Dear Yimin, many thanks for your question. Often, QCA solutions consist of several models, a phenomenon known as model ambiguity. Sometimes, there are just two of them (M1 and M2 in your case), sometimes there are thousands of models making up the solution. Model ambiguity describes a situation when there exist several models that fit your data equally well. 

Generally, the problem worsens the smaller (less heterogenous) your set of data are relative to the number of exogenous factors you include in the analysis, but model ambiguities can also arise even when you have perfect data.

This important problem has until very recently not been recognized in the QCA literature, and popular software programmes such as fs/QCA have not even fixed it yet, which is why users of these packages are not only at risk of massively over-interpreting the evidence that exists in their respective data, but also of drawing completely incorrect inferences. In fact, the only current software programme for QCA that identifies ALL equally well-fitting models by default is the QCApro package.

A fuller explanation of the problem can be found in two papers that have been published by Michael Baumgartner and myself in the Evaluation Review, Sociological Methods & Research, respectively (see links below; I'm also currently preparing a video explanation of this problem, which aims to be more digestible for applied users than a technical methods paper. I'll let you know when it is ready for release.)

What you should always do in the presence of model ambiguities is to be transparent about them. If you only have very few models, you should present all of them to your reader; if you have too many, you should at least mention the exact number.

Sometimes, as in the case of the QCA solution you present above, there are prime implicants, S*R in this case, that are common to all models. This is a significant finding! It means that you have at least found one combination of causes that seems essential to the outcome. You may then want to focus on these essential prime implicants in your substantive interpretation.

It may also be that you have reason to focus on a specific model as a whole because you have some additional case knowledge that may make it preferable to all others. Or you simply take the model with the highest product score of inclusion and raw coverage (M2 in your case) as an objective criterion.

Whatever course of action you choose, however, transparency about the degree of ambiguity is of the essence. If you follow this advice, you are already in the top 2% of QCA studies with respect to research transparency, which all of your readers will appreciate!

Best wishes,

Alrik

http://journals.sagepub.com/doi/full/10.1177/0049124115610351

http://journals.sagepub.com/doi/abs/10.1177/0193841x14550863

Dear Prof. Alrik Thiem

Thank you for your good answer and attached papers. I am looking forward to your advance of QCA theory and methodology.

I have read your paper "Often trusted but never (properly) tested: Evaluating Qualitative

Comparative Analysis" roughly and paid much attention to your causual path diagrams. It is quite evident to show the causual relationships. I hope someday you can teach us how to draw them with R. 

Best,

Yimin

Dear Yimin, you're very much welcome. There's certainly more to come from my side on QCA theory and methodology in the future. If you want, you can also follow my RG projects to keep up to date.

This is unrelated to the issue of model ambiguity, but the path diagrams in the paper "Often trusted but never (properly) tested: Evaluating Qualitative

Comparative Analysis" were produced with PSTricks, a suite of tools for producing high-quality graphics for LaTeX (http://tug.org/PSTricks/main.cgi). I may add a function to the QCApro package in the future that produces these path diagrams automatically.

Best wishes