Shape coordinates help us to carry out many familiar operations of ordinary scientific statistics. Figure 4.4, for example, shows two averages of points from the preceding scatter corresponding to the means for the subgroups of this data set in which we are the most interested: the average for the normal brains (the doctors') versus the average for the patients'.

The comparison in the figure began with 52 coordinates (x and y for each of 26 points) but lost four of these degrees of freedom when the forms were translated, rotated, and scaled to all fit the same sample average. The 48 dimensions of variability remaining are quite a bit more than the pooled sample size here, 12 + 13 = 25, so it is not at all obvious how to proceed with a conventional significance test for the improbability of the difference shown in the figure on a null hypothesis of no group difference.

Most good morphometric data sets are of this formâ€” more coordinates than casesâ€”and hence much of the time spent developing this new morphometrics was devoted to the specific problem of rigorous statistical test-

ing under these conditions (cf. Bookstein, 1996). It turns out that these problems of dimensionality are much less severe than they seemed. The 48 variables remaining are not just any set of 48 measurements; they are coordinates of corresponding points that, after the Procrustes maneuver, do not vary much in location in two-dimensional Euclidean space. Their statistical analysis is thereby susceptible to a clever maneuver that was originally laid out by Colin Goodall (1991). If, for purposes of testing, we are willing to treat all the points as equivalent and, likewise, all the ways in which the shape of their configurations can vary, then we can mount a powerful test of the presence of any group difference by paying no attention to any aspect of the sample variation except net Procrustes distances of each form from the others and from the mean.

The argument begins with a suspiciously symmetric ''null model,'' according to which each point of the data set arises from the same grand mean by pure uncorre-lated Gaussian noise of the same small variance in every direction at every landmark separately. Under this model, if one has no prior knowledge about just how the shapes are likely to differ, the best statistic for testing group differences in mean shape is exactly the Procrustes distance we have just introduced, and its distribution on the null is borne in an F-ratio that can be evaluated and tested no matter how many points there are or how few specimens per group. The F in question looks just like any other F-ratio from your introductory biostatistics course: a between-group sum of squares divided by a within-group sum of squares, multiplied by an integer fraction and looked up in an F-table. In this application the integer fraction and the degrees of freedom are functions of the number of landmarks as well as the sample sizes. The formula, which is appropriate on the assumptions opening this paragraph, regardless of the number of landmarks or cases, reads as follows: The quantity

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