QUANTITATIVE METHODS IN PALAEOECOLOGY AND PALAEOCLIMATOLOGY

Class 5

Hypothesis Testing

Espegrend August 2008

Randomisation tests – an introduction

pH changes at Round Loch of Glenhead

Assessing impacts of volcanic ash deposition on terrestrial and aquatic systems

Multi-proxy studies

Assessing potential external 'drivers' on an aquatic ecosystem

Conclusions

CONTENTS

Mandible lengths of male and female jackals in Natural History Museum

Male 120

107 110 116 114 111 113 117 114 112 mm

Female

110

111 107 108 110 105 107 106 111 111 mm

Is there any evidence of difference in mean lengths for two sexes? Male mean larger than female mean.

Null hypothesis (Ho) – no difference in mean lengths for two

sexes, any difference is purely due to chance. If Ho

consistent with data, no reason to reject this in favour of alternative hypothesis that males have a larger mean that females.

RANDOMISATION TESTS

Simple introductory example

Classical hypothesis testing – t-test for comparison of 2 means

Group 1 n1 objects Group 2 n2

x1 mean x2

s1 s2

Assume that values for group 1 are random sample from a normal distribution with 1 mean and standard deviation , and mean 2 and standard deviation

H0 1 = 2 H1 1 > 2

Test null hypothesis with estimate of common within-group s.d.

S = [{(n1 –1)S12 + (n2 – 1)S2

2}/(n1 + n2 –2)]

T = (x1 – x2)/(S(1/n1 + 1/n2))

If H0 true, T will be a random value from t-distribution with n1 + n2

– 2d.f.

Jackal data

x1 = 113.4mm s1 = 3.72mm s = 3.08

x2 = 108.6mm s2 = 2.27mm

T = 3.484 18 d.f.

Probability of a value this large is 0.0013 if null hypothesis is true.

Sample result is nearly significant at 0.1% level. Strong evidence against null hypothesis. Support for alternative hypothesis.

1. Random sampling of individuals from the populations of interest

2. Equal population standard deviations for males and females

3. Normal distributions within groups

Assumptions of T-test

Alternative ApproachIf there is no difference between the two sexes, then the length distribution in the two groups will just be a typical result of allocating 20 lengths at random into 2 groups each of size 10. Compare observed difference with distribution of differences found with random allocation.

TEST:

1. Find mean scores for male and female and difference D0 for observed data.

2. Randomly allocate 10 lengths to male group, remaining 10 to female. Calculate D1.

3. Repeat many times (n e.g. 999 times) to find an empirical distribution of D that occurs by random allocation. RANDOMISATION DISTRIBUTION.

4. If D0 looks like a ’typical’ value from this randomisation distribution, conclude that allocation of lengths to males and females is essentially random and thus there is no difference in length values. If D0 unusually large, say in top 5% tail of randomisation distribution, observed data unlikely to have arisen if null hypothesis is true. Conclude alternative model is more plausible.

If D0 in top 1% tail, significant at 1% level

If D0 in top 0.1% tail, significant at 0.1% level

The distribution of the differences observed between the mean for males and the mean for females when 20 measurements of mandible lengths are randomly allocated, 10 to each sex. 4999 randomisations.

x1 = 113.4mm x2 = 108.6mm D0 = 4.8mm

Only nine were 4.8 or more, including D0.

Six were 4.8 2 > 4.8

9 .Significance level = 5000 = 0.0018 = 0.18%

(cf. t-test 0.0013 0.13%)

20C10 = 184,756. 5000 only 2.7% of all possibilities.

Three main advantages1. Valid even without random samples.

2. Easy to take account of particular features of data.

3. Can use 'non-standard' test statistics.

Tell us if a certain pattern could or could not be caused/arisen by chance. Completely specific to data set.

Randomisation tests and Monte Carlo permutation tests

If all data arrangements are equally likely, RANDOMISATION TEST with random sampling of randomisation distribution. Otherwise, MONTE CARLO PERMUTATION TEST.

Validity depends on validity of permutation types for particular data-type – time-series stratigraphical data, spatial grids, repeated measurements (BACI). All require particular types of permutations.

ROUND LOCH OF GLENHEAD

pH change 1874-1931 (17.3-7.3cm) very marked.

Is it any different from other pH fluctuations over last 10,000 years?

Null hypothesis – no different from rates of pH change in pre-acidification times.

Randomly resample with replacement 1000 times to create temporally ordered data of same thickness as the interval of interest – time-duration or elapsed-time test. As time series contains unequal depth intervals between pH estimates, not possible for each bootstrapped time series to contain exactly 10cm. Instead samples are added in time series until depth interval equals or exceeds 10cm.

pH CHANGES IN ROUND LOCH OF GLENHEAD

Monte Carlo permutation tests

Rate (pH change per cm)

Response variable(s)

Y e.g. lake-water pH, sediment LOI, tree pollen stratigraphy

Predictor variable(s)

X e.g. charcoal, age, land-use indicators, climate

Also covariables

Basic statistical model:

Y = BX

Y X Method

1 1 Simple linear regression

1 >1 Multiple linear regression, principal components regression, partial least squares (PLS)

>1 ≥1 Redundancy analysis (= constrained PCA, reduced-rank regression, PCA of y with respect to x, etc.)

Statistical testing by Monte Carlo permutation tests to derive empirical statistical distributions

Variance partitioning or decomposition to evaluate different hypotheses.

Statistical methods for testing competing causal hypotheses

A.F. Lotter & H.J.B. Birks (1993)

J. Quat. Sci. 8, 263 - 276

11000 BP

? Any impact on terrestrial and aquatic systems

Also:

H.J.B. Birks & A.F. Lotter

(1994) J. Paleolimnology 11, 313 - 922

A F Lotter et al. (1995) J. Paleolimnology 14, 23 - 47

ASSESSING IMPACTS OF LAACHER SEE VOLCANIC ASH ON

TERRESTRIAL AND AQUATIC ECOSYSTEMS

Map showing the location of Laacher See (red star), as well as the location of the sites investigated (blue circle). Numbers indicate the amount of Laacher See Tephra deposition in millimetres (modified from van den Bogaard, 1983).

Loss-on-ignition of cores Hirschenmoor HI-1 and Rotmeer RO-6. The line marks the transition from the Allerød (II) to the Younger Dryas (III) biozone. LST = Laacher See Tephra.

Diatoms in cores HI-1 and RO-6 grouped according

to life-forms. LST = Laacher See Tephra.

Al

YD

Diatom-inferred pH values for cores HI-1 and RO-6. The interpolation is based on distance-weighted least-squares (tension = 0.01). The line marks the transition from the Allerød (II) to the Younger Dryas (III) biozone. LST = Laacher See Tephra.

Data

Terrestrial pollen and spores (9, 31 taxa)Aquatic pollen and spores (6, 8 taxa) RESPONSE VARIABLESDiatoms (42,54 taxa) % data

Biozone (Allerød, Allerød/Younger Dryas, Younger Dryas)

+/-

Lithology (gyttja, clay/gyttja) +/-

Depth ("age") Continuous

Ash Exponential decay process Continuous

= 0.5

x = 100

t = time

YD

211 years

Exp x-t

Time AL

EXPLANATORY VARIABLES

NUMERICAL ANALYSIS

(Partial) redundancy analysis

Restricted (stratigraphical) Monte Carlo permutation tests

Variance partitioning

Log-ratio centring because of % data

The biostratigraphical data sets used in the (partial) redundancy analyses

(SD = standard deviation units)

HIRSCHENMOOR CORE HI-1

Terrestrial pollen

Aquatic pollen and spores

Diatoms

Number of samples

16 16 16

Number of taxa 9 6 42

Gradient length (SD)

0.48 0.84 1.44

ROTMEER CORE RO-6

Terrestrial pollen

Aquatic pollen and spores

Diatoms

Number of samples

21 21 21

Number of taxa 31 8 54

Gradient length (SD)

0.74 0.71 1.68

RESULTS OF (PARTIAL) RESUNDANCY ANALYSIS OF THE BIOSTRATIGRAPHICAL DATA SETS AT ROTMEER (RO-6) AND HIRSCHENMOOR (HI-1) UNDER DIFFERENT MODELS OF EXPLANATORY VARIABLES AND COVARIABLES. Entries are significance levels as assessed by restricted Monte Carlo permutation tests (n = 99)

Data Set

Site Explanatory variables Covariables Terrestria

l pollenAquatic pollen & spores

Diatoms

RO-6 Depth + biozone + ash + lithology

- 0.01a 0.01a 0.01a

HI-1 Depth + biozone + ash + lithology

- 0.01a 0.10 0.01a

RO-6 Ash Depth + biozone

0.09ns 0.48ns 0.16ns Unique ash effect (no lithology)

HI-1 Ash Depth + biozone

0.28ns 0.13ns 0.01a

RO-6 Ash + lithology Depth + biozone

- 0.88ns 0.17ns Unique ash + lithology effect

HI-1 Ash + lithology Depth + biozone

- 0.10ns 0.01a

RO-6 Ash Depth + biozone + lithology

- 0.53ns 0.08ns Unique ash effect (lithology considered)

HI-1 Ash Depth + biozone + lithology

- 0.10ns 0.19ns

RO-6 Ash + lithology + ash*lithology

Depth + biozone

- 0.25ns 0.03b Unique ash + lithology + (ash*lithology) interaction effectHI-1 Ash + lithology +

ash*lithologyDepth + biozone

- 0.12ns 0.05b

a p 0.01 b 0.01 < p 0.05

The Laacher See eruption is reflected in the tree-rings of the Scots pines from Dättnau, near Winterhur, Switzerland, by a growth disturbance lasting at least 5 yr, and persisting in most of the trees for a further 3 yr. The X-ray photograph shows normal growth rings in sector (a); a very narrow tree-ring sequence in sector (b); three more rings of smaller width in sector (c); and in sector (d) after recovery, normally grown rings. The graph of the density curve shows on the vertical axis the maximum latewood densities; on the horizontal axis the tree-ring width. The latewood densities reflect a reduction in summer temperature lasting for 4 yr.

Effects on lake sediments lasted no more than 20 years.

8 winter layers contain clay and silt.

Hämelsee - annually laminated sediments

Major development in Quaternary palaeoecology in the last 10-15 years has been multi-proxy studies where several stratigraphical variables (e.g. pollen, plant macrofossils, diatoms, sediment magnetics, geochemistry, grain-size distribution) are studied on the same core.

If 'split' available data into 'reconstruction data' and 'response data', can test hypotheses about potential causes of change in the 'response data'.

MULTI-PROXY STUDIES

Sägistalsee, Bernese Oberland, Swiss Alps

A.F. Lotter et al. 2003

Sägistalsee

Sägistalsee, Switzerland

Ideal study:

1. Critical ecological situation at tree-line today; sensitive

2. One core. Many proxies (pollen, macros, chironomids, cladocera, grain size, sediment magnetics, sediment geochemistry)

3. Well dated; 18 AMS 14C dates on terrestrial plant material

4. Well co-ordinated by A. Lotter

5. High quality data:Data-set No. of

samplesNo. of

taxa/variablesPollen 212 203Plant macros 372 53Chironomids 82 30Cladocera 112 7Geochemistry 176 14Grain-size 294 6Magnetics 504 5

6. Consistent numerical methodology on all proxies

7. New approach: numerical methods used to test hypotheses about the influence of climate and catchment processes on the aquatic ecosystem in the perspective of the Holocene time-scale. (Partial redundancy analysis with restricted Monte Carlo permutation tests)

Of the catchment changes, the main ones appear to be the spread of Picea abies at about 6300 cal BP and Bronze Age and subsequent forest clearances and conversion to grazing pastures.

Hypotheses tested:1. Climate has had a significant control on lake ecosystem changes2. Catchment vegetation has played significant role on lake changes

"Responses" (proxies)

Scale Climate a significant predictor?

Catchment vegetation a significant predictor?

Terrestrial

Pollen Catchment & regional Y Y

Macrofossils Catchment - -Lake biotic

Chironomids Lake N Y Cladocera Lake N Y

Lake abiotic

Grain size Lake - Y Magnetics Lake - Y

Geochemistry Lake - (Y)* #* Tested against insolation, central

European cold phases, & Atlantic IRD record

# Veg phases: Betula-Pinus cembra; Alnus-Pinus cembra; Picea abies ~ 6300 cal BP; Pasture phases from Bronze Age to present

ASSESSING POTENTIAL EXTERNAL 'DRIVERS' ON AN

AQUATIC ECOSYSTEM

Bradshaw et al. 2005 The Holocene 15: 1152-1162

Dalland Sø, a small (15 ha), shallow (2.6 m) lowland eutrophic lake on the island of Funen, Denmark.

Catchment (153 ha) today

agriculture 77 ha

built-up areas 41 ha

woodland 32 ha

wetlands 3 ha

Nutrient rich – total P 65-120 mg l-1

Map of Dalland Sø

Multi-proxy study to assess role of potential external 'drivers' or forcing functions on changes in the lake ecosystem in last 7000 yrs.

Data: No. of samples

Transformation

Sediment loss-on-ignition % 560 None

Sediment dry mass accumulation rate

560 Log (x + 1)

Sediment minerogenic matter accumulation rate

560 Log (x + 1)

Plant macrofossil concentrations

280 Log (x + 1)

Pollen % 90 None

Diatoms % 118 None

Diatom inferred total P 118 None

Biogenic silica 84 Not used

Pediastrum % 90 None

Zooplankton 31 Not used

Terrestrial landscape or catchment

development

Bradshaw et al. 2005

Aquatic ecosystem development

Bradshaw et al. 2005

DCA of pollen and diatom data separately to summarise major underlying trends in both data sets

Pollen – high scores for trees, low scores for light-demanding herbs and crops

Diatom - high scores mainly planktonic and large benthic types, low scores for Fragilaria spp. and eutrophic spp. (e.g. Cyclostephanos dubius)

Bradshaw et al. 2005

Major contrast between samples before and after Late Bronze Age forest clearances

Bradshaw et al. 2005

'Catchment'

'Lake

'

Prior to clearance, lake experienced few impacts.

After the clearance, lake heavily impacted.

Canonical correspondence analysis

Response variables

Diatom taxa

Predictor variables

Pollen taxa, LOI, dry mass and minerogenic accumulation rates, plant macrofossils, Pediastrum

Covariable

Age

69 matching samples

Partial CCA with age partialled out as a covariable. Makes interpretation of effects of predictors easier by removing temporal trends and temporal autocorrelation

Partial CCA all variables

18.4% of variation in diatom data explained by Poaceae pollen, Cannabis-type pollen, and Daphnia ephippia.

As different external factors may be important at different times, divided data into 50 overlapping data sets – sample 1-20, 2-21, 3-22, etc.

CCA of 50 subsets from bottom to top and % variance explained

Bradshaw et al. 2005

1. 4520-1840 BC Poaceae is sole predictor variable (20-22% of diatom variance)

2. 3760-1310 BC LOI and Populus pollen (16-33%)

3. 3050-600 BC Betula, Ulmus, Populus, Fagus, Plantago, etc. (17-40%)i.e. in these early periods, diatom change influenced to some degree by external catchment processes and terrestrial vegetation change.

4. 2570 BC – 1260 AD Erosion indicators (charcoal, dry mass accumulation), retting indicator Linum capsules, Daphnia ephippia, Secale and Hordeum pollen (11-52%)

i.e. changing water depth and external factors

5. 160 BC – 1900 AD Hordeum, Fagus, Cannabis pollen, Pediastrum boryanum, Nymphaea seeds (22-47%)

i.e. nutrient enrichment as a result of retting hemp, also changes in water depth and water clarity

Strong link between inferred catchment change and within-lake development. Timing and magnitude are not always perfectly matched, e.g. transition to Mediæval Period

Bradshaw et al. 2005

Descriptive phase -

patterns are detected, described and classified

Narrative phase -

plausible, inductively-based explanations, generalisations, or reconstructions are proposed for observed patterns

Analytical phase -

falsifiable or testable hypotheses are proposed, evaluated, tested and rejected

Why is there so little analytical hypothesis-testing in palaeoecology?

MONTE CARLO PERMUTATION TESTS are valid without random samples, can be developed to take account of the properties of the data of interest, can use "non-standard" test statistics, and are completely specific to the data-set at hand. Ideal for palaeoecology.

CONCLUSIONS

Phases in palaeoecology