JAMES A. PETE A New Approach

in the Analysis of

Biogeographic Data

SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY NUMBER 107
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SMITHSONIAN CONTRIBUTIONS TO

ZOOLOGY

NUMBER 107

James A. Peters A NeW Approac h

in the Analysis of

Biogeographic Data

SMITHSONIAN INSTITUTION PRESS

CITY OF WASHINGTON

1971
ABSTRACT

Peters, James A. A New Approach in the Analysis of Biogeographic Data. Smithsonian Contributions to Zoology, number 107, 28 pages, 16 figures, 1971.—Similarity

coefficients calculated to show relationships between a number of geographic units

are ranked for each unit, in order of descending values, and these ranked columns are

then compared for all possible pairs of localities. The number of differences or discrepancies in position indicate close biotic relationship if low, or existence of a biotic

barrier if high. The ranking of similarity coefficients permits use of all relationships

known for a single locality. Earlier work established closeness of relationship on a single calculation—the highest known similarity coefficient. Tests of the method were

run on fish collections from transects in the Atlantic and Pacific oceans, and on

herpetological collections from Puerto Rico and neighboring islands. The technique

is described and illustrated. The inadequacies of the numerical taxonomic techniques in dealing with biogeographic data are pointed out. The computer program

used to analyze the data is included.

Official publication date is handstamped in a limited number of initial copies and is recorded

in the Institution's annual report, Smithsonian Year.

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James A. Peters A New Approach

in the Analysis of

Biogeographic Data

Introduction

Biogeographers have always recognized the existence

of areas of the world that can be characterized by the

presence of certain taxa and the absence of others.

The numbers of taxa that need to be present or absent, the kinds of organisms that are used as the basis

for presence-absence data, and the interpretation of

the significance of the groupings are matters of considerable debate and differences between biogeographers. Usually, there has been a rough correlation between the size of the area and the taxonomic level of

the organisms used in its definition. The largest biogeographic area is the realm, which is usually characterized by large taxonomic groups. For example, the

Australian Realm is distinguished by the diversity

and adaptive radiation of the mammalian order Marsupialia, as well as by the occurrence of the subclass

Prototheria. The smallest biogeographical area, often

called the "biotic province," has usually been defined

on the basis of presence or absence of taxa such as

species and subspecies, although genera and even

families may be either characteristic of or restricted

to a particular province.

Part of the difficulty in working with the biogeographic division and subdivision of the world has

been the nature of the data. Biogeographic information tends to come in overwhelming amounts, and

workers usually restrict themselves to a workable

amount of it, either taxonomically or through a

geographic restriction. Even so, the details of distribution patterns of individual taxa can represent almost

James A. Peters, Department of Vertebrate Zoology, National Museum of Natural History, Smithsonian Institution,

Washington, D.C. 20560

uncontrollable magnitudes of data, and short cuts are

attempted even when they recognizably obscure the

data. The advent of computers has begun to break

this kind of logjam, and recent years have seen the

first steps toward quantification of the data to permit

full utilization.

Another difficulty for biogeographers has arisen

from the revolving nature of biogeographical input

versus output. There has been a tendency to accept

earlier conformations of units, against which new

data are tested. The new data, if they fit somewhat

reasonably close to the earlier conformation, are considered as in agreement with it. Thereafter, the new

data are used as additional demonstrations of the

validity of the earlier conformation. A taxonomist who

examines particularly carefully material of a species

collected in a biotic area from which it was not

previously known, in order to discern nomenclatorially recognizable subspecies, is clearly prone to circular

reasoning. If he does give taxonomic recognition to

the material, it immediately becomes significant in

the definition of the biotic area (Peters, 1955:27).

This requires, perhaps, a rather outdated approach

to the recognition of taxa, but nonetheless must be

recognized as a danger in the analysis of biogeographic data.

These difficulties have pushed authors to attempt

to handle biogeographic data in a quantitative manner. The earliest work involved little more than a

calculation of a percentage of taxa shared or not

shared by two areas. The inability to demonstrate

clear-cut relationships using these percentages has

been recognized by most workers. Hobart Smith

(1949:227) anticipated the trend of much of the

work of the following twenty years:

1
SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

Standardization of biotic areas would be extremely difficult

by precise boundary analysis on the basis of amount of

change in fauna per given linear unit (as for instance 1

kilometer). A more reasonable method of standardization

would be the arbitrary establishment of limits of faunistic

distinctiveness of various subdivisions—including biotic provinces—of major regions.

He did not suggest how this was to be done, although

he wrote that

Formulation of such a rule is not the work for any one man.

He proposed that a committee of the Ecological Society of America do the task. While that society did

not see fit to establish a committee, and the work

has not been done as a group function, it is interesting to note that both of the methods of standardization suggested by Smith have been used, and that

individuals have not hesitated to embark on the formulation of the rules Smith felt too much for "any

one man." The work of Webb, Huheey, and Hagmeier, described in detail below, has involved the

first alternative, which Smith felt would be "extremely difficult." The computer has facilitated such

effort a great deal, and has almost eliminated the

difficulties. The advent of numerical taxonomy and

the construction of phenograms using its techniques

has made it possible for authors such as Hagmeier,

Holloway and Jardine, and Kikkawa and Pearse to

establish phenotypic levels, based on cluster analysis,

at which regions, subregions, provinces, and so on are

defined.

Review of the Use of Quantitative Methods in

Biogeography

The first steps toward quantitative analysis were

based on the calculation of various measures of faunal

resemblance between two samples. G. G. Simpson

(1960) summarized die earlier work and discussed

the variable aspects of the different coefficients that

had been used. Peters (1968) provided a computer

program that would calculate any chosen similarity

coefficient for the various combinations of a series

of localities, and Cheetham and Hazel (1969) demonstrated the identity of many of the coefficients

given by Simpson with those used by numerical taxonomists in their construction of similarity matrices.

All additional work in biogeography uses the calculation of similarity coefficients as the building blocks

for more detailed analysis.

A simple technique for using similarity coefficients

in the determination of natural areas was suggested

by Webb (1950). He prepared lists of mammals and

of snakes occurring at a series of sample points arbitrarily set at intervals of 100 miles criss-crossing the

state of Texas. Similarity coefficients were calculated

for neighboring points only, and the value was put

on a map between the two sample points used in the

calculation of the coefficient. Lines were then drawn

on the map connecting areas of equal similarity values,

as is done in constructing a contour map. Webb's

method for determining what species were found at a

particular sample point was to use overlay maps

showing known distributions for the taxa involved,

but actual collections at a series of points could be

used as easily. Webb's method has a particular value

in that it demonstrates quite clearly the existence of

broad transitional zones between the areas of high

similarity values.

A technique very similar to that of Webb was

devised by Huheey (1965). Huheey felt that one of

the major difficulties with the method used by Webb

was that "he measured similarity instead of divergence, which led him to abandon attempts to use

more refined grids and thus improve his accuracy."

Huheey measured the difference or divergence between any two samples, which he defined as "the

number of taxa occurring in either (but not both) of

two areas under consideration, divided by the total

number of taxa in both." Huheey argued that the

use of a similarity coefficient interfered with narrowing the limits between sample points, because the

similarity coefficient simply increased, while the use

of a divergence coefficient permitted more and more

sharp distinction between points as the limits were

narrowed. One might argue that all that the use of

divergence coefficients permits is perhaps easier recognition of low values, since they are identical with

similarity coefficients which are subtracted from the

value 1.00. In any event, Huheey found differences

easier to work with than similarities. Instead of basing his work on occurrence of a taxon at a specific

point, he used a grid of 11 X 20 units, each of which

enclosed approximately 400 square miles. As did

Webb, Huheey used overlays of known distributions

and included a taxon as part of the fauna of a square

if it covered more than half thereof. He then calculated his "divergence factor" for each square and its

four neighbors, and took the average of the four coef-
NUMBER 10 7 3

ficients as the value of the divergence factor for that

square. Again as did Webb, he then connected points

of equal divergence factors with lines he termed

"isometabases." Using this technique, one will find

that areas where barriers exist against species movement will appear on the map as higher areas, and

areas of considerable similarity will appear as valleys.

Huheey found a great deal of agreement between

the map of faunal areas of the state of Illinois, drawn

after using his technique, and earlier maps derived

through the use of nonquantitative techniques.

One of the first biogeographic analyses to use the

techniques of numerical taxonomy was done by Hagmeier and Stults (1964). Their methods were similar

to those of Webb and Huheey. They started with 745

sample points, for which checklists of the mammalian

fauna were prepared, and constructed a map showing number of species throughout North America. A

grid made up of blocks 50 miles square, giving a total

of 2,490 blocks, was superimposed on the first map,

and the number of species whose ranges ended in

each block was determined. An index of faunistic

change (IFC) was computed using the formula

IFC= \00h/n, with L the number of range limits in

a block, and n the number of species occurring within

the block. The result is a percentage of species which

have a range limit within the block. These IFC values

were entered on a map, and "isarithms" (which are

the same as Huheey's "isometabases") were drawn.

The result is shown in their Figure 1, with dark

areas representing areas of faunal barriers, and light

areas representing regions of faunal uniformity. At

this point, they had completed the first stage of their

analysis, in what I consider to be a successful manner.

Several recent authors (Holloway and Jardine, 1968:

155; Kikkawa and Pearse, 1969:823) have pointed

out that the selection of different "primary areas"

may produce different results in numerical analysis,

and Hagmeier himself published a second paper

(1966) in which he pointed out the error in their

results, arising from the selection of only 24 North

American mammal areas. This problem with the results of Hagmeier and Stults lies in the second stage

of their analysis, however, not the first stage, which

was not changed in the second paper.

The second stage of the analysis by Hagmeier and

Stults was an attempt to discover the degree of relationship that existed among a series of mammal

provinces. There was a certain amount of agreement

between their map of the North American continent

showing IFC values and the provinces as previously

described by Kendeigh (1961) and shown in their

Figure 3, and they used the latter as the basis for

their continued analysis. The use of Kendeigh's provinces is the error which Hagmeier (1966) was attempting to remedy. Once the provinces were delimited, however, the authors were irrevocably

committed to an analysis of the relationships between those provinces, regardless of their validity as

biogeographic entities. They did not use numerical

taxonomic techniques in the first stage, but did so

in the second stage. The provinces were subjected

to cluster analysis, using the weighted pair-group

method, and a phenogram was constructed, showing

the degree of relationships between pairs of provinces. It should be recognized that once the primary

areas are established, cluster analysis will give a

stepwise phenogram of smaller and smaller levels

of similarity, regardless of whether the original units

are valid or not.

Hagmeier recognized the difficulty, and did much

to repair it in his 1966 paper. He stated (1966:289)

that the error was corrected by "laying a transparent

overlay over the species IFC map . . . and drawing

lines through all regions of high IFC value, delimiting ultimately a total of 86 (rather than the original

24) primary areas." The 86 areas were used to calculate similarity coefficients, and then subjected to

cluster analysis. All those which clustered at values

higher than 65 percent were considered to belong

to a single province, and the author found that

enough of them combined above that value to reduce

the working number to 38. These were then used as

the primary areas for the final plotting of the mammalian provinces of the continent; they were also

used for further cluster analysis to indicate groupings

into superprovinces, subregions, and regions.

It is possible to argue that the second paper by

Hagmeier falls into the same error as the first, but

I do not think this criticism valid. Hagmeier used

the basic data from his IFC analysis throughout the

second paper, and the provinces were not drawn

from any previous author's concept, but were entirely

derived from the basic distributional data. The problem caused by the use of only a single similarity coefficient, which is discussed more fully elsewhere in

this paper, could not have been overcome by Hagmeier, because techniques did not exist previously
SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

that permitted the utilization of all coefficients known

for a locality, or, in the case of the work by Hagmeier, all primary areas.

Holloway and Jardine (1968) used numerical

analysis in an examination of the distribution patterns

of various taxa in the Indo-Australian area. They

distinguished two approaches to zoogeography. The

first of these is based on the grouping together of

geographical units into larger areas. The dendrograms

from the similarity matrices are based on pairings

of areas (shown in the authors' Figures 1—3). Preston's coefficient of dissimilarity, z (Holloway and

Jardine, 1968:156), was used. The dendrogram is

based on the matrix itself, and no shrinking of the

matrix is employed. This is the "single-link cluster

analysis" method. Once the dendrogram has been

constructed, levels of stepwise discrimination are

drawn to distinguish zoogeographic regions, subregions, and provinces.

The second approach is based on the use of faunal

elements, which are sets of taxa having similar distributional patterns (Holloway and Jardine, 1968:153,

figs. 9-11). The dendrograms are then based on

cluster analysis of the faunal elements. The coefficient of dissimilarity used for the calculation of

the matrix is \-m/n, with m the number of "primary areas" (see below) in which both taxa occur,

and n the total primary areas. The dendrogram is

derived by single-link cluster analysis, as in the first

approach. The clusters formed are designated by

numbers or letters at the base of the stem of a recognizable grouping, with numbers representing faunal

elements, and letters representing subelements.

Both of these approaches share three steps in the

six-stage process used by Holloway and Jardine

(1968:154). These are:

1. A selection of taxa of a given rank from a particular group of organisms is made.

2. Primary areas are defined.

3. The distribution of the taxa selected in (1)

among the primary areas defined in (2) are tabulated.

The second step is clearly a critical one. The

authors point out the difficulties that arose for Hagmeier (1966) when he revised the analysis done by

Hagmeier and Stults (1964) on the distribution of

North American mammals. They point out that in

this case identical numerical methods produced different results, depending upon the method of selection

of primary areas. They then stated (Holloway and

Jardine, 1968:155) that "Our selection of primary

areas for the mainland regions is arbitrary." As an

indication of what they mean by "primary areas,"

the following list gives some of their selections:

Formosa, Java, Indochina, Sula Islands, Philippines,

Australia, and so on. While the authors indicate that

islands such as New Guinea and Java are probably

not homogeneous with regard to their faunal elements, and thus are not satisfactory for this type of

analysis, "the distributional data used were not precise enough to enable us to recognize more than one

primary area for each island." Since the same primary areas were used in both approaches to zoogeographic analysis by these authors, it seems clear

that the choice of primary areas played a significant

role in their results.

Holloway and Jardine concluded that the two

approaches are complementary, and need not be

deemed arbitrary in nature. They suggested that both

intuitive and numerical approaches for grouping primary areas into zoogeographic regions are in themselves mainly of use for descriptive purposes (1968:

186).

In a numerical analysis of the distribution of the

land birds of Australia, Kikkawa and Pearse (1969)

pointed out the difficulties attendant upon the use of

primary areas by earlier authors, and based their

work in 121 selected sites on the Australian continent,

for which they prepared lists of bird taxa. They indicated that "ideally, the areas of similar fauna

should be identified as primary areas on a grid system," but the information available to them was not

accurate enough "to delimit small primary areas

contiguously over the whole continent" of Australia

(1969:823). This led them to select their sites in a

manner that permitted them to "minimize ecological

factors contributing to the resultant classification of

faunal regions." To achieve this, all niches available

at a site must be included in the sample. The authors

(1969:824) felt "if too many sites are clustered in

a small region or too few scattered in a large region,

the relative importance of each region may become

unreliable though the resultant boundaries might not

be altered in the analysis." This seems to be a consequence of the methods used for the analysis of

the data, since ordinarily a large number of wellsampled sites or localities within a single homogeneous faunal unit would be expected to show high
NUMBER 10 7

levels of relationship between each other, and uniformly low levels of relationship with other faunal

units.

Kikkawa and Pearse used the technique of "divisive information analysis" to analyze their data.

This was necessary because of the dimensions of

the data matrix, which included 121 sites and 464

species. They used a technique designed to permit

combining of sites into groups, and then the heterogeneity of each group was calculated, using the

"conventional Shannon-Weiner information content."

I do not clearly follow the technique used nor the

explanation given by the authors, and as a consequence have not been able to use it in my work.

But a final remark by the authors to the effect that

"the program termination was specified arbitrarily

to produce 20 groups for the classification of sites,

and 30 and 20 groups for the classifications of species

and genera respectively" (1969:825) leads me to feel

that there must be a loss of information content

through the use of their technique. Since they start

their final analysis of classification of sites into faunal

areas with only 20 groups of sites, and these 20

groups of sites are then divided into 11 faunal areas

(1969:826, fig. 1), it would appear that the arbitrary nature of selection of the 20 groups is reflected

in a fairly arbitrary set of faunal areas.

The authors construct dendrograms to show relationships between faunal areas, and they plot

boundaries between groups of classified sites "in such

a way as to contain all sites of each group within the

same confines" (1969:825). All of their maps and

dendrograms are based on their calculated "information statistic," but, unfortunately, they do not present

any of the data in a way to permit one to follow

the technique. Even though the reader of the paper

by Kikkawa and Pearse may be frustrated by the

omission of information that he needs to understand

the interpretations presented, it is still significant to

note that this is the first paper in which numerical

methods are used with individual sites or localities

as the basic unit for analysis, and that the results

closely conform to those derived by intuitive methods.

A New Technique for Biogeographic Analysis

In the past, authors have spent many laborious hours

calculating similarity coefficients of various sorts for

all combinations of any two geographic units under

study, whether they be localities, primary areas, biotic

provinces, or something else. At first, the calculated

coefficients were used as the basis for decision as to

the degree of similarity between two units. More

recently these values have been used as the matrix

data for cluster analysis, using the techniques of

numerical taxonomy. Anyone who has gone through

either of these processes is aware of the amount of

time one can devote to the calculation of the coefficients, and the computer is almost an indispensable

prerequisite for cluster analysis, so it is not surprising

that biogeographers continue to work with a series

of coefficients, each of which expresses the relationship between any two units. There is, however, a

large amount of additional information concerning

each of these units that is currently not used—the

data on the relationship of each unit with all the

other units involved in the analysis. The basic concept underlying the new technique described here for

estimating the degree of similarity between any two

localities is that one should take advantage of all the

available data if it is physically possible to do so.

It should also be pointed out that a serious handicap to the utilization of the value found by direct

comparison of the data for any two units is that the

result can be highly biased as a consequence of dissimilar levels of adequacy of our information concerning the biota of each unit. The absence of a

taxon from a particular unit may be a true reflection of the actual situation, but it can also simply

mean that field activity within that unit has not yet

produced a voucher specimen. Only if all field samples are made by the same individual or group can

one be sure that each is made with the same degree

of adequacy and thoroughness, under identical conditions, for the same length of time, and equivalency

of all other conditions that can affect validity of a

sample. There are few, if any, occasions when a biogeographer works with that kind of rigidly controlled

data.

If, however, one were to establish a ranked list

for each locality, based on the similarity coefficients

calculated between that locality and all others, there

would be a greater likelihood that position within

the ranking would express biotic similarity than

would the actual value of any one similarity coefficient. One might assume here that inadequacy of

sampling is basically random and would be reflected

in all coefficients calculated for the relationship of
SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

the locality with other localities. While the coefficient

expressing the relationship with the most closely

related locality might be low because of the inadequacy of sampling, one would expect closeness of

relationship to be indicated by a high position in a

rank of all localities based on coefficients, because

less closely related localities should predictably have

lower coefficients.

It should be added that if one accepts the assumption of inadequacy of sampling as a basically random

phenomenon, one must also recognize that the ranking itself can easily be affected by that randomness.

The ranking of the entire series of localities in a

diminishing series based upon their similarity coefficients with any one locality will not produce an

unchallengable sequence demonstrating diminishing

degree of relationship. It merely indicates approximate

position, and presence or absence of a comparatively

small number of taxa can elevate or reduce the level

of a specific locality to some degree. The advantage

of the ranking by similarity coefficient values is that

one is then using all available information about the

relationships of a locality, rather than only a single

expression of relationship.

Once one has a diminishing rank of similarity coefficients for every unit in his series, it is possible to

go on to the next step in the analysis. The assumption

can be made that if two localities are found within

a single biotic unit (i.e., the taxa collected at each

can be expected to be very similar), the rankings of

all the other localities under each of them should

also be very similar. If the rankings were identical,

one might feel safe in saying that the taxa living

at each were also identical, and that the two samples

had been drawn from the same fauna, flora, or biota.

Were one to find that some localities formed a group

within which they were very similar to each other in

rank comparisons and very different from all localities in another such group, the conclusion that some

kind of biotic barrier exists between these two groups

would be indicated.

The situation described above is shown in Figure 1.

In this hypothetical area, the investigator has collected samples at each of the localities one through

six. He has counted the number of taxa found at

each locality, and has recorded the number of taxa

shared between any two localities. These data have

been used to calculate a similarity coefficient for

each pair of localities. This coefficient is shown

I \ . „*„ 1 1

FIGURE 1.—A hypothetical situation with three localities

in each of two biotic units. Solid arrows connect localities

within a single unit, dashed arrows connect localities in different units. Smaller numbers within arrows are similarity

coefficients.

within a line connecting the two localities. Localities

one through three are shown as included within a

single biotic unit, and the similarity coefficients are

high, as expected. The coefficients for these localities

when compared with localities four through six are

very low, of course, because of the existence of a

biotic barrier interfering with free movement of

members of a species. All coefficients calculated for

each locality are now arranged in a descending rank.

The coefficient value is associated with the locality

number to permit its identification. There will be

five coefficients for each locality. In any comparison

of ranks only four coefficients will be useful. This is

true because each column in a pair includes a coefficient based upon its relationship with the other.

This coefficient cannot be compared with the other

column because there is no equivalent value. This

is shown in Figure 2. In this figure the actual data

from Figure 1 have been ranked under the numbers
NUMBER

CROSS-OVERS:

1 0 7

1

3--8O—

2--79

4--15—

5--12—

6--10—

2

— 3 - -81

1--79

U- -30

i 3- - 24

0--21

0

2-

1-

/>*-

6-

1

3

-81k

-23 1

-21 '

4

t>--TZ—

\Lb--bQ

Y N2--3O^

\ 3--23^

\--12—

5

5

C--68

.3-2 5

^2-2 4

—^1--15

1

6

' 5--68

v ^2--21

1--1O

1

FIGURE 2.—The localities in Figure 1 rearranged to show

relationships. Similarity coefficients involving locality one are

arranged in descending order beneath it, with second locality

number associated with each by dashed line. Solid lines

across the figure connect coefficients for a single locality.

assigned each locality, and lines have been drawn

from one column to the next to connect the series

of coefficients associated with one locality. The intersection of a pair of lines will indicate a shift in

position of two localities within the columns connected by the lines. As long as relative positions are

the same or similar, the number of intersections

should be small, but any comparisons between localities representing different biotic units should show

considerable rearrangement of position and, as a

consequence, should greatly increase the number of

possible line intersections. Sokal and Rohlf (1969:

535) demonstrated that this technique of connecting

similar points and counting the intersections can be

used as the basis for the calculation of a Kendall

Coefficient of rank correlation, although in that particular instance one of the two columns is regarded

as a standard against which the other is compared

(see elsewhere in this paper for a discussion of this

specific problem), and the Kendall Coefficient must

be recalculated for each pair of columns. The

graphic method of presentation shown in Figure 2 is

more satisfactory for presentation of the overall picture, because the area of biotic discontinuity is immediately obvious without the calculation of Kendall

Coefficients. There is nothing sacrosanct about the

sequence in which the columns are presented, of

course, and the arrangement from one to six from

left to right can be revised as much as one wishes.

The object of revision or switching about of the

columns would be to minimize the number of intersections between columns, and we shall see later that

this becomes of primary importance when one does

not have preconceived notions of the relationships

between the localities being investigated.

In the new method proposed here for analysis of

biogeographical data, the basic hypothesis is this:

if one uses as much as possible of the data available

concerning relationships between units, the conclusions drawn from the analysis will be more informative and more reliable. The technique used is to rank

all similarity coefficients for any one unit in descending order, and then compare, either visually or mathematically, the derived rank order with all other similarly derived rank orders.

Between Column Discrepancies

If one wishes to give a visual evaluation of relationships among a series of localities, the technique described above, in which the ranked columns are

listed and with lines connecting the same locality

throughout the sequence of columns, is quite satisfactory. The presentation of the data in that fashion

also facilitates the tabulation of the number of crossovers that have taken place. If, however, the number

of localities to be used is quite large, and there is

no basis for a sequential arrangement of those localities, it becomes difficult to construct the figure and

to count the crossovers. For such analyses, I have

developed a technique of tabulating the discrepancies

that exist between any two columns. If two columns

are identical in the sequence of ranked localities,

there will be no crossovers, all lines will be horizontal

and parallel, and there will be no discrepancies. If

any two adjacent localities were to have their relative

positions switched, there would be one crossover

shown in the figure, and there would be two displacements when the columns were compared. This leads

one to think that the number of crossovers should be

half the number of displacements, so one could

arrive at the number of crossovers without plotting

the figure simply by counting the displacements and

dividing the count by two. This does not work,

because the minimum number of displacements per

crossover is two, but that is not the maximum possible number. The displacement figure is in itself an

accurate measure of the total difference that exists

between two columns, however, and it replaces crossovers as a measure in a run of the computer program

JPFRF. Crossovers can only be tabulated by making

a figure of the data. Displacement data can indicate

the best possible sequence for the columns in a

figure, and such a figure will give the minimum

crossover count if drawn.
8 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

It is, I think, safe to say that this technique of

counting discrepancies is physically impossible if the

number of localities involved in the analysis is much

greater than about 20. In a run of data for 42 localities, I set up the program to print out the ranked

pair of columns for each combination, and then

attempted to count the number of discrepancies directly from the printout. I found that my accuracy

was low, and that each comparison required about

thirty minutes. Since 40 localities would involve

about 800 such comparisons, it is clearly not feasible

to try to do the task by hand. Once the computer

program had been written to do the same task,

however, all 800 comparisons could be done in less

than the time it took me to do one, and could be

printed out in a matrix form to permit direct comparison and evaluation.

The total number of discrepancies or differences

in position between two columns representing any

two localities is recorded by the computer program

JPFRF, which first ranks the similarity coefficients

calculated for any one locality in descending order,

and associates such coefficients with the identification

number of the other locality used in its calculation.

The technique used associates the identity number

with the similarity coefficient as a decimal fraction.

Thus, a value in the ranked data for the fourth

column might be 75.25, which would mean that

the similarity coefficient for columns 4 and 25 is 75.

In the ranked data for column 25 there would be a

value of 75.04, which has the same meaning. In a

comparison of columns 4 and 25, this particular

similarity coefficient would not be used in the tabulation of the total discrepancies, but it would be used

in any comparison of either of these columns with

any other column.

The computer selects two columns to be compared

and first reads the uppermost figure in each (the

procedure can be followed in Figures 4 and 5). Continuing our example, these would be 100.04 and

100.25, for columns 4 and 25. The computer examines the decimal figure of both and ignores each in

further examination of the columns. Next it reads

the first ranked value in column 4, notes its position

in the column, and separates it into its two component parts, the similarity coefficient and the locality

number. Then it searches the other column, which

is 25 in our example, for the same locality, and once

having found it, notes its position. If the position is

the same as for that locality in column 4, it records

no discrepancies, and proceeds to the next value in

column 4. If the position is different, it counts the

number of discrepancies in position. Then it compares

coefficients to see if any ties exist, or, if a deviation

is permitted, to move above or below coefficients

within the tie or the permitted deviation. This movement within ties or the limits of permitted deviation

is always such so as to minimize the total discrepancies. The total number is recorded, after the entire

two columns have been examined, either as a direct

readout of the two localities and the value, or as an

entry into a data matrix.

There are several reasons for using this discrepancy

value rather than one of the several mathematical

coefficients that are available for the comparison of

two columns of ranked data. The most commonly

used of these are the Kendall Coefficient and the

Spearman Coefficient. Either could be used with the

data calculated in this program, but there are some

problems in doing so. First, both of these coefficients

are based on the assumption that the two sets of

ranked data have something directly in common.

In the example given by Simpson, Roe, and Lewontin

(1960:255), the ranking is between the measurements of the first and second molar teeth in a series

of fossil mammals. In the example used by Sokal

and Rohlf (1969:533), the authors compared the

total length of aphid stem mothers and the mean

thorax length of their parthenogenetic offspring. The

Kendall Coefficient is often used by educational

psychologists when comparing the ranks of a group

of students as prepared by two different instructors,

or the ranks of students on two different tests. In

all such cases, one ranking can be held constant, and

the position of the same individual or source in a

second ranking relative to the constant rank can be

determined. There is, however, no such obvious basis

for determination of a "constant rank" in the biogeographic data, since both columns are based on

a set of similarity coefficients calculated on overlapping but not directly correlated data.

The second minor problem encountered in the use

of the data from this program is that the localities

are completely reshuffled in every ranking. This

makes it impossible to use a Kendall Coefficient,

which requires the tabulation of "subsequent ranks

greater than the pivotal rank R" (Sokal and Rohlf,

1969:534), unless the locality identifiers are all
NUMBER 10 7

changed to actual rank, and then in the second

column all the new ranks are identified with the

original numbers. Since, however, the data worked

with in this biogeographic problem do not bear a

direct relationship, there is little logical basis for

rearrangement of either column to check its agreement or divergence from the other. As it works out,

there is no necessity for any rearrangement of the

columns in this way. It can be demonstrated that,

if either column is rearranged or renumbered in such

a way as to give the ranks numbers from one to N,

and the other column renumbered in the same way

to agree with the first, there is no difference in the

total number of discrepancies from what would be

tabulated from the original columns of figures. It

has, however, led me to formulate a slightly different

coefficient, which will use the discrepancy data without any need to renumber the units in the column.

This coefficient will be discussed below.

A third problem in working with these data arises

from the occurrence of "ties." This term is used by

previous workers when the values of more than one

variable are identical, which means they must have

the same rank. It is mathematically accurate, and

customary, to average all the ranks covered by these

identical variables, and then to give each of them

the value of the mean of the ranks. In the biogeographical data, however, the tie cannot be given this

degree of significance. There are too many variables

involved in the preliminary data to permit me to

consider two localities identical in their relationships

because the similarity coefficient for each of them

with a third locality happens to be the same. I

have, therefore, given all tied localities complete

freedom of movement within the span of tied ranks,

and, as discussed above, the computer will move any

locality up or down, within the limits of a tie, to

minimize the number of discrepancies counted for

that locality. Again, since this minimization takes

place in every comparison for all localities, it cannot

affect the final result to any great degree.

A New Rank Correlation Coefficient

These problems with previously used rank correlation

coefficients have led me to devise a slightly different

coefficient, which will enable one to compare results

from different analyses. It is based on the relationship

of the number of discrepancies to the number of localities involved in the study.

If there are no differences between the two columns

of ranked coefficients, there will be no discrepancies

in position of localities, and the discrepancy count

will be zero. If one deliberately arranges the columns

to maximize the total discrepancies, the square of

the number of localities will be exactly twice the

number of discrepancies, if there is an even number

of localities, and the number squared will be twice

the number of discrepancies plus one, if there is an

odd number of localities. In a formula, this would be:

2D = N2 [If N is even]

2D + 1=N2 [If N is odd]

It should be recognized that the maximum number

of discrepancies will occur when one column is

exactly the reverse of the other. In such a case, the

top ranked value in the first column is the terminal

value at the bottom of the second column, and the

last value in the first column is the top value in the

second. This could make a difference in the formulation of a coefficient for the analysis of biogeographical

data, because we need to decide whether we wish to

maximize the significance of the total number of

differences, or if we want to put more emphasis on

the fact that what appears to be developing is a

negative correlation.

Of the two alternatives, I prefer the first, which

will maximize the number of differences. This is

because we are not actually trying to say that a "correlation" exists between any two localities. We are

merely trying to discover which localities should be

included in a recognized biogeographic unit. If any

two localities show a number of discrepancies approaching the maximum number possible, we do not

regard them as "negatively correlated," and we would

simply indicate that they are very different in the

taxa which inhabit them.

The following formula will give a range of values

from zero to one, with zero indicating no similarity

between two localities, and one indicating complete

agreement between two localities:

1 -

2D

N

2

The minor difference between even and odd numbers

of localities is ignored in this formula, although it
10 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

can change a value quite a bit if the number of

localities is small. It should be noted that all values

calculated for a specific set or group of localities will

be equally affected by the same factor, since all will

use the same denominator in the equation.

Test of the Hypothesis

The best test of a hypothesis is to see what happens

when it is applied to a real situation. For a test run,

I selected the data on fishes from the faunal transect

made across the Atlantic Ocean from the southeast

to the northwest, on cruise 17 by the R/V Chain,

Woods Hole Oceanographic Institution, in 1961.

The mesopelagic fish collections have been reported

on by Backus, Mead, Haedrich, and Ebeling (1965),

a group hereafter referred to as the "Chain Gang."

The Chain Gang report is particularly satisfactory

because: (1) it includes a sufficient number of

localities well separated geographically, (2) the authors have identified the fish fauna well enough to

use presence and absence of various taxa without

fear of difficulties from incomplete taxonomic data,

(3) the transect was linear, and (4) the data analysis

presented by the authors indicated that one major

biotic barrier was crossed during the transect. My

analysis is restricted to the series of localities numbered from 800 to 813 (Figure 3). There were 116

species of fishes taken at these 14 localities. The

Chain Gang analyzed the same series of localities,

reducing the number of species to 44 by removing

all but two of those known from only one locality (a

total of 50 species), and all but two of those known

from only two localities (a total of 22 species).

The data were analyzed using the computer program JPFRF, which is discussed in detail later in

this paper. The first run maintained the individual

identities of all localities, so there are fourteen columns of ranked similarity coefficients to be compared.

Since the sequence of localities was linear, there

appeared to be no need for rearrangement, and each

locality was compared only with its immediate neighbors. Many of the different similarity coefficients

described in my earlier paper (Peters, 1968) were

used for running the data. Various allowances for

errors were made, as well, to see how widely this

would affect the result. Figure 4 shows the results of

a preliminary run on all 14 localities, using the Coefficient of Community. In this run, only those

species collected at a locality were recorded as occurring there. It will be readily seen that the technique did not disclose very successfully any traces

of relationships between localities, except perhaps

between localities 11, 12, and 13. Practically identical

results were produced if the data were tested against

any other similarity coefficient.

FIGURE 3.—Stations at which fishes were collected by investigators aboard R/V Chain (from

Backus et al., 1965, fig. 3).
NUMBER 10 7 11

The Chain Gang encountered similar difficulties

in the analysis of their data, and they used a simple

device in their arrangement of the data that eliminated much of the problem, taking advantage of

the linear nature of the collection sites. In their

Table 3 (Backus et al., 1965:157), the 44 species

are listed, and the number of individuals collected

at each site is given. If no specimens were collected

at a locality, but that locality lies between two

localities where the species was collected, a dash is

inserted in the table. The Chain Gang examined

only the locality at which a species first appeared

and the locality at which the same species disappeared from their collections, referring to this as

"first-time and last-time captures" (1965:148). This

is equivalent to making the assumption that, even

though it may not have been taken at every collection site, a species occurs anywhere between the

locality of first-time capture and that of last-time

capture. In the Chain Gang table, then, any occurrence of a dash can be regarded as a locality for

that particular species. It is not difficult to imagine

situations where such an assumption would be dangerous, and it is perhaps not entirely valid for all

species in the Chain Gang data. But, since they have

analyzed their data in this way, I have done the

same in order to make our results comparable. The

results of an analysis with the presence of a species

assumed in any locality between any two known

occurrences are shown in Figure 5. This run used

Simpson's Coefficient for the calculation of similarity, although the Chain Gang mentioned (1965:148)

that they had attempted to use Simpson's index of

faunal resemblance, "but were not successful." Figure 5 shows a distinct change taking place between

localities six and eight, with a group of fairly homogeneous localities below locality six, and a second

homogeneous group above locality seven.

Based on their method of calculation, the Chain

Gang also found a change taking place around

locality 807. They say (Backus et al., 1965:150)

The peak in the interval between collections 807 and 808

(Xs = 3.79) implies a faunal boundary, for here the odds are

about 19 to 1 that the departure of the observed from the

expected is not due simply to sampling error. The value for

the interval 806-807 is also significant (P<.10), suggesting

that the boundary is not an abrupt one.

The Chain Gang was able to correlate this faunal

barrier with ecological factors. They found that

there was a deepening and divergence of the 15°

and 20° isotherms, representing a degradation of

the thermocline and thickening of the surface isothermal layer, apparently correlated with increasing

latitude. They concluded (Backus et al., 1965:152),

". . . that the faunal boundary near Collection 807

corresponds to the boundary between the South Atlantic Central Water and North Atlantic Central

Water Masses."

It was a little disturbing that, in order to achieve

the same results as the Chain Gang, it was necessary

to assume occurrence of the species in localities from

whence they had not been collected. It is an artifact

of the linear nature of this cruise that makes it possible to do so, but most tests of biogeographical data

are not likely to be linear, and such an assumption

would have a high probability of error. At the same

time, the number of species from certain of the Chain

Gang localities is so small that their relationships

with other localities tend to obscure the picture. This

is particularly true of localities 807, with only thirteen

species, and 810, with only seven species. As the map

shows, localities 800 and 801 are very close together,

and could easily be called a single locality, as can

804 and 805. Localities 809 and 810 can be combined equally appropriately, which leaves only the

problem of locality 807. As can be seen from the

map, it is quite distant from both 808 and 806, and

is perhaps inappropriately combined with either. I

have tested it in both combinations, however.

If the original data are used, and actual occurrence

only at any locality is recorded, with presence in

either of two combined localities recorded as presence

in the combination, we can eliminate the effect of

comparatively small samples, and also keep out the

bias of assumption of occurrence where a species has

not been collected. Figure 6 shows that this brings

us much closer to recognition of the distinct boundary

between the combination of 806 and 807 and the

other more uniform areas. When 807 is combined

with 808, the number of crossovers and displacements

increases considerably, indicating that the 806-807

combination is more valid.

One final step can be taken in the analysis of the

Chain Gang data. This involves the use of the combined localities, as well as assuming occurrence of a

species in any locality intermediate between two in

which the taxon is known to occur. In short, this

is the maximum amount of practical manipulation
12 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

15*12

FIGURE 4.—Diagrammatic representation of data from Chain Gang collections, using all 13

stations, with no assumption of occurrence at intermediate stations (see text). The numbers

represent the similarity coefficient, to left of decimal point, and the locality number, to right

of decimal. A value of 100 indicates the locality dealt with in that column. Since the numbers

100.03

X

. 14«S|^^^^40.0

. 08 -"^^^ p 7. 141

I37.O9I

• 15.13- -22.13

FIGURE 5.—Diagram of the relationships of all 13 Chain Gang stations, with a taxon assumed to

be present at any station between two known occurrences. See Figure 4 for an explanation of

numbers.
NUMBER 10 7

100.07

13

have been ranked and printed by the computer just as shown in the figure, the zero in station

"10" does not print, but appears as 100.1. This refers to locality ten, not locality one, which

is 100.01.
14 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

11 12 13

8v yl3 12

FIGURE 6.—Diagram of relationships after combining some

station data. No assumptions are made concerning occurrence. Station A = 800 + 801; B = 804 + 805; C = 809 + 810;

and E = 806+ 807.

that can be done with these data. Figure 7 shows

the sharpness of the distinction of a faunal boundary

at the combination of 806 and 807 (labeled "E" in

the figure). Again, if 807 is combined with 808, the

number of crossovers and displacements increases

considerably.

fc

» c—

,11—

1

12

d

U8 C

E^/1

—c 7S

IK 8

—12<< '3^' 1 3

—A *

1 1

»—

12

-13

rXT

>8—

»—

13

12

•

11

8

ft

FIGURE 7.—As in Figure 6. except occurrence of a taxon

assumed at any station between two known occurrences.

Additional Testing of the Method

As I pointed out above, the analysis of the Chain

Gang data was simplified by the fact that the cruise

transect was linear, and one did not need to be

concerned about the relationships of non-neighbor

localities. Such data are exceptional biologically and

geographically, and it cannot be regarded as typical

of the problems facing biogeographers. It is necessary

to test the method against data from non-linear localities as well. The data presented by Heatwole and

MacKenzie (1967:434) on the herpetogeography of

Puerto Rico and the neighboring islands were easily

converted for testing by the JPFRF program, because

they had already calculated the faunal similarities,

using Preston's coefficient, and published a table

showing these values.

The data presented by Heatwole and MacKenzie

deal with the "major islands of the Puerto Rican

shelf (1967:434). They included eleven islands

(Figure 8). The authors draw certain conclusions

based upon the values of the Preston Coefficient,

making it possible to compare their results with those

drawn from continued analysis of the same data. It

is not possible to start with the original presenceabsence data, because this is not given by the

authors, nor is it possible to use a different similarity

coefficient.

The data were first analyzed using the similarity

coefficient values as given in the table, except that

values of zero were converted to 0.01, and asterisks,

which represent uncalculable values in the table, were

converted to 0.99. All eleven islands in the matrix

were included in the first run, which resulted in some

comparatively high discrepancy values. These appeared to result from variation in the position of the

island called Monito, and it was then noted that

there are only two species of reptiles or amphibians

known from that island, one of which is restricted

to Mona and Monito, and the other found on all

the other islands involved in this analysis. This is

too small a sample to use successfully, and the data

were run again, leaving out the similarity coefficients

given for Monito Island. While this eliminated many

of the distorted values, it did not change the relationships expressed by the discrepancy counts.

The results of the analysis are shown in Table 1,

with the actual discrepancy counts in the lower left

half of the matrix, and the correlation coefficients in

the upper right half. The islands are arranged in

the same sequence as in the original table in Heatwole and MacKenzie (1967:434). The great similarity between the islands, St. John, St. Thomas, and

Tortola, is immediately obvious from the very low

discrepancy values. Culebra also belongs in the same

closely related group, although the values are slightly

higher in each case. Anegada, which lies at some

distance away from all the others, has intermediate

values in all cases except with Culebra. Another

grouping of islands is composed of Puerto Rico, Caja

de Muertos, Vieques, and perhaps Mona. The latter,

geographically quite isolated from the others, is nonetheless closely related to them. If the islands are
NUMBER 10 7 15

ANEGADA

MONITO

b

MONA

TORTOLA J$*

CULEBRA y ^ ^VIRGIN GORDA

ST.THOMAS

VIEQUES

CAJA DE

MUERTO

^TCROIX

FIGURE 8.—Map showing location of islands on the Puerto Rican shelf (adapted from Heatwole

and MacKenzie, 1967, fig. 1).

rearranged to put those showing low discrepancy

counts close to one another, the resulting matrix is

shown in Table 2. The change from low to high

discrepancy values as one passes from one group to

the next is obvious, and the intermediate nature of

the values for Anegada is equally clear. This is interpreted to indicate that this island is not very similar

to any of the others, rather than to indicate that its

fauna is an intermediate one.

Perhaps the most interesting observation to emerge

from this matrix concerns the island of St. Croix.

As pointed out by Heatwole and MacKenzie, this

island is characterized by a high degree of endemism

in its fauna. There are eight species found on the

island, four of which are endemic. As a consequence,

in the table of similarity coefficients given by the

original authors, St. Croix exhibits very low values,

and is sharply set off from the other islands on the

Puerto Rican shelf. This fact has long been recognized by herpetologists working on the Carribbean

fauna, and it is striking to note that the current

analysis, using the ranking technique, results in very

low counts of discrepancies between St. Croix and

the other islands—St. John, Tortola, and St. Thomas.

TABLE 1.—Matrix showing relationships among islands on Puerto Rican shelf

Puerto Rico

St. Croix

Mona .

Anegada

Tortola

St. John

St. Thomas

Culebra

Vieques

Caja de Muertos ....

ico Puerto R ^ \

23

8

12

19

21

20

16

7

4

St. Croix

.54

- \

23

14

7

4

6

9

23

20

Mona

.84

.54

18

22

25

27

19

4

6

Anegada

.76

.72

.64

10

12

13

5

13

12

Tortola

.62

.86

56

.80

^ \

1

2

4

22

20

St. John

.58

.92

50

.76

.98

2

8

24

18

aSt. Thorn

.60

.88

46

.74

.96

.96

^ \

9

25

22

Culebra

.68

.82

.62

.90

.92

.84

.82

18

15

Vieques

.86

.54

.92

.74

.56

.52

.50

.64

\ .

5

Muertos Caja de

.92

60

88

76

60

64

56

.70

.90

The values in the lower left half of the matrix are the actual counts of discrepancies between

any two localities. The upper right half of the matrix shows the values of the new rank correlation coefficients calculated, using the formula described in the text.
16 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

TABLE 2.—Matrix showing relationships as in Table 1, with islands rearranged into

groups as indicated by the discrepancy value

St. John

Tortola

St. Thomas

Culebra

Anegada

Vieques .

Caja de Muerto

Puerto Rico

Mona

Croix 4

7

fi

9

14

20

23

23

23

John -

09

1

2

5

12

24

18

21

25

«

f!0

H

2

4

10

22

20

19

22

Thoma -

9

13

25

22

20

27

lebra

3

u

5

18

15

16

19

egada

G

<

13

12

12

18

:ques

>

7

4

rtos ia de M

o

4 ^

6

0

2

o

u3

OH

The lower triangle includes four islands, the upper includes five (although St. Croix is not a

good fit), while Anegada is intermediate in both directions, and is not a good member of either

group.

This indicates to me that the ranking technique has

another potential value, in that it can give clues

as to the origin or derivation of faunas. My interpretation of the figures seen in Table 2 for St. Croix

is that the endemism that exists on the island tends

to obscure the existence of a considerable similarity

between it and its closest neighbors on the Puerto

Rican shelf. The discrepancy values permit the recognition of this similarity, and indicate that the fauna

of St. Croix is most like the highly similar faunas

shared by the three islands mentioned above, which

lie in a small group very close to one another, and

is either derived from them, or all of them have

derived their faunas from a common source.

It is worthwhile to point out that the ranking

technique appears to work quite well even though the

numbers of species involved are small. Puerto Rico itself has a total of 45 species, but none of the other

islands included in this analysis has more than fourteen, and several of them have fewer than ten. One

would certainly expect this to have a detrimental

effect on the analysis, since addition or deletion of a

single species in an island fauna would change that

fauna by about 10 percent. It is perhaps an indication of the thoroughness with which these islands

have been studied by Heatwole and MacKenzie that

such problems do not arise. One suspects that the

faunal lists are complete and accurate, so the data

is informative even though species counts are small.

It is also important to recognize that in this analysis we are working with a comparatively homogeneous

set of samples. The islands analyzed are part of a

very compact group, not too distant from each other,

and certainly not too different in their respective

faunas. Were one to run an analysis of islands from

throughout the Caribbean, the group covered in this

analysis would appear as a very uniform one, and

quite distinct from other such groups in the overall

region. It is, therefore, indicative of the sensitivity of

the new technique described here that it will pick

out and emphasize the comparatively minute differences existing between these islands.

In both the Chain Gang analysis and the Puerto

Rican study, described above, the results and interpretations of the original investigators were available

for comparison with those derived from the use of the

new method described in this paper. While it is rewarding to find that the technique can verify earlier

conclusions, it will be limited in value if it cannot

be used as an independent, original source of information concerning the biotic relationships as derived

directly from field data. In order to check the technique against previously unanalyzed data, a third test

was run on the fish collection data from cruise 13
NUMBER 10 7

17

of the research vessel Anton Bruun, made off the

west coast of South America in 1966. Giles W. Mead,

the chief scientist on the cruise, kindly made a preliminary summary of the fish collections available

to me for independent analysis. These data (which

have recently been published by Craddock and Mead,

1970: Table 6) were in the form of a table, with the

fish species listed against the stations where the vessel made Isaacs-Kidd midwater trawl hauls. There

was no indication in this table as to any sequence in

which the localities were visited, which made it necessary to base the analysis solely on the presenceabsence data, and eliminated the possibility of prior

bias concerning the results. The collections consisted

of 133 species of fishes from 42 stations.

No assumptions were made concerning occurrence.

Presence or absence at a station was the only datum

used (although the table includes the number of

specimens of each species taken at each station).

The stations were rearranged into ascending numerical sequence to facilitate their comparison using the

computer. Similarity coefficients were calculated for

each pair of stations, and these coefficients were then

arranged in descending order of rank for each station.

A total of slightly more than 800 column comparisons

were then made, and the resulting discrepancy values

were entered in a matrix in which the stations were

again arranged in numerical sequence. As was anticipated, groups of stations showing low discrepancy

values between each other and high values when

compared with stations outside the group were recognizable. The stations forming the groups were

separated within the matrix, however, with very

low-numbered stations combining with very highnumbered stations and forming a single group, while

slightly more intermediate numbers on both ends of

the list formed a second group, and so on. This continued until the stations in the middle range of

numbers formed a single group. This is not the

pattern one would expect if the numerical series

of stations were located along a straight line transect,

which would produce groups within the matrix that

would be delineated as shown in Figure 9. Each triangle includes all the discrepancy values calculated

for pairs of localities within a group. In this illustration, the passage from one biotic area to another is

sharp. In a situation where broad transitional zones

existed, the borders of the triangles would be blurred

at their adjacent corners. When the matrix of discrepancy values resulting from analysis of the fish

collections of the Anton Bruun cruise is examined,

the pattern of relationships between localities is shown

as in Figure 10. There is a group of localities forming

FIGURE 9.—Matrix of localities from a one directional transect. Each triangle includes the localities within a single

biotic area.

FIGURE 10.—Matrix of stations showing kind of situation

that appeared in the analysis of Anton Bruun Cruise 13 fish

data. See text for amplification.
18 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

a triangle of relationships at either end of the matrix,

with a square of relationship shown for this group

in the lower left-hand corner of the matrix, as well.

If one assumes that the numbering of the stations

was sequential in time, and also assumes that any

pair of stations with a very low discrepancy value

was situated in the same biotic area, the ship must

have passed through the same biotic area twice, with

the low station numbers representing an early visit,

and the high station numbers a visit late in the

cruise. Since the intermediate station numbers form

a single group, it would seem likely that the ship

turned to retrace its route while in that biotic area.

From the information derived solely from the list of

fishes as collected at each station, I was able to

determine that the ship had left some specific place,

probably a coastal port, traveled out along a path

that cut across one extremely distinct biotic boundary,

a second boundary of somewhat lesser significance,

and a third that is rather poorly defined. These three

boundaries are shown in Figure 11 as levels on a

phenogram. This phenogram is constructed using the

single-linkage method, using discrepancy values taken

directly from the matrix. The lowest values in the

table are used for the first combination of localities.

Thus, localities 4, 47, 48, 49, 50, 51, 52, and 53 are

all joined together at the level of zero discrepancies,

because each of them has that value in combination

with at least one of the others. The lowest value any

of that group of eight localities has with any other

locality is three, which number 47 has with 46, shown

by a line connecting 46 to the group at the "3 "

level. This technique has been used throughout in

constructing the phenogram. The first and second

groups are not sharply distinct, but do seem to form

groups within themselves, each of which shows

stronger relationships within the group than with

stations in the other group. The third group is more

distinctly set off from the first two, with the lowest

discrepancy value for any station within it compared

with any station outside it indicating 14 discrepancies,

while within-group discrepancy values are fairly uniformly low. The sharpness of the boundary distinguishing the fourth group is shown by the very low

values within the group, with all stations joined in

the phenogram before the level of four discrepancies

is reached, and by the high value that must be

reached for any relationship with stations outside

the group, which is 41 discrepancies. The cruise

track of the ship has been mapped by Mead (1966,

fig. 1), and the stations have been plotted by Craddock and Mead (1970, fig. 1B). Using both of these

as well as the list of stations in Mead (1966, Table

2), I have plotted the positions of their numbered

stations, and shown the four groups derived from my

analysis of the data (Figure 12). I conclude that the

first two groups lie within the area of influence of

the Humboldt Current, and possess a distinctive fish

fauna as a consequence. The third group is either

distinct in itself or transitional. The fourth group is

completely distinct, and probably represents the fauna

of a large, well-defined water mass.

Confirmation of some of my conclusions comes

from Craddock and Mead (1970:3.39), who say,

"The analyses confirmed existence of a faunal break

at about 80 W, just west of the region of maximum

motion and the high salinity, low-oxygen wedge . . ."

This is also shown graphically in their Figure 11,

which shows a sharp peak in the chi-square values

between localities 16 and 17, as well as almost as

sharp at localities 6 and 47. These points are identical

with the ones identified in my analysis.

The Use of Numerical Taxonomic Methods in

Biogeography

As was pointed out before, some authors (as, for

example, Hagmeier and Stults, 1964) have used certain techniques derived from numerical taxonomy as

the basis for analysis of biogeographic regions. One

might wonder why I have not used these numerical

taxonomic methods in my work. It would be simple

to say that I have tried them and found them inadequate and unsatisfactory. I think, however, in light

of the fact that many workers still feel uneasy about

these techniques, without knowing exactly why, I

should attempt to show why they do not work sufficiently well to be useful in biogeography.

To do this adequately, it will be necessary to

review in some detail what is done in biogeographical

data processing if numerical taxonomic methods are

used. The original data consists of records of the

presence or absence of taxa within some circumscribed

geographical unit. In my analysis of the Chain Gang

data, the unit was a single locality, or rather a single

collection site. Hagmeier and Stults ran the earlier

part of their analysis on 50 square-mile blocks, but

when they used numerical taxonomic techniques,
NUMBER 10 7 19

8 45 4Q 44 7 41 1Q 1 S | M ?6 31 30 W 23 21 20 24 2? 17 19 H .

. i i ii i i i i

f

u u u

FIGURE 11.—Phenogram of relationships of stations from Cruise 13, Anton Bruun, based on

single-linkage method. See text.

95°

30°

35°

95°

90° 85° 80° 75°

BO- 75°

70°

28 26 24

v r i

23

J n 21

29 3031

19

 7a

O —i

"x/t

i

6 V / /

| 47 S?- ^

46

3

Ml

•

30"

35°

70'

FIGURE 12.—Cruise track of Anton Bruun, Cruise 13, with stations indicated. The stations

shown here are in the same groups formed in the matrix, Figure 10, and the phenogram,

Figure 11.
20 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

they changed the basic unit to biotic provinces that

had been defined earlier. The sequence of events

involved in cluster analysis, which is the technique

used by Hagmeier and Stults, is shown in Figure 13.

One begins by constructing a matrix that will eventually show various interrelationships between the

geographic units. In Figure 13, the geographic units

are "R," "S," and "T." In the first three-by-three

matrix (or table) we see, on the diagonal, the number

of species that occur at each locality. This value is

found by going through the original presence-absence

data for each locality and counting every taxon

marked "present." There are 40 species in locality

"R," 55 in "S," and 50 in "T." Once these data are

recorded, we reexamine our original data to see how

many species are shared between each pair of localities. The second matrix has this information recorded

in its upper right half, in the larger figures. These

values are found by comparing the lists of taxa for

any two localities, and any time a taxon is found as

"present" in both, it is added to the accumulating

total. The matrix shows that 30 taxa out of the 50

found in "S" are also found in "R," and also that

40 of the 50 known from "T" are included in the

55 found in "S." The third matrix shows the results

of the first calculation involving these data. Using

the formula shown under the second matrix (the

Simpson Coefficient), a similarity coefficient is calculated. The "c" of the formula refers to the number

of taxa in common, and the "ni" refers to the number

of species found at the locality with the smaller number of taxa. There are many other similarity coefficients that could be used (Peters, 1968; Cheetham

and Hazel, 1969), but this is the simplest and, therefore, the easiest to follow through this summary. The

similarity coefficients are the enlarged figures in the

third matrix, which are calculated using the numbers

in the rest of the matrix. Thus, the value of 75 is

arrived at by using the number of species shared

between "R" and "S" (30), dividing that by the

number of species in the locality having the smaller

number ("R," with 40), and multiplying the resultant

.75 by 100. Of the three similarity coefficients calculated, the largest is 80, for localities "S" and "T,"

and is indicated by an arrow.

Up to this point, everything that has been done is

common to the JPFRF program and to cluster analysis. One refinement of cluster analysis, commonly used

in taxonomic studies, is to count the number of situations where an item is missing in both units of a

pair, as well as when it is present, which will increase

the value of a similarity coefficient considerably, but

need not concern us at the present. The next step

in cluster analysis is to "shrink the matrix," a technique commonly used in numerical taxonomic studies

for phenogram construction. The value of "80" is

recorded as the point at which "S" and "T" were

joined, and the two columns devoted to those two

localities in the three-by-three matrix are combined,

using the procedure shown in the left-hand two-bytwo matrix. The number of species shared by "S"

with "R" (30) and the number shared by "T" with

"R" (26) are added, and then divided by two for

the average value, which is 28; this is the new value

used in further work with the matrix, including

additional shrinking. The "28" is an expression of

the number of species shared between "R" and the

combined localities "S" and "T." To find the number

of taxa used in continuing cluster analysis for the

combined localities, the number of taxa found at

locality "S" (55) is added to the number found at

locality "T" (50), and the average taken by dividing

by two. This gives a new value of 53, and this new

value will be used in further shrinking of the matrix.

Were the matrix larger, this procedure would be

followed for the entire set of mutual values between

"S" and "T." Finally, then, a new similarity coefficient is calculated for the relationship between "R"

and the combined "S-T" locality. Since the averaged

value for shared taxa is 28, and the locality with the

smaller number of species is "R" (40), the new similarity coefficient is 70. This is put in the lower left,

where it would, in this example, represent the

value used for the final horizontal connection in a

phenogram.

When the matrix is larger than in my figure, which

shows only the first shrinkage, from three to two in

this case, there are two ways to combine values existing in the matrix. The first uses the averages calculated in earlier steps of shrinkage to find a new average, and is called the "weighted pair group method."

If the values found in the original, full matrix are

used, the method is called the "unweighted pair

group method." These two methods produce what I

consider to be the first failure of numerical taxonomic

methods, because they do not retain the biological

and geographical realities that exist in the original

data. First of all, let us examine the figure that rep-
NUMBER 10 7 21

resents the number of taxa found in the combined

localities "S" and "T." By averaging the known

number of species in the two localities, we arrived

at a new value of 53. But it is immediately clear that

these two localities, after they have been combined,

could not have fewer species than the larger number

found at either independently, which is the 55 found

at "S." No matter how many times "S" is combined

with other spots on a map, it is biologically accurate

to say that any geographic unit including "S" must

be recognized as having at least 55 taxa within its

limits. There are two ways to find out exactly how

many taxa occur within the new geographic unit

that includes both "S" and "T." Either the original

data can be reexamined and the taxa recounted, or

we can simply add together the two known values

and subtract the known number of shared species.

The latter procedure is carried out in the lower

right of the matrix on the right side, with the resultant value of 65. This tells us, of course, that

exactly 65 species are known from the geographic

unit we are now considering, a value quite different

from the 53 coming from the calculated average,

and one that will have a sizable effect on further

calculations of similarity coefficients.

Similarly, it is clear that the number of species

shared by the new, combined geographic unit with

"R" cannot be less than the larger number shared

with "R" by either of the original localities. This is

30 taxa, and the upper right section of the matrix

shows that the new value must be larger than that.

In general, the maximum value possible for the number of species in common would be the sum of the

species shared by "S" with "R" and by "T" with

"R," or 56. This is a very improbable figure, since

the two localities were combined because of a high

R S T R S T R S T

40

55

50

40 30

55

26

40

50

40

75

63

30

55

80

26

40

50

gxlOO

R

S+T

R

40

70

S+T

(26+30H2

28

(50+55)-2

53

B S+T

S+T

40

75

>30

50 + 55-40

65

WPGM Recalculation

D

FIGURE 13.—The first three matrices, A, B, and C, indicate the method of calculating similarity

coefficients. Matrix D shows a single step in shrinking the matrix in C, using the weighted pair

group method (WPGM). Matrix E shows the same step if one returns to the original data,

and establishes accurate figures for the combined localities.
22

similarity coefficient, and thus share many species.

In this case, there is only one way to find out the

actual number of species shared by the new geographic unit with "R." One must go back to the original

data, consolidate presence-absence data for the two

combined localities, and then retabulate the number

of species shared with all other localities. In effect,

one is saying that since these two localities are so

similar as to have the highest similarity coefficient

value within our matrix, we will henceforth treat

them as a single, recognizable unit. This seems biologically defensible to me, and is certainly the assumption made when one works with the fauna of

a "biotic province," as was done by Hagmeier and

Stults.

While doing this by hand would be extremely

tedious—and clearly is the primary reason no one has

done it in previous analyses—the computer facilitates

such data manipulations, doing them so rapidly that

one is unaware of the magnitude of the reshuffling

involved, if the number of localities is high. The twoby-two matrix on the right in Figure 13 shows the

kinds of results to be expected if one recalculates

both the number of species at a locality and the

number of species that new locality shares with others

by going back to the original data strings. The differences that result from calculation using the

WPGM method and the method of recalculation

from the original data are so great that I must consider the WPGM biologically indefensible, even

though it may be demonstrably legitimate mathematically. There is some reason to doubt the legitimacy of the latter, however, in that it involves the

use of averages to calculate further averages, a technique that results in increasing error as it is continued.

The data from the Chain Gang collections were

run through cluster analysis, using both the WPGM

and the recalculation method proposed here, to see

whether there would be significant differences. Figure

14 is the phenogram constructed through use of the

WPGM. The horizontal line connecting two or more

vertical lines is based on the selection of the highest

value in a matrix of similarity coefficients, followed

by shrinkage of the matrix through combination of

the two localities involved. If this phenogram means

anything at all, it is that there is a series of fairly

closely related localities, ranging from 802 to 813,

with 800 and 801 quite distinct from the others and

SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

0 1 2 3 4 5 6 7 8 9 ID 11 12 13

I I

FIGURE 14.—Phenogram resulting from cluster analysis ot

Chain Gang data, using the WPGM.

from each other. This bears no resemblance at all to

the faunal boundary picture derived both from the

original analysis by the Chain Gang or from the

JPFRF analysis.

If, however, the same cluster analysis program is

modified to recalculate the data concerning presenceabsence from the original information and to insert

the recalculated values in the matrix where appropriate, several changes emerge when the same data are

run through analysis. First of all, this cluster analysis

now produces subsequent similarity coefficients higher than earlier values. This is impossible when using

the WPGM, because that method is dependent upon

averaging, and any new value based on averages of

previous values must necessarily be smaller throughout the sequence. This, in fact, is vitally necessary to

permit the construction of a phenogram, as is done in

numerical taxonomy, since the reducing values are used

to indicate a lower level of connection between any

two units. Using the recalculation method, however,

it is to be anticipated that higher similarity coefficients will result as more and more localities are

combined, because the number of shared species can

only get bigger. This means that the value of "c"

in the equation shown in Figure 13 gets larger, and

the fraction gets closer and closer to unity. As a

consequence, the standard phenogram cannot be

constructed. This is shown in Figure 15. The values

given are taken directly from the computer readout,

showing the numbers of the localities which are

combined in each step (812 and 813, in the first

case), and the new value assigned to the combination (15). This is shown in A as vertical lines from

12 and 13, joining at a level of 80, which is the value
NUMBER 10 7 23

of the similarity coefficient, to form a new locality 15.

The same is true throughout the figure, except when

a new similarity coefficient is larger than a previous

one involving localities already in the phenogram.

The first of these instances is the merger of 16 and

20 at a coefficient of 64. Since the merger of "0"

and "1 " had occurred at 60, it is no longer possible

to show the further merger in the standard fashion.

I have modified the phenogram, therefore, by combining 0 to 1 into a single line, doing the same with

4 and 6, and putting the new value of 21 at the

horizontal line. As is seen in B, however, the next

similarity coefficient is still higher, and the following

higher again. In each case, I have combined all

localities previously "clustered" into a single line,

until the situation shown in C is reached. At this

point, where 19 and 24 combine to form 25, there

are two arms on the phenogram, one of which includes all localities from 0 to 7, except 3, and the

other of which includes all localities from 8 to 13,

except 10. In D, we see the final merging of these last

two with the others. While this merging of the arms of

the phenograms certainly obscures the picture and is

quite unsatisfactory as a method of demonstrating the

events taking place, it is worthwhile to point out that

two distinct groups of localities have formed, and that

the groups so formed are the same as those distinguished by the Chain Gang analysis and by the

JPFRF program. I regard this as a third verification

of the validity of the recognition of a faunal barrier at

or near locality 807, and the total absence of any

such indication in straight cluster analysis supplements my statements concerning the inadequancy of

this method of analysis.

In addition to the remarks above, there is another

fact that militates against the use of cluster analysis

for biogeographic data. This lies in the fact that the

similarity coefficients are calculated for the relationships of all localities with every other, but the only

coefficients used in any one shrinkage of the matrix

are the individual values produced by a single locality

compared with another. This requires the assumption

that a high similarity coefficient necessarily indicates a strong relationship between the two localities,

i.e., "A" and "B," which may be true in almost every

case. But it also requires the opposite assumption

12 13 9 8 11 4 6 0 1 12 13 9 8 11 4-1-6 041 2 5

90

80

70 -

60 -

-

15

18

|

19

17

16

21

V

-

-

I

V

22 „

23 ^ 1

B

0-2

4-6

25 . 24

8-9 0-2

11-13 4-7

26 25

FIGURE 15.—The "phenogram" resulting from attempting cluster analysis on Chain Gang data,

if data are recalculated, using original data after each step in matrix shrinkage. See text for

details.
24 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

that a lower similarity coefficient between two other

localities, i.e., "X" and "Y," means that this latter

pair is less closely related, and that may not be true

at all. When all localities are ranked by similarity

coefficient values under A, B, X, and Y, and discrepancy counts made, it is entirely possible that X and

Y will show no discrepancies at all, or that A and B

comparison will produce a higher discrepancy count

than X and Y comparison. This is the reason that the

technique used in the JPFRF program takes advantage

of the similarity coefficients known for all possible combinations of localities, and includes them all in the

ranking procedure. Highest degree of relationship is

correlated with number of discrepancies, not with the

calculated value of the similarity coefficient. The

failure to use all the available data in a cluster

analysis adds to its unsatisfactory nature, since the

computer is available to make it possible to use ranks.

main the same, in the same sequence, for all localities.

If a species is present in a locality, a one is entered

in the data string for that locality at the proper

point for that species in the sequence. If a taxon is

absent, a zero is entered. The data file is complete

when the string of ones and zeros has been recorded

for the last locality on the numbered list.

When the data file is properly constructed and the

file is named in JPFRF 1, the program can be run.

At the present time the program JPFRF is in two

parts, due to restrictions on its length imposed by

the computer system we are using. The first half,

READ DATA

FROM FILE

COUNT NUMBER OF SPECIES

IN EACH LOCALITY

The JPFRF Program

The computer program (JPFRF) used in this analysis

is written in the language BASIC. The initial section

of the program uses the short program published by

me in 1968, and it is still possible to use that paper for

the insertion of a choice of similarity coefficients. The

line numbers and statements remain the same.

The data to be analyzed are stored in a separate

file, to be read when called for by the program. The

name of the file, which can be any combination of

one to six letters or numbers (it is usually not advisable to start a file name with a number), is inserted

in the program by the user, by typing the following:

22 FILES FRFDTA

The "FRFDTA" is the name I assigned to my file of

data, but any name not used elsewhere in stored files

of the user's computer system can be assigned. The

data file consists of the following information: first,

the number of localities; second, the total number

of taxa recorded for all localities; and third, the

identifying numbers assigned to the localities. These

"identifiers" need not be in any numerical sequence,

but must be from 1 to 99, inclusive. Then, for each

locality, in the same sequence in which the identifying

numbers were given, a string made up of ones and

zeros, totaling the number of species recorded for

all localities, is entered. The total species list must reCOUNT NUMBER OF SPECIES

SHARED BETWEEN EACH

PAIR OF LOCALITIES

CALCULATE

SIMILARITY

COEFFICIENTS

ASSOCIATE LOCALITY

NUMBERS WITH

COEFFICIENTS

RANK ALL COEFFICIENTS

IN DESCENDING ORDER

FOR EACH LOCALITY

COMPARE EVERY COLUMN

WITH ALL OTHERS,

TABULATING DISCREPANCIES

EITHER PRINT DISCREPANCIES

LOCALITY BY LOCALITY,

OR PUT ALL INTO A MATRIX

TO BE PRINTED

Flow sequence for programs

JPFRF1 and JPFRF2.

FIGURE 16.—The flow sequence taking place in a run of the

JPFRF program.
NUMBER 10 7 25

JPFRF1, completes part of the data analysis and

records the results in an output file. It gives the opportunity to have a listing of the matrix showing the

number of species at each locality, the number of

species shared between all pairs of localities, and the

similarity coefficients that have been calculated, if

the user wishes. If one is using Peters, 1968, to

change the similarity coefficient to be used in the

analysis, the revised lines should be added to JPFRF1

before it is run, exactly as instructed in the paper.

The second half of the program, JPFRF2, is then

run. This section cannot be run prior to JPFRF1,

because it reads its data from the output file (called

FRFMAT) constructed by JPFRF1, and FRFMAT

does not exist before that run. This half of the program will produce final output either as a listing of

each possible pair of localities with the discrepancy

value tabulated for them, or as a matrix with the

localities identified across the top and down the side,

depending upon what the user wishes to have.

The sequence of events taking place during a run

of the program is shown in Figure 16. Each step in

the sequence is ended by the completion of a new

matrix based on the calculations made during that

step, and it is possible to obtain a listing of that

matrix if the user wishes. This is not built into the

program at the present time, however, and the program will run to completion without intermediate

output, unless the user modifies it.

The program itself follows:

19 DIMX(100),C(100)

20 DIM S(35,35), A(200,2), P(35)

21 LETC = O

22 FILES FRFDTA

23 READ#1,Z

24 READ#1,K

30 IFOOTHEN39

31 FORI = 1TOZ

32 LETJ = O

33 READ#1,S(IJ)

34 LETS(J,I)=S(I,J)

35 NEXTI

36 LETC = C + Z + 2

37 FOR B1 = 1TOZ

38 FILES FRFDTA

39 FORH=1TOC

40 READ#1,X

45 NEXTH

47 LETC = C + K

55 LET R = O

60 FORI = OTOK—1

65 LETJ = O

70 READ#1,A(IJ)

75 IFA(IJ)=OTHEN85

80 LET R=R+ 1

83 LET P(B1)=R

84 LET S(B1,B1)=P(B1)

85 NEXT I

90 IF Bl=ZTHEN250

95 LETB2 = B1 + 1

96 LET Q = O

100 FORI = OTOK—1

105 LETJ=1

110 READ#1,A(IJ)

115 NEXTI

120 FORI = OTOK—1

121 LETJ = O

125 LETN = A(IJ)

130 IFN = OTHEN150

135 LETM = A(IJ + 1)

140 IFNOMTHEN150

145 LETQ = Q+ 1

150 NEXTI

153 LETS(B1,B2)=Q

155 LETB2 = B2+1

160 IFB2 = Z+1THEN175

165 GOTO96

175 NEXTB1

250 LETN = O

254 LETW = Z

257 FORI = 1TOZ+1

260 LETN = N+ 1

263 IFN = ZTHEN331

264 LETB = P(N)

267 LETT = N

270 FORJ = I+1TOZ

273 IFI = JTHEN320

275 LETT = T+ 1

280 LETC = P(T)

300 LETV = S(IJ)/(B + C-S(IJ))*100

310 LETV = INT(V + .5)

315 LETS(J,I) = V

320 NEXTJ

325 LETW = W- 1

330 NEXTI

331 FILES FRFMAT

333 WRITE #1,Z

335 FORI = OTOZ
26 SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

340 FORJ = OTOZ

345 WRITE#1,S(IJ)

350 NEXTJ

355 NEXTI

505 PRINT'WANT PRINTOUT OF MATRIX?

(0 = NO, 1=YES)"

506 INPUT E

507 IFE = OTHEN 10000

510 FOR H = O TO Z—1 STEP 10

520 FORI = OTOZ—1

530 FOR J = H TO H + 9

540IFI=JTHEN570

550 PRINTS (I,J);

560 GOTO580

570 PRINTP(I) ;

580 IFJ = Z—1THEN600

590 NEXTJ

600 PRINT

610 NEXTI

620 PRINT

630 NEXTH

10000 END

JPFRF2

15 DIMS(35,35)

18 FILES FRFMAT

19 READ#1,Z

20 FORI = OTOZ

21 FORJ = OTOZ

22 READ#1,S(IJ)

24 NEXTJ

26 NEXTI

32 FORI = 1TOZ

34 FORJ=1TOZ

36 LETS(IJ)=S(J,I)

38 NEXTJ

40 NEXTI

42 LETI = O

44 FORJ = 1TOZ

46 LETA(IJ-1)=100+(S(IJ)/100)

48 NEXTJ

50 FORI = 1TOZ

52 LETH = O

54 FORJ=1TOZ

56 IFJ = ITHEN66

57 LETH = H+1

58 IFJ>ITHEN64

60 LETA(H,I-1) =S(IJ) + S(J,O)/100

62 GOTO66

64 LETA(H,I-1) =S(IJ) +S(OJ)/100

66 NEXTJ

68 NEXTI

70 FORI = OTOZ

72 FORJ = OTOZ

74 LETS(IJ)=A(IJ)

76 NEXTJ

78 NEXTI

90 FORJ = OTOZ—1

100 FORI = OTOZ— 1

105 LETX1=I

110 LETP = S(X1J)

120 LETM = X1

130 LETX1=X1 + 1

140 IFM+l=ZTHEN260

150 LETN = S(X1J)

160 IFP>NTHEN240

170 IFP = NTHEN240

180 LETX1=X1-1

190 LETS(X1J)=N

200 LETS(X1 + 1J)= P

210 IFM = OTHEN250

220 LETX1=M—1

230 GOTO100

240 LETX1=M

250 NEXTI

260 NEXTJ

270 FORJ = OTOZ —2

280 LETX2=J

285 LETD = J

290 LETC = S(O,X2)

300 LETB = INT(C)

310 LETC = INT(((C —B)*100) +.5)

320 LETA = X2

330 LETX2 = D + 1

340 LETG = S(O,X2)

350 LETH = INT(G)

360 LETF = O

370 LETG = INT(((G—H)*100)+.5)

380 PRINT

400 FORI = 1TOZ-1

405 LETX1 = I

410 LETX2 = A

420 LETN = S(X1,X2)

430 LETM = INT(N)

440 LETN=(N —M)*100

450 IFABS(N-G)<.001THEN610

460 LETT = XI

470 LETK = —XI
NUMBER 10 7 27

480 LETX1=T + K

490 IFX1>Z—1THEN580

500 LETX2 = D

510 LETQ = S(X1,X2 + 1)

520 LETP=INT(Q)

530 LETQ=(Q—P)*100

540 IFABS(N—Q) <.001THEN570

550 LETK = K+1

560 GOTO480

570 GOSUB690

580 LETK = ABS(K)

590 LETF = F + K

600 LETX1=T

610 NEXTI

620 PRINT'LOCS" ;C ;"AND" ;G;"—" ;F

630 LETD = D+1

640 IFD = Z—1THEN660

650 GOTO330

660 LETX2 = A

670 NEXTJ

680 GOTO 10000

690 IFK = OTHEN990

700 IFK>OTHEN770

710 LETR = INT(S(X1 + 1,X2 + 1))

720 IFABS(P—R)>.001THEN830

730 LETX1=X1 + 1

740 LETK = K+1

750 IFK = OTHEN990

760 GOTO710

770 LETR = INT(S(X1 —1,X2 + 1))

780 IFABS(P-R)>.001THEN910

790 LETX1=X1 —1

800 LETK = K—1

810 IFK = OTHEN990

820 GOTO770

830 LETX1=T

840 LETX2 = A

850 LETU = INT(S(X1 —1,X2))

860 IFABS(M-U)>.001THEN990

870 LETX1=X1 —1

880 LETK = K+1

890 IFK = OTHEN990

900 GOTO850

910 LETX1 = T

920 LETX2 = A

930 LETU = INT(S(X1 + 1,X2))

940 IFABS(M-U)>.001THEN990

950 LETX1=X1 + 1

960 LETK = K—1

970 IFK = OTHEN990

980 GOTO930

990 RETURN

9000 FORI = OTOZ

9005 FORJ = OTOZ

9010 PRINTS (IJ);

9015 NEXTJ

9016 PRINT

9020 NEXTI

9025 RETURN

10000 END

Literature Cited

Backus, R. H., G. W. Mead, R. L. Haedrich, and A. W.

Ebeling

1965. The Mesopelagic Fishes Collected during Cruise

17 of R/V Chain, with a Method for Analyzing

Faunal Transects. Bulletin of the Museum of Comparative Zoology, 134:139-158.

Cheetham, Alan H., and J. E. Hazel

1969. Binary (Presence-Absence) Similarity Coefficients.

Journal of Paleontology, 43:1130-1136.

Craddock, James E., and Giles W. Mead

1970. Midwater Fishes from the Eastern South Pacific

Ocean. Anton Bruun Report Number 3:3.3—3.46.

Hagmeier, Edwin M.

1966. A Numerical Analysis of the Distributional Patterns of North American Mammals. II. Re-evaluation of the Provinces. Systematic Zoology, 15:

279-299.

Hagmeier, Edwin M., and C. Dexter Stults

1964. A Numerical Analysis of the Distributional Patterns of North American Mammals. Systematic

Zoology, 13:125-155.

Heatwole, Harold, and Faustino MacKenzie

1967. Herpetogeography of Puerto Rico. IV. Paleogeography, Faunal Similarity and Endemism. Evolution, 21:429-438.

Holloway, J. D., and N. Jardine

1968. Two Approaches to Zoogeography: a Study Based

on the Distributions of Butterflies, Birds and Bats

in the Indo-Australian Area. Proceedings of the

Linnaean Society of London, 179:153-188.

Huheey, James E.

1966. A Mathematical Method of Analyzing Biogeographical Data. I. Herpetofauna of Illinois.

American Midland Naturalist, 73:490-500.

Kendeigh, S. C.

1961. Animal Ecology. Englewood Cliffs, N.J.: PrenticeHall. 468 pages.

Kikkawa, J., and Kay Pearse

1969. Geographical Distribution of Land Birds in Australia—A Numerical Analysis. Australian Journal

of Zoology, 17:821-840.
28

Mead, Giles W.

1966. Cruise Report, Research Vessel Anton Bruitn,

Cruise 13. Marine Laboratory, Texas A&M University, Galveston, Special Report Number 3:1-3,

4 tables, 4 figures, 2 appendices.

Peters, James A.

1955. Use and Misuse of the Biotic Province Concept.

The American Naturalist, 89:21-28.

1968. A Computer Program for Calculating Degree of

Biogeographic Resemblance between Areas. Systematic Zoology, 17:64—69.

Simpson, George G.

1960. Notes on the Measurement of Faunal Resemblance. American Journal of Science, Bradley

Volume, 258-A:300-311.

SMITHSONIAN CONTRIBUTIONS TO ZOOLOGY

Simpson, George G., Anne Roe, and Richard C. Lewontin

1960. Quantitative Zoology—Revised Edition, vi + 440

pages. New York: Harcourt, Brace, & World.

Smith, Hobart M.

1949. Herpetogeny in Mexico and Guatemala. Annals of

the Association of American Geographers, 39:

219-238.

Sokal, Robert R., and F. James Rohlf

1969. Biometry, xxi+776 pages. San Francisco: FreeWebb, William L.

1950. Biogeographic Regions of Texas and Oklahoma.

Ecology, 31:426-433.

* U.S. GOVERNMENT PRINTING OFFICE: 1971 »37—339/IO


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