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218 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

of length and size suggests differences in shape; in the next section we present a more direct

test for shape differences.) We also find that most of the variation in jaw length is explained

by the regression on jaw size, not by differences among localities (indicated by difference

in mean squares).

A complex trait and a categorical variable

So far in this chapter, we have discussed tests in which the dependent variable is a simple,

one-dimensional, continuous trait such as size. In such cases, there is a single total variance

to parse into its explained and unexplained components. That total was divided into

more explained components when more categorical variables were added, but the test

still only evaluated the relative magnitudes of the explained and unexplained components.

In contrast, shape is a single, complex trait described by several continuous components.

To parse the variance of this multidimensional trait, we use the same technique that would

be used to parse the variances and covariances of multiple, separately measured traits,

namely multivariate analysis of variance (MANOVA).

In the simplest MANOVA, we have a multivariate dependent variable and just one

categorical variable with two classes. Our question is whether there is a difference between

classes in the dependent variable, so we want to perform a multivariate equivalent of

the t-test. In other words, we want to evaluate the difference between the two means

on all measured variables simultaneously. The multivariate generalization of the t-test is

Hotellings T

. T

can be derived from the univariate t, using the formula introduced earlier

for the case in which samples have equal variances but different numbers of individuals

(from Equations 9.2 and 9.3):

t =

(<X

> −<X







(9.5)

Squaring this expression produces:

(<X

> −<X





(9.6)

which can be rearranged to:







(<X

> −<X



(9.7)

Now, we replace the univariate difference between means <X

> −<X

> with the vec-

tor of mean differences on all variables (<X

> −<X

, and we replace the pooled

within-group variance s

with the pooled within-group variance–covariance matrix S

This yields:





(<X

> −<X

−1

(<X

> −<X

>)(9.8)

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MULTIVARIATE ANALYSIS OF VARIANCE 219

which is distributed approximately as an F-distribution. The degrees of freedom are given

by the number of variables (V) and N

−1 −V.

When we have multiple groups, each described by a multivariate dependent variable,

we need a multivariate generalization of the F-test used in ANOVA. Recall that the uni-

variate F-test is a function of the variances within and between groups. Accordingly, the

multivariate F-test should also be a function of the within-groups and between-groups

variance–covariance matrices (W and B). Although this implies that the test statistic for

the multivariate F-test should be a function of the eigenvalues of W and B, it is not clear

what that function should be. One simple solution is the Hotelling–Lawley trace, which is

the trace of BW

−1

(the trace of a matrix is the sum of its eigenvalues). Several alternatives

to the Hotelling–Lawley trace have been proposed; all can be equated with functions of

the eigenvalues of BW

−1

. Each of these test statistics can be converted to a value that

is distributed approximately as an F-distribution, making it possible to determine the

p-value for the hypothesis that the samples were drawn from the same multivariate nor-

mal distribution. For example, a commonly used statistic is Wilks’ , formally defined as

the determinant of W(B +W)

−1

(i.e. the product of the eigenvalues of that matrix), which

is equivalent to the product



1 +θ

where θ

are the non-zero eigenvalues of BW

−1

Wilks’  can be converted to functions that approximate either the χ

(−ln  weighted

by a function of the degrees of freedom) or F distribution (W

1 −

1/H



1/H

, where H is 1 less

than the number of groups and W is a function of the degrees of freedom). Chatfield

and Collins (1980) cite several studies that compare the performance of these and other

tests, and conclude that the comparisons are indecisive. They also note that the two tests

mentioned above and a third test, Pillai’s trace (trace of B(B +W)

−1

), are asymptotically

equivalent, meaning that they approach the same value at large sample sizes and differ

little in power at small sample sizes. Furthermore, the three tests are exactly equivalent

when there is only one independent variable.

The total number of degrees of freedom differs slightly among the three test statistics,

but in all cases it is approximately the product of the number of groups (G) and the total

number of individuals in all groups (N). In all three tests, the number of degrees of freedom

for the between-groups variance is V(G −1) where V is the number of variables and G is the

number of groups. The number of degrees of freedom for the within-groups variance is the

total degrees of freedom minus the between-groups degrees of freedom. For this difference

(approximately NG −VG) to be greater than zero, the number of individuals must be

greater than the number of variables. This is consistent with the algebraic requirement

that the number of equations must be greater than the number of variables in order to

have more equations than there are variables in those equations, so N must be greater

than V.

Although MANOVA can be used to test for shape differences among groups, there

are constraints on the kind of shape data that can be used. One constraint is due to the

fact that MANOVA assumes that the measurement space is Euclidean (as discussed in

Chapter 4, shape space is not Euclidean). Another constraint is due to the fact that the

number of shape variables is smaller than the number of variable coordinates produced by

most methods of superimposition (as described in Chapter 5). By specifying the “variable

coordinates” we mean to exclude the fixed baseline endpoints produced by the two-point

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220 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Eastern Michigan

Western Michigan

Southern States

Figure 9.3 Superimposed landmarks of 119 squirrel jaws from three localities, eastern and western

Michigan, and southern states.

Eastern Michigan

Western Michigan

Southern States

Figure 9.4 Superimposed landmarks of the mean jaw shapes for the three geographic samples

shown in Figure 9.3.

registration (Chapter 2). For landmarks taken on two-dimensional images, the number of

shape variables is 2K − 4, where K is the number of landmarks. If the number of variable

coordinates is greater than the number of shape variables, then the number of variable

coordinates overestimates the true degrees of freedom.

Fortunately, we have two ways to project shapes in shape space onto a Euclidean tangent

space, thereby converting shape information to a form that satisfies these two assumptions

of MANOVA. One option is to compute Bookstein shape coordinates, which produces

the same number of variable coordinates as there are dimensions in the shape space. The

other option is to use partial warp scores for configurations obtained by Procrustes (GLS)

superimposition. The number of partial warps (including the uniform components) is the

same as the number of dimensions in the shape space.

Figure 9.3 shows the Procrustes superimposed landmarks for the squirrel jaws. At each

landmark the distributions of the position of that landmark in the three geographic samples

overlap broadly, suggesting there is little if any difference in jaw shape among localities.

Comparison to the picture of the three mean shapes (Figure 9.4) suggests that the differences

among groups are small relative to the variability within each group. Table 9.6 lists results

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MULTIVARIATE ANALYSIS OF VARIANCE 221

Table 9.6 MANOVA of jaw shape among localities using thin-plate

spline coefficients

Test statistic Value F DF p

Wilks’  0.136 5.98 52, 182 <0.000001

Pillai trace 1.25 5.87 52, 184 <0.000001

Hotelling–Lawley 3.54 6.08 52, 180 <0.000001

trace

Table 9.7 Goodall’s F test for shape differences among locations

Localities PPd F DF p

W and E 0.0265 9.83 26, 2340 <0.000001

W and S 0.0384 22.2 26, 2444 <0.000001

E and S 0.0204 3.76 26, 1248 <0.000001

All three n.a. 13.5 52, 3016 <0.000001

W =western Michigan, E =eastern Michigan, S =southern states. PPd is the par-

tial Procrustes distance between pairs of mean shapes, which does not apply to

the three-sample case.

of three tests for differences in jaw shape among the three squirrel populations. The data are

the scores of the thin-plate spline components, with the mean of all individuals (computed

by Procrustes superimposition) as the reference shape. The F-values and degrees of freedom

differ slightly, reflecting differences in the computations performed for each test; when

p-values are close to the preferred α-level, these differences can lead to different conclusions.

An alternative to performing a MANOVA on shape variables in the tangent space is to

perform the analysis in shape space, measuring deviations from means as sums of squared

Procrustes distances. The test statistic for this analysis is Goodall’s F, which is the ratio of

explained and unexplained variation in those distances. As in conventional MANOVA, the

degrees of freedom for the explained variance are given by V(G −1), where V is the number

of dimensions (variables) in shape space and G is the number of groups. However, the

total number of degrees of freedom, V(N −1), and the within-groups degrees of freedom,

V(N −G), are much higher than in a conventional MANOVA. This difference is due to

the fact that we are computing Procrustes distances for N individuals in V dimensions,

but do not need to estimate all the variances and covariances of the shape variables.

Table 9.7 shows the F-values and corresponding p-values obtained for all three pair-wise

comparisons of the three populations, and for a simultaneous analysis of all three groups.

The partial Procrustes distances between the means are also shown for the two-group

comparisons. Because the dimensionality of shape space is a factor in the total number

of degrees of freedom, the p-values can be quite small even when the categorical variable

explains very little of the shape variation in the data set. As might be surmised from Figures

9.3 and 9.4, differences among the three populations explain only a small fraction of the

variation in the Procrustes distances. Those differences can be judged significant because a

large amount of information (resulting in large numbers of degrees of freedom) was used

to estimate what those differences are.

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222 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

Other uses of the t-test

Earlier in this chapter we discussed a typical use of the t-test, namely to determine whether

the difference between two sample means is statistically significant. The t-test can also be

used to evaluate the uncertainty of a derived quantity – that is, a quantity that is not com-

puted for each specimen but is computed after analysis of several specimens. In geometric

morphometric studies, we often want to know whether the partial Procrustes distance

between one pair of mean shapes is significantly different from the distance between

another pair of mean shapes. For example, the distance between mean shapes of squirrel

jaws in the western Michigan and southern samples appears to be much greater than the

distance between mean shapes of the eastern Michigan and southern samples (0.0384 vs

0.0204). The partial Procrustes distances computed from the original data sets are the

expected values <X

> and <X

> in the formula for t (Equation 9.2, reproduced here):

t =

(<X

> −<X









−1)s

+(N

−1)s

−2







(9.2)

The sample sizes N

and N

are the combined sizes of the paired data sets (69 +27) and

(23 +27) corresponding to the distances <X

> and <X

>. Although these replacements

are straightforward, the values of s

and s

are less clear. Given only the original data

sets, we have only one estimate for each of the distances; they are single observations, not

means of multiple observations. Without multiple observations, we have no direct way

to estimate the uncertainty of the expected value. One solution to this dilemma is to use

bootstrap resampling (Chapter 8) to estimate the standard error of each distance (another

solution is to bootstrap the difference between the distances – see the next section). Because

/N, we must substitute N ·SE

into Equation 9.2, producing:

t =

(<X

> −<X









−1)N

+(N

−1)N

−2







(9.9)

Substituting the distances (0.0384 and 0.0204), standard errors (0.0030 and 0.0032) and

sample sizes (96 and 50) into this equation yields t =3.60 with 144 degrees of freedom for

p < 0.0005. Thus, the difference between the two distances is statistically significant.

Whenever using the t-test, whether comparing means of observed variables or com-

puted values of derived variables, it is important to remember that the test assumes that

deviations within the groups are normally distributed. Fortunately, the test is fairly robust

to violations of this assumption, but care should be taken when the p-value is close to 0.05

(or whatever α is chosen as the criterion for statistical significance).

Resampling-based tests

All of the statistical tests discussed so far in this chapter make assumptions about the distri-

bution of variation around the means that are being compared. As discussed in Chapter 8,

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MULTIVARIATE ANALYSIS OF VARIANCE 223

deviations of the populations from the model can lead to erroneous conclusions. Resam-

pling procedures can be used, allowing biological samples to deviate from ideal theoretical

distributions, but implementing a test using these procedures often requires the user to

write the program for that specific test. Below, we briefly demonstrate two readily available

bootstrap tests.

In the first example, we use the bootstrap to determine whether jaw shape differs

between the two samples of squirrels from Michigan. The null hypothesis is that the

observed difference could arise by chance when sampling from a single population, so

all 96 specimens were combined into a single pool. In each iteration, two bootstrap sets of

the original sample sizes (69 and 27) were drawn with replacement from the pool and the

F-ratio was computed for that pair. After the chosen number of iterations was completed

(400 in this case), the bootstrap sets with an F at least as large as the original set (9.83) were

counted. This number divided by the number of iterations is the probability of obtaining

the original samples under the null hypothesis. In this case, only the original samples had

an F as large as 9.83, so the p-value is 1/400 (0.25%).

In the second example, we use the bootstrap resampling procedure to evaluate the

uncertainty of a derived quantity: the partial Procrustes distance between the mean shapes

of the two samples. Here, the question is about the uncertainty of the distance between

sample means, which is a function of the sampling of each source population. To simulate

this by bootstrapping, we keep the samples separate, and in each iteration draw separate

bootstrap sets of each sample and compute the distance between the means of the bootstrap

sets. This set of distances is used to estimate the 95% range interval around the distance

between the population means. One use of this range interval is testing whether the distance

between one pair of samples differs from the distance between another pair of samples. In

the analysis of squirrel jaws, the partial Procrustes distances between mean shapes suggest

that the southern sample is much farther from the western Michigan sample than it is

from the eastern Michigan sample (0.0384 vs 0.0204). This is supported by the 95%

range intervals, which do not overlap (0.0338 −0.0456 vs 0.0172 −0.0295).

We can also answer the question about the difference between the two distances by

bootstrapping that difference (now the difference is the derived trait rather than one of the

distances). In this case, we draw bootstrap sets of all four samples in each iteration, com-

pute the two distances between pairs of means and the difference between those distances.

These results are used to determine the 95% range of the difference between distances over

the series of iterations. For the squirrel jaws, the distance between mean shapes of the west-

ern Michigan and southern samples (0.0384) is greater than the distance between mean

shapes of the eastern Michigan and southern samples (0.0204). The difference between

the distances is 0.0184. After 400 bootstrap iterations, the 95% range of the difference is

estimated to be 0.0070−0.0246. This range does not include zero, so we can infer that

the observed difference between distances is statistically significant.

Software

Two programs in the IMP series are available for performing the statistical analyses dis-

cussed in this chapter, CVAGen and TwoGroup. Both perform simple analyses; more

complex ones will require a commercial statistical package, or TPSRegress (at the end

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224 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

of this section we provide general instructions for using commercial packages to conduct

these analyses).

MANOVA of shape using CVAGen

As discussed in Chapter 7, CVAGen can be used to describe differences among groups. The

output for CVAGen includes a test of the hypothesis that differences among the groups

are statistically significant (for detailed instructions on running this program, look at the

discussion of CVAGen in Chapter 7). Here we discuss only the results of a statistical test

reported in the Auxiliary Results box window. These are the results of Bartlett’s test for the

number of informative CVs. Bartlett’s test is based on a MANOVA testing the hypothesis

that there are differences among the groups; the MANOVA is repeated with progressively

fewer CVs to determine how many of them are informative. Accordingly, each row in the

window shows the value of Wilk’s , the corresponding χ

, the degrees of freedom and

the p-value for the test that there are differences among groups in a progressively smaller

subset of the data. The first iteration, using all of the CVs, is the one that is relevant

here; it is the test for differences among groups using all of the available data. The null

hypothesis is that there are no differences among the groups; that the differences among

the samples are no greater than would be expected if they had all been drawn from the

same multivariate normal population. Remember, rejection of this null does not mean that

each group is different from every other. The plots of CVA scores and the classification

table (produced by selecting Show Grouping By CVA in the Statistics menu) can suggest

reasonable hypotheses of which groups differ, but these are not definitive tests.

It is possible to use CVAGen explicitly to test whether two particular groups are signif-

icantly different by including only those two groups in the data file. Then there is only one

CV (the axis maximally discriminating between the two groups). If they can be differenti-

ated on this axis, then they are significantly different. Doing this requires you to construct

a separate file for each pair of groups, so we recommend using TwoGroup, which requires

fewer files to perform the same set of tests.

Running TwoGroup

Each group must be in a separate file, in standard (X1,Y1,…CS) format. The files are

loaded separately by clicking Load Data Set 1, finding the file, then clicking Load Data

Set 2 and finding that file. As usual, you can display the data in various superimpositions,

by clicking on your choice in the Show Data field below the visualization window. Be sure

to select the correct baseline points before choosing plots or analyses that use a baseline

superimposition (i.e. BC or SBR). After the files are loaded, all of the test options are

active, so be sure that you have selected the right number of bootstraps before clicking one

of these buttons.

To test the significance of the difference between samples using Bookstein coordinates,

choose Hotelling’s T

(BC); this is the only test available for these coordinates. The results

window will report values for F, the degrees of freedom, and p. The results will also

include the distance between means. This distance is the sum of the squared distances

between corresponding landmarks, but it is not the minimized Procrustes distance because

the landmark configurations are not in Procrustes (GLS) superimposition.

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MULTIVARIATE ANALYSIS OF VARIANCE 225

To test the significance of the difference between samples using the partial Procrustes

distances, choose Goodall’s F (Procrustes). For this test, the coordinates are superimposed

using GLS with the specimens rescaled to unit centroid size (see Chapter 5). Again, values

of F, the degrees of freedom, and p will appear in the results window. The distance between

means is also reported, which is the partial Procrustes distance.

Next to the buttons for the analytic tests are buttons for two tests that use a bootstrap

resampling procedure (F-test, SBR and F-test, Procrustes). The F-test, SBR is a resampling-

based F-test for coordinates in the SBR superimposition; F-test, Procrustes is a resampling-

based version of Goodall’s F-test. Before choosing either option, select the right number

of iterations in the No. of Bootstraps box on the far left. The results, which will appear

in the results window, will include the F-value computed for the original data set. After

this is a “Significance level:..” which is the fraction of iterations (in decimal format) in

which F is greater than or equal to the value reported for the original data. The output

also includes a distance between the means of the original data sets. Again, if you selected

the Procrustes test, this is the minimized partial Procrustes distance; if you selected SBR,

it is not the minimized distance because the specimens are not in the partial Procrustes

superimposition.

The three buttons in the box labeled Bootstrapped Distances Between Means invoke

analyses in which bootstrap resampling is used to estimate the standard error and 95%

range of estimates for the distance between the group means under the indicated super-

imposition. The observed distance and the bootstrapped standard error of that difference

can be used to test whether the distance between one pair of samples is different from the

distance between another pair of samples (using TBox, described below).

Like the standard error, the 95% range is a measure of the uncertainty of the observed

distance between the two means. However, this range should not be used to test the

hypothesis that this distance is significantly greater than zero. A distance cannot be less

than zero. Even if the groups have identical means, it is unlikely that bootstrap sets will

be drawn in which the difference is exactly zero. You can demonstrate this by loading the

same file twice (i.e. using the same file as data sets 1 and 2); the lower end of the confidence

interval will be a small number, but still greater than zero.

The value of the 95% range is that it can be used to evaluate whether the distance

between one pair of groups is different from the distance between a second pair of groups.

If the ranges for the distances do not overlap, then the difference between the distances

is statistically significant. The limitation of this test is that it may be too stringent: the

probability that both distances are more similar than the adjacent ends of the ranges is

considerably less than 0.05 (in fact, it is less than 0.05

if the normal model applies). If

you have ranges that overlap, you may want to consider using the standard errors in an

analytic test (in TBox), as mentioned before.

Another option for comparing the difference between two distances is to bootstrap that

difference. Use Load Data Set 1 and Load Data Set 2 to load the first pair of samples,

then go to the File pull-down menu and use Load Group 3 and Load Group 4 to load the

second pair of samples. Now go to the More Stats pull-down menu and select Bootstrap

Distances 1+2vs3+4. When the iterations are completed, the results window will show the

partial Procrustes distance between means 1 and 2, the 95% range and the standard error.

Scrolling through the results will reveal the same information for the distance between

means 3 and 4. At the end, you will come to the 95% range for the difference between

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226 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS

the distances. If this range includes zero, then the distances are not significantly different;

sometimes one distance is larger, sometimes the other distance is larger.

In addition to the statistical tests, TwoGroup can plot the superimposed landmarks

and the superimposed means (but only for the data sets 1 and 2). These plots can be

modified using the Symbols Control pull-down menu, which allows you to change the red

and blue symbols to black or gray, fill the symbols, and increase their size. The plots of

the differences between means can be edited using the options located on the Difference

Plot Options pull-down menu. As usual, you can select from a variety of superimposition

methods and types of displays, trim the grid and rotate the reference.

Using TBox

This program can be used to perform t-tests on means of directly measured traits or on

expected values of derived quantities like the Procrustes distances between groups. Type

the values you want to compare into the boxes labeled Mean of Group 1 and Mean of

Group 2. These values could be the means of a univariate trait, like centroid size. In that

case, simply type the means, sample sizes and standard errors into the appropriate boxes.

The group with the larger mean should be entered as group 1. When you click the big

green Run Calculation button, the program will compute the t-value for the difference

between the means and determine the corresponding p-value. If the values are Procrustes

distances between samples, enter the distance between the first pair of samples as the Mean

of Group 1 and the distance between the second pair of samples as the Mean of Group 2.

The sample size for the group is the sum of the two sample sizes. Standard errors for the

distances can be obtained from the resampling tests in TwoGroup, as described above.

Note the caveats for TBox. You must have standard errors for the quantities you are

comparing; if you have variances or standard deviations, you must convert them to stan-

dard errors. The more important caveat is the assumption of normality. TBox reports the

value of p for t, when t is normally distributed, which is only expected when sample sizes

are large (>60). If your sample sizes are smaller, you may prefer to use the more con-

servative p-values reported in t-tables of more conventional statistics texts and programs.

However, even these p-values are computed under the assumption that deviations within

samples are normally distributed. If even this assumption of normality is doubtful, you

should probably use a resampling-based test.

Conducting ANOVAs/MANOVAs using other programs

Simple ANOVAs, like the test for differences in centroid size between two samples, can

often be performed using a hand-held calculator or a spreadsheet program. The hard part

is sorting the data correctly and keeping track of the number of entries. If you use a

spreadsheet program, be sure you choose the correct ANOVA or t-test options for equal

or unequal sample sizes, and for equal or unequal variances.

Complex MANOVAs, analyses in which there is more than one categorical variable

(locality and sex), or analyses in which there is a categorical variable and a covariate (sex

and size), usually must be performed in a computer program package specifically designed

for multivariate analysis. Although these analyses are not algebraically difficult, they can

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MULTIVARIATE ANALYSIS OF VARIANCE 227

be computationally intensive. Below we present some general guidelines for performing a

MANOVA on shape variables in commercial statistical packages.

The first step is figuring out how to get your data into the program you intend to use

to analyze the data. To do this correctly, you will need to understand how the analytic

program expects the data to be formatted. One part of this is determining whether the

analytic program can accept data from your database, spreadsheet or text file; another

part is determining whether the program requires particular symbols to delimit fields

(e.g. space, tab, or comma), or types of variables (e.g. “$” as the last character in the

name of a categorical variable). You should also determine whether it will be easier to add

the categorical variables to the data before or after they are read into the analytic program;

this will be a function of how easy it is to edit the data file after it has been read.

The next step is deciding whether you want to analyze the shape variables from the thin-

plate spline decomposition (scores on the partial warps and uniform components) or the

coordinates of the landmarks. If you decide to use landmark coordinates, you will need to

use CoordGen to compute the superimposition before you import the data. Whichever you

choose, make sure you import all of the shape data. For spline components, this includes all

of the partial warps and uniform components. (IMP programs output partial warps in order

of increasing spatial scale, followed by the scores for the uniform component, with centroid

size or ln(centroid size) in the last column.) If you use landmark coordinates registered to

a baseline (Bookstein shape coordinates, or sliding baseline registration), remember to

omit the invariant coordinates of the baseline points. If you do not omit them from the

input file, you will have to remember to omit them when you select the variables to be

included in the analysis. If you do not use Bookstein shape coordinates or the scores on

the spline components, you must remember that the correct number of degrees of freedom

is less than the number of variables (−4 if Procrustes superimposition, −2 if sliding baseline

registration). You must also remember that Procrustes superimposition and sliding baseline

registration do not project the specimens onto the same space as the thin-plate spline or the

Bookstein shape coordinates (see Chapter 5). Under many circumstances, these choices will

not alter conclusions about the significance of differences between groups. One situation

in which the choice can influence conclusions is when the range of shape variation across

all groups is so large that the difference between the shape space and the tangent space

become noticeable (you can check whether that is the case for your data using TPSsmall).

The other situation in which the choice can influence conclusions is when the differences

among groups are so small relative to the variance within groups that a small difference in

the number of degrees of freedom shifts the p-value of the test statistic across the preferred

α-level. In such cases, it is better to be cautious about claiming significance.

After you have resolved all of the issues relating to formatting and entering data into your

program, the next step is navigating through the program to select the right analysis. If your

program does not have a giant button labeled MANOVA, look for a menu item referring to

“linear models” or “linear hypotheses;” MANOVA will be one of the options within that

category. (You could also try searching the help menu or the index of the manual.) After you

have started the MANOVA module, you will probably be asked to select variables. Again,

remember to include all of the relevant variables in the list of dependent variables (all of the

spline components or all of the variable landmark coordinates). The independent variable

will be the categorical variable or covariate hypothesized to explain shape variation. Your

program may also require an extra step to indicate that the explanatory variable is a