Hi Stefan,

I asked a similar question some time ago (2009) in the MorphMet mailing list to which I did not receive a satisfying response. In my opinion, RV coefficients might be comparable only if the structures are subsets of the same landmark configuration. And even so, there might be some mathematical issues if the modules have a different number of LMs (Fruciano et al. 2013). I really look forward to somebody giving a more solid mathematical response to this issue.

Cheers,

Javier

Hi Stefan:

The RV coefficients are used to compare subsets of landmarks within a structure, or by separate 2 blocks that forming a overall structure. I think that you can evalute A, B, C structures A-B; B-C etc, and as result you may obtain RV for each overall structure in each subset selected (integration strengths) using PLS.  So, the results would be valid to make inferences on relative integration in differents structures in your sample. Or you can evauate by modularity the fingers on hand and testing independence. Klingenberg et al. 2003, 2008, 2010 explain more details..

Regards,

Abril.

Well, a problem is that the RV coefficient makes little sense for biological structures because it ignores spatial scale and the distribution of landmarks across a structure. 

Dear Stefan,

It is possible to perform three separate 2-Block PLS (partial least-square) analyses, one with each pair of structures (A-B, A-C, B-C) and have a look to the RV values and significations. You can also have a look to the to the patterns of covariation (see Singh et al. 2012. Journal of Human Evolution 62 :155-164). You can perform this with the "Covariation/Partial Least Square/Two separate Blocks" option in MorphoJ. What you will test in each comparison is if the two blocks of landmarks that you are comparing are associated or not (the null hypothesis is that the two blocks are independent) and with which strength (RV). The closer the RV value to 1 the bigger the strength of association. Then you can compare "qualitatively" the RV values obtained in the different comparisons. To my knowledge a test to compare the RV values among them is not available, but at least you can see if the three pairs of structures are significantly associated or not and with which strength. This will give you some information. For example, you can find that structure A is significantly associated with B, but not with C.

Regards,

Francesc

Hey you all,

thank you for your answers. Unfortunately most of those are not answering the core of my question. I already know how to calculate the RV-coefficient using various tools and what it basically describes (and I also know about Partial Least Squares analyses and the like).

However, what I do not know is whether the coefficient is comparable across different structure couples, given that the sample is the same. In so far the answer by Prof. James Rohlf comes close to what I really wanted to know. I fear that the RV is changing (with sample size, sample composition, landmark distribution and possibly relative number etc.) to such a degree, that a comparison across structures would be absolutely meaningless...

However, as J.Rohlf said, if the RV-coefficient is useless altogether, I don't know why it is used so often in various biological studies. If it is the case, that RV-coefficients are unreliable for biological structures in order to test for relative integration strengths, is there any better choice?

You can calculate a variety of variables (r2, variance of eigenvalues, mean integration, etc. see Hansen and Houle 2008 in Journal of Evolutionary Biology, and Porto et al. 2009 and Marroig et al. 2009 in Evolutionary Biology) to compare magnitudes of integration across structures or taxa, but this has only been used for distances extracted from landmark data and not landmark data itself, as far as I know. The same concerns about sample size, sample composition, etc. still apply, however, and there are some papers coming out soon (Houle and Meyer in Journal of Evolutionary Biology for example) on error estimations and sample sizes (which should be fairly large, at least over 40 and probably more). The argument for using mean integration and other calculated variables from Hansen and Houle's 2008 paper is that they are based on evolutionary/selection equations that better reflect real biological concerns. I would not use RV coefficients to compare across taxa/structures/anything except as a preliminary check to make sure things are integrated period.

For one aspect you should look at the 2015 paper by Bookstein on integration. Not an easy read however. I think it is a step in the right direction.  

The problem with RV is that the null hypothesis is too null. It does not take into account the one expects close landmarks should be more correlated.  The problem is that one cannot evenly sample the shape of a structure.  Some landmarks my be close and others far.  Clustering by distance often gives similar results as clustering by correlation. 

Hi Stefan,

I guess that one has to distinguish between the use of the RV coefficient to test the null hypothesis of absence of modularity (i.e. the procedure proposed by Klingenberg in 2009) and its use to quantify the degree of association between blocks of variables (and then compare across groups of observations or groups of structures, which seems like what you are referring to).

The use of RV to test the null hypothesis of absence of modularity has the limitation that Professor Rohlf has pointed out.

On the other hand, such limitations do not necessarily apply when one has to compare across groups of observations or groups of independent structures.

When comparing across groups of observations (i.e. I want to know if the level of association between the variables in block 1 and the variables in block 2 is different in different species) a potential problem is represented by the fact that the RV coefficient depends on sample size (i.e. the coefficient will be lower at higher sample sizes), as we show in our paper (Fruciano et al 2013 - Plos One). This obviously creates the problem that one cannot compare the coefficient between groups of observations with different sample sizes and also the difficulty of interpreting the coefficient itself (i.e. if the value of the coefficient depends in sample size, the information of, say, RV=0.7 is by itself not easily interpretable). To overcome this, we have proposed (in the same paper) different approaches, such as to test using permutations for the difference in RV between two groups of observations.

In your case, assuming that you have the same sample size in all comparisons (different sample sizes would be problematic, as explained above), you would be comparing different sets of structures. That is, you want to know if, say, blocks A and B are more strongly associated with each other than blocks C and D (I am deliberately using four different partitions so to treat them as independent). As you have rightly guessed, one potential problem could be if the different partitions had different number of variables and this would affect the value of the RV coefficient as it happens with the number of observations. Unfortunately, I don't know (or at least I cannot remember in this moment) of any study looking at this, especially in a morphometric context. I guess that to have the answer one would have to do a study similar to the one I did for the relationship between RV and number of observations.

I hope this helps...

Best,

Carmelo

What would worry me is if one one structure you were able to more or less evenly-space the landmarks but on another they were mostly in two clusters because no good landmarks could be found in between. Would the RV coefficient show that the former was "integrated" but the latter was not and had two modules? If so, is that of biological interest or an artifact of the availability of landmarks that can be digitized? if the two sets were, for example, left and right hands then the same landmarks could be found on both so this would not be a problem. Hard to imagine between pairs of different kinds of structures.

The RV coefficient is the average squared between-block covariance divided by the square root of the product of average squared within-block covariances. In this sense, it appears to resemble the squared univariate correlation coefficient, but the RV coefficient cannot be interpreted as easily in terms of explained variance as the squared correlation can be.

The main problem with all statistics of this kind is that they summarize quite complex relationships by a single number. For, say, two times 20 variables you have 400 pairwise covariances/correlations that end up being summarized by one RV coefficient. Obviously, the interpretation of this number is difficult. For example, between two structures A and B some shape features might be highly correlated whereas others are only weakly associated, resulting in an intermediate RV coefficient. Between the structures B and C all features might show an intermediate association, hence the RV coefficient would be the same for both comparisons. Note that this is all about linear relationships.

PLS is one approach to investigate the pattern of covariance, instead of the mere magnitude (however this would be defined). It finds multiple dimensions of linear combinations (shape features) with maximal covariance. In this way you can see which and how shape features are associated across structures, and how many of such shape features are worth talking about.

Note also that covariances or correlations (including the RV coefficient) cannot directly be interpreted in terms of integration. First, because in most papers the concept of integration is not at all or only very vaguely defined, vaguely enough to prevent any serious discussion of results. But in general, it is important to distinguish between a pattern of integration, e.g., how a pleiotropic gene, a growth factor, or a developmental interaction affects different parts, and how much variation in this common factor exists within the sample. Integration can be strong, but variation weak, leading to a small correlation/covariance.

To conclude, without knowing what is measured and what the patterns of integration are, no single measure of magnitude is interpretable (for detailed discussions see, e.g., Mitteroecker et al. 2013; Hallgrimsson et al. 2009). See Mitteroecker & Bookstein (2008) for an example of how pattern and magnitude can be compared separately across groups.

As Jim mentioned already, any biological interpretation of the "magnitude" of integration also requires reference to the spatial structure of measurements. I have commented on this at length in Mitteroecker & Bookstein (2007), Mitteroecker (2009), Mitteroecker et al. (2012).

Best wishes,

Philipp

Hallgrimsson, B., Jamniczky, H., Young, N. M., Rolian, C., Parson, T. E., Boughner, J. C., et al. (2009). Deciphering the palimpsest: Studying the relationship between morphological integration and phenotypic covariation. Evolutionary Biology, 36(4), 355?376.

Mitteroecker P (2009) The developmental basis of variational modularity: Insights from quantitative genetics, morphometrics, and developmental biology. Evolutionary Biology 36(4), 377–385

Mitteroecker P, Bookstein FL (2007) The Conceptual and Statistical Relationship between Modularity and Morphological Integration. Systematic Biology 56 (5), 818–836

Mitteroecker P, Bookstein FL (2008) The evolutionary role of modularity and integration in the hominoid cranium. Evolution 62 (4), 943-958

Mitteroecker P, Gunz P, Neubauer S, Müller GB (2012) How to explore morphological integration in human evolution and development? Evolutionary Biology 39(4), 536?553

A further comment on the RV coefficient and its apparent relation to sample size.

Simulation studies have shown that the RV coefficient "depends on sample size" and is "biased upwards" in small samples. This is somewhat unintuitive because sample estimates of covariance are unbiased, and their expected value does not depend on sample size. Then, how is it possible that the expected RV coefficient appears to depend on sample size?

Remember that the RV coefficient is the average SQUARED between-block covariance, divided by the square root of the product of average SQUARED within-block covariances. The strange behaviour arises from the incorrect assumption that when two blocks of variables are drawn from independent distributions, the RV coefficient is 0. For samples of finite size, the covariance between two independent variables is not exactly zero, it deviates slightly because of the (random) sampling. On average, the larger the sample size, the closer the estimated covariance will be to 0. In other words, the standard error of the covariance (its variation around the "true" value) decreases with sample size. However, the expected value of the covariance is 0, that is, the average of multiple independent estimates tends to 0.

The problem is that the RV coefficient is based on squared covariances. For two blocks of variables, sampled from independent distributions, some of the between-block covariances will be positive and others negative. Their square, however, is always positive and hence the RV coefficient can never be 0 (unless all the between-block covariances are exactly 0, which cannot occur in biology). This effect becomes smaller the larger the sample size, because the standard error decreases with sample size. The effect also decreases the more the true covariances differ from zero. If ALL the estimated between-block covariances are positive (or all negative) than there is no sample dependence (but this cannot occur in geometric morphometrics).

So the dependence on sample size is not because the RV coefficient is biased, but because it is a bad statistic that is incorrectly interpreted. This can happen if statistics are proposed without reference to any theory.

Philipp