Local disparity guru and paleontologist Randy Irmis (that’s Randall B. Irmis, Ph.D., if you go by his web page) recently posted a nice long list of recommended readings on the issue of disparity – what it is, how to calculate it, etc. As a reminder, disparity is the measure of how different species are from each other in terms of shape, size, or other discrete features (not the same as diversity, which just counts how many different species exist – once again, see Randy’s eloquent post on the topics). It just so happens that documenting disparity in ornithischian dinosaurs is at the top of our list for the ODP. Hence, I decided to buckle down and read through an important recent paper on the topic (one that Randy happened to highlight in his list, too).

In the interest of getting this post out in a timely manner, I’m mainly going to be posting my unpolished notes, taken a few weeks ago in the comfort of my bed (nothing like a little light bedtime reading). I’ve made a few adjustments here and there, but otherwise you can consider this a peek into my stream of consciousness while reading the literature. Because I was mainly interested in how the work could be applied to the ODP, I didn’t really bother with summarizing the specific analyses done by the authors. Thus, without further preface:

**The Citation:
**Brusatte, S. L., Montanari, S., Yi, H.-Y, and Norell, M. A. 2011. Phylogenetic corrections for morphological disparity analysis: new methodology and case studies.

*Paleobiology*37: 1-22. [

*unfortunately, not openly available as a PDF*] [link to abstract]

**The Main Gist:
**The fossil record just isn’t complete – and that’s particularly true for many of the early members of important ornithischian clades (like thyreophorans and marginocephalians). However, it’d be nice to interpolate some of these missing data in order to produce a more complete picture of the changes in a clade’s disparity over time and in morphospace (the multi-dimensional plot of the shape of an animal’s bones, in this case). Brusatte and colleagues, building on the work of many other authors, have formalized a method to fill in some of these gaps by producing a plausible reconstruction of missing ancestors.

**My Notes
**[as presented here, it’s a mix of to-do tasks for the ODP, a cookbook for the analysis, and how Brusatte et al.’s method will be applied; caveat emptor]

** The questions:** What is the morphospace occupied by ornithischian dinosaurs over time? How does the morphospace change? How does the morphospace occupied by specific clades differ?

**The tasks:**

- Assemble data matrix (taxon/measurement matrix)
- Reconstruct ancestoral measurements following Brusatte et al. 2011
- Calculate Euclidean distance matrix (“quantifies the pairwise dissimilarity between taxa”) – this presumably calculates dissimilarity for each taxon/measurement pair
- Apply principal coordinates analysis (PCoA) to each analysis (better handles missing data than does PCA [principal components analysis]). Can be done in R.
- PCoA produces scores for each taxon along n=#taxa axes. Can be done in R.
- Examine slope of scree plot to determine where break occurs; only examine these “interesting” axes. I think this scree plot can be done in R.
- Calculate disparity indices from the PCoAs, using different bins (categories). Can be done in R. Categories might include:
**1)**clade;**2)**time;**3)**locomotor category;**4)**combination of clade/locomotor category. - Indices include: sum of range of values along axis 1, 2, … n (i.e., range 1+range2+range3. . .); product of range of values along axis 1, 2, n (range 1 * range 2 * range 3. . .) normalized to the nth root; and same sum and products for variance in each bin.
- Rinse and repeat using ancestral values as calculated following Brusatte et al. 2011.

**Ideal results:**

- Disparity indices that can be compared statistically (using bootstrap values) for various categories. E.g., a disparity value for Ceratopsia, Ornithopoda, Thyreophora, etc. disparity value for quadrupeds vs. bipeds.
- Graphs showing point clouds for various clades along various axes (e.g., PC1 vs. PC2)
- Graphs showing trends for disparity over time, with different groups. E.g., trend line showing disparity in ornithischians as a whole, along with trend line showing disparity in thyreophorans, ceratopsians, etc. Potential sample size issues here, particularly for clades with few members or few members early in their history
- Narrative text and / or table showing what factors are loaded on which axes

It is now: http://www.miketaylor.org.uk/tmp/BrusatteEtAl2011-phylogenetic-corrections-for-disparity.pdf