Measures of the Amount of Ecologic Association Between Species

Measures of the Amount of Ecologic Association Between Species

Measures of the Amount of Ecologic Association Between Species
Author(s): Lee R. Dice
Reviewed work(s):
Source: Ecology, Vol. 26, No. 3 (Jul., 1945), pp. 297-302
Published by: Ecological Society of America
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MEASURES
OF THE AMOUNT OF ECOLOGIC
BETWEEN SPECIES
LEE
ASSOCIATION
R. DICE
Universityof Michigan
In many ecologic studies thereis need
to expressin a quantitative mannerthe
species are
degreeto whichtwo different
associated in nature. Nevertheless,no
such measurehas up to the presenttime
come into generaluse by ecologists.
Of themeasuresofassociationbetween
species previouslyproposed,the "coefficient of association" of Forbes ('07) is
useful,thoughno estimateof its statistical reliabilityhas in the past been emofassociation"
ployed. The "coefficient
of Michael ('20) is too complicated for
generaluse. The "association value" of
Hacker ('21) varies with the total numberofassociationsbetweenall the species
in a given locality and thereforeis impracticable formost fieldstudies. Van
Deventer ('36) has calculated the percentage of association between certain
species of birds,but his briefstatement
gives no details of the methodof computation. Recently Edmondson ('44) applied the chi-squaretest as a measure of
the deviationfromchance expectationof
the occurrencestogether of two given
species of rotifersin a seriesof lakes, but
he did not use any measureoftheamount
of association betweenthe species.
The coefficientof association developed by Forbes ('07) considersthe numberofassociated occurrencesoftwo given
speciesin a seriesofrandomsamplescompared to the numberof such joint occurrencesexpected by chance. In orderto
we must therecalculate this coefficient
fore count the number of samples in
whicheach of the two species occursand
also thenumberofsamples in whichboth
species occur together. Let us indicate
by a the numberofsamples in whichspecies A occurs either alone or together
with species B, by b the numberof samples in which species B occurs,by h the
numberof samples in whichboth species
occur,and by n the total numberof samples examined. The chance of species
A occurringin any particularsample in
a/n. Similarly,the
the seriesis therefore
chance ofspecies B occurringin any sample is b/n. The chance that species A
and B will both occur in any given sample is a/n X b/n. The numberof samples in the series in which species A and
B would be expected to occur togetheris
thereforea/n X b/nX n or abin. To
we now divide the
obtain the coefficient
number of samples (h) in which both
species are observed to occur by the
numberof samples (abin) in which they
would be expected to occur by chance:
hn
h
-.
Coefficient of association = a-=
abln ab'
This formulais exactlythat proposedby
Forbes, except that the symbols are
different.
A coefficientof association of 1.0
shows that the two species under considerationoccur togetherin exactly the
number of sample units expected by
chance. A coefficientsmaller than 1.0
shows that they occur togetherin fewer
samples than would be expected by
of0.5
chance. For example,a coefficient
shows that they are foundtogetheronly
halfas frequentlyas expectedby chance.
Similarly, a coefficientlarger than 1.0
indicates that the species being compared are associated more frequently
than would be expectedby chance alone.
The coefficientof association therefore
gives a readilyunderstandableindication
of the degreeto whichthe association between any two species conformsto expectationon the basis of chance.
of associaThe value of the coefficient
tionbetweenany two givenspecies varies
with the abundance of the formsin the
area studied. If the two species com-
297
298
LEE
R. DICE
Ecology,Vol. 26,No. 3
pared are both abundant, both will ap- with species A (B/A) would be 30/40
pear in a highproportionof the samples = 0.75. This indexshowsthaton threetaken. The chance of both species oc- quartersof the plots on whichspecies A
curringtogetherin any given sample is occurred,it was associated with B. If
thereforehigh,and it will be impossible we furtherassume that in the series of
forany greatdeviationfromexpectation sample plots under consideration the
to occur. Under these conditions the number (b) of plots in which species B
coefficient
of associationcan neverbe far appeared was 30, then the association
from 1.0. On the other hand, if both index of species A with species B (A/B)
species are rarein the situation,thenthe would be 30/30= 1.0. This indexshows
chance of theiroccurringtogetherin any that on all the plots wherespecies B apgivensample is also low. Any deviation peared species A also was present.
fromexpectation may result in a very
It willbe noted thattheassociationinhighor verylow value forthe coefficient dex may differdependingon whichspeof association. A highcoefficient
there- cies is used as the basis of comparison.
foredoes not necessarilymean that the Such a difference
may point out an imtwo formsare stronglyassociated, nor portantecologicrelationbetweenthetwo
does a low coefficientalways indicate species. Frequently one species is dethat the formsare stronglyrepulsed. A pendent upon another, without there
coefficientof 1.0 shows that the two being any reciprocaldependency. It is
formswereobservedto occur togetherin believedthat the associationindicesmay
exactly the numberof samples expected sometimes be of value in discovering
by chance, but it does not necessarily such unilateraldependencies.
indicatethat the formsdo not tend to be
Because the reciprocalassociation inassociated. The coefficientof associa- dices are differentit will be important
tion is thereforenot a measure of the always to indicate which species is used
amount of association between species, as thebasis ofcomparison. This may be
but onlya measureof the amount of the done by placing the name of the base
deviation of the numberof theiroccur- species second in the statement. Thus,
rences together from the number ex- for a theoreticalexample: "association
= 0.01."
pected by chance.
index gracilis/bairdii
To providea simpleand directmeasure
In some ecologic studies it will be deofthe amountofassociationbetweenany sirableto use a measureof the amountof
given species (A), taken as the basis of associationbetweenspeciesthat does not
comparison,and any otherspecies (B) I change depending on which species is
hereproposethe associationindex. This used as a base. Such a measure, here
indexis obtainedby dividingthe number called the coincidenceindex,has a value
(a) ofrandomsamplesof a givenseriesin intermediatebetweenthereciprocalassowhich species A occurs into the number ciation indicesA/B and B/A. Thus:
(h) of samples in whichspecies A and B
occur together. Therefore:
Coincidenceindex= 2h
a+b
Associationindex B/A = h/a.
In theexamplepreviouslygiventhe coinThe formulaforthe reciprocalindex is:
cidenceindexwouldbe (2 X 30)/(30 + 40)
= 60/70 = 0.86. The coincidenceindex
Associationindex A/B = h/b.
thereforeis a measure of the amount of
For example,ifspecies A should occur associationbetweenboth the two species
on a total of 40 sample plots (a = 40), compared.
The values of the association indices
and if species B should occur together
with A on 30 of these plots (t = 30), and of the coincidenceindex range from
then the association index of species B 1.0, which indicates association of the
July,1945
AMOUNT
OF ECOLOGIC
ASSOCIATION
two species in all the samples examined,
to 0.0, which indicates complete failure
of associationunderthe conditionsof observation. Intermediatevalues give the
proportionalamountofassociation. The
easy to comprehend.
indicesare therefore
No indicationof a possible deviation
fromchance expectation is given either
by the association index or by the coincidenceindex. It will thereforeoftenbe
desirable to give one or other of these
of associaindicesand also the coefficient
tion. All ofthesemay be calculated with
only slighteffortfromthe same data.
No statisticaltest of the reliabilityof
any of these measuresof association has
been devised. However, the chi-square
test (Snedecor, '40: 16-19) will show
whetherthe combinationsof species in
the samples of a particular series may
possiblybe due to chance errorsin random sampling. This test should thereforealways be applied to the data from
whichany association index,coincidence
index, or coefficientof association is
derived.
In applying the chi-square test it is
necessary to determinethe number of
unitson whicheach possiblecombination
of the two species would be expected to
occurby chance. The expectednumber
ofsamples in whichtwo species,A and B,
would be expectedto occur togetherhas
already been shown to be abin. Species
A wouldbe expectedto occuralone in the
number of samples from which it has
been recorded(a) less thenumberofsamples in which it is expected to occur togetherwith B. Similarly,B is expected
to occur alone in b samples minus the
numberofsamplesin whichB is expected
to occur along with A. The chance of
speciesA not occurringin any givensample in the seriesis (n - a)/n. Similarly
the chance of B not occurringin any
sample is (n - b)/n. The chance that
neitherA nor B will occur in a given
sampleis therefore(n - a) /nX (n - b)/n
and thetotal numberofsamplesin which
neitherspecies is expected to be present
will be (n - a)/n X (n - b)/nX n or
(n - a) (n - b)/n. From the deviations
BETWEEN
299
SPECIES
fromthese expected numberschi-square
is calculated.
As an example let us assume a seriesof
100 samples in whichspecies A has been
foundalone in 5 samples,species B alone
in 20 samples, both species A and B togetherin 20 samples,and neitherspecies
in 55 samples (table I). Then the value
Theoreticalexample of the association
betweentwospecies (A and B) shown
by 100 samples
TABLE I.
Number of samples
Class
Observed
A only
A+B
B only
Neither
Totals
Expected
5
20
20
55
15
10
30
45
100
100
Deviation
from
expecta- (deviation)2
expected
tion
-10
+10
-10
+10
0
6.667
10.000
3.333
2.222
22.222
a = 25; b =40; n = 100; coefficient
of association
2.0; association index A/B =0.5; association
index B/A = 0.8; coincidence index= 0.62; degree
of freedom=1; X2=22.222.
=
of a is 25, b is 40, and n is 100. The
numberof samples in whichA and B are
expected to occur together by chance
of associa(abin) is 10. The coefficient
tion is accordingly20/10 or 2.0, which
showsthat the two speciesin theexample
occur together twice as frequentlyas
would be expectedby chance. The association index of species A with species B
(A/B) is 20/40 = 0.5, and of species B
with species A (B/A) is 20/25 = 0.8.
The coincidence index is (2 X 20)/(25
+ 40) = 40/65 = 0.62.
In calculating the chi-square deviations in our example we used data from
two species. The numberof degrees of
freedomis thereforeone. The total chisquare is 22.222 (table I). From a table
of the values of chi-square (Fisher, '36,
table III; or Snedecor, '40, table 9.1) it
is evidentthat this amount of chi-square
with one degree of freedomis far above
the one per cent. level of significance.
The associationofspecies A and B on the
sample plots must thereforebe due to
some factorother than chance errorsin
random sampling. We may conclude
300
LEE
R. DICE
that thereis a tendencyin this situation
forspeciesA and B to be ratherstrongly
associated in their distribution. From
the differencein association indices,
moreover,it is suggestedthat species A
is somewhat more stronglyassociated
with B than species B is with species A.
The chi-squaretestis not reliablewhen
the numberofexpectedunitsin any class
fallsbelow five. If possiblethe expected
number in each class should be ten or
more. Should the number of samples
available fora particularstudynevertheless be fewand if therebe only a single
degree of freedom,Yates' "correction"
(Snedecor,'40: 168-169) may be applied.
It is better,however,in everyfieldstudy
oftheamountofassociationbetweenspecies to securea sufficient
numberofsamples so that no class willcontainless than
ten expectedunits.
The magnitudeof chi-squarevaries to
some degree with the abundance of the
species compared. Two stronglyassociated species mightoccur togetheron a
large number of sample plots and yet,
if both were abundant, the deviation
fromexpectationmightbe slightand chisquare thereforesmall and not statistically significant. Also, two species that
have only a slighttendencytoward association, if both were rare in the samples, might occur together frequently
enough to have a large and very significant chi-square. Care should therefore
be taken not to interpretthe magnitude
of chi-square as indicatingthe strength
of association betweenthe species under
consideration. On the other hand, the
degree of significanceof chi-squaremay
be an indirect but valid measure of
association.
The amount of association between
threeor morespecies can be determined
by means of the coincidenceindex and
coefficient
of associationby an expansion
of the formulas used for two species.
Also the significanceof the deviation
fromexpectationofthe observedamount
of association betweenthese species can
be determinedby the chi-square test.
Attemptsto measure the association of
Ecology,Vol. 26,No. 3
threeor morespecies will,however,very
rarelybe practical. The numberofsamples needed to securean adequate measure of the associationof only two species
is very considerable. A fully reliable
measureof theamountofassociationbetweentwo species will seldom be secured
with much less than 100 samples, and
morethan thisnumberwillnearlyalways
be desirable. It willrarelybe possibleto
secure the much greaternumberof comparable samples needed to measure in
one step the amount of association between threeor morespecies.
So farwe have discussedonlythe association between two or more given species. We may also need to measurethe
amount of association betweena species
and someinanimatefeatureofitshabitat.
For this purposewe may use the indices
of association or coincidence,the coefficient of association, and the chi-square
test of significance. We may thus compare the occurrenceof a given species
withthe simultaneouspresenceof rocks,
logs, trees,clumpsofbushes,or any other
featureoftheenvironment. It mustnot
be expected, however,that the calculation of thesestatisticswillalone solve all
theproblemsoftherelationshipsbetween
organismsand theirenvironments.
The abundance of the individuals of
either of the species concerned on the
sample unitsis not consideredin the calculation of the coefficient
of association
or of the indices of association or coincidence. Such an omissionfrommeasures
of associationhas been criticizedby Calvert ('22). Furthermore,no account
has been taken,in thesecomputations,of
the possible tendencyof the individuals
to be associated with one another in
pairs, families,or largersocial aggregations.
The abundance and also the social behavior of the individuals of any given
species obviously affect the frequency
withwhichthe formappears in any kind
of sample. Therefore the frequency
with which any species appears in a
given set of samples includes, to some
degree, an indication both of its abun-
July,1945
AMOUNT
OF ECOLOGIC
ASSOCIATION'
dance and of its social behavior. So
long as any given sample is as likelyas
any othersample to containa representative of each of the species concernedwe
may use these sample counts as a basis
for computingthe association between
the species. We will readilygrant,however, that these measuresof association
apply onlyto the samplestakenand that
if the two species involved should occur
in otherdensitiesor ifone or both should
have another type of social grouping,
such as mightbe truein otherseasons of
values forthese
the year, then different
measures of association would probably
be obtained. No-methodshave yetbeen
devised forpredictingthe values of these
measuresof associationunderotherpopulation densities or under other social
groupingsthan those exhibitedby a particularsample series.
The samples that are used as a basis
forthe measurementof association may
be any of the usual sample units employedin ecologicfieldwork. They may
be quadrats, transects,or sample plots
of any kind, or they may be sample
catches obtained by sweeps of an insect
net, secured by drags of a tow net or
dredge,or collected in any other standardized manner. The essential feature
is that the two species to be compared
actually must have been in close associationwithone anotheriftheyare recorded
in the same sample unit. Any sample
unit that covers a large area or that includes several different
types of ecologic
communities,therefore,will not usually
be a proper base for the calculations.
Sample unitsbased on timeintervalswill
be admissibleonly if theyalso are space
unitseach of whichcoversan area of not
too great extent.
The size of the units fromwhich the
samples are obtained influencesall these
measures of association. This is true
because the frequencywith which any
species appears in a series of samples is
dependentnot only upon its densityand
its social tendencies but also upon the
size of the area fromwhichthe sample is
drawn. If the sample plots are of large
BETWEEN
SPECIES
301
size, almost all the species in the local
biota, except only the rarer ones, will
appear on a largeproportionof the plots.
On sample plots ofsmall size, on the contrary,even the most abundant species
are likelyto appear on only a small fractionoftheplots. For thisreasonheterogeneous samples, such as the collections
of fishcomparedby Forbes ('07), do not
providea sound basis forthe calculation
of the amount of association between
species.
Inasmuch as the value of all these
measures of association varies with the
size of the sample units employed,it is
evident that the sample units of the
serieson whichany ofthesecomputations
is based must all be of the same size.
The size of sample unit to be employed
must of course be adjusted to the abundance and also to the size and mobilityof
the species being compared. The most
efficientmeasure of association seems
generallyto be obtained when the more
abundant one of the species under consideration occurs on about half of the
sample unitsin the series.
All sample unitson whichestimatesof
association are to be based should be
taken at random,using this termin the
statisticalsense. If any selectionis used
in locatingthe sample units the method
of selection should be fully described.
Samples taken systematically,such as
one collectionin each ofa seriesof stands
or of communities,usually cannot be
used properlyas a basis for statistical
calculations.
Any measureofassociationis of course
true only for the particular place and
time for which it is calculated. The
ecologic relations between organisms
vary fromtimeto time,fromcommunity
to community,and fromplace to place.
It will always be important,therefore,
to
specify the particular conditions upon
which any given measure is based. It
will usually be impossible to combine
data taken at different
times,in different
places, in different
typesofcommunities,
in differentmanners,or fromdifferent
sizes of samples to serve as a basis-for
302
LEE
Ecology,Vol. 26,No. 3
R. DICE
the calculation of the degree of association betweenspecies.
The fact that two species may tend to
occur together,as shownby a highassociationindex,coincidenceindex,or coefficient of association, or by a significant
chi-square deviation from expectation,
does not in any way explain the reason
fortheirassociation. It may be thatthe
two formsare mutuallyattracted to one
anotheror that one formis in some way
dependent upon the other. The two
species may, however,be associated only
because of the presence in their habitat
of certainfeaturesattractiveor essential
to both. In such a case the two associated species may have no relationship
at all to one another,otherthan selection
of the same habitat. These statisticsat
best are a measure only of the actual
amount of association between two species ratherthan of the amount of attraction betweenthem.
SUMMARY
The coefficient
of associationof Forbes
indicates the amount of association betweentwo givenspecies comparedto the
amount of association betweenthem expected by chance. In orderto providea
simple direct measure of the amount of
association of one species with another
the associationindex is proposed. If a
is the number of random samples of a
givenseriesin whichspeciesA occursand
h is the number of samples in which
anotherspecies B occurstogetherwithA,
then the association index B/A = h/a.
Similarly,ifb is the numberof samplesin
whichspecies B occurs,then the association index A/B = h/b. There is also
proposed a coincidenceindex,2h/(a + b),
whose value is intermediatebetweenthe
two reciprocal association indices. As
a measure of the statisticalreliabilityof
the deviation shown by the samples of a
given series fromthe amount of association expected by chance, the chi-square
test may be used.
LITERATURE
CITED
Calvert, Philip P. 1922. Methods for expressing the associations of different species.
Ecology 3: 163-165.
Edmondson, W. T. 1944. Ecological studies of
sessile Rotatoria. Part I. Factors affecting
distribution. Ecol. Monog. 14: 31-66, 5 figs.
Fisher, R. A. 1936. Statistical methods for
research workers; Sixth edition. Edinburgh
and London: Oliver and Boyd. xiii + 339 pp.
Forbes, S. A. 1907. On the local distribution
of certain Illinois fishes:an essay in statistical
ecology. Bull. Illinois State Lab. Nat. Hist.
7: 273-303, 15 maps, 9 pls.
Hacker, H. P. 1921. On the distribution of
anophelines in the BrickfieldsRoad area of
Kuala Lumpur, November, 1917. Federated
Malay States Malaria Bureau Reports 2:
3-28, 1 map.
Michael, Ellis L. 1920. Marine ecology and
the coefficientof association. Jour. Ecol. 8:
54-59.
Snedecor, George E. 1940. Statistical methods applied to experimentsin agricultureand
biology. Ames, Iowa: Iowa State College
Press. xiii + 422 pp., Illus.
Van Deventer, William C. 1936. A winter
bird community in western New York.
Ecology 17: 491-499, 3 figs.