Methodology is our general tool case to attack a scientific research problem; picking the appropriate research tool methodically out of that case is a matter of reach. Applied maths (stats) and logic can help us to analyze the reach and range of our problem and choose the best scientific method available. Today, I do think, computation can offer us cost-effective and time-saving methodical tools of research.
1. If you have an idea of the properties of your products (In natural sciences) or generally a hypothesis, you get informed as to what method will not damage/interfere with your results
2. The facilites (Lab, etc for natural science) or study environment in general
3. Finances may limit you to a particular method or give you flexibility of choices
4. Nature of methods at your disposal (Reliability, reproducibility, ease, difficulty level, etc)
5. Directives from stake holders (Funders, employers, partners may dictate methods to you)
6. Expertise ( Your knowledge, skills, expertise and those of your assistants)
7. Objectives
8. State/consular/religious regulations (Limitations to what you can and can not do/ and what you can and can not disseminate to the public may lead you to carefully and tactfully choose methods that wont end up putting you in a fix)
You ask an interesting questing as when the subject is consciousness, which is my field of interest, we land up with a problem when usual methods of science are applied. Like this analogy with a peach, scientists can weight it, find its chemical properties, compare it to apples, but they cannot tell us what it tastes like. For that we need to put it in our mouth and have an experience. Until phenomenology is more accepted by scientists what is known as the hard problem of consciousness, will continue to exist. So in the case of consciousness, the best method is not fully accepted by scientists (yet). Consciousness as a property of soul, is not visible, and if scientists want to really understand it and its different levels of expression, different methods based on different criteria which are not only quantitative, have to be developed.
The basis on which the appropriate research method is determined for a given research is different according to the type of research. Is it quantitative or qualitative and the research specialization is whether theoretical or applied research and the type of problem that the research is trying to solve and the equipment needed by the research and experimental design suitable for the problem of research .
Dr Ali's answer is satisfactory as it depends upon the type of the investigation. Any answer given may not yield any result because the methodologies can not be generalized.
What informs a research method for a particular study ?
Research method is dependent on research problem that a researcher tries to address / resolve, research objectives(s), research question(s) as well as conceptual framework / research model. E.g. qualitative research is embarked when the research questions are open ended & pertaining to why, how etc. Quantitative research is adopted when research model with all the hypotheses are framed. Likewise for mixed method research which mix both qualitative & quantitative methods depending on your research objective. Also within research method, what data analysis techniques to be adopted also depending on your conceptual framework e.g. whether you are performing an experimental study, analysis of variance, correlational study, mediating effect, moderating effect or structural equation modeling etc.
The tool case and the selection of the tool can never be disconnected; your extension of the methodical problem is highly welcome and diligent. However, we should view the use of a hammer or a screw driver within a set of given tools; so, my point is the selection and elaboration of tools. Thank you for adding stuff to work on.
Background of research, objectives on base of literature, your hypothesis, methodology to achieve your objectives and the what you conclude fro results.
If the study has some data. We can search for the method that fits better the data and find a mathematical model. Sometimes; it is also good to compare two distinct methods; for example to compare a qualitative to a quantitative method. If the process is probalilistic; we can try a stochastic method.