relating behavior to habitat: solutions to thefourth-corner problem

547

Ecology, 78(2), 1997, pp. 547–562q 1997 by the Ecological Society of America

RELATING BEHAVIOR TO HABITAT: SOLUTIONS TO THEFOURTH-CORNER PROBLEM

PIERRE LEGENDRE,1 RENE GALZIN,2,4 AND MIREILLE L. HARMELIN-VIVIEN3,4

1Departement de Sciences biologiques, Universite de Montreal, C.P. 6128, succursale Centre-ville, Montreal, Quebec,Canada H3C 3J7

2Ecole Pratique des Hautes Etudes, Laboratoire d’Ichtyoecologie tropicale et mediterraneenne (Centre National de laRecherche Scientifique, Unite 1453), Universite de Perpignan, F-66860 Perpignan Cedex, France

3Centre d’Oceanologie de Marseille, Station marine d’Endoume (Centre National de la Recherche Scientifique, Unite 41),rue Batterie des Lions, F-13007 Marseille, France

4Centre de Recherches insulaires et Observatoire de l’Environnement, B.P. 1013, Moorea, Polynesie francaise

Abstract. This paper addresses the following question: how does one relate the bio-logical and behavioral characteristics of animals to habitat characteristics of the locationsat which they are found? Ecologists often assemble data on species composition at differentlocalities, habitat descriptions of these localities, and biological or behavioral traits of thespecies. These data tables are usually analyzed two by two: species composition againsthabitat characteristics, or against behavioral data, using such methods as canonical analysis.We propose a solution to the problem of estimating the parameters describing the rela-tionship between habitat characteristics and biology or behavior, and of testing the statisticalsignificance of these parameters; this problem is referred to as the fourth-corner problem,from its matrix formulation. In other words, the fourth-corner method offers a way ofanalyzing the relationships between the supplementary variables associated with the rowsand columns of a binary (presence or absence) data table. The test case that motivated thisstudy concerns a coral reef fish assemblage (280 species). Biological and behavioral char-acteristics of the species were used as supplementary variables for the rows, and charac-teristics of the environment for the columns. Parameters of the association between habitatcharacteristics (distance from beach, water depth, and substrate variables) and biologicaland behavioral traits of the species (feeding habits, ecological niche categories, size classes,egg types, activity rhythms) were estimated and tested for significance using permutations.Permutations can be performed in different ways, corresponding to different ecologicalhypotheses. Results were compared to predictions made independently by reef fish ecol-ogists, in order to assess the method as well as the pertinence of the variables subjectedto the analysis. The new method is shown to be applicable to a wide class of ecologicalproblems.

Key words: behavior; coral reef; fish; fourth-corner statistic; habitat; parameter estimation; per-mutation test; tropical ecology.

INTRODUCTION

Niche theory tells us that species have ecological‘‘preferences,’’ meaning that they are found at locationswhere they encounter appropriate living conditions.This statement is rooted in the observation that specieshave unimodal distributions along environmental vari-ables, more individuals being likely to be found aroundsome value that is ‘‘optimal’’ for the given species.This has been formalized by Hutchinson (1957) in his‘‘fundamental niche’’ model, and used by ter Braak(1985) to justify correspondence analysis as a keymethod to study incidence or abundance data tables.Furthermore, Gause’s (1935) competitive exclusionprinciple suggests that in their evolution, speciesshould have developed non-overlapping niches. Thesetwo principles together suggest that species should be

Manuscript received 5 June 1995; revised 22 April 1996;accepted 1 May 1996; final version received 5 June 1996.

found to be roughly equally spaced in the n-dimen-sional space of resources. The influence of niche theoryon competition theory (Bartlett 1960, Watanabe 1984)has raised some interesting questions recently (Tilman1987). Niche theory remains limited, however, to an-swering questions such as: What is the ecological spec-ificity of a species? How do different species apportionlocal resources among themselves? And, how does thismechanism control the associations of species foundin nature?

It is understood that species have evolved biologicaland behavioral characteristics allowing them to exploitthe given combination of resources represented in theirniche. The following question, which also stems fromniche theory, has been neglected, however, by lack ofappropriate methods of analysis: How do the biologicaland behavioral characteristics of species determinetheir relative locations in an ecosystem? Observationof species in nature can lead ecologists to formulate

548 Ecology, Vol. 78, No. 2PIERRE LEGENDRE ET AL.

FIG. 1. Position of the Tiahura transect on Moorea Island,French Polynesia.

hypotheses in that respect. Testing such hypotheseswould require (1) a way of detecting associations be-tween species traits and habitat characteristics, and(2) a way of testing the significance of these associa-tions. This paper presents such a method.

The question is difficult. Ecologists often assembledata on species composition at different localities, hab-itat descriptions of these localities, and/or biologicalor behavioral characteristics of the species. These datatables are usually analyzed by pairs: species compo-sition against habitat characteristics, or against behav-ioral data, using such methods as canonical analysis.No standard statistical method can be used to directlyanalyze the relationship between the biological or be-havioral characteristics of species and the character-istics of the habitat where they are found.

To fix ideas, imagine a first table A (k 3 m) con-taining data on the presence or absence of k species atm locations (sampling stations, for instance). A secondtable B (k 3 n) describes n biological or behavioraltraits of the same k species. A third table C (p 3 m)contains information about p environmental variablesat the m locations. How does one go about relating then biological and behavioral traits to the p environmen-tal variables? In order to help find a solution, let ustranslate the problem into matrix algebra:

A (k 3 m) B (k 3 n)F GC (p 3 m) D (p 3 n)

The problem can now be stated as follows: How doesone estimate the parameters in matrix D (p 3 n) cross-ing the n biological and behavioral traits to the p en-vironmental variables? Furthermore, are these associ-ations significant in some sense, i.e., are they reallydifferent from 0 (no association), or from the valuethey could take in a randomly organized environment?Because of this matrix representation, the underlying

statistical problem can be referred to as that of esti-mating the parameters in the fourth-corner matrix D,or the fourth-corner problem. While the data in matrixA are necessarily of the presence/absence or frequencytype, data in matrices B and C can be either quantitativeor qualitative (nominal). Solutions have to be foundthat can accommodate any of these types of variables.

This paper presents statistical solutions to this prob-lem, involving groups of qualitative or quantitativevariables, or mixtures of quantitative and qualitativeinformation. Our test case, which motivated the study,concerns a coral-reef fish assemblage of 280 species.Parameters of the association between habitat charac-teristics and biological and behavioral traits of the spe-cies will be estimated and tested for significance. Theywill then be compared to predictions made indepen-dently by reef-fish ecologists, in order to assess themethod as well as the variables subjected to the anal-ysis. This is the only way to validate the approach, inthe absence of any other statistical method available atthe present time to tackle this problem. A new arrayof biological questions are now open to statistical es-timation. The method developed in this paper will beshown to be applicable to a wide class of ecologicalproblems, and to other problems found in such fieldsas sociology, marketing, and political sciences.

MATERIALS AND METHODS

Coral reef fish data

Fish-assemblage data, used in this paper to illustratethe method, were collected by R. Galzin in 1982 and1983, during daytime, along a transect extending fromthe beach to the outer reef slope, in the northwest part(called Tiahura) of the reef of the high volcanic islandof Moorea in French Polynesia (Fig. 1). Moorea(1498509 W, 178309 S) is a high volcanic tropical islandof the Society archipelago, 25 km northwest of Tahiti.Water temperature varies from 258 to 298C throughoutthe year. A coral reef, 800 m wide on average, and with12 ocean passes, surrounds the island. Because it isnarrow, this reef is ideal to study how marine organismsreact to a coast-to-sea gradient. The transect was di-vided into 22 contiguous sampling stations from thebeach (station 1) to the outer slope. On the reef, stations1 to 16 were 50 m wide and all ,8 m water depth,while station 17 was 40 m wide; stations 18 to 22, onthe outer slope descending to the Pacific floor, werelimited by depths of 3, 8, 15, 22, and 30 m (greaterdepths requiring diver safety stops). At each station,the presence/absence of 280 fish species was noted us-ing two techniques: visual observations during 45 mininside the station (snorkeling or scuba diving), and lim-ited rotenone collection around coral patches. Detailsabout this transect and the fish counting method arefound in Galzin and Pointier (1985), Galzin (1987a,b), and Galzin and Legendre (1987).

Two quantitative habitat variables were measured:

March 1997 549THE FOURTH-CORNER PROBLEM

TABLE 1. Biological and behavioral characteristics of the280 fish species used as a test case in the proposed methodfor examining the relationships between habitat, biology,and behavior of organisms. Variable 2 is coded for diurnalbehavior, since species can switch states from day to night.Frequencies of the states are given in parentheses.

Variable 1: Feeding habits (nominal)1 5 herbivorous (n 5 43)2 5 omnivorous (n 5 43)3 5 diurnal grazer on sessile invertebrates: corals, al-

cyonarians, sponges, etc. (n 5 25)4 5 carnivorous type 1 (diurnal): small crustaceans,

molluscs, echinoderms, and polychaetes (n 5 66)5 5 carnivorous type 2 (nocturnal): large crustaceans

(crabs), cephalopods, fish (n 5 58)6 5 fish eater (fish-only diet) (n 5 16)7 5 zooplankton eater (copepods) (n 5 29)

Variable 2: Ecological category (nominal)1 5 hiding in holes, cavities, etc. (n 5 58)2 5 living on the bottom (often poor swimmers) (n 5

23)3 5 circling small territories around coral heads (n 5

91)4 5 swimming above the coral heads; larger vital neigh-

borhoods (n 5 81)5 5 good swimmers, covering large distances on the

reef (n 5 8)6 5 sub-surface species (n 5 11)7 5 pelagic species (n 5 8)

Variable 3: Size class of adults (ordinal)1 5 0–15 cm (n 5 81)2 5 16–30 cm (n 5 73)3 5 31–60 cm (n 5 83)4 5 61–120 cm (n 5 33)5 5 121–240 cm (n 5 9)6 5 larger than 240 cm (n 5 1)

Variable 4: Egg type (nominal)1 5 pelagic eggs (n 5 192)2 5 benthic eggs (n 5 86)3 5 viviparous species (n 5 2)

Variable 5: Activity rhythm (nominal)1 5 diurnal (n 5 196)2 5 nocturnal (n 5 38)3 5 indifferent (n 5 46)

(1) distance (in meters) from the beach and (2) waterdepth (in centimeters). The other measured habitat vari-ables were indices of percentage coverage of the reefbottom by different materials, based on 50 observationpoints. Along the 50-m rope, 50 observations weremade at 1-m intervals. The following variables reportwhat proportion of the 50 readings pertained to eachcategory of substrate: (3) stone slab, (4) sand, (5) coraldebris, (6) turf and dead coral, (7) live coral, (8) largealgae, (9) calcareous algae, and (10) other substrate:large echinoderms (holothuroids and sea stars), spong-es, anemones, or alcyonarians. Several of these cate-gories represent biological material lying on top of,intermingled with, or attached to the mineral substrate.These variables are at an appropriate scale to be per-ceived by fish as important foraging characteristics ofthe habitat. When the 22 stations are considered glob-ally, these eight categories of substrate respectivelyrepresented 2.5%, 31.1%, 11.4%, 14.3%, 13.9%,18.7%, 7.5%, and 0.7% of the observed points.

The biological and behavioral characteristics of the280 species (Table 1) are based on data from the lit-erature (Hiatt and Strasburg 1960, Hobson 1974, Har-melin-Vivien 1979, 1989) and on personal observationsby ichthyologists M. L. Harmelin-Vivien and R. Gal-zin. One variable is ordinal and four are nominal.

Statistical methods

Comparing two nominal variables.—The first situ-ation considered implies two nominal variables, onefrom matrix B (behavior), the other from matrix C(habitat). Any nominal variable can be expanded intoa series of binary variables (0, 1), one for each state.Assume that B and C each consist of a two-state nom-inal variable, as in test cases 1 and 2 (Table 2). To fixideas, assume that the B variable describes two feeding-habit states (herbivorous, carnivorous) and that C isthe nature of the substrate at two sampling stations (livecoral, turf). We will use this example to describe theapproach for nominal variables and to introduce amethod of testing for statistical significance.

Matrices A, B, and C are all needed to obtain anestimate of the parameters in D. The simplest way tocombine them is to multiply clockwise around the setof four matrices as follows, preserving matrix com-patibility:

D 5 CA9B (1)

or, which is equivalent counterclockwise:

D9 5 B9AC9 (2)

If the states of the nominal variables in B and C are 0and 1 and mutually exclusive, or if they represent pro-portions, the sum of the values in D should be equal tothe sum of the values in A, as it is the case in Table 2.

This equation has an equivalent in traditional statis-tics. The data in A, B, and C can be combined to form

an ‘‘inflated data table’’ (Table 3a). Each row of thattable corresponds to an occurrence in matrix A; thecolumns list the states of the nominal variables in Band C corresponding to that occurrence. Matrix D, de-fined by Eqs. 1 and 2, is exactly the result of crossingthese two nominal columns; it is a contingency tablecontaining frequencies (Table 3b). So, a solution forsignificance testing is to compute a x2 statistic, usingeither Pearson’s formula, or Wilk’s (also called the Gstatistic by Sokal and Rohlf 1995). The G statistic willbe used in the present paper and forms the first typeof fourth-corner statistic.

One cannot test these G statistics for significance inthe usual manner, however, because in the general case,several species are observed at any one sampling sta-tion, and so the rows of the inflated table (Table 3) arenot independent of one another; several rows of thatmatrix have resulted from observations at the same


TABLE 2. Test cases 1 and 2 for nominal variables. In each case, matrix A is (10 species 3 2 stations), B is (10 species 32 feeding habits), and C is (2 habitat types 3 2 stations). So, D is (2 habitat types 3 2 behavioral states).

Test case 1 Test case 2

A) Stn. 1 Stn. 2 B) Herbiv. Carniv. A) Stn. 1 Stn. 2 B) Herbiv. Carniv.

Sp. 1Sp. 2Sp. 3Sp. 4Sp. 5Sp. 6Sp. 7Sp. 8Sp. 9Sp. 10

0011100000

1100011111

0000011111

1111100000

Sp. 1Sp. 2Sp. 3Sp. 4Sp. 5Sp. 6Sp. 7Sp. 8Sp. 9Sp. 10

1111111111

1111111111

0000011111

1111100000

C) Stn. 1 Stn. 2 D) Herbiv. Carniv. C) Stn. 1 Stn. 2 D) Herbiv. Carniv.

Live coral 1 0 0 2P 5 0.027E 5 0.03125

31P 5 0.494E 5 0.500

Live coral 1 0 5P 5 1.000P 5 1.000

5P 5 1.000E 5 1.000

Turf 0 1 51P 5 0.027E 5 0.03125

22P 5 0.494E 5 0.500

Turf 0 1 5P 5 1.000E 5 1.000

5P 5 1.000E 5 1.000

Contingency statistic:G 5 5.4872; P (9999 perm.) 5 0.058

Contingency statistic:G 5 0.0000, P (9999 perm.) 5 1.000

Notes: Probabilities (P) are one-tailed, assuming that the sign of the relationship is stated in the hypothesis. The hypothesisis indicated by a sign in each cell of matrix D, 1 meaning that the actual value is expected to be in the upper tail (higherthan the expected value), and 2 when it is expected to be in the lower tail. Probabilities were calculated after 9999 randompermutations following model 1. E 5 exact probabilities.

TABLE 3. Inflated data table (a); there is one row in this table for each species ‘‘presence’’(‘‘1’’) in matrix A of test case 1 (Table 2). From the inflated table, the contingency table(right) is constructed.

(a) Inflated data table

Occurrencesin test case 1

Feeding habitsfrom B

Habitat typesfrom C

(b) Contingency table (D)

Herbiv. Carniv.

Sp. 1 @ Stn. 2Sp. 2 @ Stn. 2Sp. 3 @ Stn. 1Sp. 4 @ Stn. 1Sp. 5 @ Stn. 1Sp. 6 @ Stn. 2Sp. 7 @ Stn. 2Sp. 8 @ Stn. 2Sp. 9 @ Stn. 2Sp. 10 @ Stn. 2

CarnivorousCarnivorousCarnivorousCarnivorousCarnivorousHerbivorousHerbivorousHerbivorousHerbivorousHerbivorous

TurfTurfLive coralLive coralLive coralTurfTurfTurfTurfTurf

Live coral

Turf

0

5

3

2

sampling station. To solve the problem, we propose thefollowing permutation (or randomization) test. A gen-eral introduction to randomization tests is found in So-kal and Rohlf (1995); more advanced texts are Edg-ington (1995) and Manly (1991).

1. Hypotheses.—a) H0: the species (reef fish in our examples) are

distributed at random among the sampling stations.b) H1: the species are not distributed at random

among the sampling stations. (See 5. Permutation mod-els, below, for details on the ecological contents of thenull and alternative hypotheses.)

2. Test statistic.—Compute a x2 statistic on the con-tingency table (matrix D) and use it as the referencevalue in the remainder of the test.

3. Distribution of the test statistic.—

a) Under H0, the species found at any one stationcould have been observed at any other one. Where thespecies were actually observed is due to chance alone.

b) So, a realization of H0 is obtained by permutingat random the values in matrix A, following one of themethods described below. After each permutation ofmatrix A, compute the x2 statistic on D.

c) Repeat this operation a large number of times(say, 999 or 9999 times). The different permutationsproduce a set of values of the x2 statistic, obtainedunder H0.

d) Include in this set of values the reference x2 valuecomputed for the unpermuted data matrix. This valueis considered to be one that could be obtained underH0 and, consequently, it should be added to the distri-bution (Hope 1968, Edgington 1995). The values form


FIG. 2. Permutations of matrix A can be performed infour different ways, corresponding to different null ecologicalmodels. (1) The occurrence of a species on the reef is con-stant, but positions are random; permute at random withinrows. (2) Positions of species assemblages are random; per-mute whole columns (assemblages). (3) Lottery hypothesis:the species that arrives first occupies a site; permute at randomwithin columns. (4) Species have random attributes; permutewhole rows.

an estimate of the sampling distribution of x2 under H0.(For small data sets, one could compute all possiblepermutations in a systematic way and thus obtain theexact, or complete randomization distribution of thestatistic, to be used in the next step.)

4. Statistical decision.—The decision is made bycomparing the reference value of the x2 statistic to thedistribution obtained under H0. If the reference valueof x2 is one likely to have been obtained under the nullhypothesis (which states that there is no relationshipbetween the rows and columns of matrix D), H0 is notrejected. If it is too extreme (i.e., located out in a tail)to be considered a likely result under H0, the H0 isrejected.

5. Permutation models.—Permutations can be ac-complished in different ways, depending on the natureof the ecological hypotheses to be tested against ob-servations. Technically, the fourth-corner statisticalmethod presented above can accommodate any of thepermutation models described below, as well as per-mutations constrained to accommodate spatial or tem-poral autocorrelation (e.g., Legendre et al. 1990, terBraak 1990). With our data (see Coral reef fish data,above), the random component is clearly the speciesfound at the various sampling locations; this is foundin matrix A. It is thus matrix A that should be permuted(randomized) for the purpose of hypothesis testing.This can be accomplished in various ways (Fig. 2).

Model 1: Environmental control over individual spe-cies.—The environmental control model (Whittaker1956, Bray and Curtis 1957, Hutchinson 1957) statesthat species are found at locations where they encounterappropriate living conditions. Species do that indepen-dently of one another, contrary to the assemblage modelthat has species assemblages randomly located (next).Realizations of this null hypothesis are generated bypermuting at random the values within each row vectorof table A, and this independently from row to row. Inthat model, species associations are not functional; theysimply result from the co-occurrence of species at par-ticular locations, driven by environmental control (Fa-ger 1963, Legendre and Legendre 1978, 1983). Thenumber of stations occupied by any given species in arow of matrix A (its ubiquity) is fixed because it isconsidered to reflect such characteristics of the speciesas abundance, intraspecific competition, and territori-ality, as well as ecological plasticity. If all parts of theenvironment were equally suitable for all species, asstated by H0, they could eventually all be present atany given location; this permutation model would al-low it, while the species assemblage model (next)would not. The alternative hypothesis is that individualspecies find optimal living conditions at the stationswhere they are actually found.

Model 2: Environmental control over species assem-blages.—This permutation method would be appropri-ate to test a null ecological hypothesis that the speciescomposition at any one location is as likely to have

occurred at any other location, against an alternativehypothesis that species assemblages are dependentupon the physical characteristics of the locations wherethey are actually found. In the context of the fourth-corner problem, this alternative hypothesis would bean extension of the environmental-control model (pre-vious paragraph) to species assemblages, implyingstrong biotic ties among the species that are actuallyfound together. This is the method of unrestricted per-mutations implemented in program CANOCO, for in-stance (ter Braak 1990). CANOCO is widely used tocompute canonical correspondence analysis (ter Braak1987), a technique allowing one to uncover relation-ships between a species presence-absence or abundancetable on the one hand, and a table of environmentaldescriptors on the other. The hypothesis of no rela-tionship can be tested by randomization, permuting the‘‘site’’ vectors at random in one of the tables withrespect to those in the other. The ‘‘site’’ vectors are thecolumns of our data table A.

Model 3: Lottery.—A third method consists in per-muting values within columns of matrix A, and doingthis independently from column to column. The nullhypothesis says that there is a fixed number of nichesat any one location, and that species invade themthrough some form of lottery, the identity (species) of


an individual settling at a station being a chance event.The lottery model has been advocated for coral reeffishes by Sale (1978), who argued that the main de-terminant of species composition at the various siteson coral reefs is chance, coupled with an overabun-dance of juveniles available for settlement. Instead ofassuming the ubiquity of any one species to be fixed,it is the number of niches available for settlement thatis assumed to be fixed. The alternative hypothesis ishere that species have some competitive advantage overchance settlers in given habitats.

Model 4: Random species attributes.—A fourthmethod would be to permute whole rows at randomwith respect to one another. The corresponding nullhypothesis is that species have random biological andbehavioral attributes. This model is not appropriate forthe data set analyzed in the present paper, because therelationships between species and their behavioral andbiological characteristics are fixed. It may be appro-priate to other types of problems, though.

6. Remarks.—a) In all cases, some aspects of the data are con-

sidered fixed. In our data (see Coral reef fish data,above), the species are the same in matrices A and B.So, the rows of matrix B have to remain the same asthe rows of matrix A; they cannot be permuted withrespect to A (and consequently, permutation model 4is deemed inappropriate). Furthermore, a given specieshas a fixed set of behavior states; the columns of Bcannot be permuted with respect to one another. Like-wise, the stations (column headings) are the same inmatrices A and C; the habitat characteristics at anygiven sampling station cannot be permuted with thoseof other stations.

b) We have already shown in a previous paper (Gal-zin and Legendre 1987) that fish distributions along theTiahura transect did not support the predictions of Sale’slottery model (permutation model 3). Because of thestrong coast-to-sea gradient, it is the environmentalcontrol model that we consider the most appropriate.The number of times each species has been observedis considered a parameter of the ecological situationunder description (and thus a parameter of the test), asexplained in the environmental-control randomizationmodel, and cannot be modified. To investigate possibleassociations between species traits and habitat char-acteristics, model 1 is certainly less restrictive thanmodel 2, which would assume some kind of stronglinkage among the species that are actually found to-gether. So permutational model 1 will be used in theexample analyzed in this paper.

c) Because the permutations are restricted to withinthe rows of matrix A, permutation model 1 is not equiv-alent to constructing the inflated data table (Table 3)and subjecting it to a standard x2 test. Results of thelatter would be equivalent to carrying out unrestrictedrandom permutations on one or the other of the right-hand columns of the inflated data table.

d) The observed values d in individual cells of ma-trix D can be tested for significance, in the same wayas the general association between rows and columnsof D, measured by the G statistic, is tested. In eachcell, the values obtained during the permutations arecompared to the reference value, and the number count-ed that are less than, equal to, and larger than the ref-erence value. At the end of the permutation procedure,these values are used to compute probabilities of therelationship corresponding to each cell. In Table 2 (Testcase 1), for instance, the association (correspondence)between turf and herbivores is found in the lower left-hand cell of matrix D. The probability shown in thatcell is the one-tailed probability of the null hypothesisthat herbivores are not positively associated with turf.If the actual value of the fourth-corner statistic d islower than expected in that cell of the contingencytable, the test is made in the left-hand tail of the dis-tribution, and conversely. The mean of the d valuesfound throughout the permutations is taken as an es-timate of the expected value for any given cell; thisvalue may differ markedly from the expected value oftraditional contingency table analysis, which corre-sponds to a full randomization hypothesis.

e) The permutation testing procedure allows data inmatrices B and C to be relative or absolute frequencies.This is the case with variables 3 to 10 in matrix C ofthe data on coral reef fishes. Each of these variablesrepresents the proportion of the habitat covered by oneof the eight categories of substrate. Using relative fre-quencies would have made these data unsuitable fortraditional x2 testing. With the permutation procedure,however, the probabilities remain the same under anylinear transformation of the frequency values, eventhough the value of the statistic is changed.

Let us examine how this procedure behaves whenapplied to two test data sets, each consisting of 10hypothetical reef fish species and two sampling sta-tions. The first test case of Table 2 was constructed tosuggest that herbivores are found on turf while carni-vores are more ubiquitously distributed. In matrix D,herbivores are clearly positively associated with turfand negatively with coral (P 5 0.0266, computed bythe random permutation procedure), while carnivoresare not significantly associated with either live coralor turf (P 5 0.4940, after 9999 random permutationsfollowing permutation model 1). These probability val-ues are very close to the exact probabilities calculatedfor these data, which are the values obtained from acomplete randomization procedure (E in Table 2). Val-ues of exact probabilities E are computed as follows:consider all 210 possible permutations that result fromindependently permuting the rows of matrix A in testcase 1; count how many of these would produce valuesequal to, or more extreme than, the observed value ineach given cell of matrix D. This value may differslightly from the mean of the random experimentalprobabilities. Globally, the G statistic (P 5 0.0580)


TABLE 4. Test cases 3 and 4 for quantitative or semi-quantitative (ordinal) variables. In each case, matrix A is (12 species3 4 stations), B is (12 species 3 1 biological or behavioral variable), and C is (1 habitat variable 3 4 stations). So, D is(1 habitat variable 3 1 biological or behavioral variable).

Test case 3 Test case 4

A) Stn. 1 Stn. 2 Stn. 3 Stn. 4 B) Size A) Stn. 1 Stn. 2 Stn. 3 Stn. 4 B) Size

Sp. 1Sp. 2Sp. 3Sp. 4Sp. 5Sp. 6Sp. 7Sp. 8Sp. 9Sp. 10Sp. 11Sp. 12

111000000000

000111000000

000000111000

000000000111

123456789

101112


000111000000

000000000111

111000000000

000000111000

123456789

101112

C) Stn. 1 Stn. 2 Stn. 3 Stn. 4 D) Size C) Stn. 1 Stn. 2 Stn. 3 Stn. 4 D) Size

Depth 1 2 3 4 r 5 0.97163P 5 0.0001

Depth 1 2 3 4 r 5 0.00000P 5 0.4960

Notes: Probabilities (P) are one-tailed, and computed in the tail of the sign of the coefficient, assuming that the sign of therelationship is stated in the hypothesis. Probabilities were calculated after 9999 random permutations following model 1.

indicates a marginally significant association at sig-nificance level a 5 0.10 between behavioral states andtypes of habitat. Thus, the testing procedure for theassociation between behavior and habitat behaved asexpected in this example, and the random permutationprocedure produced values quite close to the exactprobabilities.

The second test case illustrates a situation where thenull hypothesis is true in all cases, matrix A indicatingall 10 species to be present everywhere. Indeed, thetesting procedure finds all permutation statistics to beequal to the unpermuted ones, so that the probabilityof the data under the null hypothesis is 1 everywhere.The procedure once more behaved correctly.

For large contingency tables D, relationships amongdescriptor states could be visualized from a correspon-dence-analysis ordination of the contingency table.

Comparing two quantitative or ordinal variables.—Quantitative data can be handled in almost the sameway. Consider for the moment that B and C each con-tain a single quantitative variable, as in the test casesproposed in Table 4. An inflated data table can be con-structed as above; there is one row in that table foreach species ‘‘presence’’ (‘‘1’’) in matrix A. The in-flated matrices corresponding to test cases 3 and 4 hap-pen to have the same number of rows as the originalmatrix A; but in real-case studies, where several speciesmay be present at each station, the number of rows ofthe inflated matrix should be higher, corresponding tothe number of species occurrences in A. Two columnsare written in the inflated matrix, the first one givingthe values of the quantitative variable from B, the sec-ond the values of the variable from C. The cross-prod-uct of these two columns would give the fourth-cornerstatistic d 5 CA9B for quantitative variables. If the two

variables in the inflated matrix are standardized to mean0 and variance 1, and the cross-product is divided by(no. rows 2 1), the fourth-corner statistic becomes aPearson product–moment correlation coefficient r. Forreasons given above, this correlation statistic shouldbe tested using the permutation technique described inthe previous section.

One should note that the general formula of corre-lation coefficients, applicable to ordinal as well as toquantitative data, is the same as that of the Pearsonproduct-moment correlation coefficient (Kendall 1948).So B and C data tables containing either quantitativeor ordinal variables can be handled in the same way,provided that the scale used for the ordinal variablesis felt to be adequate. In case of doubt, one can calculatethe inflated data table and draw a dispersion diagram,to verify that the scales used for the variables generatea linear dispersion of data points, when a relationshipis present.

Test case 3, proposed in Table 4, is constructed insuch a way that species of increasing sizes are foundat increasing depths. The correlation coefficient of0.972 indicates positive association between matricesB and C; the associated probability P 5 0.0001 (one-tailed) shows that this association is strong. So, thepermutation testing procedure behaved as expectedwith these test data. Test case 4, on the contrary, hasbeen designed in such a way as to create no associationbetween B and C. The correlation coefficient is 0 andthe associated probability is 0.4960, which indicates alack of association between B and C. Once again, thetesting procedure performed as expected.

The unstandardized d 5 CA9B statistic should notbe used in place of the correlation statistic r, becausethe permutations of row contents of matrix A, inde-


TABLE 5. Test case 5 comparing a quantitative to a nominal variable. Matrix A is (12 species3 4 stations), B is (12 species 3 1 quantitative biological or behavioral variable), and Ccontains a nominal variable (3 states 3 4 stations). So, D is (3 states 3 1 quantitativevariable).

A) Stn. 1 Stn. 2 Stn. 3 Stn. 4 B) Size


111000000000

000100100100

000010010010

000001001001

111456789

101112

C) Stn. 1 Stn. 2 Stn. 3 Stn. 4 D) Size D) Size

State 1 1 0 0 0 d 5 0.00000 r 5 20.80472P 5 0.0141 P 5 0.0022

State 2 0 1 0 0 d 5P 5

0.105730.3420

r 5P 5

0.114960.3715

State 3 0 0 1 1 d 5P 5

0.220260.1516

r 5P 5

0.597350.0233

Notes: Probabilities (P) of the d statistics (matrix D left) are one-tailed and computed in thelower tail, to test for within-group homogeneity. Probabilities (P) of the correlation statisticsr (matrix D right) are one-tailed, and computed in the tail of the sign of the coefficient, assumingthat the sign of the relationship is stated in the hypothesis. Probabilities were calculated after9999 random permutations following model 1.

pendently of one another, can produce cross-productvalues that are higher than the reference value, whilethe association between B and C is lower. In test case3, for instance, the permutations might bring all ‘‘pres-ences’’ in matrix A to station 4, except for the sixthone found in station 3. This would result in a cross-product statistic d 5 306, higher than the actual valued 5 240 for test case 3. There is no doubt, however,that the association between B and C would be muchlower for that permutation of A than it is for test case3, indicating inadequacy of the d statistic for this per-mutation test. On the contrary, correlation coefficientsdo behave correctly; in the case of that permutation,the correlation coefficient would be 0.044, much lowerthan the reference value of 0.972.

Comparing quantitative to nominal variables.—Comparing a quantitative or ordinal variable to a nom-inal variable recoded into dummy variables (binarystates) can be done in two different ways.

a) Without standardization, the fourth-corner statis-tics d 5 CA9B are measures of within-group sums ofsquares, provided that the quantitative variable is cen-tered within each group (5 state) and the values aresquared before summing. The within-group sums ofsquares are divided by the total sum of squares to pro-vide normalized measures of within-group homoge-neity taking values between 0 and 1 (see test case 5 inTable 5, matrix D left). No value is computed when agroup contains a single, or no observation, since onecannot measure the homogeneity or heterogeneity of

such groups. Such cases, which can be produced bythe permutation procedure but are likely to occur onlyin very small problems, are left out of the referencedistribution during significance testing. The examplehas been constructed in such a way that only the firstgroup, as defined by nominal variable C, is homoge-neous. The permutation test allows one to reject thehypothesis of heterogeneity for the first group only(Table 5). The global statistic that can be derived, in-volving all groups of the nominal variable, is an anal-ysis-of-variance statistic (F or Kruskal-Wallis; F willbe used below). It can readily be computed from aninflated table, using the nominal variable as the clas-sification criterion. It is tested by permutations for rea-sons given above.

b) Consider that the multistate nominal variable hasbeen decomposed into a number of binary descriptors.If the quantitative variable and the binary descriptorsare all standardized to mean 0 and variance 1, thefourth-corner statistics become correlation coefficientsbetween the quantitative variable and the binary de-scriptors, as in Statistical methods: Comparing twoquantitative or ordinal variables, above. An exampleis presented in Table 5 (matrix D right).

These two procedures serve different purposes andtogether provide three types of information. (1) First,the F statistic is a global measure of association be-tween the two variables. In Table 5, the F test (F 59.30405, P 5 0.0138) tells us that at least one (andperhaps more) of the states of C differs from the others,


in terms of the values of B to which it is associated.A posteriori testing can be done by repeating the F testfor pairs of groups (with appropriate correction formultiple testing). (2) Secondly, the fourth-corner sta-tistics d answer the question of within-group homo-geneity. The first group (state 1) is significantly morehomogeneous than the hypothesis of random allocationof fish to sampling stations would suggest (Table 5).(3) Finally, the correlation coefficients r indicate thestrength of association of the 1s of each state of thenominal variable to small or large values of the quan-titative variable. Consider the first group in Table 5,the only one found above to be significantly homo-geneous; the significant negative correlation coefficientindicates that the presence of environmental state 1 isassociated with small values of size. By way of con-sequence, environmental state 3 is associated with largevalues of size. Interpretation of the associated proba-bilities is not without problems, though, because thestates of the nominal variable are mutually exclusive;therefore, testing the relationship of the quantitativevariable with each state of the nominal descriptor,through simple correlation coefficients, makes littlesense. A multiple-regression approach, which is theequivalent of an analysis of variance, would be moreappropriate; in test case 5, a multiple regression of theB variable on any two of the three binary state variablesprovides an R2 of 0.67401 (P 5 0.0146). The problemof interpreting tables of correlation coefficients is dis-cussed further below.

When the nominal variable consists of a series ofpercentage coverage indices, as it is for our variablesdescribing substrates (see description of the data,above), an analysis-of-variance approach cannot beused because states are not mutually exclusive (fuzzyclassification). We will be using the correlation fourth-corner statistic, although a multiple-correlation ap-proach also would be appropriate to obtain a globalstatistic linking the quantitative variable to all per-centage coverage indices.

When analyzing real data, one has to correct indi-vidual tests to accommodate the increased probabilityof committing a Type I error in the case of multiplesimultaneous tests. This can be done either by adjustingindividual probability values (P values), or by adjustingthe significance level a. A traditional way is to use theBonferroni correction, where the significance level, saya 5 0.05, is replaced by an adjusted level a9 5 a/kwhere k is the number of simultaneous tests. This isequivalent to adjusting individual P values Pi to Pi 5kPi and comparing P to the unadjusted significance′

i

level a. While the Bonferroni method is appropriate totest that the null hypothesis is true for the whole setof simultaneous hypotheses (i.e., reject H0 for the wholeset of k hypotheses if the smallest unadjusted P valuein the set is #a/k), the Bonferroni method is overlyconservative. Several alternatives have been proposedin the literature; see Wright (1992) for a review. In this

paper, we will use Holm’s (1979) procedure for ad-justing individual probabilities. It is nearly as simpleto carry out as the Bonferroni adjustment and it is muchmore powerful, leading to rejecting the null hypothesismore often. Other solutions, such as Hochberg’s (1988)and Hommel’s (1988) procedures, are even more pow-erful, but they are known to have the desired experi-mentwise error rate a only for independent tests(Wright 1992). This condition is not met in some ofour tables of results, where for instance ‘‘Other sub-strate’’ is what remains after considering all the othersubstrates in the analysis.

Holm’s (1979) procedure, which remains valid in thistype of situation, is computed as follows. (1) Order theP values from left to right so that P1 # P2 # . . . # Pk.(2) Compute adjusted probability values P 5 Pi(k 2′

i

i 1 1); adjusted probabilities can be .1. (3) Proceedingfrom left to right, if an adjusted P value in the orderedseries is smaller than the one occurring at its left, makethe smaller one equal to the larger one. (4) Compareeach adjusted P to the unadjusted a significance level′

i

and take the statistical decision. Because the adjustedprobabilities form a nondecreasing series, this proce-dure presents the property that an hypothesis in theordered series cannot be rejected unless all previoushypotheses in the series have also been rejected, and,equal P values receive equal adjusted P values.

Validation

In an attempt to validate the method, we comparedthe fourth-corner results to predictions made indepen-dently by reef fish ecologists. The difficulty was thatthere exists no other statistical method, at the presenttime, to address the problem and provide results withwhich the fourth-corner results could be compared;hence the use of traditional ecological analysis by ex-perts. Description of how this was done is saved forthe Discussion.

RESULTS

Tables 6c and 7c present relationships of the reefbottom materials to feeding habits and ecological cat-egories. Since each of these tables implies 56 simul-taneous tests, Holm’s procedure did not allow detectionat the 5% significance level after only 999 permutations(the smallest attainable probability is 0.001, which,when multiplied by 56 simultaneous tests, gives P 50.056, a value .0.05). Therefore, 9999 permutationswere used in these comparisons, as well as Table 8cwhere the same problem existed to a certain extent (24simultaneous tests for each of the two biological orbehavioral variables). In any case, considering the well-known instability of randomization probabilities com-puted from a small number of permutations (Jacksonand Somers 1989), as many permutations as practicallypossible should be used during permutation tests.


TABLE 6. Feeding habits are compared to all habitat variables. The table reports probabilities adjusted using Holm’s procedurewithin each habitat variable: Parts a and b, 999 permutations; part c, 9999 permutations, following model 1. Sign in partc indicates whether a statistic is above (1) or below (2) the estimated expected value. Results of the global tests ofsignificance (F, G) are also given.

Feeding habits

Herbivorous OmnivorousSessile

inverteb.Carniv. 1diurnal

Carniv. 2nocturnal

Fishonly

Copepodeater

a) Distance from the beach (d 5 homogeneity, r 5 correlation fourth-corner statistics)d(i,j)Pr(d)

0.203600.195

0.167722.061

0.117410.195

0.241770.048

0.121460.007

0.028982.061

0.090532.061

r(i,j)Pr(r)

20.011240.621

20.090300.007

20.051950.015

20.016690.621

0.144510.007

0.002590.621

0.033570.288

F 5 6.661, P (999 perm.) 5 0.001

b) Water depth (d 5 homogeneity, r 5 correlation fourth-corner statistics)d(i,j)Pr(d)

0.208491.580

0.115490.450

0.093520.168

0.228500.360

0.189971.964

0.026541.580

0.122101.964

r(i,j)Pr(r)

0.006500.744

20.043200.081

20.045060.056

20.046940.040

0.078190.014

0.000810.744

0.074690.014

F 5 3.544, P (999 perm.) 5 0.001

c) Materials covering reef bottom (G 5 15.426, P (9999 perm.) 5 0.0001)Stone slabP

6.2020.429

5.8410.232

3.7221.535

8.4222.650

5.1812.650

0.9612.650

2.4022.650

SandP

81.2220.039

54.2620.799

43.3420.006

94.3820.006

35.9020.006

8.9420.799

26.2620.039

Coral debrisP

34.9611.976

20.2221.976

24.3210.006

46.7410.009

25.6010.645

4.4812.650

12.0822.650

Turf, dead coralP

45.4610.207

27.8812.650

28.2810.081

57.5810.013

33.5810.029

6.2011.976

15.7612.650

Live coralP

49.8610.006

28.5011.976

29.2010.006

58.2810.006

40.8210.006

6.2211.976

21.0610.006

Large algaeP

44.6620.006

37.5012.650

28.1220.105

59.6820.048

32.2620.140

6.3422.650

19.2022.650

Calcar. algaeP

29.1210.006

16.3211.030

16.0810.079

31.0010.122

26.0210.006

4.5010.207

11.3210.036

Other substrateP

2.5210.105

1.4812.650

1.9410.006

2.9210.795

1.6411.734

0.3611.976

0.9211.976

Comparing two nominal variables

In Table 6c, the relationship between reef bottommaterials and feeding habits is globally significant(PG statistic 5 0.0001), and 20 of the fourth-corner sta-tistics d are marked as significant. According to thesesignificant d values, fish are under-represented on sandand large algae, and are unrelated to stone slab. Inaddition, herbivores are overrepresented on live coraland calcareous algae. Grazers of sessile invertebratesas well as carnivores types 1 and 2 are overrepresentedon coral debris, turf and dead coral, live coral, calcar-eous algae, and ‘‘other substrate’’ (large echinoderms,sponges, anemones, alcyonarians); this includes all ar-eas where herbivores were found. Copepod eaters areoverrepresented on live coral and calcareous algae.Omnivores and specialist piscivores (fish-only diet) donot exhibit significant associations with substrate.

In Table 7c, the relationship between reef bottommaterials and ecological categories is globally signif-icant (PG statistic 5 0.0004), and 17 of the fourth-cornerstatistics d are marked as significant. Materials avoid-ance is the same as in Table 6. In addition, fish hiding

in holes and cavities are overrepresented on live coraland calcareous algae. Fish circling small territoriesaround coral heads, as well as those with larger vitalneighborhoods above coral heads, are mostly found onlive coral, turf and dead coral, coral debris, calcareousalgae, and other substrate. Good swimmers, as well aspelagic species, were found everywhere. Subsurfacespecies are often associated with the stone slab—theonly group overrepresented in this habitat.

In Table 8c, the relationships are significant betweenreef bottom materials, on the one hand, and egg types(PG statistic 5 0.0029) and activity rhythms (PG statistic 50.0001) on the other. Except for the general under-representation on sand and large algae mentioned aboveand the lack of association with stone slab, speciesproducing pelagic eggs are significantly associatedwith all other substrate types; species producing ben-thic eggs are positively associated only with live coraland calcareous algae. Not much can be concluded aboutthe two viviparous species in the study. On the otherhand, typically diurnal species are significantly asso-ciated with all substrate types except those generally


TABLE 7. Ecological categories are compared to all habitat variables. The table reports probabilities adjusted using Holm’sprocedure within each habitat variable. Parts a and b, 999 permutations; part c, 9999 permutations, following model 1.Sign in part c indicates whether a statistic is above (1) or below (2) the estimated expected value. Results of the globaltests of significance (F, G) are also given.

Hidingin holes

Livingon bottom

Aroundcor. heads

Abovecor. heads

Goodswimmers

Sub-surfacespecies

Pelagicspecies


0.144573.795

0.055153.795

0.385543.795

0.365680.007

0.016920.186

0.015643.795

0.012043.795

r(i,j)Pr(r)

0.024520.830

20.057840.091

0.001461.776

0.001621.776

0.000401.776

20.004671.776

0.027640.810

F 5 1.020, P (999 perm.) 5 0.281


0.162251.872

0.047641.872

0.369991.872

0.384121.670

0.015771.212

0.000450.280

0.014601.872

r(i,j)Pr(r)

0.005030.836

20.030820.744

0.023060.744

20.003110.836

20.015860.777

20.053350.007

0.029230.744

F 5 1.180, P (999 perm.) 5 0.178

c) Materials covering reef bottom (G 5 11.686, P (9999 perm.) 5 0.0004)Stone slabP

5.4211.778

1.8212.918

11.2220.946

11.7820.456

0.9012.918

1.1210.006

0.4612.918

SandP

44.2620.006

20.7022.918

124.1420.006

142.5620.006

8.1421.930

1.6620.044

2.8420.946

Coral debrisP

23.7011.925

6.3421.778

60.3812.039

69.3410.009

5.3610.099

1.2222.902

2.0612.902

Turf, dead coralP

29.5211.778

10.0612.918

80.7410.021

85.1210.006

4.2422.918

2.8410.946

2.2212.918

Live coralP

35.0410.006

9.4212.918

88.0210.006

90.6010.006

5.7010.946

1.9412.918

3.2210.456

Large algaeP

33.5022.183

14.0612.183

86.2020.062

83.9820.006

4.2020.602

3.9410.946

1.8821.833

Calcareous algaeP

18.9810.009

4.1822.343

53.6210.006

51.0210.006

3.0812.039

1.2812.918

2.2010.230

Other substrateP

1.5811.980

0.4222.902

4.6810.006

4.6010.006

0.3810.368

0.0020.126

0.1212.918

avoided by fish. During daytime, nocturnal species areoverrepresented around live coral and coral debris;rhythm-indifferent species are significantly associatedwith live coral, turf and dead coral, and calcareousalgae.

Comparing quantitative to ordinal variables

The two quantitative habitat variables, distance fromthe beach and water depth, were compared to the sizeclasses of adult fishes (ordinal) using the correlationfourth-corner statistic (Table 9a). Results indicate thatadult fish size is slightly positively related to distancefrom the beach (r 5 0.05043, P 5 0.011), but not towater depth (r 5 0.02271, P 5 0.143). Size of adultfish significantly increases at stations with more coraldebris and fewer large algae (Table 9b).

Comparing quantitative to nominal variables

The relationship between feeding habits and distancefrom the beach is globally significant (Table 6a), as isthe relationship of feeding habits with water depth.Omnivores and grazers on sessile invertebrates are sig-nificantly associated with sites closer to the beach,while nocturnal carnivores and copepod eaters are sig-

nificantly associated with deeper stations and/or siteslocated farther offshore. Diurnal carnivores avoid thevery shallow stations near the beach. Nocturnal car-nivores are the only group that is strongly homoge-neous in terms of distance from the beach; analysis ofthe inflated data table shows that most sightings weremade at the nine farthest stations, farther than 650 mfrom the beach.

Ecological categories, on the other hand, are not sig-nificantly related to distance from the beach or to waterdepth (Table 7a, b). Egg types are weakly associatedwith distance from the beach, and not at all to depth(Table 8a, b). Species with benthic eggs are found atshallower stations, and those with pelagic eggs at deep-er locations; species with pelagic eggs are significantlyhomogeneous in terms of distance from the beach; anal-ysis of the inflated data table shows that the frequencyof sightings increases with distance from the beach.Activity rhythm is strongly associated with distancefrom the beach, and weakly with water depth. Diurnalspecies are found closer to the beach, while rhythm-indifferent species are significantly found farther awayand in deeper waters.


TABLE 8. Egg types (left) and activity rhythms (right) are compared to all habitat variables. The table reports probabilities(Pr) adjusted using Holm’s procedure within each habitat variable. Parts a and b, 999 permutations; part c, 9999 permutations,following model 1. Sign in part c indicates whether a statistic is above (1) or below (2) the estimated expected value.Results of the global tests of significance (F, G) are also given.

Egg type

Pelagic Benthic Viviparous

Activity rhythm

Diurnal Nocturnal Indifferent


0.724260.003

0.268421.000

0.002810.590

0.786130.832

0.119200.832

0.081320.018

r(i,j)Pr(r)

0.062170.006

20.065330.006

0.018100.244

20.077450.004

20.006340.399

0.115280.003

F 5 3.094, P 5 0.018 F 5 9.235, P 5 0.001


0.778430.873

0.218560.441

0.001460.844

0.733330.069

0.136551.622

0.125661.622

r(i,j)Pr(r)

0.037420.141

20.035090.141

20.016570.355

20.039840.096

20.009990.311

0.066730.015

F 5 1.061, P 5 0.218 F 5 3.055, P 5 0.022

c) Materials covering reef bottomG 5 4.759, P (9999 perm.) 5 0.0029 G 5 7.712, P (9999 perm.) 5 0.0001Stone slabP

24.1621.055

8.4611.527

0.1021.527

24.9220.431

4.7810.431

3.0220.767

SandP

251.7020.002

91.6020.021

1.0020.853

278.9820.002

42.2620.004

23.0620.002

Coral debrisP

133.5010.002

34.0820.216

0.8211.527

130.0810.017

22.7610.016

15.5610.767

Turf, dead coralP

163.0410.002

50.3410.726

1.3610.506

166.7010.002

24.8210.754

23.2210.002

Live coralP

180.2810.002

52.5810.018

1.0811.527

177.5010.002

29.4810.002

26.9610.002

Large algaeP

164.7620.002

62.1821.527

0.8221.527

179.9020.002

26.4220.072

21.4420.270

Calcareous algaeP

102.4410.002

31.1210.003

0.8010.899

102.5810.002

14.0810.431

17.7010.002

Other substrateP

9.1210.002

2.6411.034

0.0221.527

9.3410.002

1.4010.431

1.0410.767

Validation

Before the results of these analyses were known, reeffish ecologists (M. L. Harmelin-Vivien and R. Galzin)made the predictions reported in Table 10. They con-cern feeding habits and ecological categories, com-pared to reef bottom materials. Admittedly, these pre-dictions are not independent of the data that were usedin the analyses presented here, since the data were col-lected by R. Galzin on Tiahura in the early 1980s, butthe two analyses were carried out independently fromone another.

DISCUSSION

Doledec et al. (1996) have recently proposed a dif-ferent but complementary approach to the problem ofjointly analyzing A, B, and C. Their approach consistsin an ordination of table A under a double constraintconsisting of B and C. The two approaches differ inthat we are primarily interested here in relating thevariables in B and C and testing the significance ofthese relationships, while the purpose of Doledec et al.(1996) is primarily to obtain an ordination of sites and

species in table A under constraints, and to project thevariables in B and C onto the ordination axes for in-terpretation.

Comparison of Tables 8 and 9 indicates that the fishecologist predictions generally agreed with the resultsof the fourth-corner statistics (Table 10). For feedinghabits, there are three opposite results over 29 predic-tions, for an error rate of 10%; 15 predictions were notverified by the data (52%), possibly because insuffi-cient observations were available (poor statistical pow-er). For ecological categories, the fish ecologists dideven better; a single case is found where the fourth-corner statistics contradict ecologists’ predictions (5%error rate); a further eight predictions were not verifiedby the data (40%). All four discrepancies in Table 10(i.e., opposite predictions and statistics) concern car-nivores 1 and 2.

Confronted with the fourth-corner statistic results,the fish ecologists (M. L. Harmelin-Vivien and R. Gal-zin) checked their predictions and confirmed that theydid not think they had made any gross ecological mis-take about the habitat and behavior of these fish. Com-


TABLE 9. Size of adult fish is compared to all habitat vari-ables, using correlation fourth-corner statistics. Permuta-tions are computed following model 1. In part b, proba-bilities (999 permutations) were adjusted using Holm’s pro-cedure.

Habitat variables Size of adult fish

a) Quantitative variablesDistance from the beachPr(r)

0.050430.011

Water depthPr(r)

0.022710.143

b) Materials covering reef bottomStone slabPr(r)

20.022360.381

SandPr(r)

20.037980.220

Coral debrisPr(r)

0.060790.035

Turf, dead coralPr(r)

0.025480.381

Live coralPr(r)

0.034760.220

Large algaePr(r)

20.067400.016

Calcareous algaePr(r)

0.051740.066

Other substratePr(r)

0.023850.381

parison of the two sets of results is informative becauseit reveals conceptual differences between the two ap-proaches.

a) The fish families involved in the relationship be-tween sand and diurnal carnivores (discrepancy 1) areessentially the Mullidae (goatfishes), Labridae(wrasses), Balistidae (triggerfishes), Tetraodontidae(puffers), Lethrinidae (emperors), and Gobiidae (go-bies). The goatfishes, emperors, and gobies find theirprey preferably inside the sandy part of the reef duringdaytime (Harmelin-Vivien 1979), and therefore a pos-itive relationship was predicted.

b) The same families are involved in the relation-ship between large algae (discrepancy 2) and diurnalcarnivores. From the results of Naim (1988) for theTiahura transect, we know that macroalgae provideshelter for small motile invertebrates that are preyedupon by wrasses, triggerfishes, and puffers, suggestinga positive relationship, which is what was predicted.

c) The nocturnal carnivores involved in the rela-tionship with sand (discrepancy 3) belong essentiallyto families Lutjanidae (snappers), Holocentridae(squirrelfishes and soldierfishes), Scorpaenidae (scor-pionfishes), Serranidae (groupers), Diodontidae (por-cupinefishes), Muraenidae (morays), and Cirrhitidae(hawkfishes). Ecologists predicted a positive relation-ship essentially by considering the snappers, whichfeed at night on the invertebrates living in the sand.No such relationship is known for the other nocturnalcarnivores. Considering the scarcity of snappers on theTiahura transect, we could understand an absence ofrelationship between nocturnal carnivores and sand, butnot a negative one.

d) The prediction of a positive association betweenfish living above coral heads and large algae (discrep-ancy 4) is based on the observation that large algae arelargely found attached to hard substrates (Payri 1987).

The fourth-corner statistics produced a negative re-lationship between large algae and diurnal carnivores(discrepancy 2) as well as position above the coralheads (discrepancy 4). The reason is the following. OnTiahura, large algae are most abundant in the first sta-tions of the transect, near the beach. Since few speciesare present there, all fish types appear to avoid largealgae as well as sand or, at best, to be indifferent tothese substrates (full-transect analysis reported in Ta-bles 8 and 9). At these very shallow stations, carni-vores, in particular, are less abundant than in the restof the transect. The fourth-corner method thus hasfound a negative relationship, which is witness of anindirect effect. Fish ecologists, on the other hand, hadfirst in mind the feeding habits of the fish. On theTiahura transect, all carnivores find their prey essen-tially in or around macroalgae (Naim 1988) or insidehard substrates (Harmelin-Vivien 1979). So, consid-ering feeding habits, the situation is that of a positiverelationship between carnivores and large algae. Toreach such a conclusion, fish ecologists did not take

into account the spatial distribution of the algae and ofthe fish on the transect, and the dominance of macro-algae in shallow areas; sand (discrepancies 1 and 3)and large algae (discrepancies 2 and 4) are dominantat stations near the beach, where all fish types are notabundant because of the absence of hard substrate andof very shallow water. The fourth-corner method, onthe other hand, found negative relationships because ittakes into account all stations in the study area; it wouldprobably have found the predicted positive relation-ships had the study been conducted at a smaller spatialscale. This is an illustration that correlation does notmean causation.

The results presented in Tables 9 to 12, establishingthe linkage of ecological and behavioral characteristicsof fish species to the distance from the beach, statis-tically confirm and refine the results obtained by Har-melin-Vivien (1989), who compared and contrasted thecharacteristics of fish assemblages on the Tiahura reefflat and reef slope. (1) Diurnal fish were found to bemore abundant on the reef flat, and nocturnal fish onthe reef slope; the fourth-corner statistics indicate asimilar significant negative relationship between di-urnal fish and distance from the beach (Table 8).(2) Fish laying benthic eggs dominated on the reef flat,and fish with pelagic eggs on the reef slope; the fourth-corner method confirms these results, with a positiverelationship between distance from the beach and pe-lagic eggs, and a negative relationship for benthic eggs(Table 8). (3) In the same way, smaller fish were ob-


TABLE 10. Relations between variables C3–C10 (reef bottom, rows) and variables B1 (feedinghabits, part a) and B2 (ecological category, part b). Predictions made by coral reef ecologistsare in parentheses, followed by the results from Tables 6 and 7. Blanks indicate no relation,(1) or 1 a positive relation, and (2) or 2 a negative relation. Discrepancies are numberedto facilitate reference in the text.

a)

Reef bottom

Feeding habits

Herbivor.Omni-vorous

Sessileinverteb.

Carniv. 1diurnal

Carniv. 2nocturnal

Fishonly

Copepodeater

Stone slabSandCoral debrisTurf, dead coralLive coralLarge algaeCalcareous algaeOther substrate

2(1)(1)(1)121(2)

(1)(1)(1)(1)

(2)

(2)21

(1)1(2)

(1)1

(1)21

(1)1(1)1(1)1(1)22

(1)

(1)23

(1)1(1)1(1)(1)1

(1)

(1)

(2)(2)2

(1)1

1

b)

Reef bottom

Ecological category

Hidingin holes

Livingon

bottom

Aroundcoralheads

Abovecoralheads

Goodswimmers

Sub-surfacespecies

Pelagicspecies

Stone slabSandCoral debrisTurf, dead coralLive coralLarge algaeCalcareous algaeOther substrate

(2)2

(1)1

(1)1

(1)

(2)(2)(2)(2)(2)

(2)(2)2

(1)1(1)1

1(1)1

(2)(2)2(1)1(1)1(1)1(1)24

11

12

served on the reef flat; a positive correlation betweensize of fish and distance from shore is obtained.(4) Omnivores and sessile invertebrate feeders werefound on reef slopes. These relationships are confirmedby the fourth-corner statistics and their tests of signif-icance: a negative correlation is found in Table 6 be-tween distance from the beach and omnivores as wellas sessile invertebrate feeders, and a positive correla-tion for carnivores 2.

CONCLUSION

The fourth-corner method offers a way of analyzingthe relationships between the supplementary variablesassociated with the rows and columns of a binary (pres-ence-absence) data table. The parameters describingthese relationships can be estimated and tested for sta-tistical significance. In the example analyzed in thepresent paper, biological and behavioral characteristicsof species were used as supplementary variables forthe rows, and characteristics of the environment for thecolumns. Other ecological situations are considered be-low.

The fourth-corner method allows the detection of anew family of associations or correlations, betweenspecies traits and habitat characteristics, in a traditionaldata-analysis framework. The matrix operation leadingto estimating the parameters in the fourth-corner ma-trix, D 5 CA9B, is very general and can accommodateany type of variable (quantitative, semi-quantitative, orqualitative). The method is not a modelling technique

since it does not take into account the dynamics ofindividual species or the spatial and temporal structureof environmental variation. However, the fourth-cornerstatistics can be incorporated into causal ecologicalmodels describing the mechanisms determining the ob-served associations or correlations. On the one hand,the sign and indication of strength (value, significance)of fourth-corner statistics can be incorporated as pa-rameters in deterministic models; on the other hand,the fourth-corner correlation coefficients can be useddirectly in path analysis or other related forms of sta-tistical modelling.

Analysis of the Tiahura fish data set (280 species),which motivated the development of the method, il-lustrated that fourth-corner statistics adequately iden-tify positive or negative associations between the bi-ological or behavioral traits of animals, and the habitatcharacteristics of the locations at which they are found,much in the same way as correlation coefficients do intraditional data analysis. In many instances, the statis-tical results confirmed the intuitive analysis of fieldecologists. The statistical analysis also revealed a setof relationships that had not been predicted by the fishecologists, and which can now be embodied in ourunderstanding of the ecological determinants of theseanimals.

Currently, this method does not take into accountindirect relationships that may exist among variables,as can be done with path analysis for instance. A mul-tiple-interaction form of fourth-corner statistics re-


mains to be developed (log-linear modelling, multipleregression, path analysis, etc.) to test more sophisti-cated hypotheses. Another interesting developmentwould be to extend the fourth-corner method to a tableA of species abundance data; this extension is notstraightforward.

Other ecological problems could be studied usingthis method. In the study of feeding behavior for in-stance, consider a matrix A with rows that are indi-viduals and columns corresponding to locations. Theprey ingested by each individual are found in matrixB (in columns). Matrix C may contain either micro-habitat environmental variables, or prey availabilityvariables. One could use fourth-corner statistics toidentify relationships between prey availability andconsumption, or between prey consumption and en-vironmental variables. Problems of the same type arefound in such fields as sociology, marketing, politicalscience, and the like.

ACKNOWLEDGMENTS

Ottar Bjørnstad, Charlotte K. Hemelrijk, and Francois-JosephLapointe commented on the permutation models. The work ofP. Legendre in French Polynesia and in Perpignan in 1994 hasbeen supported by a CNRS (France) Associate Research positionin URA 1453, Universite de Perpignan. This research was alsosupported by NSERC grant No. OGP0007738 to P. Legendre.A FORTRAN program (source code and compiled versions forMacintosh and DOS) carrying out the computations of thefourth-corner statistics, with documentation, is available from P.Legendre (http://alize.ere.umontreal.ca/;casgrain/R/4thpcorner/or ftp://ftp.umontreal.ca/pub/casgrain/R/4thpcorner/). The pro-gram can easily be adapted to other operating systems for whicha FORTRAN compiler is available. The above-mentioned WorldWide Web site also makes available the data set upon whichTable 1 is based.

LITERATURE CITED

Bartlett, M. S. 1960. Stochastic population models in ecol-ogy and epidemiology. Methuen, London, UK.

Bray, R. J., and J. T. Curtis. 1957. An ordination of the uplandforest communities of southern Wisconsin. EcologicalMonographs 27:325–349.

Doledec, S., D. Chessel, C. J. F. ter Braak, and S. Champely.1996. Matching species traits to environmental variables:a new three-table ordination method. Environmental andEcological Statistics 3:143–166.

Edgington, E. S. 1995. Randomization tests. Third edition.Marcel Dekker, New York, New York, USA.

Fager, E. W. 1963. Communities of organisms. Pages 415–437 in M. N. Hill, editor. The sea. Volume Two. Intersci-ence, New York, New York, USA.

Galzin, R. 1987a. Structure of fish communities of FrenchPolynesian coral reefs. I. Spatial scales. Marine EcologyProgress Series 41:129–136.

———. 1987b. Structure of fish communities of French Pol-ynesian coral reefs. II. Temporal scales. Marine EcologyProgress Series 41:137–145.

Galzin, R., and P. Legendre. 1987. The fish communities ofa coral reef transect. Pacific Science 41:158–165.

Galzin, R., and J.-P. Pointier. 1985. Moorea island, SocietyArchipelago. Pages 73–102 in Proceedings of the Fifth In-ternational Coral Reef Congress, Tahiti. Volume One:French Polynesian Coral Reefs.

Gause, G. F. 1935. Verification experimentale de la theoriemathematique de la lutte pour la vie. Actualites Scienti-

fiques et Industrielles Number 277. Hermann, Paris, France.Harmelin-Vivien, M. L. 1979. Ichtyofaune des recifs cor-

alliens de Tulear (Madagascar): ecologie et relations tro-phiques. These de Doctorat des Sciences, Universite Aix-Marseille II, France.

———. 1989. Reef fish community structure: an Indo-Pa-cific comparison. Pages 21–60 in M. L. Harmelin-Vivienand F. Bourliere, editors. Vertebrates in complex tropicalsystems. Ecological Studies 69, Springer-Verlag, Berlin,Germany.

Hiatt, R. W., and D. W. Strasburg. 1960. Ecological rela-tionships of the fish fauna on coral reefs of the Marshallislands. Ecological Monographs 30:65–127.

Hobson, E. S. 1974. Feeding relationships of teleostean fish-es on coral reefs in Kona, Hawaii. Fisheries Bulletin 72:915–1031.

Hochberg, Y. 1988. A sharper Bonferroni procedure for mul-tiple tests of significance. Biometrika 75:800–802.

Holm, S. 1979. A simple sequentially rejective multiple testprocedure. Scandinavian Journal of Statistics 6:65–70.

Hommel, G. 1988. A stagewise rejective multiple test pro-cedure based on a modified Bonferroni test. Biometrika 75:383–386.

Hope, A. C. A. 1968. A simplified Monte Carlo significancetest procedure. Journal of the Royal Statistical Society Se-ries B 30:582–598.

Hutchinson, G. E. 1957. Concluding remarks. Cold SpringHarbor Symposium of Quantitative Biology 22:415–427.

Jackson, D. A., and K. M. Somers. 1989. Are probabilityestimates from the permutation model of Mantel’s test sta-ble? Canadian Journal of Zoology 67:766–769.

Kendall, M. G. 1948. Rank correlation methods. CharlesGriffin, London, England.

Legendre, L., and P. Legendre. 1978. Associations. Pages261–272 in Phytoplankton manual. Monographs on ocean-ographic methodology, Number Six. UNESCO, Paris.

Legendre, L., and P. Legendre. 1983. Numerical ecology.Elsevier Scientific, Amsterdam, The Netherlands.

Legendre, P., N. L. Oden, R. R. Sokal, A. Vaudor, and J. Kim.1990. Approximate analysis of variance of spatially au-tocorrelated regional data. Journal of Classification 7:53–75.

Manly, B. F. J. 1991. Randomization and Monte Carlo meth-ods in biology. Chapman and Hall, New York, New York,USA.

Naim, O. 1988. Distributional patterns of mobile fauna as-sociated with Halimeda in the Tiahura coral-reef complex(Moorea, French Polynesia). Coral Reefs 6:237–250.

Payri, C. E. 1987. Zonation and seasonal variation of thecommonest algae on Tiahura Reef (Moorea island, FrenchPolynesia). Botanica Marina 30:141–149.

Sale, P. F. 1978. Coexistence of coral reef fishes—a lotteryfor living space. Environmental Biology of Fishes 3:85–102.

Sokal, R. R., and F. J. Rohlf. 1995. Biometry, Third edition.W. H. Freeman, New York, New York, USA.

ter Braak, C. J. F. 1985. Correspondence analysis of inci-dence and abundance data: properties in terms of a uni-modal response model. Biometrics 41:859–873.

———. 1987. Ordination. Pages 91–173 in R. H. G. Jong-man, C. J. F. ter Braak, and O. F. R. van Tongeren, editors.Data analysis in community and landscape ecology. Centerfor Publishing and Documentation (PUDOC), Wageningen,The Netherlands.

———. 1990. Update notes: CANOCO version 3.10. Ag-ricultural Mathematics Group, Wageningen, The Nether-lands.

Tilman, D. 1987. The importance of the mechanisms of in-terspecific competition. American Naturalist 129:769–774.


Watanabe, J. M. 1984. The influence of recruitment, com-petition and benthic predation on spatial distributions ofthree species of kelp forest gastropods (Trochidae: Tegula).Ecology 65:920–936.

Whittaker, R. H. 1956. Vegetation of the Great Smoky Moun-tains. Ecological Monographs 26:1–80.

Wright, S. P. 1992. Adjusted P-values for simultaneous in-ference. Biometrics 48:1005–1013.

relating behavior to habitat: solutions to thefourth-corner problem

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