How to prove that one variable's change is influenced by another variable; that is to say, are they related? I major in bioinformatics. We all know that temperature changes during a year. I find that a disease incidence is really high when temperature is relatively high, while it becomes really low when the temperature is relatively low; that is to say, they are related. I want to find out a way to prove that they are related, not just intuitively feel that they are related. Any suggestions?
 A: Look into the concept of Granger causality which deals with precisely this. Basically you do a predictive model of one variable to the other. Then you run a hypothesis test to see how much of the variance in the output was explained by the input variable. If enough variance is sufficiently explained, then you can say that temperature 'Granger causes' disease. 
Note g-causation is weaker then causation because both variables could be influenced by a third unobserved variable. However, this is in principle the best you can do with only two time series....
A: Collect your data and then model it.  If you want to show a causal relationship, you should design an experiment, as their could be other confounding variables that mask the real cause and effect.    
A: I think what you are looking for is the methods of causal inference. A standard way to prove causality is through randomization. Some accessible articles on the subject are http://www-stat.wharton.upenn.edu/~rosenbap/ExperAndObsTalk.pdf and http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf
For biostats specifically, you already have a set up for this by having two groups (a treatment group and a control group) and randomly assigning people to each group.
As a generalization, it is extremely hard to prove that A causes B. I may be wrong, but I don't think that (at least legally) smoking has been proven to cause cancer, just that the probability of getting cancer is increased if you smoke.
