CHAPTER 16
Nonmetric Multidimensional Scaling
Tables, Figures, and Equations
From: McCune, B. & J. B. Grace. 2002. Analysis of
Ecological Communities. MjM Software Design,
Gleneden Beach, Oregon http://www.pcord.com
Basic procedure
1. Calculate dissimilarity matrix D.
Basic procedure
1. Calculate dissimilarity matrix D.
2. Assign sample units to a starting configuration in the kspace (define an initial X). Usually the starting locations
(scores on axes) are assigned with a random number
generator. Alternatively, the starting configuration can be
scores from another ordination.
Basic procedure
1. Calculate dissimilarity matrix D.
2. Assign sample units to a starting configuration in the kspace (define an initial X). Usually the starting locations
(scores on axes) are assigned with a random number
generator. Alternatively, the starting configuration can be
scores from another ordination.
3. Normalize X by subtracting the axis means for each axis l
and dividing by the overall standard deviation of scores:
normalized
x il =
x il  xl
k
n
  x
l 1
i 1
 xl 
2
il
/ (n  k )
4. Calculate D, containing Euclidean distances
between sample units in k-space.
5. Rank elements of D in ascending order.
6. Put the elements of D in the same order as D.
 (containing elements d , these being the
7. Calculate D
ij
result of replacing elements of D which do not satisfy
the monotonicity constraint).
Figure 16.1. Moving a
point to achieve
monotonicity in a plot
of distance in the
original p-dimensional
space against distances
in the k-dimensional
ordination space.
8. Calculate raw stress, S*
n-1
*
S =

n
2

 ( d ij - d ij )
i=1 j=i 1
Note that S* measures the departure from monotonicity. If S* = 0,
the relationship is perfectly monotonic.
Figure 16.2. Plot of distance in ordination space (dij, horizontal axis) vs. dissimilarity
in original p-dimensional space (dij, vertical axis), analogous to Fig. 16.1. Points are
labeled with the ranked distance (dissimilarity) in the original space.
Figure 16.2. Plot of distance in ordination space (dij, horizontal axis) vs. dissimilarity
in original p-dimensional space (dij, vertical axis), analogous to Fig. 16.1. Points are
labeled with the ranked distance (dissimilarity) in the original space.
9. Because raw stress would be altered if the
configuration of points were magnified or reduced,
it is desirable to standardize ("normalize") stress.
Kruskal’s “stress formula one:”
n-1
S = S /
*
n
d
i=1 j=i 1
2
ij
9. Because raw stress would be altered if the
configuration of points were magnified or reduced,
it is desirable to standardize ("normalize") stress.
Kruskal’s “stress formula one:”
n-1
S = S /
*
n
d
2
ij
i=1 j=i 1
PC-ORD reports stress as SR, the square
root of the scaled stress, analogous to a
standard deviation, then multiplied by
100 to rescale the result from zero to 100:
SR
= 100 S
“stress formula two” is:
n-1
n
S = S /   ( d ij  d )
*
i=1 j=i 1
The two formulas yield very similar configurations.
2
10. Now we try to minimize S by changing the configuration of the
sample units in the k-space.
For the method of steepest descent, we first calculate the "negative
gradient of stress" for each point i.
The gradient vector (a vector of length k containing the movement of each
point h in each dimension l) is calculated by:
g hl  S   D - D
n
n
hi
i=1 j=1
hj




d ij  x il - x jl
 d ij - d ij
 S* 2
d
ij

d ij


i, j



i indexes one of the n points,
j indexes another of the n points,
l indexes a particular dimension, and
h indexes a third point, the point of interest which is being moved on
dimension l. So ghl indicates a shift for point h along dimension l.
The Dhi and Dhj are Kronecker deltas which have the value 1 if i and h or
h and j indices are equal, otherwise they have the value 0. If the strict
lower triangles of the distance matrices have been used then the
Kronecker deltas can be omitted.
11. The amount of movement in the direction of the
negative gradient is set by the step length, a, which is
normally about 0.2 initially. The step size is recalculated
after each step such that the step size gets smaller as
reductions in stress become smaller.
12. Iterate (go to step 3) until either (a) a set maximum
number of iterations is reached or (b) a criterion of
stability is met.
At the stopping point, the stress should be acceptably
small. One way to assess this is by calculating the
relative magnitude of the gradient vector g:
magnitude of g =
2
g
 hl
/n
h,l
Mather suggested that mag(g) should reach a value of
2-5% of its initial value for an arbitrary configuration.
This is often not achievable.
Randomization (Monte Carlo) test
p = (1+n)/(1+N)
n = number of randomized runs with final stress less
than or equal to the observed minimum stress
N = number of randomized runs
Table 16.1. Stress in relation to dimensionality (number of axes), comparing 20 runs on the real data with
50 runs on randomized data. The p-value is the proportion of randomized runs with stress less than or
equal to the observed stress (see text). This table is the basis of the scree plot shown in Figure 16.3.
Axes
1
2
3
4
5
6
Stress in real data
Minimum
Mean Maximum
37.52
15.30
10.96
7.62
5.73
6.96
49.00
18.10
12.33
10.09
6.06
8.65
Stress in randomized data
Minimum
Mean Maximum
54.76
27.27
19.88
15.34
6.39
10.56
36.49
19.93
13.90
10.23
7.27
5.42
48.70
27.12
19.01
15.00
13.28
12.17
55.54
43.35
32.06
28.87
30.44
33.82
p
0.0199
0.0050
0.0050
0.0050
0.0050
0.0796
60
Real Data Randomized Data
Maximum
50
Mean
Minimum
Figure 16.3. A scree plot shows stress as a
function of dimensionality of the gradient
model. "Stress" is an inverse measure of fit
to the data. The "randomized" data are
analyzed as a null model for comparison.
See also Table 16.1.
Stress
40
30
20
10
0
1
2
3
4
Dimensions
5
6
Table 16.2. A second example of stress in relation to dimensionality (number of axes), comparing 15 runs
on the real data with 30 runs on randomized data, for up to 4 axes.
Stress in real data
Axes
1
2
3
4
Minimum
42.80
27.39
20.30
15.76
Mean Maximum
49.47
56.57
29.01
30.69
20.48
21.27
16.02
16.97
Stress in randomized data
Minimum
43.18
27.75
19.87
14.89
Mean Maximum
50.39
56.52
29.68
31.99
20.81
22.30
16.03
16.96
p
0.0323
0.0323
0.3548
0.3548
Table 16.3. Thoroughness settings for autopilot mode in NMS in PC-ORD version 4.
Thoroughness setting
Parameter
Maximum number of iterations
Instability criterion
Starting number of axes
Number of real runs
Number of randomized runs
Quick and
dirty
75
0.001
3
5
20
Medium
200
0.0001
4
15
30
Slow and
thorough
400
0.00001
6
40
50
Choosing the best solution
1. Select an appropriate number of dimensions.
60
Real Data Randomized Data
Mean
Minimum
40
Stress
Figure 16.3. A scree plot
shows stress as a function of
dimensionality of the
gradient model. "Stress" is
an inverse measure of fit to
the data. The "randomized"
data are analyzed as a null
model for comparison. See
also Table 16.1.
Maximum
50
30
20
10
0
1
2
3
4
Dimensions
5
6
2. Seek low stress.
Stress
2.5
5
10
20
<5
5-10
10-20
> 20
Kruskal’s rules of thumb
Excellent
Good
Fair
Poor
Clarke’s rules of thumb
An excellent representation with no
prospect of misinterpretation. This is,
however, rarely achieved.
A good ordination with no real risk of
drawing false inferences
Can still correspond to a usable picture,
although values at the upper end
suggest a potential to mislead. Too
much reliance should not be placed on
the details of the plot.
Likely to yield a plot that is relatively
dangerous to interpret. By the time
stress is 35-40 the samples are placed
essentially at random, with little
relation to the original ranked distances.
16
Final Stress (%)
14
12
10
8
6
4
2
0
0
10
20
30
40
50
Number of Sample Units
Figure 16.4. Dependence of stress on sample size, illustrated by subsampling
rows of a matrix of 50 sample units by 29 species.
Stress (%) or Spp count
30
25
Species remaining, count
Final Stress
20
15
10
5
0
0
20
40
60
80
100
Criterion for species retention (% of SU's)
Figure 16.5. Dependence of final stress on progressive removal of rare species from a
data set of 50 sample units and 29 species. For example, when only species occurring in
19% or more of the sample units were retained, 16 species remained in the data set and
the final stress was about 15%.
3. Use a Randomization (Monte Carlo) test.
Helpful but not foolproof.
The most common problems are:
A. Strong outliers caused by one or two extremely high values
can result in randomizations with final stress values similar to
the real data.
B. The same can be true for data dominated by a single superabundant species.
C. With very small data sets (e.g., less than 10 SUs), the
randomization test can be too conservative.
D. If a data set contains very many zeros, then most
randomizations will produce multiple empty sample units,
making applications of many distance measures impossible.
4. Avoid unstable solutions.
35
Stress
20
15
10
5
0
0
50
100
150
200
Step
Instability
Step Length
Mag(G)
0
-1
-2
log(Value)
Figure 16.7. NMS seeks a
stable solution. Instability is
the standard deviation in
stress over the preceding 10
steps (iterations); step length
is the rate at which NMS
moves down the path of
steepest descent; it is based on
mag(g), the magnitude of the
gradient vector (see text).
30
25
-3
-4
-5
-6
-7
-8
0
50
100
Step
150
200
30
25
Stress
20
Figure 16.8. NMS finds an unstable
solution; normally this would be
unacceptable instability. In this case it
was induced by overfitting the model
(fitting it to more dimensions than
required). The plotted variables are
described in Figure 16.7.
15
10
5
0
0
50
100
Step
200
Instability
Step Length
Mag(G)
0
log(Value)
150
-1
-2
-3
0
50
100
Step
150
200
50
30
20
10
0
0
50
100
150
200
Step
Instability
Step Length
Mag(G)
0
-1
log(Value)
Figure 16.9. NMS finds a fairly
stable solution, ending with a
periodic but low level of
instability. The instability is so
slight that the stress curve
appears nearly flat after about 45
steps. The plotted variables are
described in Figure 16.7.
Stress
40
-2
-3
-4
0
50
100
Step
150
200
Example
Table 16.5. Abundance of six species in each of
five sample units.
Species
SU
sp1
sp2
sp3
sp4
sp5
sp6
1
2
3
4
5
1
1
0
1
5
2
3
3
2
2
3
2
0
2
1
4
4
1
2
3
5
6
0
3
5
5
6
1
4
6
Table 16.6. Sørensen distances among the five
sample units from Table 16.5.
SU
SU
1
2
3
4
1
2
3
4
5
0
0.0952
0.6800
0.1765
0.1905
0
0.6296
0.2222
0.1818
0
0.5789
0.7037
0
0.2778
5
0
Point 1
Axis 2
Axis 2
Axis 1
Axis 1
Point 4
Axis 2
Axis 2
Point 3
Axis 1
Axis 1
50
Point 5
Stress
Stress
40
Axis 2
Figure 16.10. Migration
of points from the starting
configuration through 20
steps to the final
configuration using NMS.
Each point in the
ordination represents one
of five sample units. The
first five graphs show the
migration of individual
points (arrows indicate
starting points), followed
by the graph of the
decline in stress as the
migration proceeds.
Point 2
30
20
10
0
0
Axis 1
5
10
Step
15
20
Figure 16.10. (cont.) Migration of points from the starting configuration
through 20 steps to the final configuration using NMS. Each point in the
ordination represents one of five sample units.
Table 16.7. Focusing parameters for fitting new points to an
existing NMS ordination. The units for interval width and range
are always in the original axis units.
Focus
level
1
2
3
Range examined
Intervals
Points
80 + 2 (0.2  80) =
112
20 + 4 + 4 =
28
29
Interval
Width
112/28 = 4
2  5  4/5 = 8
2  5 (0.8/5) = 1.6
2  5 = 10
2  5 = 10
11
11
4/5 = 0.8
0.8/5 = 0.16
51.5
50.5
50.0
49.5
-20
0
20
40
60
80
100
120
Trial Score
49.90
Stress
49.85
49.80
49.75
49.70
10
11
12
13
14
15
16
17
18
19
20
Trial Score
49.737
Stress
Figure 16.11. Successive
approximation of the score
providing the best fit in
predictive-mode NMS. The
method illustrated is for one
axis at a time. Multiple axes
can be fit similarly by
simultaneously varying two or
more sets of coordinates.
Stress
51.0
49.736
49.735
13.5
14.0
14.5
Trial Score
15.0
15.5
Table 16.8. Flags for poor fit of new points
added to an existing NMS ordination.
Flag
Meaning
0
Criteria for acceptability were met
1
Final stress is too high
2
Score is too far off low end of gradient
3
Score is too far off high end of gradient
The following options were selected:
Distance measure
= Sørensen
Method for fitting = one axis at a time
Flag for poor fit = autoselect (2 stand.dev. > mean stress)
Extrapolation limit, % of axis length:
5.000
Calibration data set: D:\FHM\DATA\94\SEModel.wk1
Figure 16.12. Example
output (PC-ORD) from
“one axis at a time”
method of fitting new
points in an NMS
ordination.
Contents of data matrices:
85 Plots
and
87 Species in calibration data set
34 plots
and
140 species in test data set
85 scores from calibration data set
Title from file with calibration scores: SEModel.wk1,
rotated 155 degrees clockwise
63 species
in test data set are not present in calibration data
AXIS SUMMARY STATISTICS
-------------------------------------------------------------------Axis:
1
2
Mean stress
49.833
50.251
Standard deviation of stress
0.351
0.391
Stress used to screen poor fit (flag = 1)
50.536
51.033
Cutoff score, low end-of-axis warning flag (2)
-6.473
-12.895
Cutoff score, high end-of-axis warning flag (3)
106.473
112.895
Minimum score from calibration data
6.559
1.619
Maximum score from calibration data
93.441
98.381
Range in scores from calibration data
86.881
96.761
--------------------------------------------------------------------
BEST-FIT SCORES ON EACH AXIS
-------------------------------------------------Axis:
1
2
Item (SU) Score
Fit Flag
Score
Fit Flag
-------------------------------------------------3008281
25.673 49.713
0 90.253 50.085
0
3008367
44.440 50.058
0 74.190 50.582
0
3008372
36.099 50.141
0 80.964 50.437
0
etc.
3108526
3108564
2.563
50.000
49.505
49.953
0
0
8.199
87.543
50.007
50.491
0
0