Statistics on how a rating or winning percentage corresponds to the (opening, middle, end) game? Have their been any statistics on how your knowledge on part of the game (opening, middle, end) corresponds to your rating or winning percentage?  For example, I have found that if I play opening moves a good number of moves in, my chances of winning go way up.  I am not sure how the middle game or end game affect this though. If player A was very strong in the opening, but semi-strong in the middle-game and player B was semi-strong in the opening, but very strong in the middle-game, does one player have an advantage?  The same goes for the end-game?  I have heard that the end-game is usually where the game is won.  Do masters of the past and current have any direct correlation with these statistics?
No.  There is a rating system based on wins, losses, and draws, not for proficiency in phases of the game.  How would you quantify that?  Knowing the main line on the MCO for your openings?  People are always finding improvements to main and obscure opening lines.  This is the TN (theoretical novelty) that people search feverishly for.  So what might be sound play in the opening in January may be unsound by August.  

Further, I have seen flawless openings and middlegames come to grief because of a tiny blunder in the endgame.  As Tartakower once said, the winner is the player who makes the 2nd to last mistake.  So there is not really an accepted method of tying knowledge of component phases of the game to winning percentage, because there are too many other factors at hand.  If the game 's phases could be defined down to a formula of what is good play in the opening, middle game or ending, computers would always win.  We still have imperfections of shortsightedness and obliging opponents, and time controls, and other distractions.