A Mantel test measures the correlation between two matrices typically containing
measures of distance. A Mantel test is one way of testing for spatial
autocorrelation. Using functions in the **ade4** library, we can
perform a Mantel test in R. To download
and load this library, enter `install.packages("ade4")`

and then **
library(ade4)**. There are other Mantel test functions available in
other R libraries and our choice of this library’s should not be seen as an
endorsement in any way.

Let’s look at an example. Our dataset,
**ozone**, contains ozone measurements from
thirty-two locations in the Los Angeles area aggregated over one month. The dataset
includes the station number (**Station**), the latitude and longitude of the station
(**Lat** and
**Lon**), and the average
of the highest eight hour daily averages (**Av8top**). We will be interested
in testing if the differences in ozone measurements are smaller for stations
that are closer together than for stations that are far apart. This data, and other spatial datasets, can be
downloaded from the University of Illinois’s Spatial Analysis Lab. We can look
at a summary of our location variables to see the range of locations under
consideration.

ozone <- read.table("http://www.ats.ucla.edu/stat/r/faq/ozone.csv", sep=",", header=T) head(ozone, n=10)Station Av8top Lat Lon 1 60 7.225806 34.13583 -117.9236 2 69 5.899194 34.17611 -118.3153 3 72 4.052885 33.82361 -118.1875 4 74 7.181452 34.19944 -118.5347 5 75 6.076613 34.06694 -117.7514 6 84 3.157258 33.92917 -118.2097 7 85 5.201613 34.01500 -118.0597 8 87 4.717742 34.06722 -118.2264 9 88 6.532258 34.08333 -118.1069 10 89 7.540323 34.38750 -118.5347

To run a Mantel test, we will need to generate two distance matrices: one
containing spatial distances and one containing distances between measured
outcomes at the given points. In the spatial distance matrix, entries for
pairs of points that are close together are lower than for pairs of points that
are far apart. In the measured outcome matrix, entries for pairs of
locations with similar outcomes are lower than for pairs of points with
dissimilar outcomes. We do this using the **dist** function. The
Mantel test function will require objects of this “distance” class.

station.dists <- dist(cbind(ozone$Lon, ozone$Lat)) ozone.dists <- dist(ozone$Av8top) as.matrix(station.dists)[1:5, 1:5]1 2 3 4 5 1 0.0000000 0.3937326 0.4088031 0.6144127 0.1854888 2 0.3937326 0.0000000 0.3749446 0.2206810 0.5743590 3 0.4088031 0.3749446 0.0000000 0.5116772 0.4994034 4 0.6144127 0.2206810 0.5116772 0.0000000 0.7944601 5 0.1854888 0.5743590 0.4994034 0.7944601 0.0000000as.matrix(ozone.dists)[1:5, 1:5]1 2 3 4 5 1 0.000000 1.326612 3.172921 0.044354 1.149193 2 1.326612 0.000000 1.846309 1.282258 0.177419 3 3.172921 1.846309 0.000000 3.128567 2.023728 4 0.044354 1.282258 3.128567 0.000000 1.104839 5 1.149193 0.177419 2.023728 1.104839 0.000000

These are the two matrices which the function will be testing for a
correlation. The test consists of calculating the correlation of the
entries in the matrices, then permuting the matrices and calculating the same test
statistic under each permutation and comparing the original test statistic to
the distribution of test statistics from the permutations to generate a p-value.
The number of permutations defines the precision with which the p-value can be
calculated. The function to perform the Mantel test is **mantel.rtest**
and the required arguments are the two distance matrices. The number of
permutations can also be specified by the user, but is by default 99.

mantel.rtest(station.dists, ozone.dists, nrepet = 9999)Monte-Carlo test Observation: 0.1636308 Call: mantel.rtest(m1 = station.dists, m2 = ozone.dists, nrepet = 9999) Based on 9999 replicates Simulated p-value: 0.0312

Based on these results, we can reject the null hypothesis that these two
matrices, spatial distance and ozone distance, are unrelated** **with alpha = .05.
The observed correlation, r = 0.1636308, suggests that the matrix entries are
positively associated. So smaller differences in ozone are generally seen
among pairs of stations that are close to each other than far from each other.
Note that since this test is based on random permutations, the same code will
always arrive at the same observed correlation but rarely the same p-value.