Lecture 13
EEOB 590C
1

•Biological organisms inhabit physical space
•Several interesting questions relate to geography/space
1: How are my observations distributed?
2: Are my data associated with geography?
3: Are my data spatially autocorrelated?
4: Can I account for spatial autocorrelation in my analysis?
2

• The distribution of objects in space can be of interest
• Possible distributions: the locations are random, they are more clumped than expected, they are more regular (dispersed) than expected
Random
Clumped Regular
(Dispersed)
8
6
4
2
0
0
12
10
16
14
5 10 15
•
Other options exist
Clumps of
Regular
Regular
Clumps
• How does one statistically assess these distributions?
•
Area Partition Methods and Distance Methods
3

• Are points random, clumped, or regular?
16
14
12
10
8
6
4
2
0
0 5 10 15
• 1 st order nearest neighbor index (index of aggregation: R)
• Calculate nearest neighbor distances (NND)
•
Determine average NND: r
Obs
• Determine ratio of observed vs. expected*:
•
R < 1: clumped
•
R = 1: random
• R > 1: regular (dispersed)
•
Test for significance (convert to Z-score)
R
 r
Obs r
Exp
*see Clark and Evans (1954: Ecology ) for description of obtaining expected values
4

• Models the point pattern relative to some process
K

( ) l
E(pts) = expected # points in some area; l
= density of points
•
K describes # points in area relative to expectation
• Use
K to test hypotheses of clustering or regular dispersion
Assessed statistically via stochastic simulations
K obs
K theo
K hi r
K lo r r r
E
0.00
0.05
0.10
r
0.15
0.20
0.25
5

• Quadrat Method:
• Break into quadrats and count objects (X) in each of the n quadrats
•
Calculate Index of Dispersion (variance – mean ratio):
ID


X
2
X
ID*(n-1) follows a X 2 with ( n -1) df
•
ID Can estimate whether points differ from a random dispersion, but not HOW they differ (i.e., clumped vs. regular)
6

• 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: , where Tr is the sum r

2 T r

T
2 r of squares for the r th quadrat size
•
Test each quadrat size using ratio:
G r
(Compare to F with N/2r and N/2 df)
G
1
•
Random points: constant Gr with r; Clumped points: Gr peaks at the clump size (r); Regular points: Gr decreases and bottoms out at regular grain size (r)
•
Note: There are many other dispersion indices (see Dale, 1999; Rosenberg, 2003) 7

• 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
1st, 2nd, 3rd (etc.) nearest neighbors
•
Point-Plant Distance (X): Calculate distance between randomly chosen locations and their 1st, 2nd, 3rd (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 1st,
2nd, 3rd (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
8

•If geographic localities of objects are known, one can obtain a geographic distance matrix
•Several hypotheses can now be addressed
•Is some variable (Y) associated with geography?
•Are 2 variables (X and Y) associated while accounting for geographic patterns? morphology associated with resource use?
•Mantel tests are one way to assess these hypotheses
9

•Sokal (1988) examined European genetic and linguistic data
•27 genetic distance matrices (each data type separately)
•Geographic distance
•Linguistic ‘design’ matrix (0= same language family, 1=same language phylum but different family, 2= different language phylum)
870 European localities
From Sokal (1988). Proc. Natl. Acad. Sci., USA. 85:1722-1726.
10

•Genetic groups and Language groups are associated with geography
•Genetic groups and language groups are associated even when geography is taken into account
From Sokal (1988). Proc. Natl. Acad. Sci., USA. 85:1722-1726.
11

•What historical patterns that shaped current phenotypic diversity
•Test hypotheses of local adaptation, gene flow & vicariance
•Data: phenotypic variation at 33 sites, historical drainage patterns, habitat types: all converted to distance matrices
From Douglas et al. 1999. Evolution. 53:238-246.
12

•Pliocene drainages most highly associated with current phenotypic diversity
From Douglas et al. 1999. Evolution. 53:238-246.
13

• Spatial Autocorrelation : lack of independence of values based on spatial properties
•self-correlation: close values are more similar/different
•Test whether observed values at one locality depend (at least in part) on those at neighboring localities
•Procedure:
•Obtain data for each locality (Y)
(can be continuous or categorical)
•Determine which localities are ‘neighbors’ (or distance classes)
•Neighbors represented in a CONNECTIVITY MATRIX (or weight matrix)
•Estimate spatial autocorrelation for neighbors/distance class
14

• Regularly-spaced data: W from ‘chessboard’ methods:
• Irregularly-spaced data: Spatially-defined neighbors
(many methods)
•
Delauney tesselation/ theisson polygons
• Gabriel network
•
Relative neighborhood
• Divide distances into distance-classes: equal interval or equal frequency
15

• For categorical data (A,B)
•
Join-Count statistic: sums the number of connections of a given type: r
AA

1
2
 ij w AA ij
( ) ij
Convert to Z-score and assess vs. normal distribution w ij is the weight (0 = not connected, 1 = connected), (AA) ij is A-A connection
(divide by ½ because upper and lower diagonal
W are identical)
•
Note: negative or positive depends on how you make the connections (e.g., bishop vs. rook yield complementary -/+ values) r
BB

0 r

BW
12
• z-scores are: BB = -3.4156 WW = -2.245 BW = 3.456
• conclusion: significant negative spatial autocorrelation
16

• For continuous data, statistic is S
(weights * data)
• Moran’s I : range –1 (negative) to +1 (positive) autocorrelation
I
 n n n  i

1 j

1 w
W ij i n 


1
 y i y
 i y

 y


2 y j
 y
 w ij is the weight, y i y is data distance class, W =
S
( w ij
)
• Geary’s c: range 0 to positive values
(0 is positive autocorrelation, + values are negative)
 n

1
 i n n 

1 j

1 w ij
 y i
 y j

2 c

2 W i n 

1
 y i
 y

2
• Significance based on normal approximation
(see Legendre and Legendre for details) z
I

I ( ) obs


I
  n
1

1
•
I and c found for each distance class
( )

1
Note: Semivariograms based on unstandardized Geary’s c
(Geary’s c = semivariogram standardized by variance).
17

• Sokal and Thompson (1987) examined distribution and attributes of understory plant, Aralia nudicaulis, (fecundity, density, canopy cover, percent female)
•
Significant spatial autocorrelation for most variables (except fecundity), none of which showed the same pattern (i.e. variables exhibited autocorrelation, but not spatial association)
AN=Aralia density
CA=canopy cover
F=fecundity
PF=percent female
18

• For multivariate data, use Mantel Test Correlogram
- Calculate distances for Y
Perform Mantel test on distance class and plot
Mantel correlogram of D genetic genetic spatial autocorrelation showing
Data from Highton (1989).
19

Moran’s I is generally plotted as a discrete stepwise trend over geographic distance (bins)
Important point! In answering if data are spatially autocorrelated, one must have a model to consider. Autocorrelation is a measure of the coordinated residual size and direction of observations found at a certain geographic distance between subjects. Most models won’t be bad at large distances.
Moran’s I allows one to ask if spatial autocorrelation is perhaps intruding at small or intermediate distances.
Dormann et al. (2007) Ecography.
20

•Incorporate spatial non-independence in hypothesis testing
•In GLS, this is modeled in the error
•Standard GLM: ε   2
~

0,
 2

Y = Xβ + ε
•Spatial GLS:
Solve by:
Structured error term (
S
) = spatial covariance matrix
 
-1
β = X Σ X X Σ Y
Σ = observed correlations/distance among objects, or expected correlations based on some spatial model 21

• S describes covariance among locations, with coefficients proportional to distance
(or expected decay similarity as function of distance)
•MANY models proposed to describe spatial non-independence
Exponential
= s
2 e
r / d
Spherical
= s
2 ç
æ
è
Guassian
=
1
-
2 / p ç
æ
è r / s d
2 e
1
( r / d
)
r
2
2 d
2
+ sin
-
1 r d
÷
ö
ø
÷
ö
ø
See: Dormann et al. (2007) Ecography.
22

Adjust predicted values based on adjusting observed values, by the autoregressive scalar times the spatial weights between subjects – little effect if subjects are far apart in geographic space. W can be estimated in various ways.
Recall, W is an n x n matrix
Autoregressive models, ρ = autoregression parameter
See: Dormann et al. (2007) Ecography.
23

Adjust predicted values based on adjusting observed values and an error edition (because maybe both the explanatory and response variables are auto correlated).
Recall, W is an n x n matrix
See: Dormann et al. (2007) Ecography.
24

GLM = General Linear Model
GAM = trend-surface General Additive Model (no autocorrelation models – just adds geographic trends to the general linear model) autocov = Autocovariate model –
GLMM = GLS with mixed effects model (allows for spatial autocorrelation within groups)
CAR = Conditional Autoregressive models
SAR = Sequential Autoregressive models
GEE = Generalized Estimating Equations (like GLM, but allows for different spatial weighting for “clusters” of subjects) geese = GEE but with greater constraint on spatial correlations based on certain distances
SEVM = PCoA on Gower centered distance matrix, followed by adding eigenvectors as variables to GLM until Moran’s I is no longer significant
In all cases, GLS performed about the best!
See: Dormann et al. (2007) Ecography.
25

•
Spatial context hugely important to consider in biology
1: How are my observations distributed?
Point Pattern Analysis
2: Are my data associated with geography?
Mantel tests
3: Are my data spatially autocorrelated?
Moran’s I
4: Can I account for spatial autocorrelation in my analysis?
GLS is general and very useful model
26