Mantel Tests

Comparing Distance Matrices

Spatial Autocorrelation

• Spatial autocorrelation means that observations that are close to one another in space are more similar to one another than to other observations farther away– Nearby sites are closer in time– Nearby sites are closer in assemblage

composition

Mantel Test

• Widely used in ecology to compare two distance matrices

• Correlation is a measure of spatial autocorrelation (pearson, spearman, or kendall’s tau

• Significance must be determined by permutations of the distance matrix

Example

• Snodgrass site – does the presence of decorated/prestige artifacts show spatial autocorrelation?

• Load the data set and convert the variables to dichotomies

• Compute distance matrices and plot 91*90/2 = 4095 distance pairs

> pa <- sapply(Snodgrass[,8:39], function(x) as.numeric(as.logical(x)))> Snodgrass3 <- data.frame(Snodgrass[,1:7], pa, Snodgrass[,40:41])

> library(vegan)> S.dist <-dist(Snodgrass3[,c(26, 27, 29:38)], method="binary")> A.dist <-dist(Snodgrass3$Area, method="euclidean")> G.dist <-dist(Snodgrass3[,1:2], method="euclidean")> plot(G.dist, S.dist, pch=".", cex=2)> abline(lm(S.dist~G.dist), col="red", lwd=2)> GS.man <- mantel(G.dist, S.dist)> GS.man

Mantel statistic based on Pearson's product-moment correlation

Call:mantel(xdis = G.dist, ydis = S.dist)

Mantel statistic r: 0.1271 Significance: 0.001

Empirical upper confidence limits of r: 90% 95% 97.5% 99% 0.0282 0.0364 0.0455 0.0551

Based on 999 permutations

> str(GS.man)List of 6 $ call : language mantel(xdis = G.dist, ydis = S.dist) $ method : chr "Pearson's product-moment correlation" $ statistic : num 0.127 $ signif : num 0.001 $ perm : num [1:999] -0.01123 -0.00424 -0.003... $ permutations: num 999 - attr(*, "class")= chr "mantel"

Mantel Correlogram

• Correlation plotted by distance to see at what distances the correlation peaks

S.correl <- mantel.correlog(S.dist, G.dist)plot(S.correl)print(S.correl)

Mantel Correlogram Analysis

Call: mantel.correlog(D.eco = S.dist, D.geo = G.dist)

class.index n.dist Mantel.cor Pr(Mantel) Pr(corrected) D.cl.1 24.561714 278.000000 0.038772 0.004 0.004 **D.cl.2 53.465764 666.000000 0.065959 0.001 0.002 **D.cl.3 82.369815 848.000000 0.054817 0.002 0.004 **D.cl.4 111.273865 908.000000 0.023341 0.070 0.070 . D.cl.5 140.177916 1038.000000 0.027887 0.045 0.090 . D.cl.6 169.081966 1010.000000 -0.012919 0.221 0.221 D.cl.7 197.986017 914.000000 -0.023078 0.073 0.210 D.cl.8 226.890067 876.000000 NA NA NA D.cl.9 255.794118 640.000000 NA NA NA D.cl.10 284.698168 462.000000 NA NA NA D.cl.11 313.602219 318.000000 NA NA NA D.cl.12 342.506269 174.000000 NA NA NA D.cl.13 371.410320 56.000000 NA NA NA ---Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Partial Mantel Test

• We can control for a third variable to see if that eliminates the spatial autocorrelation

> GSA.man <- mantel.partial(G.dist, S.dist, A.dist)> GSA.man

Partial Mantel statistic based on Pearson's product-moment correlation

Call:mantel.partial(xdis = G.dist, ydis = S.dist, zdis = A.dist)

Mantel statistic r: 0.1286 Significance: 0.001

Empirical upper confidence limits of r: 90% 95% 97.5% 99% 0.0270 0.0357 0.0439 0.0542

Based on 999 permutations