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Assessment of quantitative techniques in paleobiogeography. Mar. Micro- paleontol., 7 : 213--236. A series of multivariate methods has been compared to ...
Marine Micropaleontology, 7 (1982): 213--236

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Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

ASSESSMENT OF QUANTITATIVE TECHNIQUES IN PALEOBIOGEOGRAPHY

BJORN A. MALMGREN and BILAL U. HAQ

Department of Geology, Stockholm University, Box 6801, S-113 86 Stockholm (Sweden) Woods Hole Oceanographic Institution, Woods Hole, Mass. 02543 (U.S.A.) (Revised version received September 1, 1981 ; approved September 9, 1981 )

Abstract Malmgren, B.A. and Haq, B.U., 1982. Assessment of quantitative techniques in paleobiogeography. Mar. Micropaleontol., 7 : 213--236. A series of multivariate methods has been compared to assess their effectiveness in extracting essential information out of a complex micropaleontological data-set. The data-set used for this experiment consists of relative frequencies (percentages) of Miocene coccolith taxa or groups of taxa in cores of the Deep Sea Drilling Project (DSDP) from the Atlantic Ocean. All methods tested are varieties of principal components analysis in R- and Q-mode, and "true" factor analysis. Various secondary rotational procedures ancillary to some of these methods are also tested. A test, denoted A-Test, is developed, which assesses how well principal components or factors reproduce the data-set. A-Test may be used for determining the optimum number of principal components or factors, the most relevant rotational procedure, and thus the most suitable analytical technique. The A-Test does not rely on mathematical testing, but on simple inspection of the compositions of the principal components or factors, and their relations to correlations existing in the data-set. Our experiment reveals that the most efficient methods are the maximum-likelihood factor analysis and the R-mode principal components analysis, within which the varimax (orthogonal) rotations best reproduce correlations. Of these methods, maximum-likelihood factor analysis is considered the optimum method, because of the greater simplicity of compositions. In addition to these methods, Kaiser's second generation "Little Jiffy" factor analysis was also found to be efficient. Three methods provide less sensitive reduction of the data: the "true" R-mode principal components analysis (without ~econdary rotations), the Q-mode principal components analysis, and the correspondence analysis.

Introduction During the last decade, vast amounts of data have been generated on modern and ancient distributions of various microfossil groups in the world ocean. This has been made possible through studies of relatively continuously cored sequences from the ocean floor made available primarily by the Deep Sea Drilling Project (DSDP), and the development of well-formulated taxonomies in the major microfossil groups. The avail-

ability of these data represents no less than a revolution in micropaleontological research, permitting detailed paleoenvironmental reconstructions of the Cenozoic, including clues to the past history of oceanic climates and circulation patterns. The major part of the data generated has been in the form of relative frequencies (proportions) of taxa characterizing the assemblages. Studies of variation in morphologic characteristics (size and shape) have also been attempted recently (e.g.,

0377-8398/82/0000--0000/$02.75 © 1982 Elsevier Scientific Publishing Company

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Bd et al., 1973; Hecht, 1974, 1976; Malmgren and Kennett, 1976, 1978; Healy-Williams and Williams, 1981), but here we are concerned only with the relative frequency data. Oceanic micropaleontologic data are by necessity extremely complex in nature often incorporating the effects of biogeographical provinciality and fluctuations in the province boundaries resulting from climatically induced periodical expansions and contractions of water masses, changes in surface circulation patterns, and postdepositional changes, such as selective dissolution of taxa. In certain groups, such as coccoliths, diatoms, and dinoflagellates, some portion of the variation may also be ascribed to deviations from a general trend caused by occasional local blooms. A primary problem in studies of relative frequency data is the need for reducing large data-sets by some multivariate statistical technique into a few, paleontologically meaningful variables (factors or principal components), which contain the essential information about the complex interrelations among the taxa. These variables are often interpretable in terms of the underlying environmental causes to which associated taxa may have responded in a similar manner. Q-mode principal components analysis (Imbrie and Purdy, 1962; Imbrie and Kipp, 1971), or Q-mode factor analysis as it has been often unjustifiably labeled, has been the prime method for reducing data on oceanic microfossil distributions. The reason for its popularity is that it generates simple principal components composed of one or a few taxa that normally dominate the samples in terms of absolute numbers. Also, this technique has proven useful for geographic mapping of distributions of major taxa. These features are intuitively appealing, but for more detailed delineations of the structure of a data matrix as a w h o l e , far more efficient methods are available. We initiated this study because we felt that an overview of existing multivariate techniques and their effectiveness in that

reduction of biogeographic data was desirable and timely. Earlier applications of various other analytical techniques in oceanic micropaleontology have included: Rmode principal components analysis (Blanc et al., 1972; Malmgren and Kennett, 1976b; Thunell et al., 1977); principal coordinates analysis (Malmgren and Kennett, 1973); correspondence analysis (Malmgren et al., 1978); and maximum likelihood factor analysis (Malmgren, 1981). The data-set that forms the basis of the present experiment is based on the coccolith taxa (or groups of taxa) in a series of DSDP sequences from the Atlantic Ocean (Fig. 1) spanning the Miocene epoch (24--5 Ma) (see Haq, 1980). The data-set incorporates 75°N 70 +

60 °

45 °

30 °

15 o



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15 °

30 °

45 °

60 °

o**" 70os t ~05°w

I 90 +

I,r', 75"

6o"

I 45 +

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I ~5o

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Fig. 1. Locations of DSDP sites from the Atlantic Ocean from which core material was used to gather Miocene coccolith data used in this study (see Haq, 1980, for details of sample number from each site and their biostratigraphic ages).

215 12 groups of taxa and 399 samples. The data-base thus consists of census of fossil nannoplankton placed in a time-paleolatitudinal grid. Some of the individual taxa were grouped for various reasons discussed under "taxonomic concepts" by Haq (1980). In the R-mode analyses, the percentage data were subjected to an angular transformation [arcsine (x/100) ~ ; x is a percentage]. The angular transformation has the effect of "opening u p " data arrays subject to the constant-sum constraint (Chayes, 1 9 6 0 , 1 9 7 1 ) , and it has also some desirable statistical properties (Bartlett, 1947). Among these is the tendency to make the distribution approach multivariate normality (see Malmgren, 1979). Raw data were used in the Q-mode analyses. We have tested the following quantitative techniques: R-mode and Q-mode principal components analysis, correspondence analysis, " t r u e " factor analysis (hereafter referred to simply as factor analysis), including maximum-likelihood factor analysis and Kaiser's second generation "Little J i f f y " . We also apply different secondary rotations of the principal c o m p o n e n t or factor axes (orthogonal and oblique). Our primary intention is not to provide detailed mathematical accounts or geometrical descriptions of the various methods, b u t instead to show how the methods (and their varieties) operate on a given data-set. We also present a general discussion of the characteristics of the methods and the distinctions between them that may be useful for the reader with a nonquantitative inclination. For further detailed descriptions, we refer the reader to JSreskog et al. (1976). Statistical methods All the methods we test belong to a family of multivariate methods known as canonical methods among statisticians and ordination methods among ecologists. They are all essentially techniques for data reduction, b u t certain mathematical assumptions distinguish them, and these are discussed here.

R - a n d Q-mode methods

R-mode techniques aim at examining interrelations among taxa. They operate in the so-called "taxon space", which is a multivariate space with p orthogonal axes [p is the number of taxa (or groups of taxa); 12 in our example]. Each sample represents a point in this space; its location is a function of the percentage values of each of the p taxa. The N samples (399 in our case) form a cluster of points in this space. In Q-mode analysis the objective is primarily the assessment of relations among the samples in an orthogonal N dimensional "sample space", where the coordinate axes are represented by the N samples. In practice, several of the Q-mode techniques available permit a problem to be analyzed in p space instead of in N space. This increases the cost efficiency of an analysis, since p is normally less than N. Creation of the new variables

Geometrically, the new variables (principal components or factors} are established through rotations of the original coordinate axes a b o u t the origin to new positions satisfying the requirement that one axis (denoted the first principal c o m p o n e n t or factor) is oriented so as to account for a maximum portion of variation in the data. This direction coincides with the major axis of the scatter ellipsoid, that is the cloud of sample or taxon points. It is further required that the second principal c o m p o n e n t or factor axis is associated with maximum variation in a direction perpendicular to the first axis, and that the remaining axes all in turn account for maximum variability perpendicularly to all previous axes. The orientation of the new axes is algebraically determined through computations of eigenvalues (latent roots) and eigenvectors (latent vectors) from a matrix of interrelations among the species (R-mode analysis) or a matrix of similarities among samples (Q-mode analysis). The eigenvalues

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specify the lengths of the axes and are proportional to the a m o u n t of variation that each axis accounts for. The eigenvectors, which are linear combinations of the original variables or samples, show the orientations of the new axes in relation to the original coordinate axes. The signs and magnitudes of the elements of the eigenvectors indicate the relations among taxa or samples on the basis of which the principal component or factor was formed. The axes may be subsequently subjected to a secondary rotation, usually for the purpose of easing the interpretation. Normally, because species tend to be interrelated and samples tend to show some degree of similarity, a few principal components or factors are representative of a large a m o u n t of variation in the data-set under study. These then account for major proportions of the total variability. In case no interrelations or similarities exist, the new variable axes coincide with the original variable axes and all account for the same a m o u n t of variation (in R-mode analysis, the latter is true only under certain circumstances). The usual starting point for an R-mode principal components or factor analysis of the kind of data we are dealing with here is the correlation matrix, a p X p matrix containing unit values in the diagonal and the standard Pearson correlation coefficients as off-diagonal elements. This matrix is preferential over the covariance matrix (containing variances and covariances), because it gives equal weight to all taxa. The correlation matrix is a special case of the covariance matrix in which the variances are standardized to unit values. The factor analysis models we apply here, maximumlikelihood factor analysis and Kaiser's second generation "Little J i f f y " are, however, scalefree, and, therefore, the covariance matrix may just as well be used. In our application of Q-mode principal components analysis, eigenvalues and eigenvectors are extracted from a similarity matrix, the elements of which are the cosine of the

angle between any pair of vectors of observation values in " t a x o n space". Correspondence analysis is also based on a similarity matrix, but "similarity" is defined differently (see Jbreskog et al., 1976, pp. 108-110).

Differences between models The general model underlying principal components and factor analyses may be expressed as follows (we discuss the model in terms of R-mode analysis, but the same line of reasoning applies also to Q-mode analysis): R-matrix = (A-matrix) X (A-matrix)' + ~ -matrix

(1)

The R-matrix is the correlation matrix forming the basis for an analysis, the A-matrix is a matrix of principal c o m p o n e n t or factor loadings, which are the eigenvectors in column form, the (A-matrix)' is the Amatrix with rows and columns interchanged, and the ~-matrix is a matrix of residuals (see J5reskog et al., 1976, pp. 53--59, for further explanations). If the principal components or factors are rotated obliquely, a matrix of correlations among axes has to be incorporated into this expression. The basic idea is thus to partition the variation as expressed by the correlation matrix into two parts: the one part accounted for by the model selected [(Amatrix) X (A-matrix)'], and the other part, not explained by the model, representing " u n i q u e " variation in the various taxa {diagonal elements of the ~-matrix) and residual correlations (off-diagonal elements). The " u n i q u e " portion is the sum of (1) variation in a taxon not shared with the other taxa, and (2) variation due to errors in the technical procedure of estimating relative frequencies. A fundamental distinction exists between principal components analysis and factor analysis with regard to assumptions about the "uniqueness" c o m p o n e n t and the number of new variables initially extracted. Prin-

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cipal components are defined to account for maximum variance in the taxa (diagonal elements of the R-matrix), whereas factors are established to portray optimally the interrelations among these taxa (off-diagonal elements). In this respect, principal components analysis is variance-oriented and factor analysis is correlation-oriented (JSreskog et al., 1976, p. 59). This implies that the "uniqueness" portion is assumed to be overall small in principal components analysis. In factor analysis, the "unique" variation is larger, and may be considerable in some taxa. Factor analysis thus accepts that some part of the variation in each taxon is "unique" and not shared. With one of the major purposes of analyzing relative-frequency data being to determine associations or contrasts among taxa, the factor analysis model may seem like a sensible solution. Furthermore, in factor analysis the two components of the model (A-matrix and ~matrix) are estimated separately, using different mathematical criteria. Usually an initial estimate of the "shared" portion of variation in a taxon is obtained from the squared multiple correlation coefficient (a multivariate generalization of the standard correlation coefficient) between the taxon and the other taxa. This is the method employed here. These initial estimates are later modified through an iterative procedure to final estimates, depending on the mathematical criterion used and the number of factors entering into the model. Inherent in factor analysis is the idea that there exists a small number of factors (less than the number of taxa), which reproduce the interrelations among taxa. We have employed two methods of estimating the A-matrix and the -matrix, each using a different mathematical criterion: maximum-likelihood factor analysis (Harman, 1967, pp. 211--232) and Kaiser's second generation "Little J i f f y " (Kaiser, 1970). In principal components analysis, the -matrix is initially set equal to zero, and p principal components (equal to the num-

ber of taxa) are extracted from the correlation matrix. In " t r u e " principal components analysis, no further adjustment of the model occurs, and the initial rotation is, therefore, the final solution. The number of principal components to be included in the interpretation is left to individual judgement, for example, based on the share of variation they account for (see further discussion on pp. 219--222). These principal components may subsequently be rotated secondarily (no longer " t r u e " principal components analysis), and the residuals can then, if desired, be estimated from eq. 1. The A- and S-matrices are thus not determined independently as in factor analysis. We have applied the principal components analysis without and with secondary rotations. Correspondence analysis is a variety of principal components analysis which combines the properties of both R-mode and Qmode techniques (Benz~cri, 1973; David et al., 1974}. It allows a simultaneous plot of both samples and taxa on the same coordinate axes, a feature facilitating the interpretations as to which taxon or assemblage dominates a sample or a cluster of samples. This is possible through a special scaling procedure and through the use of a particular similarity coefficient, which allows changes of scale in variables and samples (Hill, 1975). The method also uses a duality relationship between variables and samples, which enables interchanges of rows and columns in the data matrix without changing the result. Since we are here mainly interested in relations among taxa, we only interpret the R-mode part, but, naturally, this part is strongly integrated with differences among samples. Correspondence analysis is designed for count data (integers), but several applications have included quantitative (continuous} variables (discussed by Hill, 1975). We have used the raw census counts as input for the correspondence analysis. A further requirement is that data are non-negative, but this is never a problem in the t y p e of data we deal with here.

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Secondary rotations Once the principal components or factors have been determined, they may be rotated secondarily, in order to obtain "simple structure". This rotation may keep the axes orthogonal (uncorrelated) or relocate them to oblique (correlated) positions. "Simple struct u r e " implies that a few taxa load heavily on each principal c o m p o n e n t or factor, and that remaining loadings are low, thus easing interpretation. The secondary rotation procedure makes use of the so-called "indeterminacy of factors" (see discussion in JSreskog et al., 1976, pp. 60--61) to generate new orientations of the axes that equally well represent the data, but with the characteristic of being simple. Secondary rotations of axes to positions satisfying specific mathematical criteria have been a controversial matter in the scientific literature (see, for example, discussion in Temple, 1978). It has been claimed that they represent purely mathematical solutions that do not necessarily have any scientific advantage over unrotated axes. Another debatable issue in this context is the question of whether to rotate axes orthogonally or obliquely. The choice of orientation may be regarded as a philosophical question, which is ultimately dependent on one's opinion of whether underlying influences in nature are related or not. In any event, it is conceivable that an oblique rotation in certain situations may provide a better summary of the data, because of the very strict constraint imposed upon the orientations of the axes in orthogonal solutions. Thus when the first two axes have been rotated to their optimal positions, no freedom exists for rotations of the third and remaining axes, and so their positions are, in practice, fixed. This constraint may be relaxed by applying oblique rotation allowing all axes to be rotated more freely. We routinely employ varimax rotation (Kaiser, 1958), the most c o m m o n type of orthogonal rotation, to R- and Q-mode principal components analyses and maximum-

likelihood factor analysis, and direct quartimin rotation, a procedure for oblique rotation (Jennrich and Sampson, 1966), to Rmode principal components analysis and maximum-likelihood factor analysis. Kaiser's second generation "Little J i f f y " involves only an oblique rotation. It consists of Harris--Guttman image analysis followed by orthoblique rotation (Kaiser, 1970). In correspondence analysis, the initial axes are not rotated further. We also show some examples of the effects of other orthogonal and oblique procedures (Table II). The direct oblimin method of oblique rotation has a choice of a gamma ( r ) value; increasing it produces more oblique (correlated) axes. The direct quartimin method is a variety of direct oblimin rotation ( r = 0). A large gamma in orthogonal rotations (varimax and equamax methods) simplifies the loadings in any one principal c o m p o n e n t or factor (columns of loadings TABLE I Taxa included in this study (see Haq, 1980, for taxonomic details)

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

Discoaster spp. Sphenolithus spp. Cyclicargolithus floridanus Coccolithus pelagicus (s. ampl.) Dictyococcites hesslandii D. antarcticus D. minutus Re ticulofenestra pseudourn bilica R. haqii Reticulofenestra spp. Umbilicosphaera jafari Dictyococcites abisectus

TABLE II Methods of orthogonal and oblique rotations applied here (m for the equamax model is the number of rotated axes) Orthogonal rotation

Oblique rotation

Quartimax, Varimax, Equamax,

direct direct direct direct direct

r = 0.0 r = 1.0 r -- m / 2

quartimin, oblimin, oblimin, oblimin, oblimin,

r r r r r

= = = = =

0.0 0.25 0.5 0.75 1.0

219 matrix), whereas a small gamma (quartimax method) simplifies the loadings in any one taxon (rows of loadings matrix). F o r more details about these rotations, we refer to Jennrich and Sampson (1966), Harman (1967, pp. 293--341), and Kaiser (1970).

Selection of number of principal components or factors The selection of the o p t i m u m number of axes is always a problem in principal components and factor analyses, and a whole battery of methods has been developed to solve this problem (see Harman, 1967, pp. 94--99; JSreskog et al., 1976, pp. 125-127). The following rules-of-the-thumb are often employed to determine the suitable cut-off point: (1) The magnitudes of the eigenvalues (proportions of variation accounted for). A plot of the principal c o m p o n e n t or factor numbers against their associated eigenvalues, or cumulative eigenvalues, usually gives a curve which flattens-out at some point and this point may be used as a criterion for determining the appropriate numbers of axes. This method is more useful in principal components analysis than in the factor analysis, where some portion of the variance in the data set (the " u n i q u e " portion) is not involved in the derivation of the new variables. (2) x2-test of goodness of fit (JSreskog et al., 1976, pp. 82--85). Here the test only applies to maximum-likelihood factor analysis. A ×2-value is associated with any maximum-likelihood solution; it is directly related to the maximum-likelihood criterion, which is minimized by an iterative procedure to produce the estimates of the A- and $matrices. The X~-value normally decreases steadily with increasing numbers of factors, because of improvements of the fit. By looking at the magnitudes of these reductions in relation to the degrees of freedom, one best solution may be determined. The x2-test is not included in the BMDP

computer program we have used (p. 222), b u t a ×2-value can be calculated from the parameter given as "likelihood criterion to be minimized" in the printout of results (Woollcott Smith, Woods Hole Oceanographic Institution, pers. comm., 1980). Thus, the ×2-value may be obtained by multiplying this parameter with (N -- 1), where N is the number of samples. A computer program is available which performs the x:-test for a specified numbers of factors, using maximumlikelihood and other methods (JSreskog and SSrbom, 1978). (3) Residual correlations. Plots of numbers of residual correlations larger than a certain value against numbers of principal components or factors may give a clue as to the choice of the number of axes. We use two different values, the critical values at the 5% and 0.1% significance levels, to mark residual correlations. A significance test is described in JSreskog et al. (1976, p. 126).

A -Test The rules-of-the-thumb described above may be useful for an initial estimate of suitable numbers of axes, b u t they do not primarily take into account the underlying principal c o m p o n e n t or factor patterns (compositions of taxa associated with the axes) and their relations to the data-set. We introduce another criterion, the A-Test, which determines how well the patterns reproduce the actual correlations that exist among the taxa in the data-set. The A-Test is applicable both to assessments of diagnostic numbers of axes (solutions in our terminology) and comparisons of different methods and/or rotational procedures (models). The A-Test involves the following steps: (1) Determination of true correlations among taxa. The correlations are shown by the correlation matrix, and by defining some fixed value as the required lowest value, the true correlations may be determined. This value may be the lowest correlation coefficient required for significance at the 5%, 1%, or 0.1% levels, or any other value judged

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T A B L E III C o r r e l a t i o n m a t r i x s h o w i n g r e l a t i o n s h i p s a m o n g the t a x a ( n u m e r a l s refer to T a b l e I) 1 1 1.00 2 0.45 3 0.31 4 --0.17 5 --0.05 6 -0.31 7 -0.52 8 --0.14 9 -0.33 10 0.13 11 --0.11 12 0.00

2

3

4

5

6

7

8

9

I0

II

1.00 0.26 --0.18 0.04 -0.32 -0.32 -0.34 -0.31 0.18 --0.06 0.09

1.00 --0.05 0.33 --0.21 -0.62 -0.53 -0.58 0.48 --0.26 0.44

1.00 0.11 0.00 --0.24 --0.05 --0.04 --0.01 --0.21 0.21

1.00 --0.11 --0.17 --0.28 --0.24 0.18 --0.13 0.44

1.00 0.03 0.21 0.07 --0.13 --0.13 -0.13

1.00 0.10 0.36 --0.34 0.24 --0.38

1.00 0.39 --0.30 0.00 --0.26

1.00 --0.34 0.15 --0.18

1.00 --0.18 0.33

1.00 --0.13

12

1.00

C o m p u t a t i o n s are based on arcsine ( x / l O 0 ) 1/~ t r a n s f o r m e d r e l a t i v e - f r e q u e n c y data. T r u e c o r r e l a t i o n s (correlat i o n s greater t h a n 0.30) are italicized. T w e n t y t h r e e t r u e c o r r e l a t i o n s o c c u r in this data-set.

appropriate may be used as a critical value. We use a value of 0.30 to mark a correlation (Table III). We do not apply significance testing, because, with our large sample size, the critical values are too low (0.10 at the 5% level and 0.17 at the 0.1% level) to be valid estimates of a true correlation. With a value of 0.30, we obtained a good balance between interpretability and the number of meaningful correlations. Twenty three cases of correlations occur using this value (Table III). This evaluation forms the basis against which principal component or factor patterns are compared. (2) Comparison with principal component or factor patterns. The principal component or factor patterns are e~(aluated, using some suitable cut-off point as an index of a significant principal component or factor loading. In R-mode analyses, we include only loadings larger than 0.40 in our interpretation of the patterns. Experiments, using the /X-Test, showed that, this way, we keep the patterns simple, and yet including a balanced number of interpretable interrelations. Here, the eigenvectors are always scaled to have a length equal to the corresponding eigenvalue, that is, the squared loadings equal the eigenvalue. However, we do not slavishly follow this requirement. If the loading of a species is near 0.40 and its inclusion into a factor makes sense considering the

true correlations, it is judged significant. It should be noticed that we do not propose that a loading of 0.40 should generally mark a significant contribution. The "significance level" has to be determined individually in different studies. In Q-mode principal components analysis, the patterns are determined from the principal component scores. This technique is primarily intended to portray relative proportions of the taxa and thus not correlations. Despite this we chose to incorporate it in the A-Tests, because we wanted to compare the results of this widely used technique with those of R-mode techniques in terms of efficiency in reproducing true correlations. In this Q-mode analysis, the patterns are generally simple with one or a few taxa being prominent compared to the others, and this is especially true for secondary rotations. In the few cases where a component is not dominated by a single taxon, we adopt a general rule (the "2/3 rule"), which we have found to provide effective interpretations. It implies that only scores in a principal component that exceed 2/3 of the highest score are judged significant. When no score exceeds this value, the taxon associated with the highest score contributes alone. This rule can also be applied with some flexibility. The patterns shown by the respective axes

221 included in a solution are then compared with the true correlations (correlation matrix). The comparisons are made for different numbers of axes, starting with the first axis, and proceeding to the first two, the first three, and so forth. Each of these solutions will be associated with a A-value. For consistency of the terminology, we will use the term "interrelations" for relationships displayed by principal components or factors, as contrasted with "correlations" (those shown by the correlation matrix). Firstly, pairs of interrelated taxa displayed by the one-axis solution are determined. If two taxa contribute to this axis (loadings greater than 0.40), there is simply one such pair. In case three taxa are significant, three pairs of interrelations are shown. Generally, the number of interrelations is equal to n(n -- 1)/2, where n is the number of significant taxa. Naturally, no interrelation is shown by an axis being associated with a single taxon. If an interrelation between two taxa is matched by a true correlation between the same taxa, it is recorded as "correct". If this interrelation is unmatched by a true correlation, it is instead recorded as "false". The numbers of " c o r r e c t " and "false" interrelations are counted, and a third parameter, AC--F (or A to be brief), is defined as the algebraic difference between these. Technically, this is done by determining each pair of interrelated taxa and crossing out the corresponding entries in the correlation matrix. Different symbols or colors may then be used for " c o r r e c t " and "false" interrelations, which will ease the counts of these outcomes. It is important that the signs of the interrelations are taken into consideration; a positive interrelation in a factor or principal component must be matched by a positive true correlation to be "correct". In case two axes display opposite interrelations between two taxa and the corresponding correlation is n o t judged significant, these interrelations are not recorded as "false", since two opposite interrelations

may produce a near-zero correlation coefficient. The procedure is then continued for 2axes, 3-axes, . . . , x-axes solutions, whereby the interrelations displayed by all axes in a particular solution (the first two, the first three, and so forth) are evaluated in the same way. In factor analysis, the A-Test is carried out by successively extracting more factors. The maximum-likelihood m e t h o d does not allow extraction of factors beyond a certain limit, which is determined by the point where the degrees of freedom are exhausted. Similarly, in principal components analysis with secondary rotation, a continuously greater number of axes are entered into the rotational procedure. In " t r u e " principal components analysis, where the initial rotation produces the final solution of normally p orthogonal axes (equal to the number of taxa), gradually more axes are incorporated. In a one-axis solution, a secondary rotation does not change the orientation of the axis, because it is already oriented optimally. The best model or solution is chosen to have a m a x i m u m A-value (AMA X ), where the balance between the numbers of "correct" and "false" interrelations is at an optimum. The number of " c o r r e c t " interrelations alone would not be a good criterion, since, for example, a one-factor solution containing significant contributions from all taxa, would naturally account for all true correlations, but, falsely, also all of the remaining. During the evaluation procedure of A, it is advisable to simultaneously keep an eye on changes in its component parts. A A-value may be negative, but it is then an indication of an insensitive model or solution. Plots of A against number of factors or principal components generally show a curve of increasing values up to a certain point, after which values decrease again. At this point, the factors or principal components reproduce the correlation matrix at an optimum according to our criterion. The decreases for successively greater num-

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bers of axes are normally a result of unfavorable decomposition of factor or principal c o m p o n e n t patterns. Occasionally, the A-curve does not form a distinct peak, but rather a plateau for a few, adjacent solutions, indicating that one single solution is not obviously superior. If this happens, the smallest number of axes may be chosen as the optimum solution, since inclusion of fewer axes facilitates the practical utility of the result. We, however, recommend that factor or principal c o m p o n e n t patterns are inspected for all solutions within the entire interval of maxim u m /X, because a larger-dimensional solution may contain significant information not displayed in the lowest-dimensional solution. In comparisons of the effectiveness of different models, the A-values for the optimum solution of each model are compared, and that showing the highest /XMAX is selected as representing the best model. If several models have the same /XMAX, the p a t t e r n s have to be compared to determine the final choice of o p t i m u m model. In summary, the A-Test amounts to determining how well patterns shown by a series of successively larger number of axes manage to reproduce true correlations among taxa (shown by the correlation matrix). The evaluations are made by counting numbers of "correct" interrelations (interrelations shown by principal components or factors, which are matched by true correlations) and "false" interrelations (unmatched interrelations), and by plotting their difference, A c F (or /X), against number of axes. A peak or plateau in the resulting curve indicates the point at which the factor or principal c o m p o n e n t patterns best reproduce true correlations, that is the optimum number of axes. Co m p u te r p rogra ms

In this study, we used program BMDP4M (Dixon and Brown, 1977) for R-mode principal components analysis, maximum-likelihood factor analysis, and Kaiser's second

generation "Little J i f f y " factor analysis. We used program CABFAC (Klovan and Imbrie, 1971) for Q-mode principal components analysis, and program CORRES (written by J.E. Klovan and revised by R.A. Reyment) for correspondence analysis. Results R - m o d e principal c o m p o n e n t s analysis witho u t s e c o n d a r y rotation

A-values are generally low and show no distinct peak (Fig. 2). The highest value (AMAx = 6) is attained for a single-axis solution. This means that optimally only 6 out of 23 true correlations are accounted for by this model. 23 ~

C--- C I C I C - - C - - C I C I f

--CICIC

~oI c/ o

F"

15.

F

~IF--F

--F

I F I F

F

LJ

g ~

}0

-

4

J

z

1

2

3

NUMBER

Z,

5

OF

PRINCIPAL

6

7

8

q

10

ll

12

COMPONENTS

Fig. 2. R - m o d e principal c o m p o n e n t s analysis w i t h o u t s e c o n d a r y r o t a t i o n . N u m b e r s of " c o r r e c t " (C) and " f a l s e " (F) i n t e r r e l a t i o n s a c c o u n t e d for by 1, 2, 3, . . . , 12 axes, and c o r r e s p o n d i n g A-values (differences b e t w e e n " c o r r e c t " and " f a l s e " interrelations). The total n u m b e r of true correlations in t h e data set is 23.

The first axis has loadings for the majority of the taxa (Table IV). This is because it is oriented in an "average" direction (Fig. 4, upper part), despite non-existing true correlations between some of the taxa (Table III). This axis takes care of 21 of the true correlations, but, simultaneously, it displays 15 interrelations that do not exist. Inclusion of further principal components causes all true

223

TABLE IV R-mode principal components analysis (patterns shown by first six unrotated axes; a plus sign marks a positive association and a minus sign a negative association) Principal component number 1

2

Discoaster Sphenolithus C. f l o r i d a n u s C. p e l a g i c u s D. h e s s l a n d i i D. a n t a r c t i c u s D. m i n u t u s R. pseudoumbilica R. haqii R e t i c u l o f e n e s t r a spp. U. j a f a r i D. a b i s e c t u s

+ + +

+ +

+

--

+

--

Eigenvalue Percentage Cumulative percentage

3.63 30.3 30.3

1.71 14.2 44.5

5

6

+

+

+ +

1.30 10.8 55.3

÷

1.03 8.5 63.8

0.83 7.0 70.8

0.77 6.4 77.2

the seventh component; Table IV), which greatly complicates interpretations. The ninth and following principal components have no significant loadings. Reductions in eigenvalues (Fig. 3) suggest a cut-off point at five principal components, where the curve flattens out.

--J

UJ

4

+

LIJ

> Z U] 0

3

2

\

2

R-mode principal components analysis with secondary rotations

3

PRINCIPAL

/~

5

6

7

COMPONENT

B

9

10

11

12

NUMBER

Fig. 3. R-mode principal components analysis without secondary rotation. Eigenvalues for principal components 1, 2, 3 , . . . , 12.

correlations to be reproduced, but also increases the number of "false" interrelations. Increases also occur in number of taxa showing opposite relations in different principal components (from 4 in the second to 7 in

Secondary rotations of only the first axis do not affect its orientation, and, therefore, is equal to 6 for a one-axis solution (Fig. 5). Varimax rotation of t w o principal components increases the number of "correct" interrelations to 23 (all), b u t also the "false" interrelations to 26, giving a A value o f - - 3 . Fig. 4 illustrates changes in configuration of the taxa along these axes caused by the rotation. The coordinates are represented by the principal c o m p o n e n t loadings. The first axis is rotated in the direction of taxa 5, 11, and 12, and the second to coincide with taxa 1, 2, 6. Considering the deterioration of A, this rotation is not successful.

224

,2:i:

IT {1L*2 %]

F"o UNROTATED PRINCIPAL COMPONENTS

VARIMAX ROTAIED PRINCIPAL COMPONENTS

~" ~/

~ .,1 ~:

/ ) , p "



~ I L:] r~!o]

Fig. 4. R-mode principal components analysis. Configurations of taxa (numerals refer to Table I) along first two unrotated and varimax rotated axes (I and II). Configuration was determined from the loadings of the taxa. Percentages by axes numbers represent the shares of variation. Varimax rotation reorients the first axis to coincide with taxa 1, 2, and 6, and the second axis to parallel taxa 5, 12, and 11. A-Test shows that this varimax rotation is not successful, because of the drop in h (Fig. 5).

26

F

F

VARIMAX

23

ROTATION

/CX ~

DIRECT

QUARTIMIN

ROTATION 20

\C

Lq Z O ~-