How to interprete fitted covariance matrix in multivariate normal model?


#1

In a current analysis I am using multivariate normal model and I would like to make use and derive some interpretation from the fitted covariance matrix. What I have been using so far was looking at the observed correlation first and then derive the absolute change in correlation between two variables (correlation estimated by model - correlation on observed data), the idea being that if we see large drop in the absolute change this is indication that a large portion of the observed correlation was driven by factors included in the model.

To visualize this I have been using the circlize package giving the graphs that I put in attachment below:

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Briefly the top graph is the observed correlations between the response variables, red indicate positive values and blue negative ones, solid lining indicate correlation higher than 0.5 and dotted lining correlation higher than 0.25. The bottom graphs are the absolute changes for three different models that we are evaluating.

My question is: is this a relevant way to use the fitted correlation matrix? I have no idea how to include uncertainties in correlation estimates in these type of graphs … Is there some other visualization or additional analysis that could be done to extract additional infos?

Thanks!


#2

I am not sure what your final goal is - are you just exploring the data and hoping to see some interesting patterns? Or do you have a specific question in mind?

The only thing that comes to mind to visualise uncertainty in correlations is to have a grid of boxplots and/or scatterplots and display the observed correlation as a horizontal line. This lacks the elegance of the ciruclar plot, but might IMHO be more readable. Might also be of value to do this for both the fitted covariance matrix and for an “empirical” covariance matrix of simulated data from the posterior. The latter would be form of posterior predictive check and would be a good way to test that the model fits well and hence that the difference in covariance between observed data and the model is not due to poor model fit.