Completely remove all traces of an app so I can start over fresh?

This has probably been answered many times before but I can't find a satisfactory answer so maybe one of the brains on here can help!

Basically I have some older apps that I've put in my new phone (downloaded via previously purchased), there was no data stored for them in the iCloud and I'd deleted the info from Game Center yet when I open them it is still completed and how I left it on my old phone.

So my questions, where is the app pulling this save game data from as I just can't figure out where its saved, and is there a way to force them to reset if you like?

Thank you everyone!