Pros and cons of different variance covariance structures in Mixed Effects models I am working through Pinhiero and Bates' book on Mixed Effects models and have reached the section on specifying different variance-covariance structures within nlme. The book is fantastic but (i) more concerned with process than developing understanding, and (ii) pitched at an audience of mathematical statisticians (probably). 
It seems to me that the thing that makes mixed-effects models more powerful, from an analytical standpoint,  than, say, mixed ANOVA, is the ability to use different models for the patterns of correlations between observations within clusters. 
My question, as a researcher who uses statistics rather than a statistician, is what are the pros and cons of using different types of variance-covariance matrices? What types of designs and/or outcome variables are different matrices more appropriate for and why? 
In the absence of a detailed answer a recommendation for a book that discusses how and when to use different variance-covariance matrices in a way that is comprehensible for the advanced non-statistician would be much appreciated.
 A: This is a very broad question, and you start in the wrong end. The different variance-covariance matrices is implied by your model assumptions, so you should start there. 
Lets say you have many short time series (maybe different subject followed-up over some short time). We assume this time series are stationary (time trends, if they exist, taken care of by other parts of the model). If we in addition assume this time series are autoregressive of some order, then the variance-covariance matrix will have constant values on sub/super-diagonals parallel with the main diagonal. Such matrix is called a Toeplitz matrix https://en.wikipedia.org/wiki/Toeplitz_matrix
One other example: If you have a model with random intercepts for (many) small groups, then the covariance matrix will have only two different values, one on the main diagonal and another (smaller) outside the main diagonal. That is called an exchangeable covariance matrix.  I leave out the calculations. 
So: start making your conceptual model, and then the (form of) covariance matrix will follow. 
