I'm trying to find the best way to compare two representations of the same sample, but I'm struggling to pick which one would be more appropriate. Specifically I'm looking for statistics that will help be capture differences in covariance structure. Thus, I'm producing a random dataset and imputing different covariance structure on them using the Cholesky factor.
So far I have used 4 methods:
The problem thus far is that none of this methods show a high correlation with measures of covariance similarity (Krzanowski correlation or Random Skewers).
So any suggestions of additional methods would be very helpful.
why not use some tests for comparison of two covariance matrices? And of course you can compare correlation matrices too (in addition to comparing individual correlations). You can get many references in a google search.
My objective is not to compare covariance matrices per se, but to find a measure of comparison of multivariate data that is consistent with methods that evaluate the differences in covariance structure.
As I said before, I'm already comparing covariance matrices directly though RS and Krzanowski correlation. What I'm looking for is a way to analysis multivariate data that takes into account variations I see in covariance thought these methods.
John- I haven't, but I'm not sure what that would accomplish. From what I get of the method it is a path-analysis like method. I don't see how it would help me here. I'm not trying to find the influence of one set of traits on another, but the relationship between the same variables measured in two different ways.
For example, image I have two structures: the pelvis and a femur, and I'm investigating the covariance between both structures. In one dataset I have the a set of linear measurements for both structures defining their size. On the other dataset I have ensamble measures (like the geometric mean of the measures taken on each structure) and I would like to see how well both datasets reflect the same reality (thats why I used the statistics described above). But I also want to investigate if conclusions about the covariance between structures change if i change the representation. That's more tricky because i can compare covariances within a same set of variables, but not among different sets (again, sets representing "the same thing").
Could SEM help me with that?
A good measure of the degree of covariation between datasets is the PLS correlation. Based on the description of the problem, a two-block PLS might be helpful.
Dean- I was thinking more in line of a single value (like RV, correlation, etc), but maybe PLS might be helpful now that I think about it... i will have to do some tests
Thanks!
I tried Dean's suggestion and it didn't work very well. I used the PLS correlation and also the average of the correlation between all PLS vectors and they still didn't give me any result similar to covariance-comparison statistics.
Maybe these multivariate methods cannot capture subtle variations in covariance structure. In any regard, thanks for all the comments so far.
Another thought for how to compare covariance structures in two datasets is to use the Flury hierarchy. This sequentially compares PC directions and magnitudes across different datasets. Was used extensively in quantitative genetics to compare covariance patterns in different datasets.
Hello Fabio. Your description of the types of data you have and the types of analysis you want to do are unclear, so it's difficult to answer the question. You say (if I understand correctly, and I may not...) you have some data sets with individual-level data, and you can make covariance matrices from the different traits (e.g. pelvis and femur) and do statistical matrix comparisons, no problem (although I'm unclear whether you want to compare the covariance matrices of pelvis versus femur, or build a single covariance matrix including both traits, to look at among-trait associations). But then you have other data sets with "ensamble measures (like the geometric mean of the measures taken on each structure)". Are you referring to the population mean for each measure (e.g. mean femur length, mean femur height, mean femur width)? Or do you have some kind of average measure across e.g. length, height and width for each trait (e.g. average of the 3 pelvis measures, and average of the 3 femur measures) for each individual? In the former case, there's no information on population variance or covariance, so no comparison can be made with covariance matrices. In the latter case you've decomposed a multivariate measure into a single value (average across length, width and height), and so you've lost two axes of information per trait on among-trait covariance, and made it impossible to estimate within-trait covariance. It would be similar to making an among-trait covariance matrix on just PC1 from each of pelvis and femur: a total of 4 axes of information on covariance structure is lost. So unless I misunderstood your problem entirely, the answer is the ensemble measures will not give as much information as individual measures, and will therefore quite possibly lead to different conclusions.
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