1. Distance Measures and Ordination
Adapted from Ecological Statistical Workshop, FLC, Daniel Laughlin
Distance Measures and Ordination

2. Goals of Ordination
To arrange items along an axis or multiple axes in a logical order
To extract a few major gradients that explain much of the variability in the total dataset
Most importantly: to interpret the gradients since important ecological processes generated them

4. What makes ordination possible?
Variables (species) are “correlated” (in a broad sense)
Correlated variables = redundancy
Ordination thrives on the complex network of inter-correlations among species

5. Ordination helps to:
Describe the strongest patterns of community composition
Separate strong patterns from weak ones
Reveal unforeseen patterns and suggest unforeseen processes

6. “Direct” gradient analysis
Order plots along measured environmental gradients
e.g., regress diatom abundance on salinity

7. “Indirect” gradient analysis
Order plots according to
covariation among species, or
dissimilarity among sample units
Following this step, we can then examine correlations between environment and ordination axes
Axes = Gradients
In PCA, these are called “Principal Components”

8. Data reduction
Goal: to reduce the dimensionality of community datasets
(i.e., from 100 species down to 2 or 3 main gradients)
n x p
n x d
These d dimensions represent the strongest correlation structure in the data
This is possible because of redundancy in the data (i.e., species are “correlated”)

9. Ordination Diagrams Axis 2: “Biotic” Axis 1: “Abiotic”
Do not seek patterns as you would with a regression: axes are orthogonal (uncorrelated)
Know two things:
What the points represent (plots or species?)
Distance in the diagram is proportional to compositional dissimilarity
NMS Ordination
Axis 2: “Biotic”
Axis 1: “Abiotic”

10. How many axes?
“How many discrete signals can be detected against a background of noise?”
Typically we expect 2 or 3 gradients to be sufficient, but if we know that 5 independent environmental gradients are structuring the vegetation (water, light, CO2, nutrients, grazers, etc.), then perhaps 5 axes are justified

11. Two basic techniques
Eigenanalysis methods- use information from variance-covariance matrix or correlation matrix (e.g., PCA)
Appropriate for linear models since covariance is a measure of a linear association
Distance-based methods- use information from distance matrix (e.g., NMS)
Appropriate for nonlinear models since some distance measures and ordination techniques can “linearize” nonlinear associations

12. A summary table of ordination methods

13. Ecological Distance Measures

14. Distance measures Distance = Difference = Dissimilarity
Distance matrix is like a triangular mileage chart on maps (symmetric)
We are interested in the distances between sample units (plots) in species space

15. Distance measures
In univariate species space (one species), the distance between two points is their difference in abundances
We will examine two kinds of distance measures:
Euclidean distance, and
Bray-Curtis (Sorenson) distance

16. Domains and Ranges Distance Domain of x Range of d =f(x)
Euclidean all non-negative
Sorenson x ≥ 0 0<d<1
(0<d<100)

17. Which one works best?
“If species respond noiselessly to environmental gradients, then we seek a perfect linear relationship between distances in species space and distances in environmental space. Any departure from that represents a partial failure of our distance measure.”
McCune p. 51

18. Easy dataset (low beta diversity)
Figure 6.6

19. Difficult dataset (high beta diversity)
Intuitive property
Figure 6.7

20. NMS is able to linearize the relationship between distance in species space and environmental distance because it is based on ranked distances (stay tuned)

21. Theoretical basis
Our choice is primarily empirical: we should select measures that have been shown superior performance
One important theoretical basis: ED measures distance through uninhabitable, impossibly species rich space.
In contrast, city-block distances are measured along the edges of species space- exactly where the sample units lie in the dust bunny distribution!

22. Nonmetric Multidimensional Scaling (NMS, NMDS, MDS, NMMDS, etc.)

23. NMS Uses a distance/dissimilarity matrix
Makes no assumptions regarding linear relationships among variables
Arranges plots in a space that best approximates the distances in a distance matrix

24. From a map to a distance matrix
Calculate distances

25. From a distance matrix to a map
NMS
Question: How well do the distances in the ordination match the distances in the distance matrix?

26. Advantages of NMS Avoids the assumptions of linear relations
The use of ranked distances tends to linearize the relationship between distances in species space and distances in environmental space
You can use any distance measure

27. Historical disadvantages of NMS
Failing to find the best solution (low “stress”) due to local minima
Slow computation time
These concerns have largely been dealt with given modern computer power

28. In a nutshell
NMS is an iterative search for the best positions of n entities on k dimensions (axes) that minimizes the stress of the k-dimensional configuration
“Stress” is a measure of departure from monotonicity in the relationship between the original distance matrix and the distances in the ordination diagram

29. Achieving monotonicity
Fig 16.2
The closer the points lie to a monotonic line, the better the fit and the lower the stress.
If S* = 0, then relationship is perfectly monotonic
Blue = perfect fit, monotonic
Red = high stress, not monotonic

30. Instability
Instability is calculated as the standard deviation in stress over the preceeding 10 iterations
Instabilities of are generally preferred
sd = sqrt(var)

31. Mini Example

32. Landscape analogy for NMS
Global minimum
Local minimum
(strong, regular, geometric patterns emerge)

33. Reliability of Ordination
Low stress and stable solutions
Proportion of variance represented (R2)
Monte Carlo tests

34. Variance represented? “Ode to an eigenvalue”
NMS not based on partitioning variance, so there is no direct method
Calculate R2 for relationship between Euclidean distances in ordination versus Bray-Curtis distances in distance matrix
Axis Increment Cumulative R2

35. Monte Carlo test
Has the final NMS configuration extracted stronger axes than expected by chance?
Compare stress obtained using your data with stress obtained from multiple runs of randomized versions of your data (randomly shuffled within columns)
P-value = (1+n)/(1+N)
n = # of random runs with final stress less than or equal to the observed minimum stress,
N = number of randomized runs
P-value = the proportion of randomized runs with stress less than or equal to the observed stress

36. Monte Carlo tests

37. Autopilot mode in PC-ORD
Table 16.3 in McCune and Grace (2002)
PARAMETER
Quick and dirty
Medium
Slow and thorough
Maximum number of iterations 75
200
400
Instability criterion
0.001
0.0001
Starting number of axes 3 4 6
Number of real runs 5 15 40
Number of randomized runs 20 30 50

38. Choosing the best solution
Select the appropriate number of dimensions
Seek low stress
Use a Monte Carlo test
Avoid unstable solutions

39. 1. How many dimensions?
One dimension is generally not used, unless the data is known to be unidimensional.
More than three becomes difficult to interpret.
Find the elbow and inspect Monte Carlo tests.
elbow
Figure 16.3

40. 2. Seek low stress <5 = excellent 5-10 = good 10-20 = fair, useable
20-30 = not great, still useable
>30 = dangerously close to random
Adapted from Table 16.4, p 132

41. A general procedure Carefully read pages 135-136
In your papers, you should report the information that is listed on page 136
Autopilot mode works really well, but don’t publish ordinations obtained using the Quick and Dirty option! Be sure to publish the parameter settings.

42. Interpreting NMS axes Two main/complementary approaches
Evaluate how species abundances are correlated with NMS axes
Evaluate how environmental variables are correlated with NMS axes

43. Overlays
Overlays: flexible way to see whether a variable is patterned on an ordination; not limited to linear relationships
Axis 1

44. Overlays

45. Species versus Axes
Resist the temptation to use p-values when examining these relationships!
- nonlinear
- circular
reasoning
Unimodal
pattern
Linear
pattern

46. Environmental Variables
Joint plots- diagram of radiating lines, where the angle and length of a line indicate the direction and strength of the relationship

47. PerMANOVA

48. The analysis of community composition
Continuous covariates
Use ordination to produce a continuous response variable (i.e., axis)
Use covariance analysis (multiple regression, SEM) to explain variance of the axis
Categorical groups
Ordination is not required (remember, ordination is not the test)
Permutational MANOVA (PerMANOVA): can use on any experimental design
MRPP (only one-way or blocked designs)
ANOSIM (up to two factors, in R and PRIMER)

49. MANOVA Multivariate Analysis of Variance Traditional parametric method
Assumes linear relations among variables, multivariate normality, equal variances and covariances
Not appropriate for community data

50. PerMANOVA Permutational MANOVA Straightforward extension of ANOVA
Decomposes variance in the distance matrix
No distributional assumptions
Can still be sensitive to heterogeneous variances (dispersion) among groups
Anderson, M Austral Ecology

51. ANOVA
Compare variability within groups versus variability among different groups

52. Decomposing an observation (yij)
Variability of observations about the grand mean
Variability of the ith trt mean about the grand mean
Variability of observations within each treatment = +
SStotal = SSamong + SSwithin
SStotal = SStreatment + SSerror
PROBLEM: WE CAN’T CALCULATE MEANS WITH SEMIMETRIC BRAY-CURTIS

53. ANOVA
Compare variability within groups versus variability among different groups
A simple 2-D case
Unknowable with semi-metric Bray-Curtis distances

54. The key link
The key to this method is that “the sum of squared distances between points and their centroid is equal to (and can be calculated directly from) the sum of squared interpoint distances divided by the number of points.”

55. Why is this important?
Couldn’t use semimetric Bray-Curtis distance in ANOVA context because central locations cannot be found
But we don’t have to calculate the central locations anymore with this finding
The analysis can proceed by using distances in any distance matrix

56. One-way perMANOVA with two groups

57. Permuted p-values P = (No. of Fπ >= F) (Total no. of Fπ)
Fπ obtained with randomly shuffled data
Use at least 999 random permutations
I tend to use 9999 permutations

58. The link with ANOVA
This F statistic is equal to Fisher’s original F-ratio in the case of one variable and when Euclidean distances are used

59. Example: grazing effects (one-way)

60. Example: two-way factorial