Multivariate

analyses

in

microbial

ecology

Environmental

microbiology

is

undergoing

a

dramatic

revolution

due

to

the

increasing

accumulation

of

biological

information

and

contextual

environmental

parameters

This

will

not

only

enable

a

better

identification

of

diversity

patterns

but

will

also

shed

more

light

on

the

associated

environmental

conditions

spatial

locations

and

seasonal

fluctuations

which

could

explain

such

patterns

Complex

ecological

questions

may

now

be

addressed

using

multivariate

statistical

analyses

which

represent

a

vast

potential

of

techniques

that

are

still

underexploited

Here

well-established

exploratory

and

hypothesis-driven

approaches

are

reviewed

so

as

to

foster

their

addition

to

the

microbial

ecologist

toolbox

Because

such

tools

aim

at

reducing

data

set

complexity

at

identifying

major

patterns

and

putative

causal

factors

they

will

certainly

find

many

applications

in

microbial

ecology

Introduction

Microbial

ecology

is

undergoing

a

profound

change

because

structure

function

relationships

between

communities

and

their

environment

are

starting

to

be

investigated

at

the

field

regional

and

even

continental

scales

-LRB-

e.g.

Hughes

Martiny

et

al.

Ramette

Tiedje

b

-RRB-

Because

DNA

sequences

are

being

accumulated

at

an

unprecedented

rate

due

to

high-throughput

technologies

such

as

pyrosequencing

-LRB-

Edwards

et

al.

b

-RRB-

single-cell

genome

sequencing

-LRB-

Zhang

et

al.

-RRB-

or

metagenomics

-LRB-

Venter

et

al.

Field

et

al.

Gill

et

al.

-RRB-

future

challenges

will

very

likely

consist

of

interpreting

the

observed

diversity

patterns

as

a

function

of

contextual

environmental

parameters

This

would

help

answer

fundamental

questions

in

microbial

ecology

such

as

whether

microbial

diversity

responds

qualitatively

and

quantitatively

to

the

same

factors

as

macroorganism

diversity

-LRB-

Horner-Devine

et

al.

van

der

Gast

et

al.

Green

Bohannan

Hughes

Martiny

et

al.

-RRB-

Most

obstacles

encountered

by

microbial

ecologists

when

they

try

to

summarize

and

further

explore

large

data

sets

concern

the

choice

of

the

adequate

numerical

tools

to

further

evaluate

the

data

statistically

and

visually

Such

tools

which

have

been

developed

by

community

ecologists

to

work

on

distribution

and

diversity

patterns

of

plants

and

animals

could

be

readily

applied

in

microbial

ecology

Although

multivariate

analyses

of

community

diversity

patterns

are

well

described

in

the

literature

microbial

ecologists

have

used

multivariate

analyses

either

rarely

or

mostly

for

exploratory

purposes

A

brief

survey

of

the

literature

confirms

this

trend

-LRB-

Table

Fig.

-RRB-

Table

indicates

that

bacterial

studies

rank

third

after

plant

and

fish

studies

for

their

use

of

multivariate

analyses

Complex

data

sets

are

mostly

explored

via

principal

component

analysis

or

cluster

analysis

and

hypothesis-driven

techniques

such

as

redundancy

analysis

canonical

correspondence

analysis

-LRB-

CCA

-RRB-

or

Mantel

tests

are

more

rarely

used

-LRB-

Fig.

-RRB-

Axis

-LRB-

horizontal

-RRB-

clearly

differentiates

microscopic

-LRB-

bacteria

microorganisms

fungi

-RRB-

from

macroscopic

-LRB-

fish

bird

plant

insect

-RRB-

life

and

this

may

be

related

to

the

use

of

more

exploratory

methods

-LRB-

e.g.

cluster

analysis

PCA

-RRB-

in

the

first

group

It

is

important

to

state

that

the

figures

presented

in

Table

and

Fig.

have

to

be

taken

with

caution

because

many

articles

do

not

include

a

description

of

statistical

approaches

in

their

titles

or

abstracts

and

so

the

table

is

certainly

biased

and

incomplete

However

the

point

of

the

table

was

both

to

identify

some

general

trends

in

the

literature

and

to

give

one

example

of

the

usefulness

of

multivariate

analysis

to

analyze

a

data

table

Table

Usage

-LRB-

-RRB-

of

multivariate

methods

in

different

fields

Exploratory

analysis

Hypothesis-driven

analysis

Keywords

Cluster

PCA

MDS

PCoA

CCA

RDA

manova

Mantel

anosim

CVA

Total

number

Bacter

Microb

Plant

Fung

Fish

Bird

Insect

A

literature

search

was

performed

with

the

Thomson

ISI

research

tool

with

the

following

parameters

-LRB-

Doc

type

all

document

types

language

all

languages

databases

SCI-EXPANDED

SSCI

A&HCI

Timespan

-RRB-

on

December

in

the

titles

and

abstracts

of

the

articles

only

Asterisks

were

placed

at

the

end

of

each

keyword

to

accommodate

for

variations

Each

keyword

was

additionally

combined

with

the

following

technical

designations

cluster

cluster

analysis

PCA

principal

component

analysis

MDS

multidimensional

scaling

PcoA

principal

coordinate

analysis

CCA

canonical

correspondence

analysis

RDA

redundancy

analysis

Mantel

Mantel

test

or

CVA

canonical

variate

analysis

Total

number

refers

to

the

total

number

of

publications

identified

by

each

keyword

and

all

its

combinations

The

ordination

based

on

correspondence

analysis

of

the

raw

number

is

depicted

in

Fig.

Fig.

Correspondence

analysis

of

method

usage

in

various

scientific

fields

In

this

symmetrical

scaling

of

CA

scores

the

first

two

axes

explained

and

of

the

total

inertia

of

Table

respectively

The

gray

areas

were

drawn

to

facilitate

the

interpretation

Complete

row

names

-LRB-

scientific

fields

full

circles

-RRB-

and

column

names

-LRB-

methods

white

triangles

-RRB-

are

given

in

Table

Methods

-LRB-

triangles

-RRB-

located

close

to

each

other

correspond

to

methods

often

occurring

together

in

studies

The

distance

between

a

scientific

field

point

and

a

method

point

approximates

the

probability

of

method

usage

in

the

field

This

review

aims

at

presenting

some

common

multivariate

techniques

in

order

to

foster

their

integration

into

the

microbial

ecologist

's

toolbox

Indeed

it

is

no

longer

possible

to

gain

a

full

understanding

of

Ecology

and

Systematics

without

some

knowledge

of

multivariate

analysis

Or

contrariwise

misunderstanding

of

the

methods

can

inhibit

advancement

of

the

science

-LRB-

James

McCulloch

-RRB-

Such

a

review

is

ambitious

because

it

tries

to

provide

a

few

guidelines

for

a

very

vast

discipline

that

is

still

under

development

For

this

reason

it

can

not

be

exhaustive

and

does

not

pretend

to

offer

in-depth

coverage

of

all

selected

topics

The

review

is

largely

inspired

by

descriptions

comments

and

suggestions

originating

from

multiple

highly

recommended

sources

-LRB-

ter

Braak

Prentice

James

McCulloch

Legendre

Legendre

Leps

Smilauer

ter

Braak

Smilauer

Palmer

-RRB-

where

detailed

information

about

each

technique

can

be

obtained

In

the

first

part

data

type

and

preparation

are

reviewed

as

a

necessary

basis

for

subsequent

multivariate

analyses

Second

common

multivariate

methods

-LRB-

i.e.

cluster

analysis

principal

component

analysis

correspondence

analysis

multidimensional

scaling

-RRB-

and

a

few

statistical

methods

to

test

for

significant

differences

between

groups

or

clusters

are

described

focusing

on

the

methods

main

objectives

applications

and

limitations

Beyond

the

mere

identification

of

diversity

patterns

microbial

ecologists

may

wish

to

correlate

or

explain

those

patterns

using

measured

environmental

parameters

and

this

approach

is

addressed

in

the

third

part

Special

emphasis

is

placed

on

a

few

methods

that

have

proven

useful

in

ecological

studies

namely

redundancy

analysis

CCA

linear

discriminant

analysis

as

well

as

variation

partitioning

The

final

part

provides

practical

considerations

to

help

researchers

avoid

pitfalls

and

choose

the

most

appropriate

methods

Data

types

and

data

preparation

Data

sets

The

initial

multivariate

data

set

may

consist

of

a

table

of

objects

-LRB-

e.g.

samples

sites

time

periods

-RRB-

in

rows

and

measured

variables

for

those

objects

in

columns

This

table

structure

is

the

standard

used

in

the

present

review

When

the

latter

variables

are

biological

taxa

the

columns

will

simply

be

designated

as

species

thereafter

It

is

critical

to

clearly

identify

what

corresponds

to

objects

and

variables

in

the

data

set

Indeed

objects

in

one

study

may

be

species

or

operational

taxonomic

units

-LRB-

OTU

-RRB-

for

which

catabolic

profiles

gene

presence

or

polymorphism

etc.

are

measured

In

another

study

where

samples

from

different

sites

are

compared

based

on

for

instance

community

fingerprinting

techniques

objects

can

now

be

samples

and

species

variables

This

distinction

is

important

because

procedures

that

analyze

relationships

among

objects

or

among

variables

are

different

Objects

are

defined

a

priori

by

the

sampling

strategy

before

making

observations

and

variable

measurements

Besides

most

multivariate

analyses

assume

independence

between

objects

-LRB-

or

samples

-RRB-

i.e.

observations

made

on

an

object

are

not

a

priori

dependent

on

those

made

on

another

object

Variables

however

can

be

found

to

be

intercorrelated

to

various

degrees

but

this

is

not

necessarily

known

in

advance

Initial

data

sets

can

also

consist

of

distance

matrices

where

pairwise

dissimilarities

between

objects

are

calculated

The

original

table

of

raw

data

is

not

always

available

e.g.

for

DNA

DNA

hybridization

values

phylogenetic

distances

and

thus

specific

multivariate

techniques

have

to

be

considered

to

deal

with

data

matrices

Data

transformations

In

multivariate

data

tables

measured

variables

can

be

binary

quantitative

qualitative

rank-ordered

classes

frequencies

or

even

a

mixture

of

those

types

If

variables

do

not

have

a

uniform

scale

-LRB-

e.g.

environmental

parameters

measured

in

different

units

or

scales

-RRB-

or

an

adequate

format

variables

have

to

be

transformed

before

performing

further

analyses

Each

qualitative

variable

has

to

be

recoded

as

a

set

of

numerical

variables

that

replace

it

in

the

numerical

calculations

One

way

to

do

so

is

to

create

a

series

of

dummy

variables

that

correspond

to

all

the

states

of

the

qualitative

variable

For

instance

if

the

variable

season

has

to

be

recoded

four

associated

variables

will

be

constructed

and

for

each

object

the

value

will

be

given

to

the

corresponding

season

when

it

occurs

and

for

the

three

other

seasons

when

it

is

absent

Many

statistical

packages

automatically

perform

this

recoding

Standardization

provides

dimensionless

variables

and

removes

the

undue

influence

of

magnitude

differences

between

scales

or

units

A

common

procedure

is

to

apply

the

z-score

transformation

to

the

values

of

each

variable

For

each

variable

it

consists

of

-LRB-

-RRB-

computing

the

difference

between

the

original

value

and

the

mean

of

the

variable

-LRB-

i.e.

centering

-RRB-

and

of

-LRB-

-RRB-

dividing

this

difference

by

the

SD

of

the

variable

Normalizing

transformations

aim

at

correcting

the

distribution

shapes

of

certain

variables

which

depart

from

normality

One

thus

tries

to

obtain

a

homogeneous

variances

for

variables

conditions

under

which

multivariate

procedures

often

perform

better

Different

mathematical

transformations

can

be

used

to

normalize

the

x

values

of

a

variable

for

instance

the

arcsin

-LRB-

x

-RRB-

transformation

can

be

applied

to

percentages

or

proportions

log

-LRB-

x

c

-RRB-

to

variables

departing

strongly

from

a

normal

distribution

and

-LRB-

x

c

-RRB-

to

less

problematic

cases

with

c

being

a

constant

that

is

added

to

avoid

mathematically

undefined

computations

The

c

constant

is

generally

chosen

so

that

the

smallest

nonzero

value

is

obtained

by

computing

x

c

in

the

former

functions

The

constant

should

also

be

of

the

same

order

of

magnitude

as

the

variable

-LRB-

Legendre

Legendre

-RRB-

To

make

community

composition

-LRB-

either

presence

absence

or

abundance

-RRB-

data

containing

many

zeros

suitable

for

analysis

by

linear

methods

such

as

principal

component

analysis

-LRB-

PCA

-RRB-

or

canonical

redundancy

analysis

-LRB-

RDA

-RRB-

the

Hellinger

transformation

-LSB-

Eq

-LRB-

-RRB-

-RSB-

is

one

of

five

transformations

that

give

good

results

-LRB-

Legendre

Gallagher

-RRB-

The

chord

transformation

is

a

useful

transformation

that

also

gives

less

weight

to

rare

species

in

the

species

table

-LSB-

Eq

-LRB-

-RRB-

-RSB-

The

transformations

are

given

by

where

yij

is

the

original

species

value

for

site

i

and

species

j

yi

represents

the

sum

of

all

species

values

for

site

i

-LRB-

i.e.

sum

per

row

-RRB-

p

is

the

number

of

species

in

the

table

-LRB-

number

of

columns

-RRB-

and

represents

the

resulting

transformed

species

value

-LRB-

Legendre

Gallagher

-RRB-

These

transformations

are

particularly

recommended

when

rare

species

are

not

truly

rare

i.e.

when

they

mostly

occur

because

the

sampling

was

performed

blindly

as

generally

done

in

soil

or

marine

microbial

ecology

Further

data

transformations

can

be

found

in

Sokal

Rohlf

-LRB-

-RRB-

and

Legendre

Legendre

-LRB-

-RRB-

The

way

to

deal

with

missing

data

is

a

discipline

on

its

own

-LRB-

Legendre

Legendre

-RRB-

Briefly

one

can

either

delete

rows

or

columns

containing

the

missing

value

-LRB-

s

-RRB-

or

try

to

replace

the

missing

values

by

mathematical

estimates

inferred

from

values

obtained

from

other

objects

in

the

data

set

In

the

latter

case

it

is

still

difficult

to

provide

ecologically

meaningful

explanations

for

these

estimates

In

any

case

the

specific

handling

of

missing

data

should

be

reported

by

the

investigator

When

dealing

with

matrices

it

is

possible

to

change

a

similarity

matrix

-LRB-

S

-RRB-

into

a

dissimilarity

matrix

-LRB-

D

-RRB-

by

applying

the

following

transformations

D

S

D

-LRB-

S

-RRB-

or

D

-LRB-

-RRB-

To

normalize

any

D

matrix

to

the

interval

-LSB-

-RSB-

one

can

compute

D/Dmax

or

-LRB-

D

Dmin

-RRB-

-LRB-

Dmax

Dmin

-RRB-

where

Dmax

and

Dmin

represent

the

highest

and

lowest

values

of

D

respectively

-LRB-

Legendre

Legendre

-RRB-

Exploratory

analyses

Visualization

and

exploration

of

complex

data

sets

The

basic

aim

of

ordination

and

cluster

analysis

is

to

represent

the

-LRB-

dis

-RRB-

similarity

between

objects

-LRB-

e.g.

samples

sites

-RRB-

based

on

values

of

multiple

variables

-LRB-

columns

-RRB-

associated

with

them

so

that

similar

objects

are

depicted

near

from

each

other

and

dissimilar

objects

are

found

further

apart

from

each

other

Exploratory

multivariate

analyses

are

thus

useful

to

reveal

patterns

in

large

data

sets

but

they

do

not

directly

explain

why

those

patterns

exist

This

latter

point

is

addressed

in

the

third

part

of

the

review

Cluster

analysis

and

association

coefficients

Cluster

analysis

encompasses

several

multivariate

techniques

that

are

used

to

group

objects

into

categories

based

on

their

dissimilarities

The

aim

is

both

to

minimize

within-group

variation

and

maximize

between-group

variation

in

order

to

reveal

well-defined

categories

of

objects

and

therefore

reduce

the

dimensionality

of

the

data

set

to

a

few

groups

of

rows

-LRB-

James

McCulloch

Legendre

Legendre

-RRB-

This

approach

is

thus

generally

recommended

when

distinct

discontinuities

instead

of

continuous

differences

-LRB-

i.e.

gradients

-RRB-

are

expected

between

samples

-LRB-

objects

-RRB-

because

cluster

analysis

mostly

aims

at

representing

partitions

in

a

data

set

-LRB-

Legendre

Legendre

-RRB-

Because

distance

matrices

that

are

based

on

differences

in

DNA

or

amino

acid

sequences

are

commonly

used

to

describe

microbial

diversity

cluster

analysis

has

become

very

popular

in

microbial

ecology

-LRB-

Table

Fig.

-RRB-

This

is

not

surprising

because

the

grouping

of

organisms

based

on

their

phenotypic

or

genotypic

similarities

in

order

to

infer

their

taxonomic

positioning

is

generally

and

historically

based

on

cluster

analysis

-LRB-

or

at

least

based

on

a

tree-like

representation

-RRB-

and

as

such

is

central

to

biology

and

evolution

-LRB-

Avise

-RRB-

Typical

microbial

ecology

questions

that

are

addressed

by

cluster

analysis

are

whether

the

clustering

patterns

of

molecular

sequences

reflect

sample

origin

or

sampling

time

in

order

to

reveal

specific

biogeographical

or

temporal

patterns

respectively

-LRB-

Whitaker

et

al.

Acinas

et

al.

-RRB-

Those

factors

are

generally

hypothesized

to

be

of

a

discontinuous

nature

but

the

rationale

of

generally

representing

molecular

differences

as

discontinuous

clusters

in

microbial

ecology

and

microbial

genomic

studies

has

only

started

to

be

questioned

-LRB-

Konstantinidis

et

al.

-RRB-

Another

common

application

consists

of

sorting

out

clones

from

environmental

samples

based

on

specific

criteria

-LRB-

e.g.

genetic

or

phenotypic

markers

-RRB-

because

clones

or

variants

are

expected

to

form

tight

clusters

around

their

parental

strains

and

to

be

more

distinct

from

other

lineages

-LRB-

Acinas

et

al.

-RRB-

In

microarray

data

analysis

cluster

analysis

has

helped

identify

common

expression

patterns

of

groups

of

genes

which

may

shed

light

on

functionally

related

genes

or

pathways

-LRB-

Eisen

et

al.

-RRB-

Cluster

analysis

of

a

data

table

proceeds

in

two

steps

First

a

relevant

association

coefficient

has

to

be

chosen

to

measure

the

association

-LRB-

similarity

or

dissimilarity

-RRB-

among

objects

or

among

variables

Second

the

calculated

association

matrix

is

represented

as

a

horizontal

tree

-LRB-

hierarchical

clustering

-RRB-

or

as

distinct

groups

of

objects

-LRB-

k-means

clustering

-RRB-

based

on

specific

rules

to

aggregate

objects

For

ecologists

the

power

of

cluster

analysis

derives

from

the

existence

of

different

types

of

-LRB-

dis

-RRB-

similarity

coefficients

The

choice

of

appropriate

and

ecologically

meaningful

association

coefficients

is

particularly

important

because

it

directly

affects

the

values

that

are

subsequently

used

for

the

categorization

of

objects

The

analysis

of

similarities

among

objects

-LRB-

rows

-RRB-

is

designated

as

Q

mode

analysis

whereas

when

relationships

among

variables

-LRB-

columns

-RRB-

are

the

focus

of

the

study

this

is

referred

to

as

R

mode

analysis

-LRB-

Legendre

Legendre

-RRB-

Noticeably

the

two

modes

of

analysis

do

not

generally

use

the

same

association

coefficients

Although

it

is

not

possible

to

give

a

full

review

of

all

association

coefficients

here

it

is

useful

to

known

that

for

comparing

objects

-LRB-

rows

-RRB-

based

on

their

column

attributes

in

Q

mode

analysis

coefficients

may

be

chosen

as

a

function

of

data

type

-LRB-

quantitative

qualitative

ordinal

or

mixed

data

normalized

data

presence-absence

-RRB-

importance

given

to

rare

species

weight

given

to

each

object

and

calculation

of

associated

probability

levels

For

comparing

objects

in

a

sample-by-environment

table

-LRB-

e.g.

water

soil

chemistry

-RRB-

selection

of

appropriate

coefficients

generally

depends

on

data

type

and

unit

homogeneity

of

the

measured

variables

In

R

mode

analyses

in

addition

to

the

previously

cited

criteria

the

choice

of

a

coefficient

may

also

depend

on

how

the

variables

are

related

to

each

other

-LRB-

e.g.

linearly

monotonically

qualitatively

ordered

-RRB-

and

on

how

species

absence

is

handled

in

the

calculations

In

most

ecological

studies

the

absence

of

a

species

at

two

sites

being

compared

is

not

considered

as

a

measure

of

similarity

between

those

sites

Indeed

a

simultaneous

species

absence

at

two

sites

may

be

due

to

different

reasons

e.g.

the

sites

offer

different

physical

chemical

conditions

and

the

species

can

not

exist

under

both

conditions

and

so

there

is

no

straightforward

conclusion

about

site

similarity

that

can

be

drawn

in

this

case

Asymmetric

coefficients

are

coefficients

that

do

not

take

into

account

cases

of

double

absences

of

species

-LRB-

double

zeros

-RRB-

in

the

calculation

of

pairwise

similarities

among

sites

Moreover

in

microbial

ecology

where

environmental

communities

are

generally

far

from

being

exhaustively

sampled

a

double

absence

of

an

OTU

has

to

be

regarded

more

as

a

lack

of

information

rather

than

a

sign

of

common

structure

among

samples

and

asymmetric

coefficients

such

as

Jaccard

-LRB-

-RRB-

or

Sørensen

-LRB-

-RRB-

should

be

preferred

More

details

about

the

calculation

of

association

coefficients

and

their

appropriateness

can

be

found

for

instance

in

Chapter

of

-LRB-

Legendre

Legendre

-RRB-

When

an

association

matrix

is

calculated

the

relationships

between

objects

or

variables

can

be

represented

following

specific

aggregation

rules

Three

general

approaches

are

commonly

used

hierarchical

clustering

k-means

partitioning

and

two-way

joining

In

hierarchical

clustering

a

linkage

rule

to

form

clusters

and

the

numbers

of

clusters

that

best

suit

the

data

have

to

be

determined

a

priori

Clusters

which

are

nested

rather

than

mutually

exclusive

here

are

either

formed

by

progressively

agglomerating

objects

from

high

to

low

similarity

cutoff

values

-LRB-

forward

clustering

-RRB-

or

using

the

converse

strategy

i.e.

grouping

all

cases

together

and

progressing

from

low

to

high

cutoff

values

in

order

to

merge

objects

and

clusters

-LRB-

backward

clustering

-RRB-

These

two

strategies

do

not

necessarily

yield

the

same

clusters

The

merging

of

clusters

is

visualized

using

a

tree

format

-LRB-

generally

horizontal

-RRB-

and

is

successful

when

well-defined

clusters

are

identified

in

the

data

set

-LRB-

Sneath

Sokal

-RRB-

Common

linkage

rules

are

e.g.

nearest

neighbor

-LRB-

the

distance

between

two

clusters

is

the

distance

between

their

closest

neighboring

points

-RRB-

furthest

neighbor

-LRB-

the

distance

between

two

clusters

is

the

distance

between

their

two

furthest

objects

-RRB-

and

the

widely

used

unweighted

pair-group

method

using

averages

-LRB-

UPGMA

Sneath

Sokal

-RRB-

where

the

distance

between

two

clusters

is

the

average

distance

between

all

intercluster

pairs

When

within-cluster

homogeneity

is

desired

Ward

's

method

which

merges

clusters

only

if

they

increase

the

within-cluster

variation

the

least

is

recommended

-LRB-

Legendre

Legendre

-RRB-

Finally

equal

weight

can

also

be

given

to

clusters

that

are

expected

to

be

of

different

sizes

using

the

weighted

arithmetic

average

clustering

-LRB-

WPGMA

-RRB-

which

consists

of

giving

less

weight

to

the

original

similarities

of

the

largest

groups

-LRB-

Legendre

Legendre

-RRB-

In

k-means

clustering

objects

are

assigned

to

k

clusters

-LRB-

k

being

defined

in

advance

-RRB-

based

on

their

nearest

Euclidean

distance

to

the

mean

of

the

clusters

The

mean

of

the

cluster

is

iteratively

recalculated

until

no

more

assignments

are

made

and

cluster

means

fall

below

a

predefined

cut-off

value

or

until

the

iteration

limit

is

reached

Different

means

for

each

cluster

are

ideally

obtained

for

each

dimension

used

in

the

analysis

as

indicated

by

high

F-values

from

the

respective

analyses

of

variance

Unlike

hierarchical

clustering

k-means

clustering

does

not

require

prior

computation

of

dissimilarity

matrix

among

objects

and

is

therefore

more

adapted

to

large

data

sets

-LRB-

e.g.

few

thousand

objects

-RRB-

where

computing

power

is

an

issue

However

the

method

is

quite

sensitive

to

outliers

which

are

usually

removed

before

performing

the

analyses

-LRB-

Legendre

Legendre

-RRB-

Two-step

cluster

analysis

may

be

useful

to

group

objects

into

clusters

when

one

or

more

of

the

variables

are

categorical

-LRB-

not

interval

or

dichotomous

-RRB-

Objects

are

first

grouped

based

on

the

categories

which

are

themselves

hierarchically

clustered

as

single

cases

Because

neither

a

proximity

table

nor

iterative

steps

are

required

the

method

is

particularly

suited

for

the

analysis

of

very

large

data

sets

-LRB-

Eisen

et

al.

-RRB-

Principal

component

analysis

-LRB-

PCA

-RRB-

PCA

has

been

applied

to

numerous

phenotypic

and

genotypic

-LRB-

e.g.

fingerprinting

patterns

-RRB-

data

sets

and

it

is

one

of

the

most

popular

exploratory

analyses

-LRB-

Table

-RRB-

perhaps

because

the

technique

is

generally

the

first

multivariate

approach

to

be

explained

in

most

data

analysis

manuals

However

this

choice

may

not

always

be

justified

in

ecology

and

recommendations

for

appropriate

applications

are

provided

at

the

end

of

this

section

and

in

the

Practical

considerations

part

of

the

present

review

Examples

of

use

in

microbial

ecology

concern

the

identification

of

patterns

of

microbial

community

change

over

seasons

or

geographic

areas

-LRB-

e.g.

Merrill

Halverson

-RRB-

or

as

those

patterns

relate

to

different

plant

compartments

at

different

plant

developmental

stages

-LRB-

Mougel

et

al.

-RRB-

or

the

reduction

of

the

complexity

of

data

sets

involving

hydrochemistry

data

bacterial

and

archeal

community

profiles

in

order

to

visualize

and

interpret

complex

multivariate

data

sets

onto

two-dimensional

geographic

maps

of

contaminated

sites

-LRB-

Mouser

et

al.

-RRB-

The

PCA

procedure

basically

calculates

new

synthetic

variables

-LRB-

principal

components

-RRB-

which

are

linear

combinations

of

the

original

variables

-LRB-

for

instance

the

species

of

a

sample-by-species

table

-RRB-

and

that

account

for

as

much

of

the

variance

of

the

original

data

as

possible

-LRB-

Hotelling

-RRB-

The

aim

is

to

represent

the

objects

-LRB-

rows

-RRB-

and

variables

-LRB-

columns

-RRB-

of

the

data

set

in

a

new

system

of

coordinates

-LRB-

generally

on

two

or

three

axes

or

dimensions

-RRB-

where

the

maximum

amount

of

variation

from

the

original

data

set

can

be

depicted

In

practice

PCA

is

either

performed

on

a

variance

covariance

matrix

or

on

a

correlation

matrix

The

first

approach

is

followed

when

the

same

units

or

data

types

are

used

-LRB-

e.g.

abundance

of

different

species

-RRB-

The

aim

is

then

to

preserve

and

to

represent

the

relative

positions

of

the

objects

and

the

magnitude

of

variation

between

variables

in

the

reduced

space

PCA

on

a

correlation

matrix

is

rather

used

when

descriptor

variables

are

measured

in

different

units

or

on

different

scales

-LRB-

e.g.

different

environmental

parameters

-RRB-

or

when

the

aim

is

to

display

the

correlations

among

-LRB-

standardized

-RRB-

descriptor

variables

The

two

approaches

lead

to

different

principal

components

and

different

distances

between

projected

objects

in

the

ordination

hence

the

interpretation

of

the

relationships

must

be

made

with

care

-LRB-

Table

-RRB-

Indeed

for

correlation

matrices

variables

are

first

standardized

-LRB-

i.e.

they

become

independent

of

their

original

scales

-RRB-

and

so

distances

between

objects

are

also

independent

from

the

scales

of

the

original

variables

All

variables

thus

contribute

to

the

same

extent

to

the

ordination

of

objects

regardless

of

their

original

variance

Table

Interpretation

of

ordination

diagrams

Linear

methods

-LRB-

PCA

RDA

-RRB-

PCA

RDA

RDA

Scaling

Scaling

Samples

Species

ENV

NENV

Focus

on

sample

-LRB-

rows

-RRB-

distance

Focus

on

species

-LRB-

columns

-RRB-

correlation

Euclidean

distances

among

samples

Linear

correlations

among

species

Marginal

effects

of

ENV

on

ordination

scores

Correlations

among

ENV

Euclidean

distance

between

sample

classes

Abundance

values

in

species

data

Values

of

ENV

in

the

samples

Membership

of

samples

in

the

classes

Linear

correlations

between

species

and

ENV

Mean

species

abundance

within

classes

of

nominal

ENV

Average

of

ENV

within

classes

Unimodal

methods

-LRB-

CA

CCA

-RRB-

CA

CCA

CCA

Focus

on

sample

-LRB-

rows

-RRB-

distance

and

Hill

's

scaling

Focus

on

species

-LRB-

columns

-RRB-

distances

Turnover

distances

among

samples

χ2

distances

between

samples

χ2

distances

among

species

distributions

Marginal

effects

of

ENV

Correlations

among

ENV

Turnover

distances

between

sample

classes

χ2

distances

between

sample

classes

Relative

abundances

of

the

species

table

Relative

abundances

of

the

species

table

Values

of

ENV

in

the

samples

Membership

of

samples

in

the

classes

Weighted

averages

the

species

optima

in

respect

to

particular

ENV

Relative

total

abundances

in

the

sample

classes

ENV

averages

within

sample

classes

The

interpretation

of

ordination

diagrams

depends

on

the

focus

of

the

study

because

sample

scores

are

rescaled

as

a

function

of

the

scaling

choice

Approximate

relationships

between

and

among

the

different

elements

represented

in

biplots

and

triplots

as

species

-LRB-

represented

as

dots

or

arrows

-RRB-

samples

-LRB-

dots

-RRB-

environmental

variables

-LRB-

ENV

arrows

-RRB-

and

nominal

-LRB-

qualitative

-RRB-

environmental

variables

-LRB-

NENV

dots

-RRB-

A

meaningless

interpretation

-LRB-

-RRB-

happens

when

the

suggested

comparison

is

not

optimal

because

of

inappropriate

scaling

of

the

ordination

scores

Adapted

from

ter

Braak

-LRB-

-RRB-

Leps

Smilauer

-LRB-

-RRB-

ter

Braak

Smilauer

-LRB-

-RRB-

PCA

results

are

generally

displayed

as

a

biplot

-LRB-

Jolicoeur

Mosimann

-RRB-

where

the

axes

correspond

to

the

new

system

of

coordinates

and

both

samples

-LRB-

dots

-RRB-

and

taxa

-LRB-

arrows

-RRB-

are

represented

-LRB-

Fig.

-RRB-

The

direction

of

a

species

arrow

indicates

the

greatest

change

in

abundance

whereas

its

length

may

be

related

to

a

rate

of

change

Depending

on

whether

a

distance

or

a

correlation

biplot

is

chosen

different

interpretations

can

be

made

from

the

ordination

diagram

-LRB-

Table

-RRB-

The

interpretation

of

the

relationships

between

samples

and

species

differs

and

is

directly

affected

by

the

scaling

chosen

i.e.

whether

the

analysis

mainly

focuses

on

intersample

relationships

-LRB-

scaling

-RRB-

or

interspecies

correlations

-LRB-

scaling

-RRB-

For

instance

in

scaling

the

distances

between

objects

are

an

approximation

of

their

Euclidean

distances

in

the

multidimensional

space

but

this

approximation

is

not

valid

if

scaling

is

chosen

-LRB-

Table

-RRB-

Projecting

an

object

at

a

right

angle

on

a

species

arrow

in

the

ordination

diagram

approximates

the

position

of

the

object

along

that

species

descriptor

The

length

of

the

species

descriptor

indicates

its

contribution

to

the

formation

of

the

ordination

space

For

correlation

biplots

the

length

of

the

orthogonal

projection

of

a

species

arrow

on

the

axes

approximates

its

SD

on

the

respective

axes

Angles

between

species

arrows

reflect

their

correlations

e.g.

putative

interactions

between

species

-LRB-

scaling

-RRB-

An

erroneous

interpretation

of

the

biplot

would

be

to

use

the

proximity

of

an

object

point

and

the

tip

of

a

species

arrow

to

deduce

a

relationship

between

them

Indeed

only

right-angle

projections

of

samples

onto

species

arrows

are

correct

to

derive

approximated

species

abundance

in

the

samples

PCA

should

generally

be

used

when

the

objects

-LRB-

sites

or

samples

-RRB-

cover

very

short

gradients

i.e.

when

the

same

species

are

mostly

identified

everywhere

in

the

study

area

-LRB-

i.e.

when

samples

mostly

differ

in

species

abundances

-RRB-

and

when

species

linearly

respond

to

environmental

gradients

Because

those

conditions

are

often

not

met

in

ecological

studies

other

multivariate

approaches

have

been

progressively

preferred

over

PCA

-LRB-

as

also

suggested

by

Table

-RRB-

such

as

correspondence

analysis

or

multidimensional

scaling

PCA

is

successful

when

most

of

the

variance

is

accounted

for

by

the

largest

-LRB-

generally

the

first

two

or

three

-RRB-

components

The

amount

of

variance

accounted

for

by

each

principal

component

is

given

by

its

eigenvalue

The

mathematical

description

of

eigenvalue

calculation

steps

goes

beyond

the

aim

of

the

present

review

but

can

be

found

in

most

linear

algebra

manuals

Eigenvalues

derived

from

a

PCA

are

generally

considered

to

be

significant

when

their

values

are

larger

than

the

average

of

all

eigenvalues

-LRB-

Legendre

Legendre

-RRB-

The

cumulative

percentage

of

variance

accounted

for

by

the

largest

components

indicates

how

much

proportion

of

the

total

variance

is

depicted

by

the

actual

ordination

High

absolute

correlation

values

between

the

synthetic

variables

-LRB-

principal

components

-RRB-

and

the

original

variables

are

useful

to

identify

which

variables

mainly

contribute

to

the

variation

in

the

data

set

and

this

is

referred

to

as

the

loading

of

the

variables

on

a

given

axis

However

because

the

synthetic

and

original

variables

are

linearly

correlated

-LRB-

i.e.

they

are

not

independent

-RRB-

standard

tests

to

determine

the

statistical

significance

of

the

correlations

between

them

can

not

be

used

Principal

coordinate

analysis

-LRB-

PCoA

-RRB-

The

technique

is

more

rarely

used

by

microbial

ecologists

-LRB-

Table

-RRB-

despite

its

usefulness

at

reducing

and

representing

patterns

present

in

distance

matrices

displaying

dissimilarities

among

objects

-LRB-

Gower

-RRB-

Its

objectives

are

very

similar

to

those

of

PCA

in

that

it

uses

a

linear

-LRB-

Euclidean

-RRB-

mapping

of

the

distance

or

dissimilarities

between

objects

onto

the

ordination

space

-LRB-

i.e.

projection

in

a

Cartesian

space

-RRB-

and

the

algorithm

attempts

to

explain

most

of

the

variance

in

the

original

data

set

In

microbial

ecology

PCoA

has

been

used

for

instance

to

test

whether

virulence

profiles

-LRB-

i.e.

presence

or

absence

of

specific

genes

-RRB-

arising

from

pathogenic

strains

could

differentiate

either

healthy

or

contaminated

hosts

-LRB-

Chapman

et

al.

-RRB-

or

to

determine

whether

PCoA

axes

could

separate

groups

of

Staphylococcus

aureus

isolates

into

bovine

and

human

hosts

when

genetic

relationships

among

them

had

been

established

by

random

amplified

polymorphic

DNA-PCR

analysis

-LRB-

Reinoso

et

al.

-RRB-

As

opposed

to

PCA

PCoA

works

with

any

dissimilarity

measure

and

so

specific

association

coefficients

that

better

deal

with

the

problem

of

the

presence

of

many

double

zeros

in

data

sets

can

be

surmounted

Moreover

PCoA

does

not

provide

a

direct

link

between

the

components

and

the

original

variables

and

so

the

interpretation

of

variable

contribution

may

be

more

difficult

This

is

because

PCoA

components

instead

of

being

linear

combinations

of

the

original

variables

as

in

PCA

are

complex

functions

of

the

original

variables

depending

on

the

selected

dissimilarity

measure

Besides

the

non-Euclidean

nature

of

some

distance

measures

does

not

allow

for

a

full

representation

of

the

extracted

variation

into

a

Euclidean

ordination

space

In

that

case

the

non-Euclidean

variation

can

not

be

represented

and

the

percent

of

total

variance

can

not

be

computed

with

exactness

The

choice

of

the

dissimilarity

measure

is

thus

of

great

importance

and

subsequent

transformation

of

the

data

to

correct

for

negative

eigenvalues

is

sometimes

necessary

-LRB-

see

Legendre

Legendre

section

for

how

to

correct

for

such

negative

eigenvalues

-RRB-

Objects

are

represented

as

points

in

the

ordination

space

Eigenvalues

are

also

used

here

to

measure

how

much

variance

is

accounted

for

by

the

largest

synthetic

variables

on

each

PCoA

synthetic

axis

Although

there

is

no

direct

linear

relationship

between

the

components

and

the

original

variables

it

is

still

possible

to

correlate

object

scores

on

the

main

axis

-LRB-

or

axes

-RRB-

with

the

original

variables

to

assess

their

contribution

to

the

ordination

Correspondence

analysis

-LRB-

CA

-RRB-

A

basic

question

that

ecologists

may

want

to

address

when

facing

a

multidimensional

table

of

sites

-LRB-

or

samples

-RRB-

by

species

is

whether

certain

species

occur

at

specific

sites

as

a

measure

of

their

ecological

preferences

CA

has

generally

been

used

in

microbial

ecology

to

determine

whether

patterns

in

microbial

OTU

distribution

could

reflect

differentiation

in

community

composition

as

a

function

of

seasons

geographic

origin

or

habitat

structure

-LRB-

Olapade

et

al.

Edwards

et

al.

b

Kent

et

al.

-RRB-

The

overall

aim

of

the

method

is

to

compare

the

correspondence

between

samples

and

species

from

a

table

of

counted

data

-LRB-

or

any

dimensionally

homogenous

table

-RRB-

and

to

represent

it

in

a

reduced

ordination

space

-LRB-

Hill

-RRB-

Noticeably

instead

of

maximizing

the

amount

of

variance

explained

by

the

ordination

CA

maximizes

the

correspondence

between

species

scores

and

sample

scores

Several

algorithms

exist

and

the

most

commonly

described

one

is

reciprocal

averaging

which

consists

of

-LRB-

-RRB-

assigning

arbitrary

numbers

to

all

species

in

the

table

-LRB-

these

are

the

initial

species

scores

-RRB-

-LRB-

-RRB-

for

each

sample

a

sample

score

is

then

determined

as

a

weighted

average

of

all

species

scores

-LRB-

this

thus

takes

into

account

the

abundance

of

each

species

at

the

site

and

the

previously

determined

species

scores

-RRB-

-LRB-

-RRB-

for

each

species

a

new

species

score

is

then

calculated

as

the

weighted

average

of

all

the

sample

scores

-LRB-

-RRB-

both

species

scores

and

sample

scores

are

standardized

again

to

obtain

a

mean

of

zero

and

a

SD

of

one

and

-LRB-

-RRB-

steps

two

to

four

are

repeated

until

species

and

site

scores

converge

towards

stable

solutions

in

successive

iterations

-LRB-

Hill

-RRB-

The

overall

table

variance

-LRB-

inertia

-RRB-

based

on

χ2

distances

is

decomposed

into

successive

components

that

are

uncorrelated

to

each

other

as

in

the

PCA

or

PCoA

procedures

For

each

axis

the

overall

correspondence

between

species

scores

and

sample

scores

is

summarized

by

an

eigenvalue

and

the

latter

is

thus

equivalent

to

a

correlation

coefficient

between

species

scores

and

sample

scores

-LRB-

Gauch

-RRB-

The

technique

is

popular

among

ecologists

because

CA

is

particularly

recommended

when

species

display

unimodal

-LRB-

bell

shaped

or

Gaussian

-RRB-

relationships

with

environmental

gradients

-LRB-

ter

Braak

-RRB-

as

it

happens

when

a

species

favors

specific

values

of

a

given

environmental

variable

which

is

revealed

by

a

peak

of

abundance

or

presence

when

the

optimal

conditions

are

met

-LRB-

this

can

be

visualized

by

plotting

species

abundance

against

the

environmental

parameter

-RRB-

The

unimodal

model

that

supports

the

concept

of

ecological

niches

has

also

been

shown

to

be

of

the

right

order

of

complexity

for

the

ordination

of

most

ecological

data

-LRB-

ter

Braak

Prentice

-RRB-

Although

examples

of

unimodal

distributions

along

variables

or

environmental

gradients

exist

with

macroorganisms

-LRB-

ter

Braak

-RRB-

the

shape

of

the

distribution

of

the

abundance

of

microbial

species

along

environmental

parameters

or

gradients

has

not

been

extensively

investigated

-LRB-

but

see

Ramette

Tiedje

b

-RRB-

This

may

arise

from

the

fact

that

in

microbial

surveys

environmental

sampling

is

mostly

performed

blindly

in

relation

to

environmental

heterogeneity

and

the

abundance

of

target

species

is

generally

determined

without

systematically

analyzing

associated

environmental

parameters

Finally

another

important

feature

of

CA

for

microbial

ecologists

is

that

the

reciprocal

averaging

algorithm

disregards

species

double

absences

because

the

relationships

between

rows

and

columns

of

the

table

are

quantified

using

the

χ2

coefficient

that

excludes

double

absences

-LRB-

Legendre

Legendre

-RRB-

Both

samples

and

taxa

are

often

jointly

depicted

in

the

ordination

space

-LRB-

i.e.

joint

plot

Fig.

-RRB-

where

the

center

of

inertia

-LRB-

centroid

-RRB-

of

their

scores

corresponds

to

the

zero

for

all

axes

Depending

on

the

choice

of

the

scaling

type

either

the

ordination

of

rows

-LRB-

samples

-RRB-

or

the

columns

-LRB-

species

-RRB-

is

meaningful

and

can

be

interpreted

as

an

approximation

of

the

χ2

distances

between

samples

or

species

respectively

-LRB-

see

Table

for

more

details

about

interpretation

-RRB-

Sample

points

that

are

close

to

each

other

are

similar

with

regard

to

the

pattern

of

relative

frequencies

across

species

It

is

important

to

remember

that

in

such

joint

plots

either

distances

between

sample

points

or

distances

between

species

points

can

be

interpreted

but

not

the

distances

between

sample

and

species

points

Indeed

these

distances

are

not

simple

Euclidean

distances

computed

from

the

relative

row

or

column

frequencies

but

rather

they

are

weighted

distances

The

proximity

between

sample

and

species

points

in

the

plot

can

thus

be

understood

as

a

probability

of

species

occurrence

or

of

a

high

abundance

in

the

samples

in

the

vicinity

of

a

species

point

Fig.

Ordination

diagrams

in

two

dimensions

-LRB-

a

-RRB-

In

a

PCA

biplot

representation

samples

are

represented

by

dots

and

species

by

arrows

The

arrows

point

in

the

direction

of

maximal

variation

in

the

species

abundances

and

their

lengths

are

proportional

to

their

maximal

rate

of

change

Long

arrows

correspond

to

species

contributing

more

to

the

data

set

variation

Right-angle

projection

of

a

sample

dot

on

a

species

arrow

gives

approximate

species

abundance

in

the

sample

-LRB-

b

-RRB-

In

a

CA

joint

plot

representation

focusing

on

species

distance

both

samples

and

species

are

depicted

as

dots

Species

dots

correspond

to

the

center

of

gravity

-LRB-

inertia

-RRB-

of

the

samples

where

they

mostly

occur

Distances

between

sample

and

species

points

give

an

indication

of

the

probability

of

species

composition

in

samples

-LRB-

see

Table

for

more

details

about

diagram

interpretation

-RRB-

In

scaling

-LRB-

i.e.

focus

on

species

-RRB-

species

points

found

at

the

center

of

the

ordination

space

should

be

carefully

checked

with

the

raw

data

to

clarify

whether

the

species

ordination

really

corresponds

to

the

optimal

abundance

or

occurrence

of

the

species

or

whether

the

species

is

just

badly

represented

by

the

main

axes

as

it

is

the

case

when

other

axes

are

more

appropriate

to

represent

the

species

Rare

species

contribute

little

to

the

total

table

inertia

-LRB-

i.e.

they

only

play

a

minor

role

in

the

overall

table

variance

-RRB-

and

are

hence

positioned

at

the

edges

of

the

plot

next

to

the

site

-LRB-

s

-RRB-

where

they

occur

In

general

only

the

species

points

found

away

from

the

ordination

center

and

not

close

to

the

edges

of

the

ordination

have

more

chances

to

be

related

to

the

ordination

axes

i.e.

to

contribute

to

the

overall

variance

-LRB-

Legendre

Legendre

-RRB-

When

the

species

composition

of

the

sites

progressively

changes

along

the

environmental

gradient

sample

positions

may

appear

in

the

ordination

plot

as

nonlinear

configurations

called

arch

-LRB-

Gauch

-RRB-

-LRB-

or

horseshoe

in

the

case

of

PCA

-RRB-

which

may

impair

further

ecological

interpretation

In

CA

the

arch

effect

may

be

mathematically

produced

as

a

side-effect

of

the

CA

procedure

that

tries

to

obtain

axes

that

both

maximally

separate

species

and

that

are

uncorrelated

to

each

other

-LRB-

ter

Braak

-RRB-

when

the

first

axis

suffices

to

correctly

order

the

sites

and

species

a

second

axis

-LRB-

uncorrelated

with

the

former

-RRB-

can

be

obtained

by

folding

the

first

axis

in

the

middle

and

bringing

its

extremities

together

thus

resulting

in

an

arch

configuration

Further

axes

can

be

obtained

by

further

dividing

and

folding

the

first

axis

into

segments

-LRB-

Legendre

Legendre

-RRB-

To

remove

the

arch

effect

in

CA

a

mathematical

procedure

detrending

is

used

to

flatten

the

distribution

of

the

sites

along

the

first

CA

axis

without

changing

their

ordination

on

that

axis

The

approach

is

then

designated

as

detrended

correspondence

analysis

-LRB-

DCA

-RRB-

The

review

of

different

detrending

algorithms

such

as

using

segments

or

polynomials

goes

beyond

the

scope

of

this

review

but

more

information

can

be

obtained

in

-LRB-

ter

Braak

Prentice

Legendre

Legendre

-RRB-

Some

authors

have

also

argued

that

the

arch

effect

may

not

be

an

artifact

but

an

expected

feature

of

the

analysis

especially

when

species

turnover

is

high

along

environmental

gradients

-LRB-

James

McCulloch

-RRB-

In

that

case

if

the

samples

are

meaningfully

positioned

along

the

arch

the

ordination

should

be

accepted

as

a

valid

result

Nonmetric

multidimensional

scaling

-LRB-

NMDS

-RRB-

NMDS

is

generally

efficient

at

identifying

underlying

gradients

and

at

representing

relationships

based

on

various

types

of

distance

measures

Not

surprisingly

NMDS

has

found

an

increasing

number

of

applications

in

microbial

ecology

-LRB-

Table

-RRB-

The

technique

has

been

generally

applied

to

identify

patterns

among

multiple

samples

that

were

subjected

to

molecular

fingerprinting

techniques

For

instance

NMDS

was

used

to

analyze

and

to

compare

the

reproducibility

of

various

fingerprinting

techniques

such

as

ribosomal

internal

spacer

analysis

-LRB-

RISA

-RRB-

terminal

fragment

length

polymorphism

-LRB-

T-RFLP

-RRB-

and

denaturing

gradient

gel

electrophoresis

-LRB-

DGGE

-RRB-

between

different

laboratories

when

applied

to

samples

chosen

from

a

salinity

gradient

-LRB-

Casamayor

et

al.

-RRB-

NMDS

was

also

used

to

compare

diversity

patterns

of

microbial

communities

-LRB-

as

determined

by

length

heterogeneity-PCR

-RRB-

from

samples

undergoing

different

land

management

practices

-LRB-

Mills

et

al.

-RRB-

Another

example

is

the

analysis

of

the

bacterioplankton

communities

of

four

shallow

eutrophic

lakes

that

differed

in

nutrient

load

and

food

web

structure

using

DGGE

profiling

so

as

to

determine

the

specificity

of

community

signatures

in

each

lake

-LRB-

Van

der

Gucht

et

al.

-RRB-

The

NMDS

algorithm

ranks

distances

between

objects

and

uses

these

ranks

to

map

the

objects

nonlinearly

onto

a

simplified

two-dimensional

ordination

space

so

as

to

preserve

their

ranked

differences

and

not

the

original

distances

-LRB-

Shepard

-RRB-

The

procedure

works

as

follows

the

objects

are

first

placed

randomly

in

the

ordination

space

-LRB-

the

desired

number

of

dimensions

has

to

be

defined

a

priori

-RRB-

and

their

distances

in

this

initial

configuration

are

compared

by

monotonic

regression

with

the

distances

in

the

original

data

matrix

based

on

a

stress

function

-LRB-

values

between

and

-RRB-

The

latter

indicates

how

different

the

ranks

on

the

ordination

configuration

are

from

the

ranks

in

the

original

distance

matrix

Several

iterations

of

the

NMDS

procedure

are

generally

implemented

so

as

to

obtain

the

lowest

stress

value

possible

-LRB-

i.e.

the

best

goodness

of

fit

-RRB-

based

on

different

random

initial

positions

of

the

objects

in

the

ordination

space

For

sample-by-species

tables

simulations

have

shown

that

before

applying

NMDS

a

standardization

of

each

species

by

its

maximum

abundance

followed

by

the

computation

of

distances

between

samples

based

on

the

Steinhaus

or

Kulczinski

similarity

coefficients

yielded

informative

ordination

results

-LRB-

Legendre

Legendre

p.

-RRB-

In

NMDS

ordination

the

proximity

between

objects

corresponds

to

their

similarity

but

the

ordination

distances

do

not

correspond

to

the

original

distances

among

objects

Because

NMDS

preserves

the

order

of

objects

NMDS

ordination

axes

can

be

freely

rescaled

rotated

or

inverted

as

needed

for

a

better

visualization

or

interpretation

Because

of

the

iterative

procedure

NMDS

is

more

computer

intensive

than

eigenanalyses

such

as

PCoA

PCA

or

CA

However

constant

improvement

in

computing

power

makes

this

limitation

less

of

a

problem

for

small

to

medium-sized

matrices

Testing

for

significant

differences

between

groups

In

addition

to

representing

objects

in

an

ordination

plot

or

as

clusters

of

similar

objects

another

objective

may

be

to

test

whether

differences

between

groups

of

objects

-LRB-

rows

-RRB-

in

a

multivariate

table

are

significantly

different

based

on

the

set

of

their

attributes

-LRB-

columns

-RRB-

i.e.

to

test

whether

similarities

within

groups

are

higher

than

those

between

groups

Here

nonparametric

multivariate

anova

-LRB-

npmanova

-RRB-

and

analysis

of

similarities

-LRB-

anosim

-RRB-

which

are

commonly

found

in

standard

statistical

packages

are

briefly

reviewed

It

is

also

possible

to

use

canonical

analyses

-LRB-

Testing

for

significant

differences

between

groups

-RRB-

to

test

for

significant

differences

between

groups

of

objects

These

statistical

tests

however

must

not

be

used

to

assess

the

statistical

difference

among

groups

that

were

derived

from

a

previous

cluster

analysis

on

the

same

variables

because

under

those

conditions

the

two

approaches

would

not

be

independent

from

each

other

Indeed

the

groups

derived

from

cluster

analysis

-LRB-

which

are

themselves

made

to

fit

the

data

-RRB-

would

then

be

used

for

testing

the

hypothesis

that

there

is

no

difference

among

the

groups

This

hypothesis

would

then

not

be

independent

of

the

data

used

to

test

it

and

would

nearly

always

produce

significant

differences

between

the

groups

even

if

it

is

not

the

case

-LRB-

Legendre

Legendre

-RRB-

npmanova

The

method

can

be

used

to

test

for

significant

differences

between

the

means

of

two

or

more

groups

of

multivariate

quantitative

data

-LRB-

Anderson

-RRB-

The

hypothesis

of

equality

of

means

is

tested

based

on

Wilks

Λ

-LRB-

lambda

-RRB-

statistic

which

replaces

the

F-test

normally

used

in

univariate

anova

When

only

two

groups

are

compared

Hotelling

's

test

is

more

appropriate

The

latter

test

can

also

be

used

as

a

post

hoc

test

to

assess

the

significance

of

pairwise

comparisons

statistically

between

groups

following

an

overall

significant

Wilks

test

Significance

is

generally

computed

by

permutation

of

group

membership

with

several

thousand

replicates

alleviating

concerns

about

multinormality

of

the

data

Because

multiple

pairwise

comparisons

are

made

the

significance

level

of

the

pairwise

Hotelling

's

tests

needs

however

to

be

corrected

With

the

Bonferroni

correction

for

instance

the

P-value

usually

chosen

for

significant

differences

between

groups

-LRB-

i.e.

-RRB-

is

replaced

by

a

smaller

P-value

calculated

by

dividing

the

original

P-value

by

the

total

number

of

pairwise

comparisons

that

are

performed

For

instance

for

pairwise

comparisons

the

corrected

P-value

becomes

This

correction

is

often

judged

to

be

rather

conservative

as

it

leads

to

significance

for

fewer

pairwise

comparisons

-LRB-

Legendre

Legendre

-RRB-

anosim

This

nonparametric

procedure

tests

for

significant

difference

between

two

or

more

groups

based

on

any

distance

measure

-LRB-

Clarke

-RRB-

It

compares

the

ranks

of

distances

between

groups

with

ranks

of

distances

within

groups

The

means

of

those

two

types

of

ranks

are

compared

and

the

resulting

R

test

statistic

measures

whether

separation

of

community

structure

is

found

-LRB-

R

-RRB-

or

whether

no

separation

occurs

-LRB-

R

-RRB-

R

values

are

commonly

interpreted

as

well

separated

R

as

separated

but

overlapping

and

R

as

barely

separable

-LRB-

Clarke

Gorley

-RRB-

The

test

makes

fewer

assumptions

than

manova

because

it

is

based

on

the

ranks

of

distances

and

it

is

often

used

for

sample-by-species

tables

where

groups

of

samples

are

compared

All

groups

should

have

comparable

within-group

dispersion

to

avoid

finding

falsely

significant

results

-LRB-

Legendre

Legendre

-RRB-

Applications

in

microbial

ecology

include

testing

for

spatial

differences

temporal

changes

or

environmental

impacts

on

microbial

assemblages

For

instance

Kent

et

al.

-LRB-

-RRB-

determined

whether

bacterial

communities

from

the

same

lake

were

more

similar

in

composition

to

each

other

than

to

communities

in

different

lakes

The

bacterial

composition

and

diversity

of

samples

from

different

geographic

origins

habitats

and

avian

hosts

were

also

compared

using

anosim

based

on

a

length

heterogeneity

-LRB-

LH

-RRB-

PCR

-LRB-

Bisson

et

al.

-RRB-

Another

example

is

the

application

of

anosim

to

terminal

restriction

fragment

length

polymorphism

-LRB-

T-RFLP

-RRB-

generated

data

to

determine

the

impact

of

B

and

NaCl

on

soil

microbial

community

structure

in

the

wheat

rhizosphere

-LRB-

Nelson

Mele

-RRB-

Environmental

interpretation

Exploratory

analyses

may

reveal

the

existence

of

clusters

or

groups

of

objects

in

a

data

set

When

a

supplementary

table

or

matrix

of

environmental

variables

is

available

for

those

objects

it

is

then

possible

to

examine

whether

the

observed

patterns

are

related

to

environmental

gradients

Typical

objectives

may

be

for

instance

to

reveal

the

existence

of

a

relationship

between

community

structure

and

habitat

heterogeneity

between

community

structure

and

spatial

distance

or

to

identify

the

main

variables

affecting

bacterial

communities

when

a

large

set

of

environmental

variables

has

been

conjointly

collected

The

significance

of

the

relationships

between

species

patterns

and

environmental

variables

can

generally

be

assessed

by

permutation

techniques

such

as

Monte

Carlo

permutation

tests

which

infer

statistical

properties

from

the

data

themselves

The

order

of

data

-LRB-

generally

the

rows

of

one

matrix

-RRB-

is

permuted

and

the

relationships

between

the

observed

patterns

and

environmental

variables

can

be

assessed

for

randomness

This

approach

is

particularly

suitable

when

variables

do

not

follow

a

normal

distribution

-LRB-

which

is

often

the

case

with

environmental

or

ecological

data

-RRB-

as

generally

required

by

traditional

statistical

procedures

-LRB-

Legendre

Legendre

-RRB-

Indirect

gradient

analyses

Ordination

axes

or

clusters

can

be

interpreted

based

on

additional

environmental

variables

-LRB-

i.e.

variables

not

used

in

the

ordination

or

cluster

analysis

-RRB-

that

provide

ecological

knowledge

about

the

studied

sites

or

species

ecological

characteristics

When

using

exploratory

ordination

approaches

on

a

sample-by-species

table

samples

are

displayed

along

the

axes

of

main

variation

in

species

composition

These

axes

are

thus

constructed

without

reference

to

environmental

characteristics

but

they

can

be

hypothesized

to

represent

underlying

environmental

gradients

-LRB-

e.g.

environmental

parameters

spatial

or

temporal

variables

chemical

gradients

-RRB-

which

need

to

be

subsequently

identified

Such

an

approach

is

designated

as

indirect

because

synthetic

variables

-LRB-

i.e.

the

axes

-RRB-

are

first

constructed

and

thereafter

related

to

environmental

variation

For

instance

the

scores

of

the

objects

on

PCA

or

CA

main

components

-LRB-

axes

-RRB-

can

be

further

related

by

standard

statistical

procedures

-LRB-

e.g.

anova

regression

analysis

-RRB-

to

environmental

variables

Likewise

in

PCoA

or

NMDS

it

is

possible

to

statistically

compare

the

ranks

obtained

by

the

objects

on

each

axis

and

the

ranks

of

those

objects

on

additional

environmental

variables

using

Spearman

's

rank

correlation

coefficients

-LRB-

Legendre

Legendre

-RRB-

A

convenient

method

of

interpretation

is

to

represent

the

additional

environmental

variables

as

fitted

arrows

directly

on

the

ordination

diagram

These

variables

are

added

to

the

existing

ordination

by

linear

regression

of

their

values

onto

the

existing

ordination

axes

This

procedure

is

implemented

in

various

statistical

packages

-LRB-

e.g.

canoco

R

-RRB-

Hence

it

is

possible

to

assess

the

direction

and

magnitude

of

the

most

rapid

change

in

the

environmental

variables

and

to

determine

whether

they

correspond

to

the

observed

patterns

among

objects

-LRB-

Oksanen

-RRB-

In

cluster

analysis

the

magnitude

of

the

absolute

correlation

value

between

an

ordered

clustering

solution

and

environmental

variables

may

also

provide

clues

about

putative

environmental

causes

for

the

observed

discontinuities

in

the

data

set

Another

convenient

way

of

displaying

additional

information

to

help

interpret

the

ordination

is

to

use

site

symbols

whose

sizes

are

proportional

to

the

values

of

the

additional

variable

Hence

one

can

visually

assess

whether

the

ordination

of

objects

-LRB-

samples

sites

-RRB-

matches

specific

trends

in

the

additional

variable

This

strategy

was

for

instance

used

on

NMDS

ordination

plots

inferred

from

DGGE

profiles

on

which

the

values

of

five

additional

environmental

variables

were

individually

mapped

as

proportional

circles

in

order

to

identify

the

main

environmental

factors

related

to

the

bacterial

community

structure

in

four

freshwater

lakes

-LRB-

Van

der

Gucht

et

al.

-RRB-

Direct

gradient

analyses

-LRB-

constrained

analyses

-RRB-

In

constrained

-LRB-

canonical

-RRB-

ordination

analyses

only

the

variation

in

the

species

table

that

can

be

explained

by

the

environmental

variables

is

displayed

and

analyzed

and

not

all

the

variation

in

the

species

table

Gradients

are

supposed

to

be

known

and

represented

by

the

measured

variables

or

their

combinations

while

species

abundance

or

occurrence

is

considered

to

be

a

response

to

those

gradients

Constrained

ordinations

are

mostly

based

on

multivariate

linear

models

relating

principal

axes

to

the

observed

environmental

variables

and

the

different

techniques

depend

on

data

types

-LRB-

matrix

or

table

-RRB-

and

on

the

hypothesis

underlying

species

distribution

in

the

gradients

-LRB-

i.e.

linear

or

unimodal

-RRB-

Their

aim

is

to

find

the

best

mathematical

relationships

between

species

composition

and

the

measured

environmental

variables

and

to

assess

whether

statistically

such

a

relationship

could

have

been

produced

due

to

chance

alone

using

permutation

tests

The

resulting

ordination

diagrams

display

samples

species

and

environmental

variables

so

that

fitted

species

samples

and

species

environment

relationships

can

be

derived

as

easily

as

possible

from

angles

between

arrows

or

distances

between

points

and

arrows

-LRB-

Table

-RRB-

Redundancy

analysis

-LRB-

RDA

-RRB-

In

microbial

ecology

RDA

has

been

applied

for

instance

to

test

whether

the

occurrence

of

biocontrol

bacteria

with

specific

carbon

source

utilization

profiles

was

related

to

their

origin

from

different

root

samples

-LRB-

Folman

et

al.

-RRB-

to

determine

which

environmental

factors

were

the

most

significant

to

explain

variation

in

microbial

community

composition

in

undisturbed

native

prairies

and

cropped

agricultural

field

-LRB-

McKinley

et

al.

-RRB-

to

examine

the

effects

of

sampling

locations

-LRB-

longitude

latitude

altitude

-RRB-

on

genetic

diversity

of

plant

pathogenic

bacteria

-LRB-

Kolliker

et

al.

-RRB-

or

to

assess

the

influence

of

season

farm

management

and

soil

chemical

physical

and

biological

properties

on

nitrogen

fluxes

and

bacterial

community

structure

-LRB-

Cookson

et

al.

-RRB-

This

method

can

be

considered

as

an

extension

of

PCA

in

which

the

main

axes

-LRB-

components

-RRB-

are

constrained

to

be

linear

combinations

of

the

environmental

variables

-LRB-

Rao

-RRB-

Two

tables

are

then

necessary

one

for

the

species

data

-LRB-

dependent

variables

-RRB-

and

one

for

the

environmental

variables

-LRB-

independent

variables

-RRB-

Multiple

linear

regressions

are

used

to

explain

variation

between

independent

and

dependent

variables

and

these

calculations

are

performed

within

the

iterative

procedure

to

find

the

best

ordination

of

the

objects

The

interest

of

such

an

approach

is

to

represent

not

only

the

main

patterns

of

species

variation

as

much

as

they

can

be

explained

by

the

measured

environmental

variables

but

also

to

display

correlation

coefficients

between

each

species

and

each

environmental

variable

in

the

data

set

When

the

data

set

consists

of

a

matrix

of

distances

between

objects

distance-based

RDA

-LRB-

db-RDA

Legendre

Anderson

-RRB-

can

be

applied

to

determine

how

well

additional

environmental

parameters

can

explain

the

variation

among

objects

in

the

matrix

The

technique

first

applies

a

PCoA

on

the

distance

matrix

to

convert

it

back

to

a

rectangular

table

containing

rows

of

objects

by

columns

of

PCoA

coordinates

Those

new

uncorrelated

coordinates

thus

correspond

to

synthetic

species

variables

that

are

then

related

to

additional

environmental

parameters

using

a

classical

RDA

For

instance

db-RDA

was

successfully

used

to

determine

how

the

variation

in

matrices

of

genomic

distances

among

environmental

strains

could

be

explained

by

factors

such

as

soil

parameters

host

plant

species

and

spatial

scale

each

factor

being

taken

alone

or

in

combination

-LRB-

Ramette

Tiedje

-RRB-

Most

software

outputs

provide

the

total

variation

in

species

composition

as

explained

by

the

environmental

axes

the

cumulative

percentage

of

variance

of

the

species

environment

relationship

and

the

overall

statistical

significance

of

the

relationships

between

the

species

and

environmental

tables

RDA

can

be

represented

by

a

triplot

of

samples

-LRB-

dots

-RRB-

species

-LRB-

arrows

-RRB-

and

environmental

variables

-LRB-

arrows

for

quantitative

variables

and

dots

for

each

level

of

qualitative

or

nominal

variables

-RRB-

or

by

any

combinations

thereof

-LRB-

i.e.

biplots

-RRB-

-LRB-

ter

Braak

-RRB-

Depending

on

the

scaling

chosen

i.e.

whether

the

analysis

mainly

focuses

on

intersample

relationships

or

interspecies

correlations

the

interpretation

of

the

relationships

between

samples

species

and

environmental

variables

differs

-LRB-

Table

-RRB-

Canonical

correspondence

analysis

-LRB-

CCA

-RRB-

The

approach

is

very

similar

to

that

of

RDA

except

that

CCA

is

based

on

unimodal

species

environment

relationships

whereas

RDA

is

based

on

linear

models

-LRB-

ter

Braak

-RRB-

CCA

can

be

considered

as

the

constrained

form

of

CA

in

which

the

axes

are

linear

combinations

of

the

environmental

variables

CCA

uses

the

unimodal

model

to

model

species

response

to

the

environmental

variation

as

a

mathematical

simplification

to

enable

the

estimation

of

a

large

number

of

parameters

and

the

identification

of

a

small

number

of

ordination

axes

This

species

model

seems

however

to

be

robust

even

when

some

species

display

bimodal

responses

unequal

ranges

or

unequal

maxima

along

environmental

gradients

and

the

technique

is

thus

considered

to

be

the

method

of

choice

by

many

ecologists

-LRB-

ter

Braak

Smilauer

-RRB-

It

is

therefore

particularly

adapted

for

the

environmental

interpretation

of

tables

of

abundance

and

occurrence

of

species

and

accommodates

well

the

absence

of

species

at

certain

sites

in

the

data

set

CCA

is

sensitive

to

rare

species

that

occur

in

species-poor

samples

and

down-weighting

of

such

species

help

reduce

the

problem

-LRB-

Legendre

Legendre

-RRB-

Software

outputs

are

very

similar

to

those

of

RDA

and

as

for

RDA

triplot

and

biplot

representations

and

interpretation

depend

on

the

choice

of

the

scaling

type

-LRB-

Table

-RRB-

The

same

interpretation

of

the

relationships

between

sample

and

species

points

is

found

in

CA

and

CCA

Right-angle

projection

of

these

points

on

the

environmental

arrows

leads

to

the

correct

approximation

of

the

ranking

of

the

points

along

environmental

variables

CCA

has

been

used

in

an

increasing

number

of

publications

dealing

with

microbial

assemblages

in

marine

and

soil

ecosystems

Typical

questions

that

are

addressed

concern

the

identification

of

environmental

factors

that

influence

the

diversity

of

bacterial

assemblages

among

large

sets

of

candidate

environmental

parameters

measured

for

the

same

samples

when

the

diversity

is

determined

by

culture-independent

genetic

fingerprinting

techniques

such

as

automated

ribosomal

intergenic

spacer

analysis

-LRB-

ARISA

-RRB-

-LRB-

Yannarell

Triplett

-RRB-

DGGE

-LRB-

Salles

et

al.

Sapp

et

al.

-RRB-

or

T-RFLP

-LRB-

Córdova-Kreylos

et

al.

Klaus

et

al.

-RRB-

Another

interest

in

the

technique

comes

from

the

possibility

of

determining

the

specific

species

or

OTUs

that

respond

to

particular

environmental

variables

and

as

such

that

can

be

identified

as

candidate

indicator

species

Those

species

can

then

be

subjected

to

further

experiments

so

as

to

confirm

their

status

of

indicator

species

For

instance

the

relationships

as

determined

by

CCA

between

bacterial

community

composition

and

environmental

variables

for

lakes

in

Wisconsin

revealed

that

patterns

in

bacterial

communities

were

best

explained

by

regional

and

landscape-level

factors

as

well

as

by

specific

seasons

pH

and

water

clarity

-LRB-

Yannarell

Triplett

-RRB-

CCA

was

also

successfully

used

to

demonstrate

that

former

land

use

management

affected

the

composition

of

the

targeted

soil

microbial

community

-LRB-

Burkholderia

-RRB-

to

a

larger

extent

than

did

plant

species

-LRB-

Salles

et

al.

-RRB-

Another

interesting

example

in

the

marine

ecosystem

is

the

study

of

the

interactions

between

various

abiotic

parameters

and

phytoplankton

community

data

-LRB-

biotic

parameter

-RRB-

to

explain

bacterioplankton

dynamics

in

the

North

Sea

and

the

subsequent

identification

of

the

bacterial

phylotypes

responding

more

specifically

to

the

factors

-LRB-

Sapp

et

al.

-RRB-

Another

example

of

using

CCA

to

identify

some

microbial

communities

as

pollution

indicators

can

be

found

in

-LRB-

Córdova-Kreylos

et

al.

-RRB-

Partial

ordination

variation

partitioning

When

the

effects

of

a

particular

environmental

variable

need

to

be

tested

after

elimination

of

possible

effects

due

to

other

-LRB-

environmental

-RRB-

variables

partial

ordination

may

be

used

-LRB-

e.g.

partial

CCA

partial

RDA

-RRB-

Such

an

approach

is

also

referred

to

as

partialling

out

or

controlling

for

the

effects

of

specific

variables

which

are

specified

as

covariables

in

the

constrained

analysis

For

instance

in

a

study

dealing

with

the

effects

of

environmental

and

pollutant

variables

on

microbial

communities

Córdova-Kreylos

et

al.

-LRB-

-RRB-

observed

that

variation

in

microbial

communities

was

more

due

to

spatial

variation

than

to

pollutants

The

use

of

partial

CCA

to

account

for

spatial

variation

in

the

biological

data

set

revealed

that

metals

had

a

greater

effect

on

microbial

community

composition

than

organic

pollutants

This

idea

of

controlling

for

the

effects

of

specific

variables

can

be

extended

to

evaluate

the

effects

of

all

the

different

sets

-LRB-

factors

-RRB-

of

environmental

variables

present

in

a

study

so

as

to

determine

the

relative

contribution

-LRB-

amount

of

variation

explained

-RRB-

and

significance

of

each

variable

set

on

the

total

biological

variance

The

so-called

variation

partitioning

procedure

-LRB-

Borcard

et

al.

-RRB-

partitions

the

total

variance

of

the

species

table

into

the

respective

contribution

of

each

set

of

environmental

variables

and

into

their

covariations

using

both

standard

and

partial

constrained

ordinations

-LRB-

Fig.

-RRB-

Two

methods

have

traditionally

been

used

to

partition

the

variation

of

community

composition

data

i.e.

canonical

partitioning

and

regression

on

distance

matrices

based

on

Mantel

tests

-LRB-

Legendre

Legendre

-RRB-

The

canonical

approach

has

been

shown

to

be

more

appropriate

to

partition

the

β

diversity

correctly

among

sites

and

to

test

hypotheses

about

the

origin

and

maintenance

of

its

variation

-LRB-

Legendre

et

al.

-RRB-

Fig.

Partitioning

biological

variation

into

the

effects

of

two

factors

The

large

rectangle

represents

the

total

variation

in

the

biological

data

table

which

is

partitioned

among

two

sets

of

explanatory

variables

-LRB-

a

b

-RRB-

Fraction

shows

the

unexplained

part

of

the

biological

variation

Fractions

and

are

obtained

by

partial

constrained

ordination

or

partial

regression

and

can

be

tested

for

significance

For

instance

fraction

corresponds

to

the

amount

of

biological

variation

that

can

be

exclusively

explained

by

-LRB-

a

-RRB-

effects

when

-LRB-

b

-RRB-

effects

are

taken

into

consideration

-LRB-

i.e.

when

b

is

considered

as

a

covariable

-RRB-

Fraction

-LSB-

i.e.

variation

indifferently

attributed

to

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

or

a

covariation

of

-LRB-

a

-RRB-

and

-LRB-

b

-RRB-

-RSB-

is

obtained

by

subtracting

fractions

and

from

the

total

explained

variance

and

can

not

be

tested

for

statistical

significance

Applications

of

variation

partitioning

in

microbial

ecology

include

for

instance

the

study

by

Ramette

Tiedje

-LRB-

-RRB-

which

applied

the

technique

in

the

context

of

RDA

to

disentangle

the

effects

of

space

environmental

soil

parameters

and

plant

species

on

Burkholderia

community

abundance

and

diversity

By

quantifying

the

amount

of

biological

variation

that

is

left

unexplained

when

all

environmental

variables

had

been

considered

the

study

suggested

that

much

less

of

the

biological

variation

could

be

predicted

at

the

intraspecific

level

compared

with

higher

taxonomic

levels

Another

interesting

example

is

the

study

of

seasonal

changes

in

bacterial

community

composition

in

shallow

eutrophic

lakes

in

which

top-down

regulation

-LRB-

grazers

-RRB-

of

bacterial

community

composition

was

examined

after

accounting

for

bottom-up

regulation

-LRB-

resources

-RRB-

-LRB-

Muylaert

et

al.

-RRB-

Linear

discriminant

analysis

-LRB-

LDA

-RRB-

When

groups

or

clusters

of

objects

have

been

obtained

by

exploratory

analyses

for

instance

LDA

can

be

used

to

identify

linear

combinations

of

additional

environmental

variables

that

best

discriminate

those

groups

In

that

respect

LDA

can

be

seen

as

an

extension

of

manova

for

two

or

more

groups

in

which

environmental

variables

that

specifically

explain

the

grouping

of

objects

are

identified

Another

application

consists

of

assigning

new

objects

to

previously

defined

groups

for

prediction

or

classification

purposes

based

on

the

calculated

discriminant

function

For

instance

Fuhrman

et

al.

-LRB-

-RRB-

used

the

technique

to

evidence

the

existence

of

repeatable

temporal

patterns

in

the

community

composition

of

marine

bacterioplankton

over

years

The

technique

is

mostly

recommended

for

multinormal

data

for

which

attribute

data

are

linearly

related

and

for

which

variances

and

covariances

of

the

variables

are

good

summary

statistics

A

visual

representation

of

LDA

can

be

performed

and

in

the

resulting

ordination

the

axes

are

then

the

discriminant

functions

The

distances

between

objects

which

correspond

to

Mahalanobis

distances

that

take

into

account

the

correlations

among

descriptors

-LRB-

Mahalanobis

-RRB-

are

independent

of

the

scale

of

measurement

of

the

various

descriptors

and

are

mostly

used

to

compare

groups

of

sites

or

objects

with

each

other

-LRB-

Legendre

Legendre

-RRB-

Selection

of

variables

in

regression

models

In

the

previous

constrained

methods

where

linear

combinations

of

environmental

-LRB-

explanatory

-RRB-

variables

are

used

the

inclusion

of

too

many

explanatory

variables

to

describe

species

distribution

may

lead

to

difficult

ecological

interpretations

and

to

lower

predictability

of

the

models

due

to

intercorrelations

among

the

explanatory

variables

-LRB-

i.e.

multicollinearity

-RRB-

Multicollinearity

has

the

effects

of

inflating

the

variance

of

the

regression

coefficients

in

the

models

leading

to

reduced

precision

in

the

prediction

of

the

response

variables

-LRB-

Legendre

Legendre

-RRB-

In

order

to

only

include

in

the

model

the

environmental

variables

that

mostly

and

significantly

contribute

to

the

variation

of

the

species

table

automatic

selection

procedures

-LRB-

forward

selection

backward

elimination

or

stepwise

selection

-RRB-

are

often

used

The

selection

depends

on

whether

the

partial

correlation

coefficients

of

the

variables

fall

below

a

given

significance

level

the

latter

being

generally

assessed

by

Monte

Carlo

permutation

tests

In

forward

selection

the

construction

of

the

regression

model

starts

with

the

variable

that

explains

the

most

variation

in

the

dependent

variables

-LRB-

generally

the

species

table

-RRB-

What

remains

of

the

biological

variation

to

explain

after

fitting

the

first

environmental

variable

-LRB-

i.e.

of

the

residual

variation

-RRB-

is

then

used

to

choose

the

second

environmental

variable

The

process

of

selection

goes

on

until

no

more

variables

significantly

explain

the

residual

variation

In

backward

elimination

the

construction

of

the

regression

model

starts

with

all

environmental

variables

and

the

least

significant

ones

are

excluded

from

the

model

one

at

a

time

until

a

group

of

only

significant

variables

is

obtained

To

take

advantage

of

the

two

approaches

stepwise

regression

mixes

forward

selection

with

backward

elimination

by

performing

a

forward

selection

but

excluding

the

variables

that

no

longer

become

significant

after

the

introduction

of

new

variables

into

the

regression

model

Despite

the

clear

advantages

of

these

variable

selection

strategies

most

authors

still

caution

that

researchers

should

not

blindly

rely

on

automatic

selection

procedures

to

choose

the

relevant

environmental

variables

in

regression

models

because

ecologically

irrelevant

models

may

also

be

obtained

or

other

variable

combinations

could

also

yield

better

models

to

explain

species

variation

-LRB-

Legendre

Legendre

-RRB-

Noticeably

the

three

selection

strategies

do

not

necessarily

yield

the

same

set

of

significant

environmental

variables

because

they

may

be

seen

as

heuristic

methods

to

identify

a

significant

model

when

all

possible

combinations

of

significant

models

are

not

possible

to

evaluate

computationally

Another

approach

is

thus

to

combine

variables

into

biologically

or

environmentally

meaningful

sets

instead

of

relying

on

automatic

selection

procedures

and

then

to

examine

all

possible

regression

models

based

on

the

reduced

number

of

variable

sets

-LRB-

James

McCulloch

-RRB-

For

instance

before

applying

variation

partitioning

to

different

groups

of

variables

representing

spatial

scales

-LRB-

variables

-RRB-

host

species

-LRB-

four

variables

-RRB-

and

soil

parameters

-LRB-

variables

-RRB-

Ramette

Tiedje

-LRB-

-RRB-

applied

forward

selection

within

each

group

to

determine

the

variables

significantly

explaining

the

variation

of

microbial

diversity

and

abundance

at

different

taxonomic

levels

Mantel

test

This

test

is

appropriate

to

compare

two

matrices

that

were

calculated

for

the

same

objects

but

that

are

based

on

two

independent

data

sets

-LRB-

e.g.

a

species

dissimilarity

matrix

and

an

environmental

dissimilarity

matrix

for

the

same

samples

-RRB-

-LRB-

Mantel

-RRB-

It

calculates

the

correlation

coefficient

between

corresponding

positions

in

the

two

matrices

and

assesses

its

significance

based

on

permutations

of

the

objects

in

one

of

the

matrices

In

microbial

ecology

the

Mantel

test

has

become

popular

especially

for

testing

the

relationships

between

molecular

and

geographic

distance

matrixes

for

a

same

set

of

organisms

or

to

relate

community

diversity

to

environmental

heterogeneity

-LRB-

e.g.

Parker

Spoerke

Cho

Tiedje

Horner-Devine

et

al.

Scortichini

et

al.

-RRB-

Another

interesting

application

called

a

goodness-of-fit

Mantel

test

corresponds

to

the

case

where

one

matrix

is

recoded

to

represent

ecological

hypotheses

to

be

tested

on

the

other

matrix

-LRB-

Legendre

Legendre

-RRB-

For

instance

if

a

matrix

of

molecular

data

is

available

for

a

set

of

strains

and

their

habitat

of

origin

is

known

it

is

possible

to

determine

whether

the

genetic

distances

are

related

to

habitat

type

using

the

-LRB-

goodness-of-fit

-RRB-

Mantel

test

The

matrix

representing

the

ecological

hypotheses

should

then

consist

of

a

series

of

and

for

isolates

found

in

the

same

or

different

habitats

respectively

The

Mantel

test

can

thus

determine

whether

the

posited

habitat

distribution

can

significantly

explain

the

structure

of

the

molecular

matrix

This

test

can

not

be

used

however

to

test

a

hypothesis

matrix

that

would

be

based

on

the

results

of

a

cluster

analysis

for

instance

Indeed

as

indicated

in

Testing

for

significant

differences

between

groups

there

would

be

a

lack

of

independence

between

the

hypothesis

being

tested

and

the

data

used

to

test

the

hypothesis

Note

that

the

Mantel

test

is

also

used

to

compute

Mantel

correlograms

which

are

often

found

in

biogeographical

studies

-LRB-

e.g.

Mantel

correlograms

are

usually

used

to

detect

spatial

structure

in

species

assemblages

based

on

grouping

of

the

response

data

into

specific

spatial

distance

classes

-RRB-

Mantel

tests

are

then

applied

to

each

group

in

order

to

detect

significant

correlations

at

a

given

scale

i.e.

the

scales

at

which

the

data

are

autocorrelated

-LRB-

Legendre

Legendre

-RRB-

Practical

considerations

Choice

of

an

ordination

method

-LRB-

Fig.

-RRB-

Linear

methods

such

as

multiple

regression

LDA

PCA

or

RDA

are

generally

meant

to

be

applied

to

continuous

data

Their

use

is

thus

sometimes

limited

in

Ecology

where

species

generally

display

nonlinear

nonmonotone

responses

to

environmental

variables

-LRB-

ter

Braak

Prentice

Legendre

Legendre

-RRB-

Different

approaches

can

be

undertaken

to

choose

the

most

appropriate

ecological

model

Plot

of

species

abundances

along

ordination

axes

or

explanatory

variables

-LRB-

also

called

coenocline

-RRB-

may

help

visualize

whether

species

responses

are

linear

or

unimodal

-LRB-

ter

Braak

Smilauer

-RRB-

Besides

the

choice

of

linear

-LRB-

PCA

RDA

-RRB-

or

unimodal

-LRB-

CA

CCA

-RRB-

species

response

models

can

be

made

on

the

basis

of

whether

the

underlying

gradient

length

is

short

or

long

respectively

Gradient

length

as

measured

in

SD

units

along

the

first

ordination

axis

can

be

estimated

by

DCA

for

unconstrained

ordination

and

by

detrended

CCA

-LRB-

DCCA

-RRB-

for

constrained

ordination

in

e.g.

the

software

canoco

-LRB-

ter

Braak

Smilauer

-RRB-

It

is

recommended

to

use

linear

methods

when

the

gradient

length

is

SD

unimodal

methods

when

it

is

SD

and

any

method

for

intermediate

gradient

lengths

-LRB-

ter

Braak

Smilauer

-RRB-

Fig.

Relationships

between

numerical

methods

Exploratory

tools

such

as

PCA

CA

PCoA

NMDS

or

cluster

analysis

can

be

applied

to

a

sample-by-species

table

to

extract

the

main

patterns

of

variation

to

identify

groups

or

clusters

of

samples

or

specific

species

interactions

Sample

scores

on

the

main

axes

of

variation

can

be

related

to

variation

in

environmental

variables

using

indirect

gradient

analyses

When

a

constrained

analysis

is

desired

-LRB-

i.e.

direct

gradient

analysis

-RRB-

RDA

db-RDA

CCA

or

linear

discriminant

analysis

can

be

used

as

extensions

of

the

unconstrained

methods

Mantel

tests

are

appropriate

to

test

the

significance

of

the

correlation

between

two

distance

matrices

-LRB-

e.g.

one

based

on

species

data

and

the

other

on

environmental

variables

-RRB-

Raw

data

may

be

transformed

normalised

or

standardised

as

appropriate

before

analysis

Data

type

is

also

another

important

criterion

To

represent

absolute

abundance

values

linear-based

methods

-LRB-

PCA

RDA

-RRB-

which

produce

weighted

summations

are

appropriate

whereas

unimodal

techniques

-LRB-

CCA

CA

-RRB-

are

rather

used

to

model

relative

abundances

-LRB-

because

species

scores

are

weighted

averages

of

the

samples

scores

and

vice

versa

-RRB-

i.e.

they

model

the

dissimilarities

between

samples

-LRB-

β

diversity

-RRB-

They

also

accommodate

well

the

presence

of

many

zeros

in

the

species

table

in

contrast

to

linear-based

methods

for

which

double

zeros

lead

to

inadequate

estimates

of

sample

distances

Cluster

analysis

is

the

method

of

choice

when

relationships

between

objects

are

expected

to

be

discontinuous

and

where

defined

categories

or

groups

of

objects

are

expected

On

the

contrary

ordination

would

be

more

useful

when

the

variation

between

objects

is

posited

to

be

continuous

Although

NMDS

is

more

computer

intensive

than

PCoA

it

is

generally

better

at

compressing

the

distance

relationships

among

objects

into

a

few

dimensions

This

is

because

NMDS

can

always

lead

to

a

Euclidean

representation

even

for

non-Euclidean

embeddable

distances

-LRB-

Legendre

Legendre

-RRB-

NMDS

and

PCoA

can

be

compared

using

Shepard

diagrams

to

decide

which

technique

better

represents

the

original

distances

If

one

assumes

that

species

do

not

have

a

linear

response

to

environmental

gradients

NMDS

is

more

appropriate

than

PCA

CA

may

also

be

an

alternative

to

PCA

when

many

zeros

populate

the

data

set

and

one

strong

gradient

is

present

With

long

ecological

gradients

however

CA

may

produce

the

arch

effect

that

can

be

corrected

for

using

DCA

In

terms

of

the

underlying

species

model

the

main

difference

between

DCA

and

NMDS

is

that

the

former

is

based

on

a

specific

model

of

species

distributions

-LRB-

unimodal

model

-RRB-

while

NMDS

is

not

Thus

DCA

may

be

favored

by

ecologists

who

assume

that

the

niche

theory

better

fits

their

data

set

while

NMDS

may

be

a

method

of

choice

if

species

composition

is

determined

by

factors

other

than

position

along

a

gradient

-LRB-

for

instance

if

the

habitat

is

known

to

be

fragmented

-RRB-

In

constrained

and

unconstrained

ordinations

all

species

are

posited

to

react

to

different

extents

to

the

same

composite

gradients

of

environmental

variables

whereas

in

a

multiple

regression

approach

a

different

gradient

could

be

modeled

for

each

species

separately

Because

most

species

do

not

respond

linearly

to

environmental

gradients

fitting

nonlinear

models

to

individual

species

may

be

difficult

especially

when

dealing

with

a

huge

data

set

Constrained

ordinations

thus

provide

a

good

summary

of

species

environment

relationships

and

can

be

very

successful

in

ecological

data

analysis

-LRB-

ter

Braak

Prentice

-RRB-

It

is

also

useful

to

note

that

RDA

is

very

similar

to

manova

but

in

contrast

to

the

latter

RDA

allows

the

consideration

of

any

number

of

species

-LRB-

columns

-RRB-

-LRB-

Legendre

Legendre

-RRB-

Constrained

and

unconstrained

-LRB-

exploratory

-RRB-

methods

should

be

used

in

parallel

-LRB-

Fig.

-RRB-

because

with

the

former

only

the

biological

variation

that

can

be

explained

by

the

available

environmental

variables

is

represented

on

the

main

axes

whereas

with

unconstrained

methods

the

highest

amount

of

variance

is

extracted

from

the

biological

data

alone

and

represented

on

a

few

axes

If

the

constrained

and

unconstrained

approaches

yield

the

same

ordination

of

the

samples

-LRB-

objects

-RRB-

it

thus

means

that

the

measured

environmental

variables

explain

most

of

the

biological

variation

In

order

to

compare

the

results

of

different

ordinations

a

useful

technique

is

Procrustes

analysis

-LRB-

Gower

-RRB-

which

estimates

the

concordance

of

scores

in

two

ordinations

after

rotating

translating

and

dilating

them

in

order

to

obtain

the

best

fit

A

permutation

procedure

can

also

be

used

to

test

for

the

significance

of

the

concordance

between

ordinations

or

matrices

-LRB-

Peres-Neto

Jackson

-RRB-

Cluster

analysis

and

ordination

techniques

can

be

combined

to

provide

powerful

visualization

tools

For

instance

hierarchical

clustering

can

help

obtain

a

better

interpretation

of

ordination

diagrams

-LRB-

Fig.

-RRB-

Because

ordination

diagrams

represent

most

of

the

data

set

variation

into

a

dimensionally

reduced

space

some

relationships

among

objects

can

be

distorted

because

only

a

few

projection

axes

are

considered

The

addition

of

linkage

results

obtained

from

cluster

analysis

may

help

identify

objects

belonging

to

the

same

clusters

even

if

their

relative

position

in

the

ordination

diagram

is

not

ideal

-LRB-

Legendre

Legendre

-RRB-

Fig.

Combination

of

ordination

and

cluster

analysis

On

a

same

distance

matrix

NMDS

or

PCOA

can

be

applied

to

represent

the

major

axes

of

variation

among

objects

in

a

two-dimensional

space

The

superimposition

of

the

results

of

cluster

analysis

-LRB-

primary

connections

-RRB-

onto

the

ordination

diagram

can

help

identify

the

structure

in

the

data

set

as

discontinuities

-LRB-

clusters

-RRB-

into

a

continuous

space

-LRB-

ordination

-RRB-

Adapted

from

Legendre

Legendre

-LRB-

-RRB-

Ordination

and

diversity

indices

The

measurement

of

diversity

is

generally

performed

using

indices

such

as

the

Shannon

or

Simpson

indices

The

latter

are

often

applied

to

measure

different

components

of

the

diversity

such

as

α

β

and

γ

diversity

corresponding

to

diversity

within

a

particular

site

or

ecosystem

to

change

in

species

composition

from

site

to

site

-LRB-

i.e.

species

turnover

-RRB-

and

to

the

diversity

at

the

landscape

scale

respectively

-LRB-

Whittaker

-RRB-

The

ordination

approach

sounds

similar

in

that

variation

among

samples

is

compared

based

on

their

within-sample

composition

in

species

assemblages

and

so

some

of

the

α

and

β

diversity

should

be

depicted

on

ordination

diagrams

Because

diversity

indices

pool

the

multispecies

information

into

a

single

value

for

each

observation

before

comparing

them

it

is

not

surprising

that

complex

diversity

patterns

may

not

be

identified

sometimes

For

instance

Hartman

Widmer

-LRB-

-RRB-

did

not

find

significant

changes

in

soil

bacterial

communities

submitted

to

various

soil

managements

when

using

diversity

indices

while

community

structures

were

shown

to

have

changed

using

community

fingerprinting

analysis

To

obtain

a

consistency

between

ordination

techniques

and

diversity

index

measurements

two

numerical

strategies

have

been

proposed

for

species

occurrence

data

the

CA-species

richness

strategy

adapted

for

data

set

rich

in

rare

species

and

the

Nonsymmetric

CA

Simpson

strategy

which

is

more

appropriate

for

tables

dominated

by

abundant

species

-LRB-

Pelissier

et

al.

-RRB-

These

strategies

attribute

specific

weights

to

the

species

data

so

that

simple

or

constrained

ordinations

of

the

new

species

table

represent

the

total

inertia

as

α

and

β

diversity

and

would

thus

be

consistent

with

the

measures

obtained

by

common

diversity

indices

Misconceptions

about

multivariate

analyses

It

is

essential

to

reiterate

that

multivariate

statistical

procedures

may

suggest

causes

or

factors

but

investigators

should

bear

in

mind

that

the

synthetic

variables

axes

or

clusters

derived

do

not

necessarily

correspond

to

biological

or

ecological

entities

in

nature

-LRB-

James

McCulloch

-RRB-

One

should

thus

not

overinterpret

the

data

by

relying

on

unjustified

causality

especially

in

the

absence

of

real

experimentation

In

theory

it

would

be

necessary

to

validate

the

inferences

and

models

made

about

pattern

formation

and

putative

causes

by

analyzing

new

data

but

this

is

rarely

performed

in

practice

Moreover

whether

the

originally

collected

data

are

typical

of

the

situation

to

be

described

is

most

of

the

time

not

even

questioned

Another

common

misconception

is

that

multivariate

analyses

alone

can

sort

out

all

solutions

of

complex

multivariate

studies

Although

exploratory

analyses

may

help

reveal

interesting

patterns

in

data

sets

the

interpretation

and

explanation

of

the

observations

ultimately

rely

on

the

researcher

's

hypotheses

and

previous

knowledge

of

the

ecological

situation

Microbial

ecologists

themselves

need

to

formulate

ecologically

sound

hypotheses

and

test

them

Conclusions

Exciting

questions

in

Ecology

typically

consist

of

determining

whether

community

patterns

are

structured

across

space

or

time

of

explaining

how

those

patterns

can

be

related

to

environmental

heterogeneity

and

of

quantifying

how

much

still

remains

unexplained

when

all

significant

measured

variables

have

been

considered

Such

questions

can

now

start

to

be

addressed

in

microbial

ecology

because

numerical

tools

may

help

explore

and

test

such

ecological

hypotheses

These

are

indeed

exciting

times

because

even

larger

and

more

complex

databases

are

being

created

and

in

parallel

computing

power

gradually

becomes

less

of

an

issue

If

microbial

ecologists

want

to

test

numerical

methods

develop

new

ecological

theories

or

validate

existing

ones

for

the

microbial

case

access

to

diversity

data

and

above

all

to

the

relevant

associated

environmental

parameters

becomes

a

central

issue

It

would

thus

be

of

great

interest

to

make

such

complex

data

sets

publicly

available

such

as

microbial

ecological

databases

so

that

microbial

diversity

can

be

studied

in

its

environmental

context

This

would

indeed

be

a

step

toward

making

microbial

ecology

a

central

discipline

in

Ecology

ordination multivariate statistics gradient