Experiments

evaluated using multivariate methods

Separating the effect of (correlated) explanatory variables

Variation partitioningA B

A in addition to B

B in addition to A

A or B

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?

Analysis of Variance; DV: NSP (fertenv.sta)

Sums ofSquares Df

MeanSquares F p-level

Regress. 382.9215 2 191.4607 57.7362 2.98E-18

Residual 394.6195 119 3.31613

Total 777.541

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

BETASt. Err.of BETA B

St. Err.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

Multiple regression: test of the complete model & test of partial (conditional) effects [plus possible test of marginal (simple) effects]

-1.0 +0.7

-0.5

+0.9

dose

cover

Veronica persica

Thlaspi arvense

Fallopia convolvulus

Medicago lupulina

Galium aparineMyosotis arvensis

Veronica arvensis

Arenaria seryllifoliaAnagalis arvensis

Vicia angustifolia

Stellaria media

-1.0 +1.0

-1.0

+1.0

dose

cover

Note: in this Figure, CaseR scores are used instead of CaseE

Variation partitioning

Dose

Dose in addition to Cover

Cover in addition Dose A

Cover or Dose

Cover

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

Effect of dominant species, moss and litter on seedling germination

Randomized complete blocks

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.

Just of historical interest (the FORTRAN format etc.)

Case1 Case2 1 10 5 50 7 70 10 100 3 30

Standardization by cases

Grubb theory of regeneration niche: importance of standardization - the standardization fundamentally changes the ecological interpretation of results

If “standardize by case norm” is used, the two cases are identical

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]

Permutations of the whole-plots

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

Carex pulicaris C. hartmanii

14 Carex species

Summary of all Effects; design: (ohrazenv.sta)

1-MOWING, 2-FERTIL, 3-REMOV, 4-TIME

dfEffect

MSEffect

dfError

MSError F p-level

1 1 65.01041 16 40.83333 1.592092 0.225112

2 1 404.2604 16 40.83333 9.900255 0.006241

3 1 114.8438 16 40.83333 2.8125 0.112957

4 3 87.95486 48 7.430555 11.83692 6.35E-06

12 1 0.260417 16 40.83333 0.006378 0.937339

13 1 213.0104 16 40.83333 5.216582 0.036372

23 1 75.26041 16 40.83333 1.843112 0.19342714 3 75.53819 48 7.430555 10.16589 2.69E-05

24 3 174.2882 48 7.430555 23.45561 1.72E-09

34 3 41.48264 48 7.430555 5.58271 0.002286

123 1 6.510417 16 40.83333 0.159439 0.694953

124 3 14.67708 48 7.430555 1.975234 0.130239

134 3 11.48264 48 7.430555 1.545327 0.214901

234 3 2.565972 48 7.430555 0.345327 0.792657

1234 3 3.538194 48 7.430555 0.476168 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 3

Control 0 0 0 0 0

Treatment 1 0 1 2 3

Time

Control

Treatment

Note: expl. variables (incl. interactions) are centered and standardized, but only after calculation of interactions)

Baseline: time=0

Time as A.D.

  Time 2000 2001 2002 2003

Control 0 0 0 0 0

Treatment 1 2000 2001 2002 2003

Time

Control

Treatment

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!

Analysis Explanatory

variables Covariates

% expl. 1st axis

r 1st axis

F ratio

P

C1 Yr, Yr*M,

Yr*F, Yr*R PlotID 16.0 0.862 5.38 0.002

C2 Yr*M, Yr*F,

Yr*R Yr, PlotID 7.0 0.834 2.76 0.002

C3 Yr*F Yr, Yr*M,

Yr*R, PlotID 6.1 0.824 4.40 0.002

C4 Yr*M Yr, Yr*F,

Yr*R, PlotID 3.5 0.683 2.50 0.002

C5 Yr*R Yr, Yr*M,

Yr*F, PlotID 2.0 0.458 1.37 0.040

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

Original data After „subtraction“ of the covariate effect

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

anthodorbrizmed

descespi

festovin

festrubr

nardstri

poa triv

siegdecu

carepale

carepani

carepilu

carepuli

careumbr

luzumult

lychflosplanlanc

poteerec

prunvulg

ranuacer

ranunemo

scorhumi

selicarvsuccprat

aulapalu

brachyte

climdendhylosple

rhitsqua

pseupuru

Principal response curves

YEAR

-0.6

0.8

PRC1

MR

M

R

F

FMFMR

FR

1994 1996 1998 2002 2004 20062000

triangles - mown circles unmown

full symbol - fertil. open symbol - unfert.

solid line - control broken l. - removal

Further use of ordination scores

Do we need PIC here?