As we finish up combining the data, it’s time to start thinking about the specific analyses that we’re going to do. What are the specific questions we’re asking? What are the techniques that we need to address the questions? Some excellent discussions between ODPers have been happening in one of the recent posts, and I was hoping to continue that here. In particular, I wanted to refocus the discussion on the project’s essential questions, and consider the types of analyses that we can use to answer each question. I’m just thinking out loud here (this is open notebook science, after all), and invite suggestions and discussion in the comments section. In particular, I’m referencing the “big questions” outlined in one of our first posts.
Why did ornithischians evolve quadrupedality multiple times?
I think this one is going to have to simply rely upon our interpretation of the data. After all, we can perhaps answer “how,” but the “why” can’t really be answered in this setting. So, it’s something to consider in the “discussion” section of the paper. But, see the next question. . .
Was the evolution of quadrupedality consistently associated with an increase in body size?
Here, we’re basically looking at evolutionary trends. In other words, can we detect a trend in body size within various ornithischian lineages? The more I think about this, the less I’m convinced we can directly answer the question (if you disagree, and have a solution, pipe up in the comments, please). One problem is the difficulty in knowing whether or not certain taxa were truly quadrupedal. So, where do you make the cut-off for quadrupedal vs. bipedal vs. both? In many cases we just don’t know. There’s a danger in circular reasoning, too (the limb bones look like it’s quadrupedal, so we call it quadrupedal, and then use it as an example of a quadrupedal taxon for analysis of limb bones).
But, I think we can detect trends across Ornithischia as a whole, and within specific lineages. For instance, is there a trend for increasing body size across Ornithiscians? Is there a trend for increasing body size within Ornithopoda? Ceratopsia? Thyreophora? In fact, Matt Carrano found a consistent and statistically significant increase in body size within ornithischians (and indeed, within most dinosaurs) when considering femoral measurements (go here to download a free PDF of Carrano, 2006). So, that makes this question a little less interesting (and indeed, less publishable, because it’s already been done). Do you think we should move it to the back burner? Or should we spin it in another way? Thoughts are welcome.
Did different groups of quadrupedal ornithischians arrive at this body form in similar ways, or did they have different strategies?
Here (as far as I know) is a genuinely novel question, and I think it’s the core of the ODP’s current phase. What we’re really saying (I think) is this: We know that thyreophorans, hadrosaurs, and ceratopsids independently evolved quadrupedal locomotion. Did each group have similar limb proportions, or were they different? I think this is where we’ll want to look at principal components analysis, at least as a starting point for data visualization. And, we’ll have to do that within a phylogenetic context. A recent paper by Liam Revell (2009) addressed how to do this (thanks to ODPer Randy Irmis for bringing up this paper; you can download it for free here – it’s well worth a read).
A second way to look at this question is to look for trends in certain structures – for instance, do the metacarpals tend to get elongated in each group (relative to the rest of the arm) as different clades became quadrupedal? Here, we might use a simple non-parametric correlation of the ratio with patristic distances (see the Carrano paper, again, and references therein, for a brief introduction to this method), to investigate that question within different lineages. Basically, patristic distance estimates the distance of a particular species from the base of the tree (by the number of branching points leading up to it). A taxon that split off early in a group’s evolution would have a low patristic distance, and vice versa for one that split off late in a group’s evolution. So, we might look at the correlation of metacarpal:arm length ratio to patristic distance for thyreophorans, hadrosaurs, and ceratopsians.
I think I’ll end here for now! Please add thoughts, suggestions, corrections, and anything else you think relevant in the comments. Next time, I’ll move on to the final issue, quantifying morphological disparity in ornithischian evolution.
Carrano, M. T. 2006. Body-size evolution in the Dinosauria. In M. T. Carrano , R. W. Blob, T. J. Gaudin & J. R. Wible (eds.), Amniote Paleobiology: Perspectives on the Evolution of Mammals, Birds, and Reptiles. University of Chicago Press, Chicago:225-268. Freely available here.
Revell, L. J. 2009. Size-correction and principal components for interspecific comparative studies. Evolution 63: 3258-3268. Freely available here.
Image at beginning of post by Pavel Riha, licensed under the Creative Commons Attribution ShareAlike 3.0 License.
Andy et al.,
I’ve actually been looking into this over the past few weeks. I started simply with humerus:femur length ratios as a potentially objective method of separating bipeds from quadrupeds. The rationale being quadrupeds *should* have similarly-sized proximal limb elements. However this did not work out so well. I tried to take into account the mechanical requirements for quadrupedality by using the width measurements to calculate the approximate volume of the humerus and femur (for the femur I used both width and depth measurements to calculate an ellipsoidal volume). Of the small number (N=19) of taxa with all those measurements, the division between what we call bipedal and quadrupedal was pretty consistent with expectations , even with things like Iguanodon falling into an intermediate zone we might call a facultative biped. I also ran a PCA on these data (though not accounting for phylogeny) and 97% of the variance in PC 1 was due to increasing length of the humerus and femur. The remaining 3% in PC 2, though small, showed a difference between stegosaurs and ceratopsians. Early representatives of each group almost overlap on the PCA plot, however more derived (and larger bodied) members of each group seemed to achieve a quadrupedal stance differently. Stegosaurs appear to have lengthened the femur and widened the humerus, while ceratopsians lengthened the humerus more relative to the femur. I am very excited by these preliminary results and would be happy to share the plots. There is a slight problem with allometric scaling though, since some very small “quadrupeds” like Protoceratops look more bipedal but due to their small size they have generally more gracile limb proportions. I would also recommend the following two older papers by Carrano, which may help with additional analysis methods as things move forward:
Carrano, M. T. 1999. What, if anything, is a cursor? Categories versus continua for determining locomotor habit in mammals and dinosaurs. Journal of Zoology 247:29-42.
Carrano, M. T., C. M. Janis, and J. J. Sepkoski. 1999. Hadrosaurs as ungulate parallels: Lost lifestyles and deficient data. Acta Palaeontologica Polonica 44(3):237-261.
Looking forward to your thoughts.
I can think of one way that might get at the question of “Did different groups of quadrupedal ornithischians arrive at this body form in similar ways, or did they have different strategies?” I’m not an expert on dinosaurs (or any animal with endoskeletons for that matter), so please point out mistakes in my argument.
I assume that quadrapeds generally have longer forelimbs relative to their hindlimbs than do bipeds. If so, then a frequency distribution of our species’ forelimb-to-hindlimb ratio should show a bimodal distribution rather than a normal distribution. Furthermore, because I’d expect forelimb-to-hindlimb ratios to be taxon-dependent to some degree, we could calculate frequency distributions for each clade (at whatever clade level is deemed appropriate). Then we could determine whether each clade’s species fit a normal or bimodal distribution in forelimb-to-hindlimb ratio, suggesting the clade is comprised of strictly one type of pedalism or both bipeds and quadrapeds, respectively. A priori hypotheses about clades that are believed to be bipeds, quadrapeds, or both could be made prior to analyses and their predictions tested with frequency distribution shape results.
To me, this is a simple approach that would be intuitive for readers to understand. But I may be way off too. I’d like to hear what others think of this approach.
I’m extremely naive about doing phylogenetic analyses but I just wanted to mention some ideas about data visualization.
If I understand correctly, you already have a phylogenetic tree which you believe in, and aren’t going to try to create a new cladogram with this data (especially since if I understand correctly, people generally seem to use discrete characters rather than numerical measurements to construct cladograms). Perhaps you could combine this cladogram with principal component information to create an informative graphical display.
It should be possible to represent each species a colored shape, whose characteristics are determined by various variables. For example, you might have colored circles whose diameters are determined by log femur length and whose color on a red-to-blue scale is determined by humerus-to-femur ratio. (Maybe another variable could be represented by the size of an X inside the circle, or something.) You might also be able to use principal components in this way, although I’m not sure because it might be hard to get reliable component scores for each species (since you don’t have the scores for each species on each of the original variables).
The advantage of this might be that you might see, say, the ankylosaurs get bluer or redder or wider as they branch off the basal thyreophorans (or whatever). So you get a cladogram with little balls like Christmas ornaments hanging on it! This would not provide a hypothesis test in itself but it would perhaps provide useful summary information and show where to look next.
Excellent thought, John. These sorts of plots are almost ridiculously easy to produce within the program Mesquite, and a figure like that of some sort would probably be useful for the paper. We might be able to do this on a trimmed version of the tree (removing unneeded species, or those lacking info); Mesquite will also calculate/infer ancestral character states, and color branches accordingly. So, it could be a *very* powerful tool for data exploration and visualization.
With a ratio we’ll get an inherently skewed distribution no matter what (I think Biometry by Sokal and Rohlf has a whole section on dealing with ratios). But perhaps you’re on the right track – rather than skew, how about focusing on the “spread” of the data points. The only downside of this track is that we would need some sort of validation of the method; i.e., we can’t necessarily just *say* that we expect this pattern; we should be able to point to some modern animals as support. Alternatively, we can look at trends in ratio of forelimb length : hindlimb length (this is called the intermembral index). Does it trend in one way or another within a group, or is it random? This might tell us something along the lines of what you’re thinking about. When I penned the post last night, I was thinking just ratios within a limb (and these are probably important). . .but I think looking at an intermembral index will be just as useful!
I guess I’m not quite as surprised that different quadrupeds do different things – after all, stegosaurs are just so stupidly weird! Your mechanical approach is an interesting one, and maybe one we should look into more. Would circumference work too, or are we missing too many of those?
The issue of allometry that you bring up is also an important one – perhaps this is where information other than cross-sectional properties is most important? E.g., limb ratios. . . Of course, allometry within different lineages is an interesting question in its own right. 🙂
I like the idea with the coloured ‘character’-mapping like in Padian et al. (2001): dinosaur growth rates and bird origins. Nature.
I want to ask what phylogenetic hypothesis should prefer as the base for the evolutionary scenario. I think the tree from Butler et al. (2007): The phylogeney of ornithischian dinosaurs. JoSP is maybe qualified, because it has a huge taxon sampling. But we should mapp the results on different hypotheses to verify the first scenario. do anybody of you know another qualified phylogeny??
I know it’s a trivial observation, and that I am not telling you anything you don’t know, but we really want to avoid using words like “strategies” here — the implication is that the dinosaurs had an evolutionary goal in mind.
On whether size is correlated with quadrupedality: a more straightford question we could look at, that is obviously related, is how the relative proportions of the bones vary with size. Some of that is fairly straightforward — e.g. to what extent do the long-bones follow the allometry rules predicted by theory, where the diameter is positively allometric with respect to the length. But more interesting would be to run pairwise comparisons between the lengths of all the bones we know about, plotting e.g. femur length vs. tibia length, and seeing whether the allometry constants for any of the pairs are notably higher or lower than the others. I guess we have at least six bones of interest (humerus, ulna, MCIII, femur, tibia, MTIII) so that gives us 30 pairs. If nearly all the allometry constants are in the range 0.9-1.1 but two of them come out at 1.3, then that is telling us something.
I think the mechanical approach was one attempt at this allometry problem, however there are very few taxa with long bone width measurements in the dataset (at least the version I am using right now). There are a few with a circumference measurement as well. As to Mike’s suggestion of looking at relative limb proportions with increasing size, there will also be the likely problem of locomotor adaptations fudging with the results, i.e. cursorial vs. graviportal adaptations. This may disappear at very large body sizes, and it would be neat to see if there’s a size threshold in a lineage above which taxa have similar limb proportions.
You’re correct Mike – “strategies” was used only in the most colloquial of senses.
If we want to move towards correcting for overall body mass, I would recommend Seebacher’s paper, which includes regression-derived mass equations for the major clades of dinosaurs. I think the only problem would be having accurate length estimates for certain taxa.
Seebacher, F. 2001. A new method to calculate allometric length-mass relationships of dinosaurs. Journal of Vertebrate Paleontology 21(1):51-60.
I like this, and I think it’s important. Of course, we should not shy away from complex methods, but I always think a project that restricts itself to sausage-machine methods is the weaker for it. (By “sausage machine”, I mean methods where you pour the data in one end of a complex piece of software that you don’t understand, turn the handle, and believe that whatever comes out the other end is true.)
Along those lines, I very much like the idea of simply plotting forelimb vs. hindlimb length and seeing whether the result is bimodal. That would be a result that anyone could understand, see and sort of feel the truth of.
Wow! I did not know that there was software available to do this so easily. It might not be easy to get a score for each dinosaur of interest on each component (because of missing data) but you could certainly construct your own indices using smaller numbers of variables so that more cases would be available. I think it would be really nice to have one of these colorful plots for humerus-femur ratio, maybe one for (humerus/radius)/(femur/tibia) or something (if you don’t mind a ratio of ratios), etc.
Here’s a plot I did a little while ago of Femur vs Humerus length:
This was only using examples with both measurements available (and prior to when we started combining specimens together).
The intermembral index sounds very useful.
I’ve seen the tibia/metatarsal length ratio used, both in theropods and in mammals (here’s a paper with a whole bunch of limb-bone measurement ratios in mammals – it’s actually about a notoungulate, but they did measurements for 39 extant mammal species for comparison; http://palaeo-electronica.org/2008_1/138/analysis.htm). That might be an useful one to use.
I’m not sure a bimodal distribution or lack thereof would mean much, though, because quite a few of our dinosaurs might have been able to walk both bipedally and quadrupedally – it may not really be binary.
(wish we could edit comments here)
Also, the tibia/femur length would be good to do; it’s been done with theropods (this paper: http://www.mnhn.ul.pt/geologia/gaia/21.pdf has a bunch of ornithomimid and tyrannosaur tibia/femur ratios – which would provide an outgroup comparison, and are pretty clearly entirely bipedal.)
This is a bit more readable:
This is using only combined data with the points labeled by Genus:
I don’t know if this shows anything that would be of any interest though.
Here’s a plot of (Femur L)/(Tibia L) using the combined data that has both Femur and Tibia measurements available
Some other plots:
1) (Femur L)/(Humerus L) of combined data where both the Femur and Humerus were present (one Lambeosaurus outlier was thrown out):
2) Here’s the IM index plot for combined data with the Humerus, Radius, Femur and Tibia all present:
Inverse of the other plot (so tibia/femur):
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It is wonderful how much detailed analysis you guys are doing – in addition to collecting the data!
I apologize, I had an error in the IM plot above. Here’s the correct one:
Does intermembral index include the length of the entire limb (H+R+MCIII/ F+T+MTIII) or can it use (humerus+radius)/(femur+tibia)? I looked at the available data and so far only a few taxa (genera mostly) have complete limb element measurements (about 34 by my count). Being able to include taxa lacking manus and/or pes data would probably boost sample size considerably.
As far a ratios are concerned, I thought Rohlf’s primary concern about ratios was that often people end up comparing the ratio to one of the original measurements used to create the ratio, for example comparing the intermembral index to humerus length or even body size. I also like ratios because they reduce the nasty problem of dealing with body size. If comparing measurement data, perhaps a logarithmic transformation of the data would be appropriate. Alternatively, since a ratio represents the proportional difference between elements, an arcsin transformation of the data could be used.
Sorry, I meant to blockquote the first paragraph of the post but it didn’t work.
[Fixed — Mike.]
I’m not sure if there’s an exact standard for intermembral index – and I don’t think there’s anything to stop us from running it both ways. At the least, it would be important to know that leaving out the metatarsals or metacarpals isn’t too dumb of a thing!
I don’t know that ratios completely eliminate the problem of body size (after all, we might have allometric variation in ratios), but they certainly do help.
A Nonnegative Matrix Factorization seems to confirm (some of) Chris’s PCA results. I found that there was a ‘Coelophysis (or Lambeosaurus?) Direction’ and a ‘Sauropelta Direction’. The proportion of these two directions that optimally reconstruct a specimen’s limb proportions separates out the different clades fairly well (with overlap of the certain clades). The dimensionality of the ‘classification’ is actually only one here because the vectors of each specimen can be normalized to, e.g., unit 1-norm. Then it’s only the angle in the 2-D plane that’s important.
Plot is here:
I wonder what little Leptoceratops is doing hanging around with Sauropelta and the other hard-core quadrupeds at the left end of Dr. Dreisigmeyer’s new plot.
And from Dr. Dreisigmeyer’s first Feb. 25 plot, it’s also interesting that Scelidosaurus has a femur-to-humerus ratio that looks like the Coelophysises (sp.?).
Sorry — I made a mistake on my last comment — it wasn’t his first Feb. 25 plot, but the first plot mentioned in his comment #20 on Feb. 25. And there seemed to be two Scelidosaurus specimens shown, one that looked bipedal and one that looked quadripedal. I think that’s really interesting. (Was one a juvenile??)
I think, evolutionarily speaking, it is relatively “easy” for a lineage to adopt a quadrupedal stance if they are small-bodied. The limb bones do not necessarily need to be rescaled to deal with the new forces placed upon them because the size threshold requiring allometric change in bone shape has not been crossed. In my own analysis Scelidosaurus, Protoceratops, and Leptoceratops (and even Avaceratops to a certain extent) group with the bipeds in a PCA plot. It will be difficult to test this on data from another group because mammals all derived from quadrupedal ancestors. Would limb bone scaling in primates be a useful comparison? They have many large-bodied taxa that seem to have secondarily adopted a more quadrupedal stance, like mandrills or baboons. But then again brachiation is a problem. Just a thought.
In the IM plot above one of the Leptoceratops has a value of ~1. The next closest is Sauropelta with a value of ~.85. It may be picking that relationship up. The others seem to be closer to the middle of the graph.
For the Scelidosaurus, both had the same humerus length (168) but very different femur lengths (226 vs. 399). The humerus length for NHM R1111 may have been entered incorrectly though. If so, only BCM Ce 12785 is correct, which would be the quadripedal one.
(Very naive question alert) I was wondering: other than crocodilians, is there anything alive today that evolved quadrupedality or at least partial quadrupedality, from a bipedal ancestor? Maybe something in a kangaroo family or some kind of sloth? (I don’t know anything about those groups except that they seem to be kind of in between bipedal and quadrupedal.)
This is an interesting questions. Did crocodiles evolve from quadrupedal species to bipedal and then back again? I’m thinking of Protosuchus here.
Actually I’m not sure. I thought that some of the early archosaurs like Euparkeria were supposed to be at least facultatively bipedal, but I might have misunderstood. I later found out that Kubo and Benton (2007) published an article about the “Evolution of Hindlimb Posture in Archosaurs,” but they were talking about erect vs. sprawling posture within the context of quadrupedality.
I’m not sure if this was brought up before, but wouldn’t an age-stamp (e.g., Early Jurassic) on the specimens be useful for analysis?
It could, depending on what kinds of trends we are interested in. As Andy mentioned earlier, a program like Mesquite could be used to examine evolutionary trends within lineages that may be of interest regardless of the age of the taxa. However, ages, or age ranges, may be useful in looking for evidence of any “directional” trends over time. Age estimates (in MY) for taxa can be downloaded from the Paleobiology Database, but I would strongly suggest reading the method on how they were obtained before using them.
The issue of noting age range estimates for each taxon in our data set was first addressed in an ODP blog entry from 24 November entitled Big Tasks Ahead. Some excellent suggestions followed in the comments area. I’ve actually been working sporadically on this task for the last few weeks, and have at least something down for (nearly) all taxa beginning with the letters A-K, I think. Humble beginnings, really – I’ve mostly just been consulting Dinosauria II, and plugging the formations and age range estimates that appear near the beginning of each chapter into a spreadsheet (or otherwise going to the initial descriptions for species too new to have been included there).
If anyone is interested in helping out, I’m happy to share some of the remaining work along with my progress to date. (And note that even if I make it through my initial pass, there will still be plenty of opportunities for others to contribute. No doubt there will be different published age estimates out there for some of these taxa, and I’ve mostly been recording just one entry per taxon thus far.)
I’d be happy to help. You can reach me via email at firstname.lastname@example.org
Hi Rob – would it be worth putting this up there as a Google Doc editable by anyone? I think it should be OK, as long as we make folks cite references.
@Andy – Indeed it would, particularly since I’ve been progressing at a snail’s pace! A couple small changes necessary to make the list fit for public consumption. Will render those tomorrow and forward it on to you for posting at earliest opportunity.
@Chris – Thanks for the offer to assist!
Andy had written:
“One problem is the difficulty in knowing whether or not certain taxa were truly quadrupedal. So, where do you make the cut-off for quadrupedal vs. bipedal vs. both? In many cases we just don’t know. There’s a danger in circular reasoning, too (the limb bones look like it’s quadrupedal, so we call it quadrupedal, and then use it as an example of a quadrupedal taxon for analysis of limb bones).”
One thing that would work in a perfect world would be to separate the different limb bone measurements into the classes of: 1) Definitely bipedal, and; 2) Definitely quadrupedal (and maybe 3) Species that we think can use either mode). Then a Generalized Linear Discriminant Analysis could be done on these to find a single direction that optimally classifies a species into one of these classes. Now you bring in a species that you’re not sure about and project its measurements onto the classification direction. Is it closer to the bipedal or quadrupedal group? The problem is that we’re missing many measurements, so the world is not perfect.
Also, from the rough analysis I’ve done, I’m not seeing a consistent pattern of limb measurements across clades. But, the above could be extended by adding in classes, say dividing by clade (and maybe in within clades), or only analyzing in a single clade. But we still run up against the missing data problem and maybe a lack of many samples. But maybe not.
@David in #43: For some types of analyses, we should have plenty of data. I did a quick tally of how many species have individuals with humerus, radius, femur, and tibia all represented, and got 68 species. . .not too shabby.
1) Here’s the PCA using every species with humerus, radius, femur, and tibia present (85 total specimens. I use Ceratopsia quadrupedal and bipedal loosely.):
and where I scaled all femur lengths to 1 (feedback on this scaling choice would be appreciated):
NMF analysis on scaled data has pretty much the same result:
Notice that there’s still that Sauropelta ‘outlier’ that is giving us one of our NMF ‘limbs’. Should that sample be retained in this type of analysis? Its specimen AMNH 3032. Ditto for Leptoceratops specimen AMNH 5205.
2) GLDA won’t work here because the data is already in such a low dimensional space — essentially we’d just end up with the means of the clusters. I should have remembered this. Since NMF is closely related to K-means, it’s a more appropriate tool in this case for clustering than GLDA.
Cool! For the PCA, what are the percentages of variance on each component? Just looking at it quickly, it looks like PC1 for the first plot is primarily tracking “size”.
For the second PCA on scaled data, it looks like hadrosaurs are doing one thing, and thyreophorans+ceratopsids are doing another thing, and the little bipedal ornithischians+ceratopsians are doing their own thing. Very cool! Of course, we’ll also have to run this analysis while accounting for the effects of phylogeny.
Re: the scaling, I might recommend a geometric mean of humerus, radius, femur, and tibia instead of just femur. This would help to “smooth out” any odd proportions, and perhaps better reflect the true “size” of the animal.
We’ll have to look at AMNH 3032 and 5205 a little more closely, including going back to the original literature.
Raw PCA has PC1=~94% of the energy and PC2=~98%
Scaled PCA has PC1=~61% of the energy and PC2=~30%
Here’s the geometric mean normalized PCA. PC1=~46% of the energy and PC2=~34%.
Raw PC2=~4% (not ~98%)
Instead of doing the above projections, the humerus, radius, and tibia (femur length normalized to 1) can just be examined in 3D directly. There seems to be some good clustering here with significant overlap for Ankylosauria and quadrupedal Ceratopsia. Huayangosaurus (CV 00720), Tethyshadros (SC 57021) and Camptosaurus (USNM 2210 — juvenile?) seem to be outliers (along with the above Sauropelta and Leptoceratops). (I’m not saying that the outlier status is not necessarily from bad data.) The PCA plots above seem to be throwing the “Radius L” dimension away (for the normalized data, whether “Femur L” = 1 or the geomean method). The “Humerus L”-“Tibia L” pair provide pretty darn good discriminatory power. So what to make of that, if anything?
Data is here (as ODP_plot_data.Rdata):
Command to plot is (needs rgl package — sorry couldn’t figure out how to add a legend to this thing. If you do please let me know how. The plots below are in the order “Ankylosauria” , “Ceratopsia (quadrupedal)” , “Ceratopsia(bipedal)” , “Hadrosauridae” , “Ornithopoda” , “Stegosauria”):
plot3d(Proj[1,1:5],Proj[2,1:5],Proj[3,1:5],xlim=c(0,1.5),ylim=c(0,1),zlim=c(0,1.5),pch=8,col=”blue”,size=5,xlab=”Humerus L”, ylab=”Radius L”,zlab=”Tibia L” ); points3d(Proj[1,6:16],Proj[2,6:16],Proj[3,6:16] ,pch=8,col=”red”,size=5); points3d(Proj[1,17:27],Proj[2,17:27], Proj[3,17:27],pch=8,col=”green”,size=5); points3d(Proj[1,28:53],Proj[2,28:53],Proj[3,28:53],pch=8,col=”yellow”,size=5); points3d(Proj[1,54:76],Proj[2,54:76],Proj[3,54:76],pch=8,col=”black”,size=5); points3d(Proj[1,77:85],Proj[2,77:85],Proj[3,77:85],pch=8,col=”cyan”,size=5);
Huh? What I meant to say is the data may be good and the outlier status is real.