Overview

• Point Patterns

  • What is spatial association?

• Mantel

  • Association between response and geography

• Partial Mantel

  • Association between response and ecology, given geography

• GLS

  • Estimating ecology effects wth respect to geography

• Comparisons: Multiple methods - which is best?

• Summary: Areas for future exploration

Geography in Biological Data

• Biological organisms inhabit physical space

• Several interesting questions relate to geography/space

  • How are my observations distributed?

  • Are my data associated with geography?

  • Are my data spatially autocorrelated?

  • Can I account for spatial autocorrelation in my analysis?

Point Patterns

• The distribution of objects in space can be of possible interest

• Possible distributions: locations are random, they are more clumped than expected, they are more regular (dispersed) than expected

Point Patterns

• The distribution of objects in space can be of possible interest

• Possible distributions: locations are random, they are more clumped than expected, they are more regular (dispersed) than expected

• Other options depend on the grain scale (the scale at which the pattern is evaluated)

• One statistically assesses these distributions using Area Partition Methods and Distance Methods

Point Patterns: Aggregation Index

• Are points random, clumped, or regular?

• \(\small{1}^{st}\) order nearest neighbor index (index of aggreggation: R)

  • Calculate nearest neighbor distances (NND)
  • Obtain average NND: \(\small{\overline{r}_{obs}}\)
  • Determine ratio of observed vs. expected: \(\small{R}=\frac{\overline{r}_{obs}}{\overline{r}_{exp}}\)

see Clark and Evans (1954: Ecology) for description of obtaining expected values (based on bivariate random data: expected value obtained analytically: see their Appendix)

Point Patterns: Aggregation Index

• Are points random, clumped, or regular?

• \(\small{1}^{st}\) order nearest neighbor index (index of aggreggation: R)

  • Calculate nearest neighbor distances (NND)
  • Obtain average NND: \(\small{\overline{r}_{obs}}\)
  • Determine ratio of observed vs. expected: \(\small{R}=\frac{\overline{r}_{obs}}{\overline{r}_{exp}}\)

see Clark and Evans (1954: Ecology) for description of obtaining expected values (based on bivariate random data: expected value obtained analytically: see their Appendix)

• \(\small{R<1}\): clumped

• \(\small{R=1}\): random

• \(\small{R>1}\): dispersed

• Test for significance via effect size conversion: \(\small{Z-score}\)

Point Patterns: Ripley’s K

• Models the point pattern relative to some process

\[\small{K=\frac{E(pts)}{\lambda}}\]

where \(\small{E(pts)}\) is the the expected number of points in some area, and \(\small\lambda\) is the density of points

• \(\small{K}\) describes number of points in some area relative to expectation

Point Patterns: Ripley’s K

• Models the point pattern relative to some process

\[\small{K=\frac{E(pts)}{\lambda}}\]

where \(\small{E(pts)}\) is the the expected number of points in some area, and \(\small\lambda\) is the density of points

• \(\small{K}\) describes number of points in some area relative to expectation

• May be used to test hypotheses of clustering or regular distribution (assessed via stochastic simulation)

Point Patterns: Area Partition Methods

• Quadrat method:

  • Break area into \(\small{n}\) quadrats and count \(\small{X}\) objects in each

  • Calculate Index of Dispersion: \(\small{ID=\frac{\sigma^2_X}{\overline{X}}}\)

• ID can estimate whether points differ from a random dispersion, but not HOW they differ (i.e., cannot distinguish between clumped & regular)

Point Patterns: Area Partition Methods

• Contiguous quadrat method

  • Break area into successively smaller quadrats nested one within the other, such that each quadrat is half the size of its ‘parent’

  • Count objects in each quadrat

  • For each quadrat size (r) calculate: \(\small{G=2T_r-T_{2r}}\); where \(\small{T_r}\) is SS for the \(\small{r}^{th}\) quadrat size

  • Test each quadrat size using: \(\small\frac{G_r}{G_1}\); tested against \(\small{F_{N/2r,N/2}}\)

• Random points: constant \(\small{G_r}\) with r; Clumped points: \(\small{G_r}\) peaks at the clump size (r); Regular points: \(\small{G_r}\) decreases and bottoms out at regular grain size (r)

Note: There are many other dispersion indices (see Dale, 1999; Rosenberg, 2003)

Point Patterns: Distance Methods

• Use distances among objects to determine whether they are farther apart, or closer together than expected by chance (randomness)

• Plant-Plant Distance (W): Calculate distance between objects and their \(\small{1}^{st}\), \(\small{2}^{nd}\), \(\small{3}^{rd}\) (etc.) nearest neighbors

• Point-Plant Distance (X): Calculate distance between randomly chosen locations and their \(\small{1}^{st}\), \(\small{2}^{nd}\), \(\small{3}^{rd}\) (etc.) nearest neighbor objects

• Point-Plant-Plant Distance (Y): choose random location, find the nearest object to it, and calculate the distance from that object to its \(\small{1}^{st}\), \(\small{2}^{nd}\), \(\small{3}^{rd}\) (etc.) nearest neighbors

• Determining whether the distribution of objects is different from Poisson random is basically asking the probability that another object is within an area radius W (or X, Y)

• MANY different tests for randomness have been proposed based on these measurements

• One can also compare observed data to randomly generated point patterns

When Traits are Influenced by Spatial Gradients

• Here we see data that are influenced by spatial location. Such patterns exhibit spatial autocorrelation.

Spatial Autocorrelation: General Considerations

• Here we see data (\(\small\mathbf{Y}\)) that are influenced by spatial location. Such patterns exhibit spatial autocorrelation.

• Here are some questions one may consider:

  • How are my observations distributed in space? (point patterns: discussed previously)

  • Are the data (\(\small\mathbf{Y}\)) associated with geography? Or more precisely, are changes found among subjects associated with changes in geography?

  • Are the data (\(\small\mathbf{Y}\)) spatially autocorrelated? (NOTE: not the same as associated: a more precise meaning)

  • If my data (\(\small\mathbf{Y}\)) have spatial autocorrelation, how can I assess ecological hypotheses while taking this into consideration (i.e., without geography becoming a counfounding variable)?

  • Of the various methods to address the previous concerns, which are suited for what kind of circumstances?

Association with Geography

• Let us again examine this pattern. Here we see that species diversity and geography seem associated.

• One could assess this using the Mantel Test using distances between subjects for geography \(\small\mathbf{X}\) and species diversity \(\small\mathbf{Y}\)

Association with Geography

• Let us again examine this pattern. Here we see that species diversity and geography seem associated.

• One could assess this using the Mantel Test using distances between subjects for geography \(\small\mathbf{X}\) and species diversity \(\small\mathbf{Y}\)

  • Obtain: \(\small{z}_M= \sum{\mathbf{X}_{i}\mathbf{Y}_{i}}\) where \(\small\mathbf{X}\) & \(\small\mathbf{Y}\) are the unfolded distance matrices

  • Estimate the Mantel correlation coefficient: \(\small{r}_M = \frac{z_M}{[n(n-1)/2]-1}\)

  • Assess significance of \(\small{r_M}\) via permutation, where R/C of distance matrix are permuted.

Mantel Test: Example

mantel(dist(g), dist(y), permutations = 999)  

## 
## Mantel statistic based on Pearson's product-moment correlation 
## 
## Call:
## mantel(xdis = dist(g), ydis = dist(y), permutations = 999) 
## 
## Mantel statistic r: 0.4739 
##       Significance: 0.001 
## 
## Upper quantiles of permutations (null model):
##    90%    95%  97.5%    99% 
## 0.0624 0.0869 0.1008 0.1247 
## Permutation: free
## Number of permutations: 999

plot(dist(g), dist(y))

• There is an association of species diversity with geography (locations that are geographically-proximate are similar in their levels of species abundance)

Mantel Tests: Basic Design

• One can think of this implementation of the Mantel test as evaluating:

• This is one way of evaluating whether there is spatial association in one’s variables

Accounting for Geography: Partial Mantel Tests

• If \(\small\mathbf{X}\) and \(\small\mathbf{Y}\) both covary with geogaphy \(\small\mathbf{Z}\), one can consider partialing out the effect of geography

• Conceptually: \(\small\mathbf{Y~X|Z}\)

• Assesses relationship of \(\small\mathbf{Y}\) and \(\small\mathbf{X}\) holding \(\small\mathbf{Z}\) constant

mantel.partial(dist(t), dist(y), dist(g), permutations = 999)

## 
## Partial Mantel statistic based on Pearson's product-moment correlation 
## 
## Call:
## mantel.partial(xdis = dist(t), ydis = dist(y), zdis = dist(g),      permutations = 999) 
## 
## Mantel statistic r: 0.02606 
##       Significance: 0.111 
## 
## Upper quantiles of permutations (null model):
##    90%    95%  97.5%    99% 
## 0.0273 0.0436 0.0532 0.0700 
## Permutation: free
## Number of permutations: 999

• There is an association between species diversity and ecotype while partialling out effects of geography

Mantel Tests: Cautions

• Mantel tests seem appropriate for all manner of spatial hypothesis testing. But remember

  • They suffer from inflated type I error, low power, and can have significant bias, particularly with autocorrelated error

  • Other approaches should be used for these sorts of hypotheses

See: Oden and Sokal 1992. J. Classif.; Legendre 2000. J. Stat. Comp. Simul.; Harmon and Glor 2010. Evolution; Guillot & Rousset 2013. Methods Ecol. Evol.

Spatial Autocorrelation: Digging Deeper

• What can produce spatial autocorrelation (SAC)?

  • Biological processes (e.g., assortative mating)
  • Ecological processes/Natural Selection/Adaptation/Phenotypic plasticity
  • Non-linear relationships between species and environment modeled as linear
  • An undetected environmental covariate, which is, itself, SAC (Spatial dependency, but not SAC: Legendre et al. 2002. Ecography 25: 601-615)

Spatial Autocorrelation: Digging Deeper

• What can produce spatial autocorrelation (SAC)?

  • Biological processes (e.g., assortative mating)
  • Ecological processes/Natural Selection/Adaptation/Phenotypic plasticity
  • Non-linear relationships between species and environment modeled as linear
  • An undetected environmental covariate, which is, itself, SAC (Spatial dependency, but not SAC: Legendre et al. 2002. Ecography 25: 601-615)

• In general statistical terms, we can consider the problem as a linear model:

\[\small\mathbf{Y}=\mathbf{X{\hat{\beta}}+\epsilon}\] Here, \(\small\epsilon\) is not iid, as these assumptions are not met (there is SAC). Thus, we should model \(\small\epsilon\) something like: \(\small\sim\mathcal{N}(0,\mathbf{\Sigma})\). where \(\small\mathbf{\Sigma}\) embodies the expected spatial covariation between subjects.

Spatial Autocorrelation: Digging Deeper

• What can produce spatial autocorrelation (SAC)?

  • Biological processes (e.g., assortative mating)
  • Ecological processes/Natural Selection/Adaptation/Phenotypic plasticity
  • Non-linear relationships between species and environment modeled as linear
  • An undetected environmental covariate, which is, itself, SAC (Spatial dependency, but not SAC: Legendre et al. 2002. Ecography 25: 601-615)

• In general statistical terms, we can consider the problem as a linear model:

\[\small\mathbf{Y}=\mathbf{X{\hat{\beta}}+\epsilon}\] Here, \(\small\epsilon\) is not iid, as these assumptions are not met (there is SAC). Thus, we should model \(\small\epsilon\) something like: \(\small\sim\mathcal{N}(0,\mathbf{\Sigma})\). where \(\small\mathbf{\Sigma}\) embodies the expected spatial covariation between subjects.

• This leads to two steps:

  • Determining if geography could influence variation in \(\small\mathbf{Y}\) (i.e., is there spatial autocorrelation?)
  • Describe the spatial autocorrelation in some way (i.e., estimate \(\small\mathbf{\Sigma}\))

Spatial Autocorrelation

• Spatial Autocorrelation: lack of independence of values based on spatial properties (self-correlation)

• Can test whether observed values at one locality depend (at least in part) on those at neighboring localities.

• Knowing the spatial auto-correlation ‘structure’ informs one how to model ecological factors

Spatial Autocorrelation

• Spatial Autocorrelation: lack of independence of values based on spatial properties (self-correlation)

• Can test whether observed values at one locality depend (at least in part) on those at neighboring localities.

• Knowing the spatial auto-correlation ‘structure’ informs one how to model ecological factors

• Procedure:

  • Obtain data \(\small\mathbf{Y}\) for each locality (can be continuous or categorical data)
  • Determine which localities are ‘neighbors’ (or distance classes)
  • Neighbors represented in a CONNECTIVITY MATRIX (or weight matrix)
  • Estimate spatial autocorrelation for neighbors/distance class

Connectivity (W): Neighbors and Distances

Measuring Autocorrelation: Join Counts

• For categorical data (A,B)

  • Join-Count statistic: sums the number of connections of a given type:

Measuring Autocorrelation: Continuous Data

• For continuous data, one still measures \(\small\sum{(weights \times{data})}\)

• Moran’s I: \(\small{I}=\frac{n\sum\sum{w_{ij}(y_i-\hat{y})(y_j-\hat{y})}}{W\sum(y_i-\hat{y})^2}\)

  • where \(\small{w_{ij}}\) is the weight of influence of site \(\small{j}\) over \(\small{i}\), \(\small{y_i}\) is the data, \(\small\hat{y}\) is the estimate for that distance class, and \(\small{W=\sum{w_{ij}}}\)
  • range –1 to +1 (- are negative autocorrelation, + are positive autocorrelation)

Measuring Autocorrelation: Continuous Data

• For continuous data, one still measures \(\small\sum{(weights \times{data})}\)

• Moran’s I: \(\small{I}=\frac{n\sum\sum{w_{ij}(y_i-\hat{y})(y_j-\hat{y})}}{W\sum(y_i-\hat{y})^2}\)

  • where \(\small{w_{ij}}\) is the weight of influence of site \(\small{j}\) over \(\small{i}\), \(\small{y_i}\) is the data, \(\small\hat{y}\) is the estimate for that distance class, and \(\small{W=\sum{w_{ij}}}\)
  • range –1 to +1 (- are negative autocorrelation, + are positive autocorrelation)

• Geary’s c: \(\small{c}=\frac{(n-1)\sum\sum{w_{ij}(y_i-y_j)^2}}{2W\sum(y_i-\hat{y})^2}\)

  • range 0 to positive values (0 is positive autocorrelation, + values are negative)

• For both, significance based on normal approximation (see Legendre et al. 2012): \(\small{z}_I=\frac{I_{obs}-E(I)}{\sigma_I}\) where \(\small{E(I)}= -\frac{1}{n-1}\). Also, \(\small{E(c)=1}\)

Note: Semivariograms are based on unstandardized Geary’s c

Spatial Autocorrelation: Semivariogram

The semivarigram for the simulated data, and with a Gaussian model

Spatial Autocorrelation: Example

• Sokal and Thompson (1987) examined distribution and attributes of Aralia nudicaulis (an understory plant), including: fecundity, density, canopy cover, and percent female

• Found significant spatial autocorrelation for most variables (not fecundity), none of which displayed the same pattern

• Thus, variables exhibited autocorrelation, but not the spatial association with one another

Measuring Spatial Autocorrelation: Importance of Model

• Evaluating spatial autocorrelation is not just ‘procedural’. One is measuring the degree of pattern across spatial scales. That requires a model!

Modeling Spatial Non-Independence

• We now return to the GLS model:

\[\small\mathbf{Y}=\mathbf{X{\hat{\beta}}+\epsilon}\]

Here, \(\small\epsilon\) is not iid, as these assumptions are not met (there is SAC). Thus, we should model \(\small\epsilon\) something like: \(\small\sim\mathcal{N}(0,\mathbf{\Sigma})\). where \(\small\mathbf{\Sigma}\) embodies the expected spatial covariation between subjects.

Thus, given an estimate \(\small\mathbf{\Sigma}\), we can fit the model as: \(\small\hat{\mathbf{\beta }}=\left ( \mathbf{X}^{T} \mathbf{\Sigma}^{-1} \mathbf{X}\right )^{-1}\left ( \mathbf{X}^{T} \mathbf{\Sigma}^{-1}\mathbf{Y}\right )\)

The question is: what is \(\small\mathbf{\Sigma}\)?

Modeling Spatial Non-Independence

• We now return to the GLS model:

\[\small\mathbf{Y}=\mathbf{X{\hat{\beta}}+\epsilon}\]

Here, \(\small\epsilon\) is not iid, as these assumptions are not met (there is SAC). Thus, we should model \(\small\epsilon\) something like: \(\small\sim\mathcal{N}(0,\mathbf{\Sigma})\). where \(\small\mathbf{\Sigma}\) embodies the expected spatial covariation between subjects.

Thus, given an estimate \(\small\mathbf{\Sigma}\), we can fit the model as: \(\small\hat{\mathbf{\beta }}=\left ( \mathbf{X}^{T} \mathbf{\Sigma}^{-1} \mathbf{X}\right )^{-1}\left ( \mathbf{X}^{T} \mathbf{\Sigma}^{-1}\mathbf{Y}\right )\)

The question is: what is \(\small\mathbf{\Sigma}\)?

It turns out there are various ways to parameterize \(\small\mathbf{\Sigma}\), but it is always an \(\small{n\times{n}}\) matrix with values estimating the expected covariance between subjects as based on the geographic distance between them.

Note: this is conceptually identical to what we discussed with phylogenetic comparative methods, where \(\small\mathbf{\Sigma}\) was the phylogenetic covariance matrix (\(\small\mathbf{V}\))

Modeling Spatial Non-Independence

• In general, \(\small\mathbf{\Sigma}\) describes the expected covariance among locations, with coefficients proportional to distance (or the decay of expected similarity as a function of distance)

• MANY models have been proposed to describe spatial non-independence. Here are a few:

\[\small{Exponential}= \sigma^2e^{-r/d}\]

\[\small{Gaussian}= \sigma^2e^{(-r/d)^2}\]

\[\small{Spherical}= \sigma^2(1-2/\pi(r/d\sqrt{1-r^2/d^2}+sin^{-1}r/d))\]

where \(\small{r}\) describes the expected covariance (correlation) between a pair of subjects, and \(\small{d}\) is the distance between them over which this corralation decays.

NOTE: There are many other models of spatial dependence! (see Dormann et al. (2007) Ecography.)

Spatial Dependence: Autoregressive Models

• Some approaches use ‘autoregressive’ approaches to account for spatial dependence

NOTE: There are many other models of spatial dependence! (see Dormann et al. (2007) Ecography.)

Spatial Dependence: Autoregressive Models

• Some approaches use ‘autoregressive’ approaches to account for spatial dependence

see Dormann et al. (2007) Ecography

Spatial Dependence: Comparing Models

• Dormann et al. (2007) compared numerous models

Results imply that GLS approach (i.e., a spatially-weighted model) performs quite well!

Spatial Statistics: Summary

• Having a spatial context is hugely important to consider in biology

• How are my observations distributed?

  • Point Pattern Analysis

• Are my data associated with geography?

  • Mantel tests (and others)

• Are my data spatially autocorrelated?

  • Moran’s I & Geary’s c

• Can I account for spatial autocorrelation in my analysis?

  • GLS is general and very useful model