# Correlations of correlations using 3 data sets

I'm interested in how 14 environmental parameters co-vary with other environmental parameters and whether they show the same patterns in 3 regions. So 1st, I made a pairwise correlation table of the 14 environmental variables for each region. Then, I made another pairwise correlation table of correlation values from 3 regions (i.e. region1 vs region2 & region1 vs region3 & region2 vs region3). I found a good correlation (a similar pattern) between region2 and region3 (or called "sector" in graphs).

What I want to do from here is to put the all correlations values from the 3 regions together in one graph and also calculate correlation values (i.e. correlations of correlation values) because I want to know which environmental variables are correlated well in all 3 regions. I thought of using PCA, but it's not really what I imagined to be the best way to present a pattern. Maybe a 3D graph would be good for presentation. I'm a R user. If you could instruct me with R language on how to make that kind of graph and also how to calculate 3-way correlations, that would be much appreciated.

Thanks!

• I think you should give us some sense of what you are trying to achieve or what you think these correlations would reveal because “3-way correlations” or “which environmental variables are correlated well in all 3 regions” is somewhat vague. Also how many measures do you have in each region?
– Gala
Commented Jun 7, 2013 at 8:22
• Thanks, @GaëlLaurans. Please see comments I made in response to Nick Cox below. I have 14 environmental variables and 91 combinations (=correlation coefficients). Commented Jun 10, 2013 at 1:57
• This I understood but each variable has to be measured several times to be able to compute a correlation coefficient. What I wanted to know is how many observations on each variable you have. This is not the most important however, you should really heed the advice you received.
– Gala
Commented Jun 10, 2013 at 6:58
• Hi @GaëlLaurans, sorry I didn't understand your comments at first. Observation numbers for sector1, sector2 and sector3 are 55, 43 and 113, respectively. Thank you for clarifying your question. I appreciate it! Commented Jun 10, 2013 at 7:23

In very broad terms I'd question the value of this. It is easy to concoct examples in which correlations are similar but the relationships between variables are different -- and in which correlations are different but the relationships between variables are similar. I write not only as someone interested in statistics but also as someone whose main applications are with environmental data.

Also, what you are proposing to do doing puts enormous weight on correlations as a catch-all summary measure, which necessarily cannot do justice to nonlinearities, clustering, outliers, etc., which are commonplace with environmental data. An analysis of analyses is not out of the question, but the great risk is that each analysis step is a step away from the data you are trying to understand.

Yet another negative: It is difficult to make sense of your graphs without labelling which correlation is which using the names of the variables. You have presumably 91 correlations, but labelling them all will just be confusing; labelling none of them will just be uninformative.

Suggesting a positive alternative would depend on a deeper acquaintance with your scientific objectives, but if these were my data I would start with a single pooled multivariate analysis of three regions and then see whether regions cluster in some low-dimensional subspace. PCA does indeed spring to mind if your variables are mostly or all measured variables.

You name yourself as a R user, but your graphs look like to me like Excel defaults. I suggest that your graphs should show bounds of $[-1,1]$ on both axes; shift the $x$ axis with its numeric labels away from the middle of the graph; and use open or hollow symbols such as "o" rather than solid symbols.

P.S. In statistics, parameters and variables are not alternative terms. Your parameters are all variables.

Whether you are a student or a professional, you might benefit from finding a friendly local statistician, or someone in your field with more statistical experience, to talk to.

(LATER) If you are determined to do this, an extension of @Dualinity's approach to parallel coordinate plots might help.

• Thanks for your comments, @Nick Cox. Sorry I didn't put much descriptions for the graph.... I made a pairwise plot of correlation coefficients of 14 environmental variables from 3 different geographic sectors. Please see this link link. As values are correlation coefficients, a value ranges from -1 to 1. 1 and -1 indicate the highest correlations and the closer the value is to zero, the less correlated two variables are. Commented Jun 10, 2013 at 1:49
• Yes. You are right, @Nick Cox. I have a big assumption here that different environmental variables are correlated in a linear fashion. But my interest here is to see if there is some degree of similarity in environmental patterns "among 3 sectors". Commented Jun 10, 2013 at 1:51
• If I get some good correlations between some variables (which I did) when I put my environmental values from some region in some model (a linear regression here: 1st correlation test), then I expect to see similar correlation coeffient between the variables in other region (2nd correlation test is used to see if the two regions are similar in patterns) if the two regions are similar in environmental patterns (i.e. geographic pattern of variables in relation to other variables). Commented Jun 10, 2013 at 1:52
• If I use other model (maybe logistic model?), values might be different, but I guess correlation values or whatever values that indicate a similarity in geographic pattern of environmental variables may be similar among different regions if the variables vary in a similar fasion in those regions. If there are a clustering and outliers, correlation coefficients (from 2nd correlation test) would differ among regions. That's exactly what I want to see. But if there is any stats model that you recommend for environmental data, I will definitely try that! Commented Jun 10, 2013 at 1:52
• Sorry if it doesn't make much sense. Please see my data set link. Hope this helps. Commented Jun 10, 2013 at 2:03

Is it correct you mean this:

sector1-sector2: cor(s1var1, s2var1), cor(s1var2, s2var2) ....

sector1-sector3: cor(s1var1, s3var1), cor(s1var2, s3var2) ....

sector2-sector3: cor(s2var1, s3var1), cor(s2var2, s3var2) ....

Then plot the correlations for each combined sector?

You'd get correlations on the y axis, and the combinations of sectors (3 of them) on the x axis.

If this is what you mean indeed, I'll update the answer for code.

• Hi @Dualinity, thanks! But this is a little different from what I'm looking for. Please see comments I made in response to Nick Cox above. Commented Jun 10, 2013 at 1:59