without knowing the makeup or politics of how your institution works and how connected your supervisor is, its hard to give concrete tips. So take the answers here with a pinch of salt.
Which field are you in? From your question it seems like you are in a field that uses quantitative data (primarily), thus the emphasis on right and wrong results. What is the kind of feedback the other guy gets when (or if) he publishes his work?
When you say that the other guy "did not get the correct result", is he being:
- fraudulent - wrongfully analysing data on purpose or not following correct procedures
- ignorant - the guy is clueless about the 'right' way to analyse the data and needs some serious help
- just different - are you sure that your way is the right way and the other guy's way is the wrong way? Can you be confident of this (academically speaking)? Are there research papers you can send to the other guy and your supervisor about your approach being valid and the other guy's approach being invalid.
There are a number of things you can do in each of these situations:
- reporting fraudulent practices (anonymously and only if it is safe to do so for you!) to an ethics/review board
- seek opinion from others in your field outside of your institution (are there institutions that work in the same research field in your city/region? make connections with them and get a mentorship role form a senior academic there - this will take a few months so no quick fixes unfortunately).
- Additionally have you been to a conference/workshop of people from your field (again outside your institution)? If yes, then recall those connections and make those ties stronger (via skype/coffee) or visit a conference soon and make those connections happen.
In my experience, the PhDs who progress and learn the most are those who are connecting and learning from those in their field, regardless of whether they are from your institution or not - so this is valuable (irregardless of the troubles you are having).
You can write up your work to a workshop/conference and gain feedback there? Might be valuable in backing up your claims to your supervisor.
Is there a 'third-way'? A compromise between your approach and your supervisors's preferred approach? Unfortunately, the PhD-Supervisor relationship is filled with power dynamics which mean that your PhD will never be 'purely academic' as we might want it to be..