I think the stat was win/draw%... but both work.
Well, there are a few factors affecting why you send messages:
(a) Time alive and relevant in the game
(b) Number of people you message on a regular basis
(c) Number of things you have to discuss
In (a), the correlation is obvious; the longer you're alive and relevant, the better your odds of gaining more GR from a game and the better your odds of winning or surviving to the final draw. It's not very common for people who are on the verge of death to get a lot of press back from other players, meaning their message rates are likely to drop off a bit; but even still, the only way a 1-piece power is going to get into the final draw is if the player negotiates his/her ass off to get in.
In (b), the correlation is still relatively clear. If you're talking to more people, your messages are going to increase, and by more than a mere linear amount (your conversation with Russia as France, for instance, often leads you to write more messages to Turkey than you normally would, which might reveal something to tell Austria, etc.). And clearly talking to another power you otherwise wouldn't throughout the game is unequivocally a good thing, as it gives you more potential units to manipulate, and if the relationship is well-fostered then it gives you another voice supporting your bid for the final draw.
(c) is vague, but when explained the correlation again fits. If you're simply going "alliance?" to your neighbors, "yeah/nope" to any question from a distant power, "alliance against <ally>?" to those distant powers and "vote draw vote draw! vote draw!! VOTE DRAW YOU ASSHOLE" to the guy about to solo over your imminent corpse, you're probably not going as well as the guy who messaged everyone to figure out alliance structures, made a point to talk to everyone once a turn until he or the other power(s) were defeated, and just generally outworked you on the diplomatic front. The game is host to myriad opportunities; one just has to be able to take advantage of them, which can only be done in context of talking to everyone early and often.