Is it better for me to audit or enroll in a class for independent research?
In general, auditing a class provides somewhat close to zero information for anyone considering your application in the future. At the very best, it shows you were interested in the subject even though you would get no credit for it. Its not much different than taking an online course with no grade, studying on your own, or just showing up to a class without being enrolled. At least in the US, auditing doesn't even require you show up to class, much less do any work, thus why it has little signal meaning.

So I would suggest that if you audit the class, do it for some other reason than what people looking at your application will think of it. It is reasonable to try to audit if you want to improve your training, prepare for a more advanced class (or just taking the same class next semester), just want to do it for fun (I personally plan to audit stats classes in the future, but that's because I'm at a point that no one cares what classes I take so its just for my own utility and interest), want to get to know the professor a bit better to see if you'd like to work more with them or take other classes by them, etc. 

If programs you are planning to apply to strongly value or require quantitative/statistical training, though, auditing a class generally won't do it. This completely depends on what sort of program you apply to, as some won't really care about stats preparation anyway, while others will demand a lot more than 1 class in stats and research methods/design. Now would be a good time to look ahead at what sort of programs you are applying to, what they say they like to see, and plan your courses accordingly.