Experiments
evaluated using multivariate methods
Separating the effect of (correlated)
explanatory variables
Variation partitioning
A
A in
addition
to B
B
A
or
B
B in
addition
to A
In fact, more often used in observational studies - in experiments, we
try to avoid correlated predictor (however, in ecology, we are not able
to control everything)
Effect of nitrogen fertilization on
weed community
Dose of
fertilizer
Cover of
barley
Weed community
Pysek P. & Leps J. (1991):Response of a weed community to nitrogen fertilizer: a
multivariate analysis. J. Veget. Sci. 2: 237-244.
Effect of nitrogen fertilization on
weed community
Dose of
fertilizer
Cover of
barley
Weed community
Basic questions:
* Is there an effect of fertilization on the structure of the weed
community? (either direct, or mediated through cover of the
crop)
Problem of correlated predictors:
* Is there a direct effect of fertilization (i.e., which could not be
explained as mediated through the cover of the crop)?
* Is there an effect of the crop that can not be explained by the
direct effect of fertilizer?
Multiple regression: test of the complete model & test
of partial (conditional) effects [plus possible test of
marginal (simple) effects]
Analysis of Variance; DV: NSP (fertenv.sta)
Regress.
Residual
Total
Sums of
Squares Df
382.9215 2
394.6195 119
777.541
Mean
Squares F
191.4607 57.7362
3.31613
p-level
2.98E-18
Regression Summary for Dependent Variable: NSP (fertenv.sta)
R= .70176746 R2= .49247757 Adjusted R2= .48394778
F(2,119)=57.736 p<.00000 Std.Error of estimate: 1.8210
St. Err.
St. Err.
BETA
of BETA
B
of B
t(119)
p-level
Intercpt
9.423662 0.388684 24.24506 0
DOSE
-0.02342 0.099678 -0.08501 0.361781 -0.23498 0.814629
COVER
-0.68390 0.099678 -0.06174 0.008999 -6.86113 3.28E-10
+0.9
dose
Stellaria media
cover
Veronica persica
Galium aparine
Myosotis arvensis
-0.5
Anagalis arvensis
Arenaria seryllifolia
Fallopia convolvulus
Vicia angustifolia
Veronica arvensis
Medicago lupulina
Thlaspi arvense
-1.0
+0.7
Note: in this
Figure, CaseR
scores are used
instead of
CaseE
dose
cover
-1.0
+1.0
Variation partitioning
Dose
Cover
Dose in Cover Cover in
addition or
addition
to Cover Dose Dose A
adjusted
Variation partitioning - n.b.
• In linear methods, trace (all the eigenvalues
together) sum up to 1, so the eigenvalue
corresponds to the proportion of explained
variability
• In unimodal methods, trace is higher than
one, so the eigenvalue has to be divided by
trace to get the proportion of explained
variability
Variation partitioning - n.b.
• The variation could be partitioned among
more than 2 variables (however, for more
than 3 the clarity of the result is lost)
• More useful: partitioning between groups of
variables
• The amount of explained variability is
positively dependent on the number of
explanatory variables in a group
Spacková I., Kotorová I. & Leps J. (1998): Sensitivity of seedling recruitment to moss, litter
and dominant removal in an oligotrophic wet meadow. Folia Geobot. Phytotax. 33: 17-30.
Effect of dominant species, moss and litter on
seedling germination
Randomized complete blocks
Ohrazeni 1994-seedlings, design of experiment
(i3,8(f3.0))
8
1 1 0 0 0 1 0 0 0
2 0 1 0 0 1 0 0 0
3 0 0 1 0 1 0 0 0
4 0 0 0 1 1 0 0 0
5 1 0 0 0 0 1 0 0
6 0 1 0 0 0 1 0 0
7 0 0 1 0 0 1 0 0
8 0 0 0 1 0 1 0 0
9 1 0 0 0 0 0 1 0
10 0 1 0 0 0 0 1 0
11 0 0 1 0 0 0 1 0
12 0 0 0 1 0 0 1 0
13 1 0 0 0 0 0 0 1
14 0 1 0 0 0 0 0 1
15 0 0 1 0 0 0 0 1
16 0 0 0 1 0 0 0 1
control litter-rnardus-rlit+mossblock1
rel1
rel2
rel3
rel4
rel5
rel11
rel12
rel13
rel14
rel15
Just of historical
interest (the
FORTRAN
format etc.)
block2 bl.
rel6
re.
rel16
Figure Chyba! V dokumentu není žádný text v zadaném stylu.-1: Environmental data characterizing the
design of the experiment (Canoco file in full format).
Standardization by cases
Case1
Case2
1
10
5
50
7
70
10
100
3
30
If “standardize by case norm” is used, the two cases are
identical
Grubb theory of regeneration niche: importance of
standardization - the standardization fundamentally changes
the ecological interpretation of results
If there are very different eigenvalues of the two displayed axes,
then the “Focus scaling on” really plays a role!
Note: centroids are scaled as cases
on interspecies
correlation
on intercase distances
Hierarchical structure
each whole-plot is subdivided into 25
split-plots
Seedlings - nested design [seme96su.spe, seme96su.env]
1
6
11
16
21
2
7
12
17
22
3
8
13
18
23
4
9
14
19
24
5
10
15
20
25
26
31
36
41
46
27
32
37
42
47
28
33
38
43
48
29
34
39
44
49
30
35
40
45
50
Permutations of the whole-plots
1
6
11
16
21
2
7
12
17
22
3
8
13
18
23
4
9
14
19
24
5
10
15
20
25
26
31
36
41
46
27
32
37
42
47
28
33
38
43
48
29
34
39
44
49
30
35
40
45
50
Repeated observations from a factorial experiment
fertilization, mowing, dominant removal]
3 replications, i.e. 24 plots together
Ohrazení (http://mapy.atlas.cz)
Molinia caerulea
Nardus stricta
Species diversity and “interesting plants” (e.g. red list species)
concentrated in “traditional”, i.e. mown, unfertilized
Dactylorhiza majalis
Senecio rivularis
14 Carex species
Carex pulicaris
C. hartmanii
Summary of all Effects; design: (ohrazenv.sta)
1-MOWING, 2-FERTIL, 3-REMOV, 4-TIME
1
2
3
4
12
13
23
14
24
34
123
124
134
234
1234
df
Effect
1
1
1
3
1
1
1
3
3
3
1
3
3
3
3
MS
Effect
65.01041
404.2604
114.8438
87.95486
0.260417
213.0104
75.26041
75.53819
174.2882
41.48264
6.510417
14.67708
11.48264
2.565972
3.538194
df
Error
16
16
16
48
16
16
16
48
48
48
16
48
48
48
48
MS
Error
40.83333
40.83333
40.83333
7.430555
40.83333
40.83333
40.83333
7.430555
7.430555
7.430555
40.83333
7.430555
7.430555
7.430555
7.430555
F
1.592092
9.900255
2.8125
11.83692
0.006378
5.216582
1.843112
10.16589
23.45561
5.58271
0.159439
1.975234
1.545327
0.345327
0.476168
p-level
0.225112
0.006241
0.112957
6.35E-06
0.937339
0.036372
0.193427
2.69E-05
1.72E-09
0.002286
0.694953
0.130239
0.214901
0.792657
0.700348
Time
• In repeated measures – time is categorial
(but, linear or polynomial trends – contrasts
– can be used)
• In CANOCO, we can decide and use time
either as a categorial or as a quantitative
variable
• If quantitative – we expect a trend!
Interaction – just multiplication of
the two values
Time
0
1
2
Control
0
0
0
0
Treatment
1
0
1
2
Baseline: time=0
Note: expl.
variables
(incl.
3
interactions)
0
are centered
3
and
standardized,
but only after
Treatment
calculation of
interactions)
Control
Time
Time as A.D.
Time
Control
Treatment
2000 2001 2002 2003
0
0
0
0
0
1 2000 2001 2002 2003
Treatment
Control
Time
Time vs. Time * Treatment
• Time: 0, 1, 2, 3 and 2000, 2001, 2002,
2003 – after centering and standardization,
both series are identical
• Time * treatment interaction – the results
are very very different!
C2
Explanatory
variables
Yr, Yr*M,
Yr*F, Yr*R
Yr*M, Yr*F,
Yr*R
C3
Yr*F
C4
Yr*M
C5
Yr*R
Analysis
C1
Covariates
% expl.
1st axis
st
r
1 axis
F
ratio
P
PlotID
16.0
0.862
5.38
0.002
Yr, PlotID
7.0
0.834
2.76
0.002
6.1
0.824
4.40
0.002
3.5
0.683
2.50
0.002
2.0
0.458
1.37
0.040
Yr, Yr*M,
Yr*R, PlotID
Yr, Yr*F,
Yr*R, PlotID
Yr, Yr*M,
Yr*F, PlotID
After „subtraction“ of
the covariate effect
Original data
Plot
time1
time2
time3
time4
mean
time1
time2
time3
time4
1
5
3
2
2
3
2
0
-1
-1
2
17
12
10
8
11.75
5.25
0.25
-1.75
-3.75
3
22
26
20
15
20.75
1.25
5.25
-0.75
-5.75
4
6
4
0
0
2.5
3.5
1.5
-2.5
-2.5
4.0
Principal response curves
aulapalu
3.0
0.8
MR
2.0
M
1.0
R
PRC1
FMR
rhitsqua
nardstri
anthodor
prunvulg
carepilu
brizmed
luzumult
carepale
hylosple festovin poteerec
selicarv carepani climdend
ranuacer
succprat
planlanc
pseupuru
scorhumi lychflos careumbr
brachyte
carepuli
ranunemo
siegdecu
triangles - mown
circles unmown
FM 0.0
-1.0
FR
F
full symbol - fertil.
open symbol - unfert
festrubr
poa triv solid line - control
descespi
-2.0
-0.6
1994
1996
1998
2000
2002
2004
2006
YEAR
broken l. - removal
Further use of ordination scores
Do we need PIC here?