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

Подождите немного. Документ загружается.

chap-05 4/6/2004 17: 23 page 118

118 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(A)

(B)

(C)

Figure 5.9 Superimpositions of forms with an axis of symmetry – ontogenetic changes in a rodent

skull. Dotted lines connect landmarks on the sagittal plane. (A) Landmark displacements inferred

from GLS, which appears to indicate translation and rotation of the sagittal plane; (B) landmark

displacements inferred from SBR, which does not appear to suggest translation and rotation of the

sagittal plane; (C) landmark displacements inferred from GLS on symmetrized and back-reflected

configurations, which also does not appear to suggest translation and rotation of the sagittal plane.

chap-05 4/6/2004 17: 23 page 119

SUPERIMPOSITION METHODS 119

(We should note that all the analytic software we provide uses the coordinates obtained

by GLS, but other types of superimposition are available for depicting the results – although

GLS is usually fine for that purpose as well.)

Resistant-fit superimposition

Like GLS, resistant-fit superimposition methods minimize differences between configu-

rations by minimizing differences at corresponding landmarks over all landmarks. In

recognition of this general similarity, the resistant-fit methods have also been characterized

as “Procrustes methods” (see Chapman, 1990). However, the crucial difference between

resistant-fit methods and GLS is that the former do not use Procrustes distance as the cri-

terion for optimal superimposition. The resistant-fit methods also differ from each other

in the optimization criteria they do use. Below we examine the rationale for rejecting the

Procrustes distance metric and the general objective of the alternative optimization criteria,

then we focus on the oldest and most well-known of these methods, resistant-fit theta-rho

analysis (RFTRA; Siegel and Benson, 1982) and examine its optimization criterion in

somewhat greater detail.

The general objection is that least squares optimizations like GLS are very sensitive

to large displacements at few landmarks. In statistical procedures like regression, a few

cases with unusually large deviations from the general pattern (“outliers” or “influential

observations”) can have a large effect on the results because the procedure minimizes the

sum of the squared deviations. In shape analysis, a large change limited to one or a few

landmarks is sometimes called the Pinocchio effect (however, the influential landmarks

need not be at the tip of a long process). Figure 5.10A shows a hypothetical example in

which the only shape change in a tree squirrel scapula is the ventral displacement of the

three most ventral landmarks. When GLS is used to superimpose landmark configurations

(Figure 5.10B), the Pinocchio effect can have a large effect on the superimposition. The

least squares criterion distributes the displacement of the few landmarks across all the

other landmarks. In graphical displays, the Pinocchio effect appears to be “smeared out”

over all landmarks, which can be unsettling for some workers. The more severe conse-

quence is that the least squares criterion causes the variances of the influential landmarks

to be allocated to other points, inducing covariances (Walker, 2000). Ironically, mini-

mizing induced covariances was one of the reasons for using GLS rather than a baseline

method.

Resistant-fit methods reduce the influence of the Pinocchio effect by taking a “robust”

approach to superimposition. In statistics, “robust” means that the method is relatively

insensitive to outliers in the data. Similarly, a robust superimposition method is relatively

insensitive to a few landmarks with large relative displacements. A wide variety of error

functions have been used as criteria for robust fitting procedures, the interested reader

is referred to Press et al. (1988) for a discussion of several alternatives. None of them

allow analytic solutions for the rotation and scaling parameters needed to carry out a

superimposition; instead they use numerical methods (simplex searches) to find the rotation

and scaling necessary to minimize the error function.

The robust approach implemented by RFTRA uses the method of “repeated medians”

to determine the scaling and rotation necessary to superimpose one shape on another

chap-05 4/6/2004 17: 23 page 120

120 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(

)(

)

Figure 5.10 Hypothetical example of the Pinocchio effect as exemplified by ventral displacements

of the three most ventral landmarks of a tree squirrel scapula: (A) configurations superimposed by a

resistant-fit method (RFTRA); (B) configurations superimposed by GLS. The outline of the scapula

is approximated by lines connecting the landmarks.

(Chapman, 1990). We describe the steps used to find the scaling factor in some depth;

then more briefly describe the steps to find the rotation. For the scaling factor:

1. Compute the pairwise interlandmark distances in both shapes and then compute the

ratio of each pair of corresponding distances.

2. For each landmark, find the median of ratios for all segments radiating from that

landmark. This will yield one ratio for each landmark.

3. Find the median of the medians generated by step 2. This median of medians is the

scaling factor used in the superimposition – in other words, all coordinates of the

second shape are scaled by this factor.

After scaling the second form, the rotation angle used by RFTRA can be determined in

a similar fashion from the same set of line segments. The first step is to compute the angles

between the corresponding segments; the remaining steps find the median angle associated

with each landmark and then the median of the medians. (As in GLS, an iterative procedure

is used to compute a reference shape and superimpose all the specimens on it.)

RFTRA is robust because medians are relatively insensitive to outliers, and, conse-

quently, large changes at one or a few landmarks will not appreciably alter the median

scaling factor or the median rotation angle. This makes RFTRA resistant to the Pinoc-

chio effect, which helps to highlight the region where the effect occurs, as in Figure 5.10.

However, in the absence of the Pinocchio effect, superimpositions produced by resistant-

fit methods usually do not differ greatly from those produced by GLS. Figure 5.11 shows

GLS and RFTRA superimpositions of the real squirrel scapulae that were the basis of

the hypothetical example. The real scapulae differ in the relative length of the ventral

chap-05 4/6/2004 17: 23 page 121

SUPERIMPOSITION METHODS 121

(

)(

)

Figure 5.11 Comparison of GLS (A) and RFTRA (B) superimpositions for data that lack a Pinocchio

effect: real differences in scapula shape between two squirrels. The circles denote the two landmarks

that undergo large relative displacements, in addition to the three most ventral landmarks. The

outline of the scapula is approximated by lines connecting the landmarks.

process, as in the hypothetical case, but they also differ in the shape of the anterior edge

of the scapula (producing large relative displacements of the two circled landmarks). The

difference between the superimpositions is subtle; it is most noticeable at the ventral end,

where RFTRA attributes somewhat greater anterior displacements to the more ventral

landmarks.

The principal advantage of RFTRA and other resistant-fit methods lies in their ability

to address the Pinocchio effect. Their principal disadvantage lies in their departure from

the Procrustes distance metric. Comparisons performed under these superimpositions are

not covered by the theory developed in Chapter 4.

If you are considering a resistant-fit method, we recommend that you compare several

alternative methods. Not all rely on the median of medians, as RFTRA does; some employ

an explicit distance metric (or error function), not just the Procrustes distance. Methods

that use different metrics will produce different superimpositions, which may lead to dif-

ferent biological inferences as a consequence of the difference between metrics. It is usually

difficult to justify using any particular metric, so it is important to compare results based

on a variety of methods. That will increase the likelihood that your conclusions reflect

the structure of your data and do not depend on the type of software that happened to be

available. The program SuperPoser (Liebner and Sheets, 2001) carries out robust resistant-

fit methods using a variety of error functions. A different robust approach, one based on

identifying and excluding highly variable landmarks and then carrying out the superimpo-

sition on the remainder (using either generalized least squares or a resistant-fit method),

has been developed by Dryden and Walker (1999), but the software for this approach does

not seem to be readily available at present.

chap-05 4/6/2004 17: 23 page 122

122 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Summarizing the advantages and disadvantages of different methods

Before summarizing the advantages and disadvantages of the methods, we must admit to

our own preference. In general, we favor GLS because of its most notable advantage over

all other methods – it is the one consistent with the general theory of shape. The Procrustes

distance is the function minimized by this method, so it is the one most consistent with

theory (and therefore with our agreed-upon definition of shape). Even the pictures embody

that measure. When you look at the pictures, you can see the distance between shapes – it

is the square root of the summed squared lengths of the vectors displayed in the graphics.

At this point you still may have difficulty appreciating how important this is, but it is the

primary advantage of the method and the reason these coordinates are the ones preferred

for statistical analyses. Accordingly, we regard it as the “default” method – the one to

use unless there is a good reason to choose an alternative. We thus use it as the basis for

comparing all other methods.

Another important advantage of GLS over the baseline methods (BC and SBR) is shared

with the resistant-fit methods. The advantage is that GLS and the resistant-fit methods do

not have landmarks constrained to lie on the baseline, so there is no transfer of variance

from these points to all other landmarks. In general, this reduces the induced covariances

among landmarks and produces smaller variance ellipses around most landmarks. How-

ever, GLS enjoys less of an advantage when there is a strong Pinocchio effect. In this case the

resistant-fit methods will have a greater advantage, particularly if the user is not troubled

by their departure from the Procrustes distance metric and the associated shape spaces.

One disadvantage of GLS is that the number of variable coordinates exceeds the number

of shape variables by four. If the data are analyzed by conventional statistical tests, four

randomly selected coordinates must be dropped – which could have a substantial impact

on results. Fortunately, there are three generally accepted ways of getting around this

problem. One is to replace the landmark coordinates with a set of shape variables that

convey the same information but use the correct number of variables (the partial warps

scores discussed in Chapter 6). The second is to use resampling based methods that do

not require estimates of the number of degrees of freedom (these methods are discussed in

Chapter 8). The third option is to use statistical tests specifically adapted to the coordinates

produced by GLS (such as Goodall’s F-test, discussed in Chapter 9). Thus the discrepancy

between the number of variable coordinates produced by the GLS superimposition and

the number of shape variables is an obstacle that is easily removed.

A potentially more serious problem is that the Procrustes method freely rotates forms

to maximize their similarity. This is consistent with the definition of shape that forms the

basis of geometric morphometrics, in which rotations do not alter shape. From a bio-

logical perspective, however, this attitude toward rotations might seem unreasonable. As

Bookstein (1996) observes, the method “happily” rotates shapes around axes of bilateral

symmetry. The effect is illustrated in the analysis of rodent skulls (Figure 5.9), in which

the Procrustes superimposition appears to rotate and translate the midline, even though

that is biologically unrealistic. The effect is not due to asymmetry in the skulls because the

data for each individual were “symmetrized” (as described earlier); instead, the rotation

reflects the fact that posterior landmarks generally undergo larger medio-lateral displace-

ments than anterior landmarks (this might be most visible in the depiction by the sliding

baseline, Figure 5.9B). The change in the relative width of the two ends forces the GLS

chap-05 4/6/2004 17: 23 page 123

SUPERIMPOSITION METHODS 123

superimposition to rotate the skull, and the midline with it. The GLS is free to perform that

rotation because the orientation of the skull is not information relevant to an analysis of

shape. In contrast, our interpretation that the midline has been rotated, and especially our

unease with that interpretation, reveals a perspective in which orientation is information

relevant to an analysis of skulls.

The conflict between the two views outlined above raises an important conceptual issue.

If we regard rotation as inappropriate under some conditions, we need to clarify what we

actually mean by equivalent shapes. In particular, if we would not consider two forms to

be equivalent when they differ by rotation (or translation) of an axis of symmetry, then

we should not place them in the same class or claim that there is no distance between

them. This view of rotation is not consistent with the Procrustes metric, or with the idea

that rotations do not alter shape. Either we must adopt a different definition of shape

(and a new theory of shape analysis to go with it), or we must recognize that sometimes

the difference of interest is not purely a difference in shape. The latter approach seems

more productive; not only does it retain a well-established theory of shape analysis, but

it also recognizes that there is more to morphology than shape. In the case of the rodent

skulls, the rotations that were used to reveal shape differences removed an important

component of information about skull morphology – namely skull orientation. Usually

orientation is viewed as a “nuisance” parameter because it only refers to the orientation

of a specimen on a digitizer, but when dealing with axes of symmetry, orientation has

a biological significance. The loss of information about orientation is what makes the

pictures difficult to interpret.

Fortunately, the conflict between biological and geometric perspectives can be addressed

by judicious choice of graphical styles; we do not need to give up one perspective for the

other. The statistical analyses can be performed on the symmetrized data (the half skull)

to evaluate shape differences, regardless of the graphical representation. One option is to

use SBR to depict the results, although this method will not convey the actual Procrustes

distances among shapes (the coordinates obtained by SBR can be analyzed statistically,

using a resampling method, to check that the results are consistent with those based on

the coordinates obtained by GLS). Another option is to duplicate the coordinates of the

symmetrized landmarks, reflect them back across the midline to create whole symmet-

rical shapes, and then perform a GLS superimposition on the reconstructed whole skulls

(Figure 5.9C; see also Zelditch et al., 2003). This approach allows us to use coordinates that

are consistent with the Procrustes distance metric (hence directly depict the results of any

statistical analyses that are done) while avoiding the problem of interpreting inappropriate

translations or rotations of the baseline.

Although none of the limitations of GLS are particularly burdensome, there are still

times when a baseline superimposition method might be preferred. Generally, these are

cases when it is useful to have all shapes aligned to a standardized or conventional ori-

entation. Alignment of the skulls along the midline (as above) is just one such example.

Frequently, fixing the baseline simplifies the interpretation of complex shape changes by

making it possible to begin with an analysis of each landmark’s displacement relative to

the baseline (although it remains difficult to talk about changes in all the free landmarks

relative to each other). To the extent that this is an advantage, it is a bigger advantage

for BC than SBR because BC fixes all four coordinates of the baseline endpoints and

SBR fixes only two. In some cases this advantage might be cancelled out by the fact that

chap-05 4/6/2004 17: 23 page 124

124 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

SBR coordinates are scaled by centroid size rather than baseline length. Not only does

SBR have the advantage of using a concept of scale that is consistent with the theory of

geometric morphometrics; the implied shape distances are also closer to the Procrustes

distances.

Interpreting graphical results: resolving apparent inconsistencies

The graphical representation of results is one of the main reasons why geometric methods

are so useful. Because the graphics can influence the interpretation of results, it is impor-

tant to understand exactly how the superimposition methods influence the graphics. To

illustrate these effects, we use the four superimposition methods that have been the focus

of this chapter (BC, SBR, GLS and RFTRA) to depict the results of a single analysis of the

ontogeny of body shape in the piranha S. gouldingi (Figure 5.12).

Perhaps the most obvious difference among the four panels is the degree to which

postcranial landmarks are vertically displaced. It might appear that Bookstein shape coor-

dinates either exaggerate the degree to which the postcranial body is deepened, or else

that the other superimpositions understate it. However, this is not the case; all the other

superimpositions show a relative shortening of the body, which is equivalent to a relative

deepening. Both mean exactly the same thing. Relative body depth is a ratio between depth

and length, so it is just as reasonable to think of it as a decrease in length relative to depth as

to think of it as an increase in depth relative to length. Increasing body depth increases the

ratio by increasing the numerator; decreasing body length also increases the ratio, but by

decreasing the denominator. Because we come to the pictures informed by our knowledge

(A) (C)

(

)(

)

Figure 5.12 Ontogenetic change in body shape of S. gouldingi depicted by vectors of relative land-

mark displacement computed from four superimpositions: (A) BC coordinates from the 1–7 baseline;

(B) SBR to the same baseline; (C) GLS; (D) RFTRA.

chap-05 4/6/2004 17: 23 page 125

SUPERIMPOSITION METHODS 125

that body length increases over ontogeny, it may be difficult to grasp that it decreases

relative to depth. We would probably avoid saying that body length decreases relative to

depth, simply because that phrasing is disconcerting to biological intuition; instead, we

would say that depth increases relative to length. When pictures show a relative decrease

in a feature that is increasing in absolute length, readers may need some explanation of the

unexpected contrast. In particular, it is important to explain that the decrease is in relative

(not absolute) length.

Other apparent inconsistencies between pictures can also be reconciled, usually by con-

centrating on the changes in relative positions of landmarks rather than on the vectors at

individual landmarks. It may take a lot of practice before this is easy. For example, look at

the circled landmark in Figure 5.13. If you look only at this landmark, the results from the

different superimpositions appear to be inconsistent. That landmark appears to “move”

quite far anterodorsally in the BC superimposition (Figure 5.13A), but much less and in

three different directions in the other superimpositions: anteriorly in SBR (Figure 5.13B);

anteroventrally in GLS (Figure 5.13C), and almost entirely ventrally in RFTRA (Figure

5.13D). However, none of these statements actually reflect what the pictures show. None

of the pictures show the independent movement of any one point in isolation; rather, what

they show is the relative displacements of all points.

We get a better indication of the displacement of the circled landmark relative to neigh-

boring landmarks by “connecting the dots” – drawing line segments between landmark

locations to approximate the profile of S. gouldingi’s head. In Figure 5.14 we show the

same superimpositions and ontogenetic displacements of landmarks as in Figure 5.13, and

we add lines to show the relative positions of the landmarks early in ontogeny (dotted

lines connecting the bases of the arrows) and late in ontogeny (solid lines connecting the

tips of the arrows). Now we can see that the profile of the head is initially fairly shallow

(A) (C)

(B)

(D)

Figure 5.13 Ontogenetic change in body shape of S. gouldingi, highlighting the landmark at

the epiphyseal bar. Displacements are shown in four superimpositions: (A) BC coordinates from

the 1–7 baseline; (B) SBR to the same baseline; (C) GLS; (D) RFTRA.

chap-05 4/6/2004 17: 23 page 126

126 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

(B)

(A) (C)

(D)

Figure 5.14 Ontogenetic changes in dorsal head profile are highlighted by drawing in line segments

between the locations of landmarks at two different ages: early in ontogeny (dotted line) and late in

ontogeny (solid line). Displacements are shown in four superimpositions: (A) BC coordinates from

the 1–7 baseline; (B) SBR to the same baseline; (C) GLS; (D) RFTRA.

(nearly a straight line across all three points), and becomes much steeper (particularly

between the tip of the snout and the second landmark – the one that was circled). All

four superimpositions show this same change in profile. Despite apparent discrepancies

in the displacements of individual landmarks, the relationships among the landmarks are

consistently represented. Before we can interpret the results in terms of these vectors of

relative landmark displacement, we must become accustomed to what these vectors rep-

resent. The individual vectors do not show changes at landmarks; rather, the differences

between vectors show changes between the landmarks.

Because the different superimpositions show the same shape change, the primary cri-

terion for choosing which to use to display results is simply the ease of interpretation. In

some cases, it is easier to understand the results depicted by Bookstein shape coordinates

because the fixed baseline provides a simple, straightforward line of reference. That line

makes it easier to interpret and to verbalize the information contained in the pictures;

aiding both your ability to understand your results and your ability to communicate your

results to others.

Recommendations regarding superimposition methods

1. Unless there is a good reason for choosing an alternative, use GLS.

2. When an axis of symmetry is present, and it is rotated by GLS, decide whether you

think that shape is unaltered by rotation. If you are willing to redefine shape, so that

it actually is altered by rotation, use SBR, and use resampling-based statistical tests.

Alternatively, use GLS for statistical analyses and depict the results by back-reflecting

chap-05 4/6/2004 17: 23 page 127

SUPERIMPOSITION METHODS 127

the data. However, if choosing this second approach, realize that ordination methods

(like principal components and canonical variates analysis, discussed in Chapter 7) do

not necessarily give the same results for the symmetrized and back-reflected data. If

using ordination methods, do the analysis on the symmetrized data, save the PC or

CV scores, and regress the back-reflected data on those scores to obtain the pictures.

3. When conventional statistical tests are the only option, either use Bookstein’s shape

coordinates or use the variables explained in Chapter 6 (partial warps). These are

the only methods that produce 2K −4 variables, equal to the dimensionality of shape

space.

4. When the major objective is to depict the differences between forms, use several meth-

ods to see how each represents that difference. Only the GLS method displays the

difference in terms that are exactly commensurate with the Procrustes distance. If

your primary concern is having a picture that unambiguously represents the measured

distance between shapes, use the GLS to depict the results. If ease of interpretation

or communication is of greater importance, use whatever method produces the most

easily interpreted graphics.

5. When there is a great deal of noise that does not respond to alternative choices of

baselines, or at least not to alternatives that permit straightforward interpretations,

SBR or GLS will be more useful than Bookstein shape coordinates (regardless of

interpretability).

6. If you insist on a robust fitting method, look beyond RFTRA. Determine which of

the many possible error functions is most appropriate for your data, and use that in

weighting the variances.

Software

The program CoordGen, introduced in Chapter 3, can also be used to obtain coordinates

by SBR, GLS, and RFTRA, in addition to BC. A variety of resistant-fit coordinates can be

calculated using SuperPoser. Virtually all the software in the IMP series can read coordinate

data produced by any kind of superimposition and convert it to GLS for analysis, so you

can input the BC coordinates you saved earlier and use them in all subsequent analyses.

Each program gives you the option to display your results (based on the GLS) using any

of the other available superimposition methods.

When running CoordGen, you will follow the same general procedure to obtain super-

imposed coordinates regardless of the method of superimposition, so most of the directions

for using CoordGen are given in Chapter 3 (which introduced BC). There is one major

difference that applies to GLS and RFTRA. Unlike BC and SBR, GLS and RFTRA do not

depend on baseline, so you do not need to enter the endpoints of the baseline. However,

during the input of a new dataset, CoordGen automatically aligns the specimens to the

default baseline. This will be the approximate orientation of your specimens after GLS,

unless you change it. To select a more convenient orientation, enter a baseline that is closer

to your preferred orientation, click Show BC, then click Show Procrustes. Another option

for orienting your GLS superimposition is to click the Show Procr (PA) button. This will

align the reference configuration so that its principal axis (its long axis, approximately)

is aligned with the X-axis. You can also click on the Vertical Axis (Procrustes PA) before