Spiders on Mazurian lake islands: Wigry –Mikołajki, Nidzkie, Bełdany)

Analysis of variance

Photo: Wigierski Park Narodowe Photo: Ruciane.net

Araneus diadematus

Salticidae

Photo: Eurospiders.com

Spider species richness on Mazurian lake islands

Does species richness differ with respect to the degree of disturbance?

High Medium Low Pristine33 51 6 2534 43 28 2732 75 2238 47 1929 60 21

49 4664 31

302531253493242857

T-TESTMedium Low Pristine

High 0.145265 0.172254 0.931288Medium 1 0.081749Low 0.211812

If we use the same test several times with the same data we have to apply

a Bonferroni correction.

Single test

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signpsigp

signp

sigpsigp

sigpnsigp

testExp

test

ntestExp

ntestExp

n independent tests

Bonferroni corrected

T-TEST

Medium Low Pristine

High 0.857544 0.862042 0.988548Medium 1 0.846958Low 0.868635 n

n

Test

TestExp

05.0

05.0

Island Disturbance SpeciesGórna E High 33Kopanka High 34Kopanka N High 32Piaseczna High 38Górna W High 29Królewski Ostrów

Medium 51

Wygryńska Medium 43Maleńka Low 6Ruciane - ląd Low 28Mikołajki - ląd Low 75Wierzba Low 47Kamień Low 60Mysia Wigry Low 49Ordów Low 64Koń Pristine 25Mała Wierzba Pristine 27Ośrodek Pristine 22Śluza Pristine 19Bryzgiel Pristine 21Bryzgiel - ląd Pristine 46Brzozowa L Pristine 31Brzozowa P Pristine 30Cimochowski Grądzik C

Pristine 25

Cimochowski Grądzik N

Pristine 31

Cimochowski Grądzik S

Pristine 25

Krowa Pristine 34Ostrów Pristine 93Rośków Pristine 24Walędziak Pristine 28Wysoki Pristine 57

Spider species richness on Mazurian lake

islands

sH2

sM2

sL2

sP2

sT2

If there would be no difference between the sites the average within

variance sWithin2 should equal the

variance between the sites sBetween2 .

One way analysis of variance Sir Ronald Aylmer Fisher(1890-1962)

Hx

Lx

Mx

Px

sBetween2

22

2

2

2

Between

Between

Within

Between

ss

s

s

sF

T

We test for significance using the F-test of Fisher with k-1

(Between) and n-k (Within) degrees of freedom.

Island Disturbance SpeciesGórna E High 33Kopanka High 34Kopanka N High 32Piaseczna High 38Górna W High 29Królewski Ostrów

Medium 51

Wygryńska Medium 43Maleńka Low 6Ruciane - ląd Low 28Mikołajki - ląd Low 75Wierzba Low 47Kamień Low 60Mysia Wigry Low 49Ordów Low 64Koń Pristine 25Mała Wierzba Pristine 27Ośrodek Pristine 22Śluza Pristine 19Bryzgiel Pristine 21Bryzgiel - ląd Pristine 46Brzozowa L Pristine 31Brzozowa P Pristine 30Cimochowski Grądzik C

Pristine 25

Cimochowski Grądzik N

Pristine 31

Cimochowski Grądzik S

Pristine 25

Krowa Pristine 34Ostrów Pristine 93Rośków Pristine 24Walędziak Pristine 28Wysoki Pristine 57

n-1 = n-k + k-1df Total df Within df Between

Between

Between

k

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Between dfSS

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Within

Withink

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Total

n

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total between withinSS SS SS total between withindf df df

dfSS

MS

Within

Between

MSMS

F

2

22

1

21

11

ns

ns

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Welch test

The Levene test compares the group variances using the F distribution. Variances shouldn’t differ too much (shouldn’t be heteroskedastic)!!!

The Tuckey test compares simultaneously the means of all combinations of groups. It’s a t-test corrected for multiple comparisons (similar to a Bonferroni correction)

Observations A B C D1 0.08 0.19 0.83 2.80 0.404 0.109 0.220 2.0592 0.71 1.21 0.71 2.69 0.404 0.109 0.220 2.0593 0.19 1.97 1.10 1.93 0.404 0.109 0.220 2.0594 0.51 0.19 0.11 2.57 0.404 0.109 0.220 2.0595 0.73 0.19 0.30 2.58 0.404 0.109 0.220 2.059Group mean 0.445 0.750 0.611 2.515

0.131 0.319 0.046 0.0810.070 0.216 0.010 0.0320.065 1.484 0.244 0.3420.004 0.314 0.250 0.0040.082 0.312 0.096 0.004

Total SSwithin 4.11Total SSbetween 13.96Grand mean 1.08

1.00 0.80 0.06 2.960.14 0.02 0.14 2.610.79 0.79 0.00 0.720.32 0.79 0.94 2.230.12 0.79 0.61 2.24

Grand SS 18.07SSbetween+SSwithin 18.07

F 18.14F-test 2.118E-05

Treatments

SSwithin

Grand SS

SSbetween

Between

Between

k

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Between dfSS

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Within

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Within dfSS

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Total

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n

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Total dfSS

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Island Complex Disturbance SpeciesGórna E NBM High 33Kopanka NBM High 34Kopanka N NBM High 32Piaseczna NBM High 38Górna W NBM High 29

Królewski OstrówNBM Medium 51

Wygryńska NBM Medium 43Maleńka NBM Low 6Ruciane - ląd NBM Low 28Mikołajki - lądNBM Low 75Wierzba NBM Low 47Kamień Wigry Low 60Mysia Wigry Wigry Low 49Ordów Wigry Low 64Koń NBM Pristine 25Mała WierzbaNBM Pristine 27Ośrodek NBM Pristine 22Śluza NBM Pristine 19Bryzgiel Wigry Pristine 21Bryzgiel - lądWigry Pristine 46Brzozowa L Wigry Pristine 31Brzozowa P Wigry Pristine 30

Cimochowski Grądzik CWigry Pristine 25

Cimochowski Grądzik NWigry Pristine 31

Cimochowski Grądzik SWigry Pristine 25

Krowa Wigry Pristine 34Ostrów Wigry Pristine 93Rośków Wigry Pristine 24Walędziak Wigry Pristine 28

Wysoki WęgiełWigry Pristine 57

Island Complex Disturbance SpeciesMaleńka NBM Low 6Ruciane - ląd NBM Low 28Mikołajki - lądNBM Low 75Wierzba NBM Low 47Kamień Wigry Low 60Mysia Wigry Wigry Low 49Ordów Wigry Low 64Koń NBM Pristine 25Mała WierzbaNBM Pristine 27Ośrodek NBM Pristine 22Śluza NBM Pristine 19Bryzgiel Wigry Pristine 21Bryzgiel - lądWigry Pristine 46Brzozowa L Wigry Pristine 31Brzozowa P Wigry Pristine 30

Cimochowski Grądzik CWigry Pristine 25

Cimochowski Grądzik NWigry Pristine 31

Cimochowski Grądzik SWigry Pristine 25

Krowa Wigry Pristine 34Ostrów Wigry Pristine 93Rośków Wigry Pristine 24Walędziak Wigry Pristine 28

Wysoki WęgiełWigry Pristine 57

We include the effect of island complex (Wigry – Nidzkie, Bełdany, Mikołaiki)

There must be at least two data for each combination of groups.

We use a simple two way ANOVA

total A B AxB errorSS SS SS SS SS

Main effects Secondary effects

ComplexSS eDisturbancSS eDisturbancComplexSS The significance levels have to be divided by the number of tests (Bonferroni correction)

Spider species richness does not significantly depend on island complex and degree of disturbance.

y = 33.431x0.1917

R² = 0.7215

0

20

40

60

80

100

0 10 20 30 40 50

Spec

ies

Area

Island Complex Disturbance Area [ha] SpeciesGórna E NBM 1 0.7 33Koń NBM 4 0.5 25Kopanka NBM 1 0.69 34Królewski Ostrów NBM 2 6.15 51Maleńka NBM 3 0.0003 6Mała Wierzba NBM 4 0.4 27Kopanka N NBM 1 0.18 32Ośrodek NBM 4 0.09 22Piaseczna NBM 1 0.63 38Ruciane - ląd NBM 3 15 28Mikołajki - ląd NBM 3 20 75Śluza NBM 4 0.48 19Górna W NBM 1 0.44 29Wierzba NBM 3 0.78 47Wygryńska NBM 2 0.67 43Bryzgiel Wigry 4 0.2 21Bryzgiel - ląd Wigry 4 16 46Brzozowa L Wigry 4 3.81 31Brzozowa P Wigry 4 2.32 30Cimochowski Grądzik CWigry 4 0.15 25Cimochowski Grądzik NWigry 4 0.14 31Cimochowski Grądzik SWigry 4 0.76 25

Kamień Wigry 3 3.13 60

Krowa Wigry 4 4.49 34Mysia Wigry Wigry 3 1.55 49Ordów Wigry 3 8.69 64Ostrów Wigry 4 38.82 93Rośków Wigry 4 0.56 24Walędziak Wigry 4 0.76 28Wysoki Węgieł Wigry 4 18 57

Correcting for covariates: Anaysis of covariance

Instead of using the raw data we use the residuals.

These are the area corrected species numbers.

The conmparison of within group residuals and between group residuals

gives our F-statistic.

Disturbance does not significantly influence

area corrected species richness

SStotal = SSbetween + SSerror

Within group residuals

Total residuals

We need four regression equations: one from all data points and three within groups.

Repetitive designsIn medical research we test patients

before and after medical treatment to infer the influence of the therapy.

We have to divide the total variance (SStotal) in a part that contains the variance between patients (SSbetween) and within the

patient (SSwithin). The latter can be divided in a part that

comes from the treatment (SStreat) and the error (SSerror)

k2

jj 1treat error

k n2error treat

ij i jj 1 i 1

n (T x)SS df (n 1)(k 1)

FSS df k 1(x P T x)

total between within between treat errorSS SS SS SS SS SS

total between within between treat errordf df df df df df

kn 1 n 1 n(k 1) n 1 k 1 (n 1)(k 1)

SStotal

SSbetween SSwithin

SSErrorSStreat

Medical

treatment

Before After

SSwithin

SSbe

twee

n

2

1 1

1

2

2

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Before – after analysis in environmental protection

In the case of unequal variances between groups it is save to use the

conservative ANOVA with (n-1) dferror and only one dfEffect in the final F-test.

2

1 1

1

2

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xTPxSS

xTnSS

ji

n

i

k

jijerror

k

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Island Spring Summer AutumnGórna E 26 14 22Koń 19 10 16Kopanka 21 17 15Królewski Ostrów 50 46 47Maleńka 6 5 4Mała Wierzba 25 19 21Kopanka N 28 17 23Ośrodek 16 15 12Piaseczna 34 25 29Ruciane - ląd 22 15 13Mikołajki - ląd 43 39 26Śluza 12 10 7Górna W 19 10 11Wierzba 29 25 23Wygryńska 26 18 26Bryzgiel 15 11 14Bryzgiel - ląd 44 23 28Brzozowa L 22 20 13Brzozowa P 29 17 23Cimochowski Grądzik C 19 15 17Cimochowski Grądzik N 29 25 29Cimochowski Grądzik S 14 8 14Kamień 37 21 37Krowa 19 11 13Mysia Wigry 32 16 29Ordów 37 25 25Ostrów 77 50 57Rośków 21 14 17Walędziak 14 8 13Wysoki Węgieł 32 19 19Mean P 27 19 21Grand Mean 23

SStreat 1115.30

df 2

Mean P SSError

21 15.235015 9.793518 11.420148 8.07765 29.2908

22 4.807323 4.528814 31.656929 0.644917 10.904236 120.532210 19.222213 5.088125 13.552923 18.365814 16.576832 98.008918 45.491423 6.883317 9.578128 17.343412 13.284732 93.925314 0.090426 60.167629 18.788961 193.469817 2.583512 9.107624 28.6667

SSerror 917.0866df 58

dftreat = k-1

dfError = (n-1)(k-1)

Mean P 27 19 21 SSerror 917.0866Grand Mean 23 df 29

SStreat 1115.30SStreat/

SSerror1.2161338

df 1 F 35.26788p(f) 1.885E-06

SStreat/

SSerror1.2161338

F 70.53576p(f) 2.953E-09

Bivariate comparisons in environmental protectionIsland Complex Area[ha] Species Predicted_SpeciesResidualGórna E NBM 0.7 33 31.22156 1.778435Koń NBM 0.5 25 29.27129 -4.27129Kopanka NBM 0.69 34 31.13556 2.864436Królewski Ostrów NBM 6.15 51 47.35619 3.643813Maleńka NBM 0.0003 6 7.060143 -1.06014Mała Wierzba NBM 0.4 27 28.04557 -1.04557Kopanka N NBM 0.18 32 24.06496 7.935042Ośrodek NBM 0.09 22 21.07064 0.929363Piaseczna NBM 0.63 38 30.59729 7.402711Ruciane - ląd NBM 15 28 56.18315 -28.1831Mikołajki - ląd NBM 20 75 59.3686 15.6314Śluza NBM 0.48 19 29.04312 -10.0431Górna W NBM 0.44 29 28.5627 0.437301Wierzba NBM 0.78 47 31.87601 15.12399Wygryńska NBM 0.67 43 30.9605 12.0395Bryzgiel Wigry 0.2 21 24.55595 -3.55595Bryzgiel - ląd Wigry 16 46 56.88256 -10.8826Brzozowa L Wigry 3.81 31 43.20288 -12.2029Brzozowa P Wigry 2.32 30 39.28379 -9.28379Cimochowski Grądzik CWigry 0.15 25 23.23839 1.761609Cimochowski Grądzik NWigry 0.14 31 22.93307 8.066934Cimochowski Grądzik SWigry 0.76 25 31.71767 -6.71767

Kamień Wigry 3.13 60 41.60497 18.39503Krowa Wigry 4.49 34 44.58461 -10.5846Mysia Wigry Wigry 1.55 49 36.36101 12.63899Ordów Wigry 8.69 64 50.60104 13.39896Ostrów Wigry 38.82 93 67.41729 25.58271Rośków Wigry 0.56 24 29.91417 -5.91417Walędziak Wigry 0.76 28 31.71767 -3.71767Wysoki Węgieł Wigry 18 57 58.18153 -1.18153

y = 33.431x0.1917

R² = 0.7215

1

10

100

0.0001 0.01 1 100

Spec

ies

Area

The outlier would disturb direct comparisons of species richness

Due to possible differences in island areas between the two island complexes we have to

use the residuals. A direct t-test on raw data would be

erroneous.

00.010.020.030.040.050.060.070.08

0 0.2 0.4 0.6 0.8 1

Freq

uenc

y

t-values

NBM Wigry NBM Wigry NBM Wigry NBM Wigry1.778435 -3.55595 0.929363 -10.0431 1.761609 1.778435 -10.5846 -3.55595-4.27129 -10.8826 -10.5846 2.864436 7.402711 -3.55595 15.12399 1.7616092.864436 -12.2029 25.58271 3.643813 25.58271 2.864436 -12.2029 12.03953.643813 -9.28379 1.778435 12.0395 -12.2029 -10.5846 -4.27129 -10.8826-1.06014 1.761609 -3.55595 15.12399 -5.91417 18.39503 -1.18153 0.929363-1.04557 8.066934 1.761609 8.066934 -10.8826 -9.28379 0.437301 1.7784357.935042 -6.71767 15.6314 -4.27129 3.643813 -6.71767 25.58271 8.0669340.929363 18.39503 -5.91417 12.63899 15.6314 13.39896 -6.71767 -28.18317.402711 -10.5846 -3.71767 -28.1831 0.929363 7.935042 15.6314 2.864436-28.1831 12.63899 -1.04557 -1.18153 -1.18153 0.437301 -9.28379 12.6389915.6314 13.39896 13.39896 7.402711 12.63899 -28.1831 -1.06014 18.39503-10.0431 25.58271 -9.28379 -10.8826 -4.27129 8.066934 7.402711 13.398960.437301 -5.91417 18.39503 0.437301 -1.04557 -10.0431 -3.71767 7.93504215.12399 -3.71767 -1.06014 7.935042 12.0395 -3.71767 -10.0431 -1.0455712.0395 -1.18153 -12.2029 -6.71767 -1.06014 15.12399 3.643813 -5.91417

t 0.118799 0.34257 0.766559 0.346264

Randomized valuesObserved values

Permutation testingObserved P(t)

Upper 2.5% confidence limit.

10000 randomizations of observed values gives a null distribution of t-values and

associated probability levels with which we

compare the observed t.

This gives the probability level for our t-test.

Bivariate comparisons using ANOVA

01412.011884.0 22 Ft

2tF

t and F tests can both be used for pair wise comparisons.

Repeated measures

Plot Before Leaf-litter free Mean

1 52 34 432 58 39 48.53 10 1 5.54 50 52 515 49 45 476 15 6 10.57 32 33 32.58 14 12 139 52 28 4010 19 1 1011 29 35 3212 22 7 14.513 18 33 25.514 11 7 915 15 9 1216 15 10 12.517 2 3 2.518 3 7 5T-Test 0.027271

Mean 25.88889 20.11111111Grand Mean 23

SSEffect 16.69136

df 1

Mean SSError

43 74.6913648.5 87.413585.5 5.19135851 30.2469147 1.580247

10.5 5.19135832.5 22.9691413 7.13580240 166.024710 74.6913632 69.35802

14.5 42.5246925.5 215.858

9 1.58024712 0.024691

12.5 0.3024692.5 22.969145 47.80247

Sum 875.5556df 17

SSEffect/

SSError0.019064

F 0.324083P(F) 0.576609

Species richness of ground living Hymenoptera in a beech forest

Photo Tim Murray Photo Simon van Noort

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Advices for using ANOVA:

· You need a specific hypothesis about your variables. In particular, designs with more than one predicator level (multifactorial designs) have to be stated clearly.

· ANOVA is a hypothesis testing method. Pattern seeking will in many cases lead to erroneous results.

· Predicator variables should really measure different things, they should not correlate too highly with each other

· The general assumptions of the GLM should be fulfilled. In particular predicators should be additive. The distribution of errors should be normal.

· It is often better to use log-transformed values

· In monofactorial designs where only one predicator variable is tested it is often preferable to use the non-parametric alternatives to ANOVA, the Kruskal Wallis test. The latter test does not rely on the GLM assumptions but is nearly as powerful as the classical ANOVA.

· Another non-parametric alternative for multifactorial designs is to use ranked dependent variables. You loose information but become less dependent on the GLM assumptions.

· ANOVA as the simplest multivariate technique is quite robust against violations of its assumptions.