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```Constrained ordinations
Dependence of multivariate response
on one or many predictors
Linear regression
MSRgr
F
MSresid
Covariables
Just another type of explanatory
variables, effect of which is not
interesting at this moment and
should be removed from the analysis
Analogy with covariates in ANCOVA
Marginal and partial effects in Multiple
regression
Partial analyses:
The effect of covariable(s) is first subtracted
from the data, and then the analysis (usually
constrained, but also unconstrained) is carried
out on the residual variability.
If X is
covariable,
the analysis
is then
carried out
on e.
Two main reasons to apply partial analyses.
1. Covariable is real nuisance (e.g. meadows
were sampled in the course of 3 weeks,
because we did not managed to sample faster.
The date of sampling is used as covariable.)
2. We want to separate effects of several (often
correlated) predictors. Each explanatory
variable will be used in different analyses both
as “environmental variable” and “covariable”.
Linear regression
MSRgr
F
MSresid
Test on the first axis and on the trace
(=on all canonical axes)
F1 
1
RSS /(n  p  q)
p=no. of expl. var.
p
Ftrace 
 / p
i 1
n=total no. of axes
i
q=no. of covariab.
RSS /(n  p  q)
If there is a strong univariate variation in the
data, the test on the first axis is stronger than
test on the trace.
Monte Carlo permutation test
nx  1
P
N 1
What is permuted
Reduced model: Residuals after fitting
covariables
Full model: Residuals after fitting all
variables
Permutation types
Permutation within blocks
original
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
block
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
Permutations within blocks
perm1
2
4
3
1
7
8
5
6
11
9
10
12
14
15
16
13
perm2
4
3
2
1
5
8
7
6
9
12
10
11
16
13
15
14
perm3
1
4
2
3
7
6
8
5
9
12
11
10
14
15
13
16
perm 4
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
etc.
perm 5
Line transects or grids - spatial
dependence
Problem of autocorrelation
N
N
Permutation restrictions
Time series or line transect
ORIG
1
2
3
4
5
6
7
8
PERM1
6
7
8
1
2
3
4
5
PERM2
3
4
5
6
7
8
1
2
MIRORP
6
5
4
3
2
1
8
7
Remove the trend by using position as a covariable
30
25
20
Roses
N
15
Regression
(correlation):
r=0.48, P=0.0073
10
5
0
1
3
5
7
R2=0.230 F=8.368
9 11 13 15 17 19 21 23 25 27 29
Position on the transect
S
c
a
t
t
e
r
p
o
l
t(
t
r
a
n
s
e
c
t2
v
*
3
0
c
)
R
o
s
e
s
=
1
.
7
7
7
3
+
0
.
8
9
8
7
*
x
2
6
Permutation test:
2
4
2
2
2
0
1
8
1
6
1
4
Rose
Test of significance of all
canonical axes : Trace
= 0.230
1
2
1
0
8
F-ratio = 8.368
6
4
2
P-value =
0.1540
0
2
0
2
4
6
8 1
0 1
2 1
4 1
6
N
:
R
o
s
e
s
: r=
0
.
4
7
9
7
,p
=
0
.
0
0
7
3
;y
=
1
.
7
7
7
2
7
3
2
4
+
0
.
8
9
8
7
1
1
6
9
4
*
x
N
1
8
2
0
2
2
We do remove the trends consequently, this is a good
methods for testing of patchily
distributed characteristics.
Not to be used for trends.
(If there is a trend, the data are
necessarily autocorrelated.)
Repeated measurements are
analysed using split plot design.
With trend removal
Test of significance of all canonical axes : Trace
F-ratio = 8.134
P-value = 0.2340
= 0.231
Hierarchical desing (split plot)
1
2
5
6
3
7
8
4
The main plots are permuted
Treatment
0
0
0
0
1
1
1
1
ORIG
1
2
3
4
5
6
7
8
PERM1
3
4
7
8
1
2
5
6
PERM2
3
4
5
6
7
8
1
2
PERM3
7
8
3
4
5
6
1
2
The subplots of the same main plot have always
the same treatment level of the variable tested.
Stepwise selection (forward)
FS summary
Variable
Ca
conduct
Mg
pH
Corg
Na
NH3
Si
SO4
K
Fe
Cl
slope
NO3
PO4
Marginal Effects
Var.N
Lambda1
1
0.35
14
0.32
2
0.3
13
0.24
12
0.24
5
0.18
10
0.15
6
0.12
7
0.12
4
0.12
3
0.1
11
0.1
15
0.1
9
0.09
8
0.08
Variable
Ca
conduct
Corg
Na
NH3
Fe
Cl
pH
Si
Mg
NO3
SO4
K
PO4
slope
Conditional Effects
Var.N
LambdaA P
1
0.35
14
0.13
12
0.11
5
0.12
10
0.1
3
0.09
11
0.1
13
0.08
6
0.08
2
0.08
9
0.08
7
0.06
4
0.06
8
0.06
15
0.06
F
0.002
0.002
0.002
0.002
0.02
0.018
0.082
0.056
0.126
0.188
0.332
0.488
0.842
0.812
0.814
4.79
1.79
1.6
1.58
1.45
1.34
1.39
1.25
1.17
1.11
1.07
0.99
0.86
0.85
0.83
```
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