I try to learn and follow the rules of this site. The moderators are given very broad authority to edit responses, close questions, and move answers to comments. This is done sparingly and most of the time seems to be appropriate to me. However sometimes I disagree and say so openly in discussion. Is this frowned upon? If so why? I found that my latest criticism led to a moderator contacting me directly to admonish me for this. What is the community's opinion on this. I welcome comments from anyone ioncluding moderators.
Absolutely. How else can the moderators improve?
This question should remind us that criticism of ideas and actions is an essential part of our site, but criticism of people is not. It is important to keep the two separated. Everyone in our community is dedicated, knowledgeable, and well-meaning--I firmly believe that as an article of faith, which I find daily confirmed in all the posts on this site--but sometimes we disagree about things and sometimes our actions have unintended consequences. Let's take care not to offend each other in our efforts to resolve differences and make the site better.
This may also be a good place to clarify some assumptions in the question:
All community members of sufficiently high reputation, not just moderators, have authority to edit answers, close questions, and even delete answers.
Community members (again depending on reputation) can vote to undo the preceding actions.
Moderators acquire their authority by means of community elections; it is not something that is just arbitrarily given.
Discussion about how we all interact with the site belongs here on the meta pages. Discussion per se within the replies and even the comments on the main pages is pretty strongly discouraged by SE policy, except when it aims to clarify a particular question. (This is a policy honored more in the breach than in practice, I admit, but it's still an ideal to aim at.)
As someone who has been involved with Cross Validated since day one, I have always found the moderators on Cross Validated to be highly competent, diligent, and reasonable. This is an impressive feat given the very open nature of Stack Exchange sites, and the many varied challenges that can arise.
Naturally questioning moderator action is welcome and encouraged. The site has many mechanisms to encourage discussion of site policies and decisions. The effective way to do this is in a respectful way. Moderators typically have a great deal of knowledge about how the Stack Exchange sites work and the many challenges involved. Thus, when a new user dislikes a decision made by a moderator, it is often worth considering why the decision was made and how that fits into the broader goals of the stack exchange network.
Yes, you have the right to criticize our actions: I'm all ears for criticism and suggestions to improve my daily work on this site. Three moderators have been elected to ensure that this site remains live and esteemed, and we pay a particular attention to questions about site management, our FAQ, Q&A appropriateness, tagging or voting activity. Those questions belong here, on our Meta, because this is the best way to gather ideas or opinions from the whole community, in a collegial manner. We all want this site to be a nice place to ask questions and to provide our users and visitors with a strong and enduring knowledge base for "statisticians, data miners, and anyone else doing data analysis or interested in it as a discipline."
Moderator messages are never sent without valid reasons, and they have to be taken with consideration, especially when they concern posting rules which apply to all registered users on Stack Exchange sites.