity retains predictive value once the predictor has been
controlled, showing that there is an additional unmeasured cause with strong SAC. All four results are
good because they are in line with how the data were generated.
4. A different kind of spatial autocorrelation
The previous examples were easy in the sense that the data pretty clearly contained clusters of points.
One could have easily classified the cases into the regional/island clusters and used a more traditional
approach to the question: multi-level analysis. In simple terms, one would look whether relationship
found in the pooled data holds when analyzing the data within each cluster. Table 2 shows these results.
Cluster Dataset 1 Dataset 2
1 -0.176 0.511
2 -0.094 0.615
3 -0.037 0.5
4 0.117 0.511
5 0.162 0.617
Table 2: Correlations inside clusters in datasets 1-2.
Clearly, the datasets are markedly different in that there is a near-zero relationship inside the clusters in
the first, but there are fairly strong relationships in the second.
When results give discrepant results across levels, it is termed Simpson's paradox. Not understanding
this concept has lead to bogus claims of sex discrimination at least twice (Albers, 2015; Kievit,
Frankenhuis, Waldorp, & Borsboom, 2013). An interactive visualization of the phenomenon can be
found at: http://emilkirkegaard.dk/understanding_statistics/?app=Simpson_paradox
Clearly we did not need to use SAC to spot the difference between datasets 1-2. However, the datasets
are unusual in that they permit very easy clustering of the cases. Real life datasets are often more
difficult to deal with. Consider datasets 3-4 shown in Figures 9-12.
These datasets are variations of datasets 1-2, yet now the clusters are not easy to spot. One would not
be able to use a simple clustering algorithm on the spatial data to sort them into 5 clusters correctly. If
one cannot do that, one cannot use the standard multi-level approach because it requires discrete levels.
The correlations between the predictors and outcomes is in fact identical to before: .66 and .68
respectively. These correlations are generated in the same way as before, meaning that the first is a
spurious, SAC induced correlate, while the second is a true cause. The datasets still contain a sizable
amount of SAC as was seen in Figure 8, so one might still wonder how this affects the results. Using
the semi-partial SAC control approach, the results are .15 and .46 respectively. Should the number .15
be closer to 0? Not exactly.
To understand why, I need to be more explicit about how the data were simulated. In the odd numbered
datasets, the outcome and predictor variables are both a function of the clusters, which serve is a SAC
variable (SACV). In the even numbered datasets, the predictor is a function of the clusters + noise and
the outcome is a function of the predictor + noise. Figure 13 shows the path models.
The difference between datasets 1-2 and 3-4 is solely that the within cluster spatial variation was
increased from standard deviation=6 to 15. This weakens the SAC in the data and thus the effect of
controlling for SAC. The results are thus in line with expectations.
Still, we would like paired examples that appear similar – i.e. has similar r (predictor x outcome) – but
where the first effect is spurious and the second is real and that our methods correctly identify this. To
do this, I created two more datasets. This time there are no a priori clusters.
The generation proceeded like this:
1. For each case:
a) Sample two values from the uniform distribution of numbers between 1 and 100 and use
these as x and y coordinates.
b) Sample a number from a standard normal distribution and use it as the score for the SACV.
2. Induce SAC in the SACV using a KNSN-type approach.
After this, I proceeded similarly to before, namely by creating two variables, predictor and outcome,
according to the path models shown in Figure 13. Noise were added to hit a target uncorrected
correlation of .66, similar to datasets 1-4. Because the SACV variable has strong SAC, controlling for
SAC decreases the correlation in the spurious cause scenario because path goes thru the SACV, but not
in the true cause scenario where causation is directly from predictor to outcome.
It is worth going into more detail about how SAC was induced. The algorithm works as follows:
1. For each iteration:
a) For each case:
1. Find the k nearest spatial neighbors.
2. Find the mean value of the chosen variable for these neighbors.
b) For each case:
1. Change the value of the datapoint to (1-w) times its current value and w times the mean
value of its neighbors.
(a) and (b) must proceed in this fashion otherwise the order of the cases would affect the algorithm. The
algorithm requires values for the three parameters: i, k and w. i determines the number of iterations the
algorithm goes thru. k control the number of neighbors taken into account and w controls the relative
weight given to the value from the neighbors. Smaller k means the SAC pattern will be more local,
while both i and w makes the SAC stronger but in different ways. The parameters used for datasets 5
and 6 were i=20, k=10, w=1/3.
Figures 14 to 17 show the spatial layout as well as predictor and outcome values for datasets 5-6.
Figure 1: Illustrations of spatial autocorrelation. From (Radil, 2011).
Figure 7: Comparison of r, CD and r2.
Figure 2: Dataset 1. Flatland and outcome.
Figure 3: Scatter plot of distance scores for spatial and outcome.
Figure 4: Dataset 1. Flatland and predictor.
Figure 5: Dataset 2. Flatland and outcome.
Figure 6: Dataset 2. Flatland and predictor.
Figure 8: The amount of SAC in datasets 1-6 according to KNSNR at a k 1-100.
Figure 9: Dataset 3. Flatland and outcome.
Figure 11: Dataset 4. Flatland and outcome.
Figure 12: Dataset 4. Flatland and predictor.
Figure 13: Path models for the example datasets.
Figure 14: Example 5. Flatland and outcome.
Figure 15: Example 5. Flatland and predictor.
Figure 16: Example 6. Flatland and outcome.
Figure 10: Dataset 3. Flatland and predictor.

From just inspecting the spatial plots, it is not easy to see a marked difference between datasets 5 and 6.
Both datasets clearly show some degree of SAC. Figures 18 and 19 show the regressions between
predictor and outcome.
Altho not identical, there do not seem to be marked differences between the datasets. Thus, one could
not tell the situations apart by simple inspection. There are also no obvious clusters one could classify
cases into and use multi-level analysis.
However, if we examine the strength of SAC in the datasets, they are fairly different as shown in Table
3.
Variable Dataset 5 Dataset 6
SACV 0.998 0.998
x 0.999 0.999
y 0.998 0.998
predictor 0.758 0.632
outcome 0.794 0.365
outcome_predictor_resid
s 0.28 -0.035
Table 3: SAC in datasets 5 and 6. SAC calculated using k=10 because data were generated using
k=10.
Unsurprisingly, the coordinate variables and the SACV show extreme levels of SAC. The predictors
diverge a bit in SAC, while the difference in SAC for the outcome variables is fairly large. This is as
expected given the way the data were generated. In dataset 6, the outcome variable is 2 steps away
from SACV, while it is only 1 step away in dataset 5. Also worth noting is the SAC in the model
residuals in dataset 5, which was discussed by (Hassall & Sherratt, 2011) as indicating a problem. This
indicates that one or more causes not included in the model are SAC.
The correlations with the predictor after controlling the outcome for SAC are .08 and .49 for datasets 5
and 6, respectively showing that the method can successfully detect which association is due to an
uncontrolled common cause which is SAC.
4.1. Control outcome, predictor or both for SAC?
So far I have only presented results where the outcome was controlled for SAC. However, one might
also control the predictor or both. It is not immediately clear to me which one is the best. For this
reason, I tried all three side by side. Table 4 shows the results.
Dataset k3_o k3_p k3_b k10_o k10_p k10_b k50_o k50_p k50_b
1 0.17 0.17 0.07 0.04 0.04 -0.01 0.01 0.01 0.02
2 0.57 0.57 0.58 0.45 0.45 0.53 0.42 0.42 0.54
3 0.3 0.3 0.27 0.19 0.19 0.24 0.15 0.15 0.24
4 0.6 0.6 0.61 0.52 0.52 0.58 0.46 0.46 0.56
5 0.18 0.18 0.07 0.08 0.08 0.03 0.27 0.27 0.36
6 0.57 0.57 0.57 0.49 0.49 0.55 0.51 0.51 0.59
Table 4: KNSN regression results, controlling either or both variables with three different k values. The
number is the value of k used, the letter indicators which variable was controlled for SAC: p=predictor,
o=outcome, b=both.
We see some interesting patterns. When k is 3 or 10, controlling both variables tends to reduce the
correlation further in the datasets with spurious causes, but not in the datasets with real causes. When
using k=50, controlling both actually tends to make the results stronger for both true and spurious
causes.
The most important comparison is that between datasets 5 and 6 because these are the most realistic
datasets. When using an appropriate k (3 or 10), we see that controlling both reduces the spurious
correlations (.18 to .07; .08 to .03) but either leaves unchanged or increases the true cause correlation
(.57 to .57; .49 to .55). For this reason, it would seem best to control both predictor and outcome
variables provided one has a reasonable idea about the correct k to use.
4.1.1. Multiple regression
There is another way to control for a variable, namely multiple correlation. However, because the
predicted values from KNSNR were often very similar (r>.7) to the predictor, this was not very useful.
Therefore, if one does use the MR approach one should pay attention to multicollinearity in the models,
e.g. by means of the variable inflation factor (Field, 2013; Gordon, 2015).
5. Spatial local regression
Local regression is a method related to the k nearest neighbor regression. The idea of local regression is
that instead of fitting a model to all the predictor data, one fits the model for every case and weigh the
nearby cases more than the distant cases. Afterwards we combine the models. The result of this is a
smooth fit to the data. Figure 20 shows as illustration of this.
In the above case, the distance measure for the cases is their absolute difference on the predictor
variable. However, one could also use spatial distance, a method one might call spatial local regression
(SLR). In this way, one would fit a regression for each case and use only the cases that are spatially
nearby and finally aggregate/meta-analyze the results. The rationale here is that SAC will have only a
weak influence when one examines cases near each other.
For the regression betas, one could use a weighted mean with unit-weights or inverse distance weights.
The latter is theoretically superior, but I implemented both options. The implementation can also output
the vector of beta values in case someone wants to experiment using other central tendency measures.
The results are surprisingly similar to those from KNSNR. Table 5 shows results for the predictor x
outcome relationship in datasets 1-6 using SLR.
Dataset slr_3_none slr_10_none slr_50_none slr_3_inverse slr_10_inverse slr_50_inverse
1 -0.017 -0.033 0.047 -0.008 -0.021 0.03
2 0.442 0.493 0.551 0.436 0.493 0.551
3 0.102 0.156 0.316 0.1 0.159 0.278
4 0.457 0.533 0.583 0.456 0.535 0.588
5 0.141 0.225 0.494 0.129 0.194 0.452
6 0.51 0.555 0.623 0.501 0.55 0.625
Table 5: Spatial local regression results for the predictor x outcome relationship in datasets 1-6. The
number refers to the chosen k value. The word refers to the use of weights.
Recall that datasets with odd numbers are spurious causes while those with even numbers are true
causes. For k=3 and 10, we see strong contrasts between the results from the even and odd numbered
datasets, but not for k=50. It appears that using k=50 is sufficient to induce a strong spurious effect.
The use of weights does not seem to have much effect, which is somewhat surprising. It does have a
small effect on reducing the size of the spurious cause correlations.
To further investigate this, calculated the same numbers as shown in Table 5 but for k = [3-50] using
only inverse weighing. Results are shown in Figure 21.
We see that at lower k values, all correlations decrease, but the spurious ones decrease more. As k
increases towards N of the sample, the spurious and true cause correlations converge.
6. Comparison of measures of SAC
It is worth comparing the two measures of SAC examined in this paper to the more standard methods in
the field. Two methods are widely used to measure the amount of SAC in a dataset: Moran's I and
Geary's C (Gelade, 2008; Hassall & Sherratt, 2011; Radil, 2011). These two measures are described as
approximately inversely related and often only the first is used. I also used only the first because I was
unable to understand the implementations of the latter in R (e.g. in spdep package). Both Moran's I and
Geary's C are global measures of SAC, altho local variations exist (Radil, 2011).
Because KNSNR tends to hit a ceiling quickly (e.g. in Table 2), the analysis here is focused on
variables where this doesn't happen. For this purpose, the outcome, predictor and residuals from
predicting the first with the latter are examined with all three methods in across all datasets. Results
from different datasets with the 'same' variable were appended. Because KNSNR requires a tuning
parameter, 3 values were chosen for this (3, 10, and 50). A square root version of CD was also added to
see if the non-linearity inherent in the CD correlations would substantially change the relationship to
the other measures.
To increase sample size and diversity of datasets, more datasets were created. Dataset 7 is a variant of
dataset 6, but where SAC is also induced into the outcome and predictor variables. Dataset 8 uses the
distance to the center as the basis for the predictor values and the outcome values are a noisy version of
the predictor. Datasets 9 and 10 are variants of dataset 8 but where the distance to the points (75;75)
and (100;100) are used, respectively. Dataset 11 had no SAC induced and functions as a null dataset.
Dataset 12 has datapoints perfectly distributed in a grid pattern, perfect negative SAC for the predictor
and uneven SAC for the outcome. The supplementary materials contains scatter plots for these
additional datasets. Table 6 shows the SAC values according to the 5 measures and Table 7 shows their
intercorrelations.
Dataset and variable Morans_I cd cd_sqrt knsn_3 knsn_10 knsn_50
ex1_outcome 0.403 0.452 0.673 0.741 0.789 0.798
ex1_predictor 0.432 0.547 0.739 0.764 0.812 0.819
ex1_outcome_predictor_resids 0.074 0.062 0.249 0.227 0.283 0.325
ex2_outcome 0.146 0.172 0.415 0.3 0.453 0.468
ex2_predictor 0.298 0.309 0.556 0.598 0.671 0.685
ex2_outcome_predictor_resids -0.002 0.02 0.141 -0.02 -0.021 -0.107
ex3_outcome 0.248 0.378 0.615 0.646 0.716 0.735
ex3_predictor 0.278 0.468 0.684 0.698 0.758 0.756
ex3_outcome_predictor_resids 0.038 0.054 0.233 0.16 0.271 0.303
ex4_outcome 0.09 0.176 0.419 0.268 0.386 0.445
ex4_predictor 0.18 0.263 0.512 0.507 0.601 0.635
ex4_outcome_predictor_resids -0.004 0.04 0.199 -0.018 0.014 -0.059
ex5_outcome 0.136 0.066 0.258 0.737 0.794 0.671
ex5_predictor 0.14 0.055 0.234 0.693 0.758 0.662
ex5_outcome_predictor_resids 0.021 0.026 0.162 0.14 0.28 0.243
ex6_outcome 0.031 0.041 0.202 0.246 0.365 0.323
ex6_predictor 0.105 0.042 0.204 0.54 0.632 0.565
ex6_outcome_predictor_resids -0.013 0.023 0.152 -0.092 -0.035 -0.101
ex7_outcome 0.176 0.065 0.256 0.98 0.994 0.729
ex7_predictor 0.219 0.101 0.317 0.987 0.997 0.831
ex7_outcome_predictor_resids 0.174 0.088 0.296 0.976 0.99 0.676
ex8_outcome 0.126 0.017 0.131 0.599 0.651 0.642
ex8_predictor 0.27 0.028 0.167 0.994 0.993 0.972
ex8_outcome_predictor_resids -0.007 0.002 0.04 0.018 -0.01 -0.171
ex9_outcome 0.188 0.201 0.448 0.595 0.645 0.65
ex9_predictor 0.395 0.483 0.695 0.997 0.997 0.985
ex9_outcome_predictor_resids -0.007 0 0.015 -0.02 -0.199
ex10_outcome 0.2 0.25 0.5 0.6 0.652 0.66
ex10_predictor 0.414 0.565 0.752 0.998 0.998 0.993
ex10_outcome_predictor_resids -0.007 0 0.015 0.016 -0.016 -0.176
ex11_outcome -0.002 -0.002 -0.027 -0.023 -0.04
ex11_predictor -0.006 0.001 0.037 0.019 -0.004 -0.126
ex11_outcome_predictor_resids -0.002 -0.002 -0.027 -0.023 -0.04
ex12_outcome 0.004 0.001 0.032 0.722 0.702 -0.609
ex12_predictor -0.006 0 -1 0.978 -0.969
ex12_outcome_predictor_resids 0.004 0.001 0.032 0.722 0.702 -0.609
Table 6: SAC measures of three variables in 11 datasets.
Morans_I cd cd_sqrt knsn_3 knsn_10 knsn_50
Morans_I 0.89 0.884 0.724 0.718 0.824
cd 0.882 0.968 0.501 0.493 0.639
cd_sqrt 0.876 1 0.487 0.533 0.736
knsn_3 0.822 0.652 0.523 0.689 0.761
knsn_10 0.767 0.591 0.587 0.879 0.547
knsn_50 0.931 0.831 0.809 0.793 0.721
Table 7: Intercorrelations among 6 SAC measures in 12 datasets. Pearson correlations above diagonal.
Spearman's rank-order below.1
All correlations between the measures were sizable as one would expect. The CD measure showed
downward-biasing non-linearity because the rank-order correlations were noticeably stronger.
Behind the strong intercorrelations lie some surprisingly inconsistencies. Consider ex1_outcome vs.
ex7_outcome_predictor_resids. CD gives 0.452 and 0.088 while knsn_3 gives 0.741 and 0.976. What
this seems to mean is that there is a strong local SAC (which knsn_3 measures) in the residuals but a
weak global SAC (which CD measures).
The KNSNR results for dataset 12 need special attention. Because the points are distributed in an exact
grid pattern, many pairs of points are equidistant. This poses a problem for KNN-type approaches
because they have to picked k nearest neighbors. In this case, the computer chooses one based on the
position in the distance matrix which is an arbitrary choice. If one is examining data distributed in a
perfect grid pattern, extra attention should be given to the choice of k. Notice also that cases close to
the edges have no neighbors in that direction (this isn't Pacman world), so the algorithm must choose
neighbors further away. This can give strange results.
7. Comparison of correction methods
In the previous sections, three different methods for controlling for SAC were discussed. It is useful to
do a straight up comparison with the best choice of parameters. Table 8 shows the uncontrolled and
SAC-controlled correlation in datasets 1-6 using correlation of distances (using semi-partial, partial and
multiple regression sub-methods), KNSNR (using all three parameters with regards to which
variable(s) to control for SAC, k=10) and SLR (using inverse weights, k=3).
Dataset Uncontrolled CD_d_sqrt CD_p_sqrt CD_b_sqrt CD_mr_sqrt KNSNR_d_k10 KNSNR_p_k10 KNSNR_b_k10 SLR_k3
1 0.664 0.356 0.367 0.356 0.402 0.042 0.046 -0.012 -0.008
2 0.684 0.622 0.633 0.622 0.65 0.447 0.481 0.532 0.436
3 0.664 0.445 0.456 0.445 0.485 0.194 0.199 0.235 0.1
4 0.684 0.628 0.635 0.628 0.646 0.518 0.533 0.583 0.456
5 0.661 0.611 0.61 0.611 0.611 0.08 0.089 0.034 0.129
6 0.662 0.626 0.626 0.626 0.626 0.487 0.514 0.548 0.501
Table 8: SAC corrected results across methods.
We see that for datasets 1-4, the CD approaches fail to correct for the spurious correlation since the
correlations after control are still sizable. For the same datasets we see that KNSNR and SLR perform
ok to well. For the most important datasets, 5-6, the CD approaches barely make a difference, but the
other two methods make a very large difference.
8. Discussion and conclusion
Table 9 shows an overview of SAC methods and their uses.
Approach Can induce SAC? Can measure SAC? Can control for SAC?
Correlation of distances ? Yes Yes
k nearest spatial neighbor Yes Yes Yes
Spatial local regression No No Yes
Moran's I and similar No Yes No
Table 9: Overview of SAC methods and their uses.
Correlation of distances (Piffer, 2015) is a clever idea, and a potentially powerful approach because it
can be used to both measure and control for SAC and possibly induce SAC too. However, the version
of it examined in this paper (“simple CD”) seems to be fairly useless in the sense that it failed to correct
for spurious causation in simulated datasets.
It is worth examining whether limiting CD to the nearest k neighbors or neighbors within a radius of a
given number of kilometers makes it a better measure. Another option is to weigh the datapoints by
weights that diminish as the distance increases (e.g. inverse). Both suggested modifications attempt to
deal with the problem that results from the globalness of the simple CD.
A second problem with CD approaches is that because distance scores are based on two datapoints, any
measurement error in these scores will have a compound effect on the difference scores, but the matter
is complicated (Trafimow, 2015). I'm not sure if this has an effect on the conversion back to an r-type
measure as described in this paper, but it might.
A third problem with CD approaches is that the number of distances quickly become very large. E.g.
for a dataset with 10,000 cases, the number of distance scores is 49,995,000. This makes for fairly
resource-intensive calculations. Random sampling of datapoints could probably be used to offset this
problem at the price of some sampling error.
KNSNR approaches are powerful since they can be used to all three ends. The main weakness seems to
be that they require a tuning parameter. Probably some approach could be find this through re-sampling
methods similar to those used to find tuning parameters for e.g. lasso regression (James et al., 2013).
One problem with KNSNR is that it tends to fairly quickly reach a ceiling.
SLR approaches seem more limited because they can only be used to correct for SAC. They do
however seem to give similar results to KNSNR approaches, so they could probably be used along
KNSNR in a multi-method approach to controlling for SAC. Like KNSNR approaches SLR requires a
tuning parameter, but it seems to always be better to use k=3. The smaller the k, the smaller the effect
of SAC in the data on the estimate but the more sampling error.
Still, the present findings call for further research into all three approaches.
8.1. Limitations
Only 12 test datasets were examined. Testing the methods against each other on a larger and more
diverse set of datasets might change conclusions somewhat.
Of the traditional measures of SAC, only Moran's I was used in the comparisons. Further research
should compare the proposed measures with other already developed measures such as Geary's C, Gi,
Gi* and local Moran's I (Radil, 2011, p. 19).
Another method used by (Hassall & Sherratt, 2011), spatial eigenvector mapping, to correct for SAC
could not be used because I could not find a suitable R implementation.
Supplementary material and acknowledgments
The R source code for this paper is available at https://osf.io/f2pnr/files/
Thanks to Davide Piffer for helpful comments.
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1 The reader with a sense for detail might wonder how the rank-order correlations for CD and CD_sqrt differ. Taking the
square root of a vector of numbers does not change their order, so how can the rank-order correlations be different? The
explanation is that sometimes the square root is undefined because the original number was negative. The correlations
are based on pair-wise, not case-wise complete data, which results in a slightly different set of cases used for the
correlations for CD and CD_sqrt.
Figure 17: Example 6. Flatland and predictor.
Figure 18: Scatter plot for predictor and outcome in dataset 5.
Figure 19: Scatter plot for predictor and outcome in dataset 6.
Figure 20: Local regression. At each point, a model is fit. Only nearby cases are
used to fit and they are weighted by their distance such that the more distant cases
weigh less. Copied from (James et al., 2013, p. 285).
Figure 21: Spatial local regression results for datasets 1-6 for varying
values of k. Inverse distance used as weight.