Field or method to understand what independent variable affect the response variable the most Say I have several bio-reactors (systems) and a bunch of independent variables which can be for example flux-intensity, rotational velocity in the bioreactor, nutrition etc. I also have my response variable or dependent variable which is the weight of the microbes I'm growing. How do I figure out what variable(s) is/are affecting the weight the most? What is the name of the method?
Many thanks
 A: Your question could be dealt with as a type of variable-selection or feature-selection problem. This is a very broad issue, as you can tell by following that tag on this site. 
With only 5 to 10 independent variables, starting with a regression approach would make sense. There are tools for examining variable importance in regression models; for example, the anova function of the rms package in R provides useful measures and plots. (These are essentially formalizations of the general leave-one-out approach suggested by @spdml.) What's complicated is how to apply these tools intelligently in a way that adequately answers your underlying question.
Design of your regression model to start with is really key. The crucial choices depend a lot on your understanding of the subject matter. Will the first-order linear approximations inherent in linear regressions be adequate, or do you need to consider more complicated relations? Should you be working in the scales that the variables are usually expressed in, or should they be transformed (e.g., logarithmically) to fit linear relations better? Which independent variables do you expect to have effects on outcome more-or-less independent of the levels of other variables, and which may need to be included together via interaction terms in the model? A good deal of preliminary data exploration might be needed to help design the model. And after you've designed and fit a model, you will need to examine the results to see if the assumptions you made were valid.
Another complication is your reliance on a particular set of data. Yes, you can estimate the relative importance of predictor variables on a particular data set--but would they be the same variables chosen based on a different sample of data from the same systems? In many circumstances a model fit manages to match a particular data set but fails miserably when applied to another similar situation. Validation techniques like bootstrapping can be very important to make sure that the "important" variables maintain their importance beyond the data that you have already collected.
In this particular application, it seems that you have the ability to design controlled experiments to get at the underlying issues. Use these estimates of importance of variables, based on existing data, as guides to designing solid experimental tests for optimizing your processes.
