I am not sure what you are asking for exactly. Please elaborate on what quasi-statistics are or more specifically what you think they are. Are you speaking about quasi-experimental designs?
If numbers were used it couldn't really be qualitative research so your question essentially is: How do we make qualitative research quantitative research?
The general practice is to use qualitative research in the investigation of a relatively unknown phenomena and then qualitatively gather information about it. This research then, in theory, leads to a greater understanding of the phenomenon and allows researchers to form more specific hypotheses.
At the same time, qualitative research itself is quite valuable as it can attempt to gain insight into questions that are not quantifiable, lack random sampling, probability testing etc. This research can also bring a depth to an already studied phenomena. So qualitatively researching a topic doesn't necessary have to be seen as an inferior precursor to a more refined quantitative study.
Therefore, while quantifying some features of a qualitative study may seems like great way to make the study more scientifically viable, I am not so sure that this would actually be true. though certainly there are some cases that could be thought of in which quantitative analyses would definitely make a study more interesting and easier to remember.
Sorry to not actually give any real answer to this question; but I maintain that qualitative research does not necessarily need quantitative analyses to give it value.