Zelditch M.L. (и др.) Geometric Morphometrics for Biologists: a primer

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258 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Each group must be in a separate file. When you load them, the data will be plotted in

the visualization window (the plots will not make sense if the data are traditional morpho-

metric measurements or partial warp scores). Before doing the analysis, choose whether

to regress on the values in the last column (where CS is usually located) or its log (LCS).

Also before doing the analysis, determine the number of bootstrap sets you wish to use;

the default is 100. To perform the analysis, click on Compute Growth Vector. The results

will be displayed in the results window; it will give the angle between vectors, and the

range of angles that can be obtained by resampling each. They can be saved to a file using

the Save File option.

To test whether the angles between pairs of vectors are significantly different, load the

first pair of data sets using the Load File 1 and Load File 2 buttons. Load the second pair

of data sets by going to the File pull-down menu and selecting Load Data Set 3 then Load

Data Set 4 (for a three-way comparison, A-B vs A-C, load A as data sets 1 and 3, load B

as 2 and C as 4). Next, select Regression Function (CS or LCS) and Number of Bootstrap

Sets Used in those control windows. Now start the calculation by going to the More Stats

pull-down menu and selecting Bootstrap Test of Difference in Angle. The results will be

displayed in the results window; they can be saved to a file using the Save File option.

Running ShuffleAllometry

Unlike all other programs in the IMP series, this one reads data in column vector for-

mat. The input data should be a single column of regression coefficients. These can be

regression coefficients from any sort of data, such as allometric coefficients of traditional

data. The original purpose of the program was to shuffle allometric vectors of traditional

morphometric data, so the program can read these as well as column vectors of geometric

shape variables. The reason for formatting the data as column vectors is because most

journals publish vectors of allometric coefficients (or principal component coefficients) as

column vectors. If your vectors are in rows, you can transpose them by opening the file

in Excel, copying the vector, then, pasting it, using the Paste Special menu to select the

option “transpose.”

Once you have loaded the data, you need to select the desired number of bootstraps;

the default is 400 and can be altered by typing the desired number in the box. Clicking on

Shuffle runs the program. The results will appear in the results window, where you will find

the mean value for the correlation between the input vector and 400 randomly permuted

versions of it, along with the upper and lower 2.5 and 5 percentiles of the distribution.

You will also obtain the maximum value found over the chosen number of bootstraps.

Running VecDisplay

This program is intended for purely graphical purposes. Its function is to provide pictures

of deformations that cannot be obtained by the conventional software but can be expressed

in terms of partial warp scores. You can use it to display a hypothetical vector, such as the

expected change in shape under a model (so long as you can frame the model in terms of

partial warp scores). You can also use it to display the difference between two deformations

obtained from other software, such as the difference between two regressions (those files,

properly formatted for input into VecDisplay, are produced by Regress6 and can be saved

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REGRESSION 259

either in normalized (the Growth vector) or unnormalized (the Deformation vector) form).

The program will display each vector individually, and the sums and differences between

them, using all the standard display options. In addition to the usual displays, there is also

an option to show the difference between two deformations by pairs of vectors at each

landmark.

The input files each contain a row vector of partial warp scores, ordered from greatest

to least bending energy, including the two uniform components (the last column, as usual,

is centroid size). For VecDisplay to be able to interpret the partial warp scores, it needs the

reference form that was used in calculating them. The same reference must have been used

in calculating the partial warp scores for both files. If you want to input a hypothetical

vector, you need to produce the partial warps describing it. One method you could use for

this purpose is to draw expected changes in landmark locations (as vectors of landmark

displacement from a given shape, which will serve as the reference). You can then digitize

the coordinates located at the endpoints of the vectors and use those coordinates in any

program that outputs partial warp scores. In some cases a biological model might corre-

spond to a particular partial warp – for example, one partial warp might describe a global

growth gradient. If that is the case, you can input a vector of zeros for all scores except

for that one.

After loading the reference form and selecting the superimposition method, load the

one or two vector files. If using only one file, select the Zero Vector option for the second.

You may display (1) each vector separately (these are the first two options), or (2) the sum

of the two vectors (which represents an average between them, omitting only the step of

dividing the scores by two), or (3) the difference between them, either by subtracting the

second from the first or the first from the second.

The options for drawing the deformations, as well as for editing them, are as described

in Chapter 7, with the exception that you can also draw the difference between the two

groups as pairs of vectors at each landmark (the final option). We should note that if

the input vectors are the normalized growth vectors produced by Regress6, you probably

will need to rescale them because the changes will be so large that they will be virtually

unreadable. To reduce the scale, use the Deformation Multiplier function.

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chap-11 4/6/2004 17: 27 page 261

Partial least squares analysis

Partial least squares (PLS) is a method for exploring patterns of covariation between two

(and potentially more) blocks of variables. It can be used to study covariation between two

blocks of shape variables, making it potentially useful for studies of morphological integra-

tion (e.g. Bookstein et al., 2003; Bastir et al., 2004), and also for synthesizing information

about three-dimensional morphologies sampled by two two-dimensional views (e.g. Rohlf

and Corti, 2000). The method can also be used for analyzing the relationship between shape

and other variables, including traditional morphometric or meristic variables, or measures

of biomechanical, ecological factors or behaviors (e.g. Corti et al., 1996; Lundrigan, 1996;

Rüber and Adams, 2001). A particularly creative use of the method examines the relation-

ship between patterns of fluctuating asymmetry and variation (Klingenberg et al., 2001).

An important feature of the method is that the blocks are taken as a given – the data

are partitioned into blocks before the analysis begins. For that reason, PLS is not useful

for finding the blocks in the first place, which means it is not suited for dissecting complex

morphologies into modular units. Even so, it may be useful for building integrated units out

of the modules, if we accept that the blocks are indeed modules. We can ask whether one

block is more highly correlated with another than either of those is with a third block (e.g.

Bastir et al., 2004). Although that is part of the question normally addressed in studies of

integration, the other part is the identity of the blocks. PLS may eventually prove useful for

that second question as well, but no current implementation of the method has attempted

to address it. Thus, when using PLS to study morphological integration, it is important to

remember that the method is useful for analyzing correlations between units but not for

subdividing the whole into units.

PLS is probably unfamiliar to many biologists, although it has been used extensively

in the social sciences (see Bookstein, 1982; Jöreskog and Wold, 1982) and in clinical

studies (e.g. Sampson et al., 1989; Streissguth et al., 1993; Lowe et al., 1997). Thus,

a large part of this chapter discusses similarities and differences between PLS and more

familiar methods, including regression, principal components analysis (PCA) and canonical

correlation analysis (CCA). Because of the potentially wide range of applications of this

method, it is important to understand how it is related to other methods that address

similar questions. As well as understanding the relationships of PLS to other methods it

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262 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

is also important to understand the limitations of PLS, especially when another method

might be equally appropriate (both conceptually and mathematically).

Before exploring the relationships among methods, we should note that partial least

squares analysis employs a mathematical technique called singular value decomposition

(SVD), which has not been introduced yet in this text. SVD is related to the more familiar

decomposition by eigenanalysis used to extract principal components (from the variance–

covariance matrix) and partial warps (from the bending-energy matrix). Because PLS

uses SVD, the vectors generated by PLS are often called singular axes (SAs); in studies

of covariances among geometric shapes, they are also sometimes called singular warps.

PLS compared to regression

Both regression and PLS examine the relationship between two sets of variables, but they

differ in that the (Model I) regression model casts one set of variables as dependent on the

other whereas PLS treats them symmetrically. That is, PLS does not assume that one set

of variables causes the other, but rather views both sets as jointly (and linearly) related to

the same underlying causes. Linear combinations of measured variables that are thought

to reflect responses to underlying (unobserved) variables sometimes are called “latent vari-

ables,” and the terminology of latent variables often is used in PLS (in this, PLS resembles

factor analysis).

PLS seeks the latent variables in one block that are maximally correlated with the latent

variables of another block. Thus, there is assumed to be at least one latent variable within

each block. These linear combinations are constructed to account for as much of the

covariation as possible between the two blocks of variables. The method yields two sets of

vectors, one consisting of the coefficients of the latent variable underlying the first block,

the other of the coefficients of the latent variable of the second block. The number of

interblock linear combinations is equal to that of the smaller block, P

min

. So, for example,

if we have two blocks, one of which has four variables and the other of which has three,

the first linear combination will have four coefficients, the other will have three, and there

will be three interblock linear combinations.

Linear regression is based on a linear statistical model, which assumes that the indepen-

dent variable is measured without error (hence all the error is ascribed to the dependent

variable). No model underlies PLS, and no error is ascribed to any variables (in either

block). For this reason alone, we would not expect to obtain the same coefficients from

PLS as we obtain from regression. However, there is a more important reason for expecting

that the two methods will yield different results when there is more than one independent

variable. In that case, we would compare PLS to multiple regression, and the coefficients

of PLS and multiple regression have very different interpretations. The vectors obtained by

multiple regression express the dependence of the dependent variables on just one of the

independent variables, with all others held constant. The coefficients produced by multiple

regression do not assess the covariance between the two blocks of variables, a distinction

that becomes particularly important when several independent variables are highly cor-

related. Under those conditions, most of the variance in the dependent variables will be

associated with one independent variable, leaving little to be explained by the others. Even

though all the independent variables might affect the dependent variables, only one would

be accorded a high weight, so the others might appear to have trivial explanatory power.

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PARTIAL LEAST SQUARES ANALYSIS 263

That is because they are explaining the residual variance – i.e. the variance not already

explained by the one with the large coefficient. Consequently, interpreting the coefficients

of multiple regression can be difficult, and the method is poorly suited to cases in which the

independent variables are uncorrelated with each other. In those cases, no latent variable

accounts for covariances among the independent variables because they do not covary.

PLS treats the variables of both blocks symmetrically, and therefore we obtain variables

within one block most relevant for predicting the variables in the other block and vice

versa. These coefficients are called saliences because they indicate which variables in one

block are most relevant (salient) for explaining covariation with the other block.

PLS compared to PCA

PLS and PCA greatly resemble each other in the definition of axes. As we saw in Chapter 7,

PCs are extracted from a variance–covariance matrix (by eigenanalysis), producing a set

of mutually orthogonal dimensions (eigenvectors) ordered according to the amount of

variance each one explains. Similarly, PLS decomposes a matrix into mutually orthogonal

axes, but the matrix is an interblock variance–covariance matrix and the components

are ordered according to the amount of covariance between blocks explained by each

one. The mathematical difference is that, instead of using an eigenvalue decomposition

of the variance–covariance matrix, PLS uses a SVD of the interblock variance–covariance

matrix. The reason for using SVD instead of an eigenvalue decomposition is that the

covariance matrix between blocks need not be a square, symmetric matrix (a square,

symmetric matrix is one in which the number of rows equals the number of columns, the

first row of the matrix is the same as its first column, and every other row is also the

same as its corresponding column). Variance–covariance matrices are always square and

symmetric, but interblock variance–covariance matrices need not be (because the numbers

of variables in each block can differ). Therefore, we need a different method to decompose

the matrix. SVD yields pairs of singular axes (SAs), one per block. Each pair is associated

with a singular value (SV), which is a relative measure of the covariance explained by the

paired axes (we should note that singular axes “explain” covariance in the same sense that

principal components “explain” variance). Consequently, one of the primary differences

between PCs and SAs is that SAs come in pairs. For each singular value there is a pair of

axes that, taken together, accounts for the patterns of covariances between blocks.

Just as we can calculate scores for individuals on PCs, and explore the patterns of

variance in their plots, we can also calculate scores for individuals on SAs and explore the

patterns of covariance between blocks in their plots. Scores on SVs are computed the same

way as scores on PCs (i.e. by the dot product between either the PC or SA and the data for

an individual). Also, SAs, like PCs, can be depicted graphically by the deformation along

an axis, aiding the interpretation of their biological meaning. In the case of SAs, we would

plot SV1 for one block against SV1 of the other block. When one of the blocks is not a set of

landmark coordinates, no deformation can be drawn for the associated SA, but we can still

interpret the axis using the loadings of the variables on it. Rather than having a picture, we

will have a list of numerical values expressing the correlation between the variable and the

SA. The plots of the scores, as well as the depictions of shape transformations or numerical

values, provide the information about the nature of the covariance between blocks.

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264 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Unlike the situation for PCA, there is no analytic statistical test of the significance of

SAs, nor for the significance of the correlation between blocks. However, resampling-based

approaches can be applied to test these hypotheses. A permutation test, discussed by Rohlf

and Corti (2000), determines if the singular values are larger than could be produced by

a random permutation of associations among variables between blocks (keeping within-

block associations intact). We can ask whether the covariances between blocks exceed

those we would expect by chance. We can also ask if the correlation between singular axes

is significant using a permutation test – this determines whether the correlation between

the scores for each block exceeds what we would expect by chance. Both tests indicate

whether the observed patterns of covariance between blocks are statistically significant.

The similarity between PLS and PCA is important to understand because both impose

a similar constraint on the analysis: both define axes to be mutually orthogonal. SV2 is

defined to be orthogonal to SV1, just as PC2 is defined to be orthogonal to PC1. This

becomes important when biological factors are not orthogonal, which may be the general

rule. Even though the axes (both PCs and SAs) provide a useful, simplified space in which

to explore patterns in the data, the axes themselves need not correspond to any biological

factors. It is likely that PC1 and SA1 have a biological interpretation when they account

for a very large proportion of the variance or covariance, but the remaining axes are,

by definition, constrained to be orthogonal to them, making their interpretation more

dubious. This same issue arises when using PCA for explanatory or even comparative

purposes (see Rohlf and Corti, 2000, pp. 747–748; Houle et al., 2002). It is possible that

no useful (interpretable) axes will emerge from the PLS analysis, and that no significant

correlations between blocks will be found, particularly when the structure of the variation–

covariation within each block is especially complex.

Another important similarity between the methods, which should also inspire a cautious

approach to interpreting results, is that PLS extracts linear combinations of variables

(like PCA), but the relationship between blocks may be non-linear. In such cases, the

first dimension may represent the dominant linear trend, and others represent orthogonal

deviations from linearity. Thus, we would need to interpret SV1 together with SV2 to

understand the relationship between the two blocks, recognizing that a single non-linear

factor accounts for both. Of course, the issue of linearity is also important whether we are

analyzing the data by PCA/PLS, by regression, or by the method discussed in the following

section, CCA. However, most workers recognize that linearity is an important assumption

of regression; non-linearity might not seem so important in studies using PCA or PLS

because neither method is explicitly based on a linear model, so the impact of non-linear

relationships among variables might not seem to violate assumptions of the method.

PLS compared to CCA

Canonical correlation analysis, like multiple regression and PLS, examines the correlation

between blocks of variables. CCA closely resembles multiple regression, although, like

PLS, both blocks are treated symmetrically (there is no presumption that one block of

variables comprises causes and the other comprises effects). Nevertheless, the coefficients

produced by a CCA are interpreted like partial regression coefficients. This means that each

coefficient indicates the contribution made by an independent variable when the effects of

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PARTIAL LEAST SQUARES ANALYSIS 265

the others are held constant, as discussed above in the context of regression. In this way,

CCA, like multiple regression, differs from PLS.

CCA and PLS also differ in the quantity being maximized by the procedure. CCA seeks

pairs of axes (canonical axes) that are maximally correlated with each other; in contrast,

PLS seeks axes that maximally account for the covariance between the blocks (for a more

detailed comparison between CCA and PLS, see Rohlf and Corti, 2000).

Multigroup PLS: using PLS to compare patterns of covariance

PLS is usually used to examine patterns of covariances between blocks of variables mea-

sured in a single sample, but we can also use it to compare those covariances between

samples, as in a comparative analysis of morphological integration. Such comparisons rely

on the same logic (and methods) used in comparative analyses of regression equations or

PCs. In all of these, we are asking if the biologically corresponding vectors point in the same

direction. To answer that question, we can compute the angle between comparable SAs,

then test it statistically (using, for example, a bootstrapping procedure). In a similar fash-

ion, we can also compare SAs to PCs, asking whether the major dimension of covariance

between blocks is equivalent to the dominant dimension of variation within blocks. For

example, when our data come from an ontogenetic series, the major dimension of covari-

ance between blocks of variables may be their developmental correlations and the major

dimension of variance within each block might also be explained by ontogeny. Comparing

SAs to PCs can be especially useful for understanding causes of variance when PLS indi-

cates a significant relationship between morphology and some collection of environmental

variables; that same relationship may also explain the within-sample variation.

Mathematical details of two-block PLS

Suppose we have a matrix Y of P variables measured on N specimens, and that these

observations can be split into two blocks, Y

and Y

, which have P

and P

variables

respectively. We can now compute the variance–covariance matrix, R, which can be

thought of as comprising the variance–covariance matrices within blocks Y

and Y

and R

respectively) and the covariance matrix between the two blocks R

, giving:

R =





(11.1)

where R

is the transpose of R

. We then perform a SVD of R

= USV

(11.2)

which produces a P

×P

diagonal matrix S, whose diagonal entries are the P

min

singular

values, λ

. As mentioned above, there are as many singular values as there are variables in

the smaller block (P

min

). The matrices U and V have dimensions P

×P

min

and P

×P

min

respectively; their columns are called the singular axes (SAs). The first columns of U

and V comprise the paired SAs corresponding to the first singular value λ

, just as the

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266 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

first principal component (PC1) is the axis corresponding to the first eigenvalue of the

variance–covariance matrix. The SAs are ordered by decreasing singular values, just as

PCs are ordered by decreasing eigenvalues. Also, as mentioned above, scores on SAs are

calculated just like scores on PCs, i.e. by taking the dot product between an SA and the

data for a specimen (scores are calculated for each block separately). The fraction of the

total covariance of the two blocks expressed by the ith pair of singular axes is given by:



min

j=1

(11.3)

Whether a singular value is larger than we would expect from randomly related blocks is

determined by comparing the observed singular value to the distribution produced by ran-

domly permuting the covariance structure between blocks. In such a permutation test, the

vectors of P

observations, each representing a specimen in the first block, are randomly

associated with vectors of observations from the second, thereby randomizing the covari-

ance structure between blocks without altering the variance–covariance structure within

the blocks. If the observed singular value lies outside the 95% confidence interval obtained

from the permuted data sets, the observed SA is judged to be statistically significant. The

correlation between the scores on the two blocks on the ith SA is also a measure of the

statistical significance of the axis, and this correlation may also be tested via a permutation

test in exactly the same manner.

Using PLS to examine ontogenetic integration between cranial and

postcranial regions

One of the most promising applications of PLS is in studies of morphological integration,

although, for the reasons discussed in the introduction to this chapter, it does not fully

address one of the basic questions asked in such studies: which parts comprise integrated

units or modules? PLS takes the blocks as a given rather than testing the hypothesis that a

given block is, in fact, a coherent unit. The value of PLS lies in its ability to test a different

sort of hypothesis – the integration between blocks. Granting its limitations, the method

can be useful for analyzing relationships among morphological parts that are chosen

a priori.

Studies of morphological integration usually focus on covariances among variables mea-

sured in a single homogenous sample (for studies of this sort using PLS, see Hingst-Zaher

et al., 2000; Bastir et al., 2004). It also has been used to analyze the relationship between

blocks in heterogeneous samples such as ontogenetic series and purported evolutionary lin-

eages (e.g. Bookstein et al., 2003). Used in that second way, PLS examines the covariance

between blocks over age, or “evolutionary divergence” or “geological time.”

When applied to studies of heterogeneous samples, regression is often a feasible alterna-

tive to PLS – so long as the factor explaining the heterogeneity of the sample is separately

measured. For example, if the sample comprises an ontogenetic series, the dominant fac-

tor explaining the heterogeneity of the sample is age. Age is not a factor in a causal

sense, but it explains the variation in shape within the sample. Because studies of inte-

gration are usually concerned with hypotheses of causality, age is not an explanation for

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PARTIAL LEAST SQUARES ANALYSIS 267

morphological integration. However, often we want to know whether two or more struc-

tures are integrated through their ontogenies, and this is the sort of question we can answer

using either regression or PLS. Using PLS to explore the integration of ontogeny is an alter-

native to describing that same ontogeny by multivariate regression. Regression does not

require breaking up the data into multiple blocks a priori (and that subdivision is one of

the more problematic features of PLS), but it does require that we measure age (or a proxy

for it, like size). Furthermore, regression does not fully answer the question about inte-

gration because it only tells us whether shape covaries with age; it does not tell us which

blocks most covary with each other. The two questions, while at least subtly different,

are also interrelated – the block that covaries least with the others also covaries least with

the measured factor. Considering that regression is a feasible alternative to PLS for some

questions, it is important to understand how the two methods differ when describing the

same ontogeny. Additionally, because age might account for most of the variance in the

sample, it is also important to understand how PLS might differ from PCA.

We will use PLS to address three sorts of questions about the ontogenetic integration

between cranial and postcranial morphology:

1. How are cranial and postcranial regions integrated over ontogeny?

2. Do species differ in these patterns of ontogenetic integration?

3. Are cranial landmarks more highly integrated with those that measure the position of

median fins, or with those of the caudal peduncle?

A major objective of the first two analyses is to compare results based on PLS to those

based on regression and PCA. The primary objective of the third is to explore the use of

PLS to test explicit and conflicting hypotheses about integration.

Ontogenetic integration between cranial and postcranial shapes: results from

PLS, PCA and regression

The landmark configuration and its subdivision into two blocks are shown for the data

of Pygopristis denticulata (Figure 11.1). The two blocks do not correspond to halves of

the body, because the pectoral fin landmark (landmark 11) is topographically part of

an “anterior” block but is included in the postcranial block because it is not a cranial

feature. Thus, the cranial and postcranial blocks partly overlap on the body. Analyzed

by PLS, we find that the covariance between blocks is substantial; the first singular value

(0.0536) explains 67.1% of the total covariance, significantly more than expected by

chance (p < 0.01). The correlation between the two blocks is a very high 0.862, and that

correlation is also significant (p < 0.01). No other singular value is statistically significant,

so there is only one dimension of covariance between the blocks. The plot of the scores of

postcranial SA1 on cranial SA1 shows the pattern we expect from such a high correlation,

although the relationship between blocks is not strictly linear (Figure 11.2).

Having found that there is a dimension of significant covariation, we need to display

and interpret it (Figure 11.3). To aid in this interpretation, we first ask whether the SA1

of each block is correlated with size as expected – and they are; both cranial SA1 and

postcranial SA1 are indeed highly correlated with size (R =0.86, R =0.72, respectively).

Size thus appears to be a plausible interpretation of the biological factor responsible for the

covariance between regions. Given that result, we might expect to find a similar depiction