Z. zool. Syst. Evo1ut.-forsch. 29 (1991) 87-96 0 1991 Verlag Paul Parey, Hamburg und Berlin ISSN 0044-3808

Received on 10. June 1990

Institut fur ZooIogie der Universitat Wien, Osterreicb

Character coupling for taxa discrimination: a critical appraisal of quadratic assignment procedures (QAP)'

By H. L. NEMESCHKAL

Abstract

Taxa can be characterized by character coupling represented in similarity matrices. The custom- ary methods of testing equality of variance-covariance matrices are based u on the multinormal- ity assumption which is, however, frequently unacceptable in reality. Quagat ic assignment pro- cedures (QAP) have proved to be an alternative. They represent a type of computer-based test and utilize a random-permutation stratefy to ,discover significant f t tern correspondences between matrices. A comparison of the app icability of both testing met ods requires an example with underlying multinormality. The samples of two species of land snails (Pulmonata, Helicidae), i. e. Arianta arbustorum (n = 104) and Arianta chamaeleon (n = 36), fulfil this require- ment. Four parameters of shape and two parameters of spiral change were determined in each shell. The data serve as the basis for similarity matrices (variance-covariance, product-moment and rank order correlations).

The inspection of methods reveals that Q A P are suitable for correlation matrices, but can be applied for variance-covariance matrices with limitations only. Nevertheless, they are recom- mended procedures in taxonomy and evolutionary biolo y. Straightforward application, inde- pendence from distributional assumptions, and the possi%ility to test hypotheses of character coupling are advantageous features.

The snail species are significantly discriminated by character coupling. Also, their parameters of shape and spiral change are morphologically integrated in a different way.

Key words: Assignment statistics - Morphological integration - Taxonomy - Land snails - Arianta -Shell characters

Introduction

Either below or immediately above the species level, taxa classifications as well as indi- vidual identifications occasionally depend o n quantitative characters. Since these charac- ters are quite frequently represented by a considerable number of variables, multiple treat- ments are necessary. Although the techniques of multivariate statistics, e. g. analyses of discrimination and identification, seem to be adequate methods, there are certain reserva- tions against their unlimited application.

On o n e hand, linear discriminant analyses always require multivariate normal distribu- tions as well as equality of the variance-covariance structures in the groups of individuals being compared (FLURY and RIEDWYL 1983; MORRISON 1986; SNEATH and SOKAL 1973). As is well k n o w n from taxonomic practice, we must stress that in most cases either the assumption of multinormality does not hold or the assumption is not testable. Proof of multinormality is a rare exception (NEMESCHKAL and KOTHBAUER 1988; NEMESCHKAL 1990). If we assume an underlying multinormal distribution, however, a specific taxon is defined by its centroid and the variance-covariance matrix (NEMESCHKAL 1990). Neverthe- less, two supposed taxa can be very similar in their means; therefore it may be doubtful

Dedicated to 0 . Univ. Prof. Dr. FRIEDRICH SCHALLER on the occasion of his 70th birthday in August 1990.

U. S. Copyright Clearance Center Code Statement: 0044-3808/91/2902-0087/$02.50/0

88 H. L. Nerneschkul

whether an unambiguous assignment of an individual to either of the taxa is possible. In such cases differences between taxa often emerge exclusively from the different linkage of characters. The variance-covariance matrices immediately reflect these differences. In summary, linear discriminant analyses are only applicable in taxonomic use with some restrictions.

O n the other hand, a complicated theoretical framework is considered to be the main obstacle for a more frequent employment of multivariate statistical techniques in biology. Although recently an increasing number of software packages have become available, even now good results are rarely achieved without an extensive knowledge of statistics.

Quadratic assignment procedures (= QAP) are an obvious alternative (HUBERT 1987; MANTEL 1967; MANTEL and VALAND 1970). In the following - being compared with the model of multinormality - the application of QAP in taxonomic use is critically demon- strated using the example of European land snails (Mollusca, Gastropoda, Helicidae). Shell characters in two species, i. e. Arianta arbustorum and Arianta chameleon, provide the basis for the approach.

I I

C P C I 69,5%tv PCIL 11,3%tv

I

la l b

Fig. 1. Shell shape and sha e parameters in Ariantu. a: The curved line between the oints A, N, C and B is designated as skape. The shell axis is marked by a vertical broken line. 1: The shape parameters are represented by 4 princi a1 components ex lainin about 93 % of total variance (tv). Participations of components in $ape are indicated &y thicaened lines. PCI = lower seg- ment, PCII and PCIII = mixed participations, PCIV = upper segment

Materials and methods

The analyzed snail shells belong to the Naturhistorisches Museum Wien. The museum collection of these s ecies was sampled random1 (n - 104 individuals of Ariantu urbustomrn and n = 36 individuai of Ariantu churnadeon). Tie shells were drawn using a stereo microscope and the image data were subsequently transmitted by a digitizer to a personal computer (Commodore Amiga 1000) for calculations (further detailed informations in NEMESCHKAL and KOTHBAUER 1988, 1989).

The followin chonchological characters are the basis for the analyses: a. shell shape (aEbreviated as ‘shape’): The curve between the points A, N, C and B (Fig. la) is

designated as shape. It represents the projection of a complex three-dimensional curved line (NEMESCHKAL 1990). The shapes were scanned with high precision to cover as much image information as ossible. Nevertheless, a suitable method was re uired to simplify the re resentation of &a structures produced b scanning. With regar] to character intercorreE1 tions, a PCA (= rincipal component anahsis) may substitute a new set of few variables for the original set ofnumerous variables without a considerable loss of information.

The reduction of original data by PCA resulted in the extraction of 4 new axes re resent- ing shell shape parameters. Together they explain about 93 % of total variance. &o axes

Character coupling for taxa disuimination 89

c5

almost perfectly correspond with two geometrically different segments of shape (Fig. 1 b): Whereas PCI corresponds with its ‘lower segment’ (= silhouette segment), PCIV corres- gonds with its ‘upper segment’. PCII and PCIII, however, consist of mixed participations in

b. spiral ckange: The spiral change is represented by the vertical (VESPI) as well as horizontal (HOSPI) spiral alterations, i. e. the alterations in the direction of the shell axis and normal to it (NEMESCHKAL and KOTHBAUER 1989). VESPI and HOSPI are shell parameters.

In the following, the parameters of shape (PCI, PCII, PCIII and PCIV) are equated with the symbols c l to c4, the parameters of spiral change (VESPI and HOSPI) with c5 and c6. These six parameters will be used as variables for matrix calculations (Tables 1-4).

0th se ments (Fig. 1 ; further details in NEMESCHKAL 1990).

c6

The example

European land snails are predominantly classified according to their shells (KERNEY et al. 1983). Certain taxa, e. g. Arianta arbustorum and Arianta chamaeleon, are characterized in particular by an outstanding variability of shell features (KLEMM 1973). In such cases it is not surprising that numerous varieties have been picked out by taxonomists based on con- spicuous individuals. The varieties have frequently been assigned the rank of subspecies (races) or species. Most have been defined in typological manner, based to a great extent on the personal impressions of specialists (MILDNER 1981). According to a biological species concept, however, such conventional classificatory splitting must be abandoned (GIUSTI et al. 1986). In comparing conchological characters within A. arbustorum, many of the previously designated subspecies have proven to be merely varieties of the single species (NEMESCHKAL and KOTHBAUER 1988, 1989). Furthermore, neither A. arbustorum nor A. chamaeleon can be separated as a whole, nor are distinct individual assignments possible using shell features. Up to now only a slight tendency of separation as indicated by differences in variance-covariance structures of shell shape has been observed (NEMESCHKAL 1990).

In the following, A. arbustorum and A. chamaeleon, abbreviated as ‘ARBU’ and ‘CH’, are the units of investigation.

Table 1. Variance-covariance matrices

c l

12.18805

c l 16.81001 c2 1.89235 c3 0.46223 c4 0.16027 c5 0.10784 ~6 - 0.59260

J L . 2 . - 1.32397 7.161 12

12.35596 3.13029

- 1.29755 - 1.11342

0.13932

Ic3 1.38147

- 1.12163 3.71490

10.38331 0.18561

- 1.14364 0.77048

c4 - 0.31289

- 0.05546 0.45065

2.53957

7.60960 - 0.08774 - 0.01609

2.45455 0.13164 0.49961

I c l c2 c3 c4 c5 c6

I CHvc:

Note that for correct variance-covariance values the table entries must be multiplied by Table 1 up to Table 4 are identical in the following respects: The coefficients of similarity matrices are calculated between shell characters, i. e. c l to c6 (further explanations see ‘mate- rials and methods’). Two semimatrices are joined together in one table. The upper half corres- ponds to ARBU, the lower half to C H .

90

150-

f -

100-

H. L. Nemeschkal

6 m OD_

0)

0 I

E (D

T

Principles of QAP

Although primarily elaborated for handling distance matrices (DIETZ 1983; Dow and CHEVERUD 1985; SOUL 1979), quadratic assignment procedures are equally applicable to similarity matrices, since the QAP universally detect pattern correspondences between matrices (CHEVERUD 1989; CHEVERUD et al. 1989; WAGNER 1990). Furthermore, some ex- tensions already exist for multiple treatments (Dow et al. 1987; SMOUSE et al. 1986).

In the present case a simple Pearson product-moment correlation serves as the measure of correspondence between two similarity matrices, e. g. variance-covariance matrices of ARBU and C H (Table 1, Fig. 2). The correlation coefficient crma is calculated between identical pairs (= matrix elements) from both initial matrices, and is called the observed correlation.

urma Fi . 2. Distribution of permutation correlation values, comparing variance-covariance matrices ofARBU and CH. The following a plies to Fig. 2 through Fig. 4. : The histogram combines the results of 10 QAP-runnings. Solid gars represent means of cell frequencies, thin bars over cell midpoints mark standard errors. The observed correlation value is indicated by an arrow. For abbreviations see 'appendix'

In generating a distribution of permutation correlations by Monte Carlo methods, the significance of an observed correlation can be tested. The null hypothesis assumes that there is no correlation between the two matrices being compared. A distribution of permu- tation correlations is then obtained as follows: One matrix, e. g. variance-covariance matrix CH, is repeatedly modified by random permutations of the rows as well as by simultaneous permutations of the columns; the other matrix, e. g. variance-covariance matrix ARBU, remains unmodified. After each permutation, a coefficient of correlation, i. e. prma, between matrices is calculated and participates in accumulating the distribution. As a result the probability level (= P-value, P) of the observed correlation is determined. P is the final proportion of permutation correlations for values greater than or equal to crma. For the significance testing of crma, P is crucial inasmuch as the null hypothesis will be supported if more than 5 % (1 "A) of permutation correlations exceed the observed correla- tion. Otherwise the null hypothesis is rejected, and consequently matrix similarity must be accepted as the alternative.

The calculations are based on a random selection of permutations, since complete per- mutational series of matrices with numerous dimensions are too tedious for computation. Typically, 500 to 1000 selected permutations are sufficient to estimate the probability level of crma (CHEVERUD et al. 1989). Although in the present case a complete enumeration of all 6! permutations only amounts to 720, 1000 permutations has been randomly selected and employed each time for the purpose of demonstration.

Character coupling for taxa discrimination

c i i cz

91

c3 c4 c5 c6

Results

c l

Limitations of QAP

The variance-statistical relations between characters are represented in different ways (Tables 1-3), i. e. by covariation (CV), Pearson’s product-moment correlation (PM) as well as Spearman’s rank order correlation (RK).

In order to determine whether ARBU and C H are significantly different in their char- acter couplings, we may test the equality of variance-covariance matrices by means of a generalization of the Bartlett test (MORRISON 1986, p. 252) . Multinormal distributions, however, are required for this purpose. As already proved (NEMESCHKAL 1990), the as- sumption of multinormality is tenable for both groups. Furthermore, the employment of variance-covariance matrices makes sense because the shell parameters are ratios without physical dimensions; consequently they are mutually comparable variables. All things considered, the Bartlett test statistic is x2 = 59.2% As this result exceeds by far the upper 1 percent critical value of chi-square distribution with 21 degrees of freedom, we must accept the hypothesis of significant difference between matrices. This outcome unexpec- tedly contradicts the results obtained by QAP.

In comparing the variance-covariance matrices of ARBU and CH, the assignment pro- cedure results in the acceptance of matrix similarity (Table 5 ; Fig. 2 ) , whereas for the corre- lation matrices (Table 5; Fig. 3) the null hypothesis of no matrix similarity must be ac- cepted.

Due to the introduction of a certain amount of noise into the observed correlation crma by estimation error, the null hypothesis of no matrix similarity will be accepted more easily (CHEVERUD et al. 1989). In the present case, however, the assignment procedure lays stress upon the opposite: it suggests matrix similarity even though the multinormality model - as a more specific statistical instrument - favors matrix dissimilarity. If the pairs of similarity matrices, either in ARBU or in CH, are compared within the same matrix, the

c2

Table 2. Pearson’s product-moment correlation matrices

c5 c6

c6

c4 I_ c3

0.22207 - 0.05183 - 0.30093 0.15248

0.06939 0.16396

0.24273 0.08391 - 0.13090 - 0.03269

92 H. L. Nemeschkal

variance-covariance matrices prove to be quite different from correlation matrices by the ranges of pair values. Wide ranges in variance-covariance matrices seem to generate a certain pattern which is dominant enough for the recognition of matrix similarity by QAP. Perhaps the assignment procedures are oversensitive to pattern similarity due to the lack of any distributional assumption.

To overcome these difficulties we may first minimize the proportion of noise in crma. The extent of estimation error decreases with decreasing numbers of characters and with increasing individual numbers (CHEVERUD et al. 1989). In the case when matrix dissimilar- ity is indicated although matrix similarity actually exists, we may increase the number of individuals to minimize objections. In the reverse case, however, when similarity is suggested instead of an actually existing dissimilarity, the effect of pattern dominance must we weakened by a standardization of matrix pairs, e. g. by using correlation coefficients as measures between characters (Table 2, 3).

Morphological integration

Character coupling means that within a similarity matrix, complexes of variables (‘P-sets’ in CHEVERUD 1982; ‘MKOVAGs’ in NEMESCHKAL and ELZEN 1990) exist which are com- posed of two or more, very strongly intercorrelated traits. Mutual correlations between characters from inside a certain complex are stronger than correlations with characters from outside the complex. Characters linked on a high level of correlation are said to be

prwa Fig. 3. Distribution of permutation correlation values, comparing pro- duct-moment correlation matrices of ARBU and CH

- ‘integrated‘ (see ‘morphological integration’ in OLSON and MILLER 1958; CHEVERUD 1982). Al- though in most cases it is quite difficult to provide an immediate interpretation of integrated sets (RISKA 1986), sometimes these sets can be inter- preted as cofunctional units (ELZEN et al. 1987; NEMESCHKAL and ELZEN 1990) using biological, e. g. ecological, chorological and ethological argu- ments (LEISLER and WINKLER 1985; WINKLER 1988). 2003

f -

150-

T

prmr Fi . 4. Distribution of permutation correlation vafues, comparing the product-moment correlation matrix of ARBU with the theoretical matrix of ARBU

Character coupling for taxa discrimination

Table 4. Theoretical matrices

c l

93

c2 c3 c4 c5 c6

ARBUT:

ARBUvc A R B U ~ M ARBURK A R B U ~ M A R B U ~ M C H ~ M CHPM

~

crma 0.88991 0.90406 0.89180 0.76859 0.61635 0.82969 0.56168 Meanof P-value 0.01040 0.18640 0.38970 0.00150 0.27480 0.00180 0.39660 SE of P-value 0.00124 0.00570 0.00800 0.00034 0.00479 0.00047 0.00786

1000 permutations per QAP-running and 10 QAP-runnings per matrix comparison have been performed; d. f . = 9. The subscript of matrix name indicates the type of similarity matrix. For abbreviations see ‘avvendix’.

First of all, taxon discrimination by character coupling has priority in taxonomic prac- tice. Beyond the simple finding of a universal taxon difference, it is of interest to determine the exact reasons for different matrix patterns.

In order to elucidate an underlying coupling pattern, the display of a slightly exagger- ated hypothetical pattern proposed in a so-called theoretical matrix (matrices ARBUT and CHT in Table 4) seems to be very useful. Such a theoretical matrix is arranged as a special correlation matrix. A proposed linkage between traits is indicated here by ‘I’ or by ‘-1’, whereas a proposed lack is marked by ‘0’. In the following, the matrices with product- moment correlations (Table 2, 4) are the basis for the composition of theoretical matrices and for interpretations.

As the assumption of multinormality is tenable for ARBU as well as for CH, the statis- tical significance of a bivariate correlation coefficient between characters can be tested (ARBU: d.f. = 102, p = 0.05, critical value of r = 0.19; CH: d.f. = 34, p = 0.05, critical value of r = 0.325). Whereas in ARBU four linkages, i. e. cl/c6, cl/c5, c2/c3 and ~ 3 1 ~ 6 , exceed the critical value of r, in C H only one linkage, i. e. c3/c6, is significant. Neverthe- less, correlations close to the critical value are also accepted as essential in the matrix struc- ture of CH.

Both groups, ARBU and CH, agree that there is neither a significant correlation between the parameters of spiral change (see also in NEMESCHKAL and KOTHBAUER 1989) nor a dependence of the fourth shell parameter (= c4, upper segment of shape; Fig. 1) on any other trait. In both groups, too, the shape parameter c3 is closely and similarly con- nected with the parameter of horizontal spiral change, i. e. c6, as indicated by significant correlations with a positive sign.

O n one hand, the connections of c3 with ct as well as with c2 can be specified as main differences between ARBU and CH with respect to shape parameters. O n the other hand, the groups additionally differ by the linkage of shape parameters with parameters of spiral change. Whereas in ARBU the ‘silhouette’-parameter (cl) is connected with vertical spiral

94 H. L. Nernescbkal

change (c5) , in CH c l is connected with horizontal spiral change (c6). The couplings c2/c5 and c3/c5 are listed as further incongruities.

In the next step, the hypothetical patterns in the theoretical matrices must be tested for significant correspondences with the observed matrices. For that reason A R B U ~ M is com- pared with ARBUT (Fig. 4) and C H ~ M with CHT. In both cases the assignment procedure results in the acceptance of matrix similarity (Table 5 ) ; consequently the hypothetical pat- terns must be accepted as accurate copies of the observations.

Finally, since the detected patterns are apparently responsible for taxa discrimination, we must expect dissimilarity between matrices in crosswise comparisons. In the compari- sons of ARB& with CHpM and CHT with A R B U ~ M these expectations are fulfilled (Table 5 ) . Consequently, we can conclude that the theoretical matrices truly represent major differences of morphological integration between both groups.

Practical comments

The general acceptance of new methods and modern theories alike frequently depends on the condition that their fundamental principles are intelligible and their procedures appli- cable without great difficulty. Although the assignment methods satisfy these require- ments, the practical application calls for some additional comments.

The original programs have been written and carefully tested in Amiga-Basic (Commo- dore Amiga 1000) as well as in GW-Basic (Sharp PC-4502). For users of personal com- puters the QAP routines are made available as disk copies in ASCII-code either for AMIGA-DOS (3.5") or for MS-DOS (3.5" or 5.25") format. Alternatively, printed program listings may be obtained from the author. There is no doubt that the package can be very comfortable modified for individual applications by moderate know-how in com- puter programming, since many helpful instructions and information on program struc- ture are included. As assignment procedures are time-consuming in large matrices, it is recommended to employ fast computers. Compilation of programs will be a further veloc- ity support.

Conclusions

Quadratic assignment procedures are undoubtedly a noteworthy alternative to multinor- mal techniques; they are superior to these customary techniques in certain respects. They will enable taxonomists as well as evolutionary biologists to compare patterns of variation, covariation and correlation between several groups of individuals. The patterns serve as group characteristics. Independence from distributional assumptions, ease of application, and last but not least the possibility of testing hypotheses of character coupling are the main advantages of QAP.

It is to be hoped that frequent employment of QAP will intensify the focus on one of the most conspicuous features in organisms, the variability.

Appendix of abbreviations

The investigated groups ARBU Ariantu urbustorurn CH Arianta charnueleon

Coefficients of correlation between matrices Crmz observed correlation prm, permutation correlation

Matrices of relationship between characters PM product-moment correlation d. f . degrees of freedom RK S earman's rank order correlation f frequency T tfeoretical relarionship PC principal component VC variance-covariance SE standard error

Further abbreviations

tv total variance

Character coupling for taxa discrimination 95

Acknowledgements

I am much indebted to Mrs. S. U. NEULINGER for the preparation of figures and to Dr. M. STACHOWITSCH for translation assistance.

Zusammenfassung

Merkmdskopplung als Mittel ZUY Taxaunterscheidung: Eine kritische Bewertung der ,,quadratic assignment "-Verfahren (QAP)

Taxa lassen sich durch Angaben uber Merkmalskopplun in Form von Ahnlichkeitsmatrizen cha- rakterisieren. Den herkommlichen Verfahren zur Prufung auf Gleichheit von Varianz-Kova- rianz-Matrizen liegt die Multinormalitatsannahme zugrunde, welche in der Natur jedoch oft- mals nicht erfiillt ist. Als Alternative dazu bieten sich ,,quadratic assignment procedures" (QAP) an: Es handelt sich um computergestiitzte Testverfahren, die nach einer Zufalls-Permutations- strategie darauf abzielen, signifikante Ubereinstimmungen in den Mustern von Matrizen zu ent- decken. Fur einen Leistungsvergleich der beiden unterschiedlichen Testmethoden ist ein Beispiel vonnoten, dem Multinormalitat zugrunde liegt. Die Stichproben zweier Landschneckenarten (Pulmonata, Helicidae), und zwar von Arianta arbustorum (n = 104) und Arianta chamaeleon n

ermittelt und gingen als Variablen in die Ahnlichkeitsmatrizen (Varianz-Kovarianz, Produkt- Moment-Korrelation und Ran korrelation) ein.

Die Methodenpriifung ergi%t, dai3 sich die Q A P zwar fur Korrelationsmatrizen, jedoch nur bedingt fur Varianz-Kovarianz Matrizen ei nen. Dennoch empfehlen sie sich als Verfahren fur den Einsatz in Taxonomie und Evolutionskrschung. Dafiir sprechen vor allem die leichte An- wendung, deren Unabhangigkeit von Annahmen iiber zu rundeliegende Verteilungen und schliei3lich die Moglichkeit, Hypothesen uber Merkmalskoppfungen testen zu konnen.

Die Stich roben der beiden Schneckenarten lassen sich signifikant nach der Merkmalskopp- lung untersckeiden. Sowohl Form- als auch Schalenzuwachsparameter sind bei beiden Arten ver- schieden morphologisch integriert.

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Author’s address: Dr. HANS L. NEMESCHKAL, Institut fur Zoologie, Universitat Wien, Althan- strage 14, A-1090 Wien, Osterreich