Sampling Strategies for the Assessment of Ecological Diversity

Lorenzo Fattorini

Dipartimento di Metodi Quantitativi, Università di Siena,P.za S. Francesco 8, 53100 Siena (Italy)

fattorini@unisi.it

ISSUES

• ECOLOGICAL DIVERSITY QUANTIFICATION • ESTIMATION OF DIVERSITY INDEXES • ECOLOGICAL DIVERSITY ORDERING

DESIGN-BASED INFERENCE

• no assumptions about the population under study • “Design-based inference is objective, nobody can challenge that the sample was really selected according to the given sampling design. The probability distribution associated with the design is real, not modelled or assumed” Sarndal et al (1992)

ECOLOGICAL DIVERSITY QUANTIFICATION

ECOLOGICAL DIVERSITY (??)

• a formal definition is still lacking • traditionally ED relies on the apportionment of abundance (or related quantities such as biomass or coverage) into the animal or plant species forming the ecological community under study

ECOLOGICAL COMMUNITY (??)

all the organisms in a delineated study area belonging to a taxonomic group of level higher than species (taxocene, Pielou, 1977, p.269)

e.g. ecological diversity of the snakes in a delineated tropical area ecological diversity of the trees in a nature reserve

QUANTIFICATION OF DIVERSITY BY INDEXES (evenness, dominance and rarity of species)

huge literature detailed reviews : Dennis et al, 1979, Magurran, 1988, Frosini, 2003

most relevant contributions

Patil and Taille (1979, 1982) - average rarity diversity indexes Rao (1982) – quadratic entropy

NOTATIONS

A size of the study area N number of individuals in the community (total abundance) k number of species in the community (species richness)

lN number of individuals (abundance) of species l ),,1( kl

NNp ll / relative abundance of species l ),,1( kl

T1 ,, kNN N abundance vector

)/(,, TT N1Np kpp 1 relative abundance vector

AVERAGE RARITY DIVERSITY INDEXES

• a community is diverse when there is a large number of rare species (Patil and Taille, 1979,1982)

)( ll pR rarity of species l depending on its relative abundance

k

llll pRp

1)(

family is obtained when )1(/)1()()( llll ppRpR

0 (Shannon index) 1 (Simpson index)

RAO’S QUADRATIC ENTROPY INDEXES

• a diversity index is perfect if splits diversity between and within subpopulation determined by means of hierarchical classifications of any order (Rao, 1982) Lau (1985)

DppT

lhdD , lhd difference (usually in biological sense) between species l and h

advances: Solow and Polasky (1994, Gore’s discussion), Champely and Chessel (2002), Izsak and Szeidl (2002)

ESTIMATION OF DIVERSITY INDEXES

ABUNDANCE ESTIMATION

)(p )(N

• knowledge of requires knowledge of N

• knowledge of N requires a census of the ecological community

N is unknown and must be estimated to estimate

)ˆ(ˆ p

SIMPLE RANDOM SAMPLING (with replacement)

• most papers devoted to estimation of diversity indexes are based on the assumption that individuals are selected from the community by means of simple random sampling with replacement n number of independent drawings

lx number of sampled individuals of species l T,, kxx 1x

),( px nMNd x minimal sufficient statistic for p

Good (1953), Blyth (1958), Basharin (1959), …, Baczkowsky et al (2000), Chao and Shen (2003)

• animals and plants cannot be selected from the populations just like balls from an urn !!! “Sampling of plant is usually by line traverse or quadrats: sampling of animals is either by nests or traps or visually along traverses or in quadrats. For the most usual types of sampling, then, the sampling distribution of the index of diversity depends, possibly strongly so, on the spatial distributions and associations of the various species, and the characteristics of the sampling procedure” (Zhal, Ecology, 1977) “Generally, when a population is sampled, the individuals in separate sampling units (quadrats) are not random or independently drawn from the parent community” (Helthse and Forrester, Biometrics, 1983)

• Pielou (1966) was the first to account for the actual field conditions proposing the estimation of diversity indexes by means of a pooled quadrat sampling plan. • Heyer and Berven (1973) extended Pielou’s method to improve efficiency and to estimate the sampling variance • Zahl (1977) proposed the use of the jackknife in plot sampling in order to estimate the Simpson index • Heltshe and Bitz (1979), Heltshe and Forrester (1983), Gove et al (1994) investigated jackknifing procedures in plot sampling by means of simulation studies or field work • theoretical results of general validity were lacking Barabesi and Fattorini (1998) give general results on the estimation of diversity indexes when abundance is estimated by means of plots, points or transects

SAMPLING ECOLOGICAL COMMUNITIES

• an ecological community on a delineated study area constitutes a without-frame population of N organisms spread over the area

• owing to the lack of frame, the most effective schemes for sampling ecological populations differ from the traditional ones

• their choice is mainly determined by practical considerations on the nature of the community to be sampled

ENVIRONMENTAL SCHEMES the selected units are those encountered from points or within plots or along transects randomly thrown onto the study area tree communities plot sampling , Bitterlich sampling shrub communities line intercept sampling animal population line transect sampling , point transect sampling

(problems related to the elusive behaviour of animals)

De Vries (1986), Thompson (1992), Schreuder et al. (1993), Overton and Stehman (1995)

N,,, 21U population of N units

US sample

S family of the possible N2 samples

S SS)Pr( sampling design

• when the population frame is available, the inclusion probabilities may be determined from the design

• when handling with ecological communities, the sampling design is unknown • the encounter schemes must be strictly ruled to determine (directly or by field measurements) the first-order inclusion probabilities at least for the selected units (computation of the Horvitz-Thompson estimate)

FLOATING PLOT SAMPLING

a point is randomly thrown onto the study area and the selected units are those included in a circular or square plot of a pre-fixed size a centered at the sample point

• all the inner units have first-order inclusion probability Aaj / • edge effects can be removed by suitable modifications of the sampling scheme (e.g. Gregoire and Valentine, 2008).

FLOATING PLOT SAMPLING

a point is randomly thrown onto the study area and the selected units are those included in a circular or square plot of a pre-fixed size a centered at the sample point

• all the inner units have first-order inclusion probability Aaj / • edge effects can be removed by suitable modifications of the sampling scheme (e.g. Gregoire and Valentine, 2008).

FLOATING PLOT SAMPLING

a point is randomly thrown onto the study area and the selected units are those included in a circular or square plot of a pre-fixed size a centered at the sample point

• all the inner units have first-order inclusion probability Aaj / • edge effects can be removed by suitable modifications of the sampling scheme (e.g. Gregoire and Valentine, 2008).

FLOATING PLOT SAMPLING

a point is randomly thrown onto the study area and the selected units are those included in a circular or square plot of a pre-fixed size a centered at the sample point

• all the inner units have first-order inclusion probability Aaj / • edge effects can be removed by suitable modifications of the sampling scheme (e.g. Gregoire and Valentine, 2008).

BITTERLICH SAMPLING (variable circular plot sampling)

a point is randomly thrown onto the study area and a tree is selected if its bole at breast high subtends an angle greater than a pre-fixed angle onto the point

• the first order inclusion probability of each tree is proportional to the bole area at breast height (which can be readily determined in the field by measuring the bole circumference)

BITTERLICH SAMPLING (variable circular plot sampling)

a point is randomly thrown onto the study area and a tree is selected if its bole at breast high subtends an angle greater than a pre-fixed angle onto the point

• the first order inclusion probability of each tree is proportional to the bole area at breast height (which can be readily determined in the field by measuring the bole circumference)

BITTERLICH SAMPLING (variable circular plot sampling)

a point is randomly thrown onto the study area and a tree is selected if its bole at breast high subtends an angle greater than a pre-fixed angle onto the point

• the first order inclusion probability of each tree is proportional to the bole area at breast height (which can be readily determined in the field by measuring the bole circumference)

LINE INTERCEPT SAMPLING (fixed length)

a transect of fixed length is randomly thrown onto the study area and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the perimeter length of the minimum convex polygon enveloping the unit to the perimeter length of the minimum convex polygon enveloping the whole area (Kaiser, 1983)

LINE INTERCEPT SAMPLING (fixed length)

a transect of fixed length is randomly thrown onto the study area and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the perimeter length of the minimum convex polygon enveloping the unit to the perimeter length of the minimum convex polygon enveloping the whole area (Kaiser, 1983)

LINE INTERCEPT SAMPLING (fixed length)

a transect of fixed length is randomly thrown onto the study area and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the perimeter length of the minimum convex polygon enveloping the unit to the perimeter length of the minimum convex polygon enveloping the whole area (Kaiser, 1983)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE INTERCEPT SAMPLING (fixed direction)

a transect of fixed direction is randomly thrown onto the baseline and the selected units are those intercepted by the transect

• the inclusion probability of a unit is the ratio of the length of the shadow cast by the unit onto the baseline to the baseline length (e.g. Thompson, 2002)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

LINE and POINT TRANSECT SAMPLING

a line or a point is randomly thrown onto the area and the selected units are those spotted from it

• the inclusion probabilities are evaluated on the basis of some simplifying assumptions adopted to model the sighting process (Buckland et al, 1993) • inference cannot be considered entirely design-based)

HORVITZ-THOMPSON ESTIMATION

kee ,,1 standard basis of kR

Nyy ,,1 population vectors

lj ey if individual j belongs to species l

N

jj

1yN

j known at least for each Sj

Sj j

jy

N HT estimate of N

VARIANCE ESTIMATION PROBLEMS

N unbiased ΣN )ˆ(V

• Σ depends on first and second-order inclusion probabilities determined by characteristics of the ecological community (spatial distribution of the individuals over the study area)

• an unbiased estimator for Σ exists iff the second-order inclusion probabilities are invariably positive

• in encounter schemes individuals that are very far apart cannot be encountered jointly by the same plot, point or line • Σ cannot be unbiasedly estimated by a unique sample S

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

REPLICATIONS

• a study area cannot be adequately sampled using one plot or one line or one point only the sampling scheme is independently replicated n times (n plots, lines or n points randomly and independently thrown onto the study area)

n samples nSS1 ,, n estimates nNN ˆ,,ˆ 1 iid ΣNNN )ˆ(,)ˆ( ii VE

n

iin 1

1 NN ˆ

nVE /)(,)( ΣNNN

T)ˆ()ˆ( NNNNS

in

iin 11

1

ΣS )(E

central limit theorem ),()(/ I0NNΣ kd

Nn 21

ESTIMATION OF RELATIVE ABUNDANCE

)/(ˆ T N1Np

delta method ),()ˆ(/ I0ppΞ kd

Nn 21

2)/()()( TT N11pIΣp1IΞ T variance-covariance matrix

2)/()ˆ()ˆ(ˆ TTT N1p1IS1pIΞ consistent estimator

• the species richness k is often unknown • rare species are not present in the n samples.

SO number of observed species •N and p are actually SO-vectors plus an unknown number of zeros ( SOk )

0,,0,,,1T SONNN 0,,0,ˆ,,ˆˆ 1

T SOppp •S and Ξ are square matrices of order SO plus zero-matrices of unknown order

GENERAL RESULTS

),,,,,(),,( 0011 kk pppp ,

• estimation does not require the knowledge of k since the missing species may be ignored when computing )ˆ(ˆ p

• )(ˆ N univariate transformation of the mean of the n iid estimates nNN ˆ,,ˆ 1

• xxx /)()(g exists in a neighbourhood of p, is non-null and continuous at p

),(ˆ 101 Nnd ΣggT2 )(Ng g

JACKKNIFE (deleting one replication at time)

• is asymptotically unbiased but a bias occurs for finite samples

jack 2jackv

standard results on jackknife (Shao and Tu, 1995):

if T/)( xxx 2 exists and is continuous at N

has a )( 1nO asymptotic bias

jack has )( 1no asymptotic bias

),()ˆ( 101 Nvd

jackjack .

the result holds for the most familiar diversity indexes

ECOLOGICAL DIVERSITY ORDERING

PROBLEMS

• inconsistent rankings : different indexes may lead to different rankings (Hurlbert,1971) Gove et al. (1994)

Species C1 C2 C3 Pinus strobus 0.50 0.25 0.35 Quercus rubra 0.30 0.25 0.35 Tsuga Canadensis 0.10 0.25 0.30 Acer rubrum 0.05 0.25 0.00 Betula papyrifera 0.05 0.00 0.00 Shannon index 1.24 1.39 1.10 Simson index 0.65 0.75 0.67

INTRINSIC DIVERSITY ORDERING (Patil and Taille , 1982)

the community 2C is intrinsically more diverse than )( 211 CCC if 2C is obtained from 1C through a finite sequence of the following operations:

(a) transferring abundance from more to less abundant species without reversing the rank-order of the species;

(b) transferring abundance to a new species; (c) re-labelling the species.

INTRINSIC DIVERSITY ORDERING (Patil and Taille , 1982)

the community 2C is intrinsically more diverse than )( 211 CCC if 2C is obtained from 1C through a finite sequence of the following operations:

(a) transferring abundance from more to less abundant species without reversing the rank-order of the species;

(b) transferring abundance to a new species; (c) re-labelling the species.

INTRINSIC DIVERSITY PROFILES

mT - relative abundance of the mk rarest species ( 0,10 kTT ) The plotting of mT versus m gives a profile which turns out to be convex and decreasing from 1 to 0

0

0,2

0,4

0,6

0,8

1

0 1 2 3 4

if 21 CC then the intrinsic diversity profile of 2C is everywhere above that of 1C if the two profiles intersect one or more times, no intrinsic ordering of the two communities is possible

0

0,2

0,4

0,6

0,8

1

0 1 2 3 4

0

0,2

0,4

0,6

0,8

1

0 1 2 3 4

DIVERSITY PROFILE ESTIMATION

• any intrinsic diversity profile T,, 11 kTT T may be viewed as a function of N, )(NT t

• any intrinsic diversity profile can be estimated by )(ˆ NT t Fattorini and Marcheselli (1999)

0N )(JhlNN hl

(the jacobian matrix of the function t at N exists and differs from the null matrix)

),()ˆ(/ I0TTΩ 121

k

dNn

T)()( NΣNΩ JJ

• jackknife procedure is not suitable for estimating T since jackT may fail to be convex

• jackknife variance estimator jackV consistent for Ω

PROFILE CONFIDENCE BAND

• any sample intrinsic diversity profile is obtained by joining the 1k points kmTm m ,,),ˆ,( 0 • any sample intrinsic diversity profile may be viewed as a set of linear combinations of the component of T • T is asymptotically normal with expectation T and consistent estimates of its variance-covariance matrix are available

RICHMOND’S PROCEDURE (Richmond, 1982) asymptotically )( 1 conservative confidence band (Fattorini and Marcheselli, 1999)

the band is bounded from below by joining the )( 1k points ),(),ˆ,(,),ˆ,(),,( 01110 11 kLkL k and from above by joining ),(),ˆ,(,),ˆ,(),,( 01110 11 kUkU k

mmkmm vTL ˆˆˆ , 1 mmkmm vTU ˆˆˆ , 1

,1k - upper 1 quantile of the studentized maximum modulus distribution with parameter 1k and degrees of freedom

2mmv - ),( mm -element of jackV

• the species richness k, which determines the profile length and the order of jackV as well as the quantile ,1k , usually constitutes an unknown parameter • T is of length SO and jackV is of order 1SO • since SO underestimates k, sample diversity profiles tend to be shorter than their population counterparts rule of thumb (conservative): since ,g slowly increases with g (the ratio 21500 ./ ,, gg for )15g choose k as the maximum number of species which might be present in the community (no excessive enlargements of the confidence bands)

HYPOTHESIS TESTING Patil and Taille (1979): method for assessing hypotheses on diversity profile “this procedure must be viewed as only an approximate test because it involves difficult and unresolved questions of simultaneous inference” (Gove et al. , 1994) Fattorini and Marcheselli (1999): asymptotically conservative procedure for comparing couples of diversity profiles by means of some methods previously adopted to make inference on Lorenz curves (Bishop et al.,1991) - dominance, equivalence and crossing assessment

1T - 1C profile ; 2T - 2C profile

211

210TTTT

::

HH

1210 kH )dim()dim( TT

1

121000

k

mmmmm TTHHH :

1T ( 1n replications) variance covariance matrix 11 n/Ω ( jack1V )

2T ( 2n replications) variance-covariance matrix 22 n/Ω ( jack2V )

under 0H and as 21 nn ,

),()ˆˆ(/ I0TTΨ 12121

k

dN

2211 nn // ΩΩΨ

1122

21

21

km

vvTTZ

mmmm

mmm ,,

ˆˆ

ˆˆ

BISHOPet al. (1991) CONSERVATIVE RULE accept 0H if ,1 kmZ for any 11 km ,, ; reject 0H and accept the dominance of )( 21 TT if there is at least one significant positive (negative) difference and no significant negative (positive) differences; reject 0H and accept the crossing of the two profiles if there is at least one significant positive difference and one significant negative difference

k is unknown

1SO , 2SO number of species observed in the two communities Fattorini and Marcheselli (1999) consider 1T and 2T of order ),max( 21 00 SS (setting at 0 the last component of the vector with the lower number of observed species) choose k as the maximum number of species which might be present in the two communities (conservative critical value ,1k )

Fattorini and Marcheselli (1999): mutual comparison of diversity for the avian populations settled in 11 parks in Milano and Pavia (Italy).

ADVANCES Marcheselli (2003, Annals): conservative inference based on a generalization of the Delta method when hl NN for some hl (rare species with unit abundance)

FUTURE DEVELOPMENTS - 1

• the complete random selection of n points, plots or transects over the study area (replications) gives rise to straightforward theoretical results but it is likely to produce unsuitable voids (undetected parts) in the study area • more complex sampling schemes should be adopted in order to ensure a systematic search over the study area (e.g aligned and unaligned systematic samplingpseudo-replications)

• the use of pseudo-replications instead of genuine replications requires more refined methodological tools, because the estimates derived from these plots, points or lines cannot be considered equally distributed and in the aligned case they are even dependent extension to the diversity index estimation of the results by Barabesi (2003), Barabesi and Marcheselli (2005, 2008), Gregoire and Valentine (2008, Chap. 10), Mandallaz (2008, Sec 4.2),

FUTURE DEVELOPMENTS – 2

Dardanoni and Forcina (1999): generalization of the Bishop et al. (1991) for comparing more than two Lorenz curves

comparison of more than two intrinsic diversity profiles

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