I took every player since 1974 who was between 24 and 34 years old and played at least 1,000 minutes in back-to-back seasons. I then sorted those players by the absolute change in their possession usage %, and took the top quartile as my sample of players who definitely changed roles. For these 1,036 players, I ran correlation coefficients on their year-to-year performances in these offensive rate stats: True shooting % (PTS / (2 * (FGA + 0.44 * FTA))), Assist Rate (% of teammate FG assisted when on the court), Turnover Rate (TOV/Possessions used), Free throw rate (FTA/FGA), Offensive rebounding % (% of available OReb pulled down while on the court).The conclusion:
True shooting % is easily the least consistent stat when a player changes roles, which seems to back up the concept of skill curves. When a player has a high TS% and a low possession %, it may be that his efficiency is inflated by taking relatively easy shots, attempts that comprise a smaller proportion of his shot selection when he is asked to increase his usage. Along the same lines, turnover rate was the 2nd-least consistent offensive rate stat when changing roles, suggesting that not only is shooting % dependent on the player's usage, but the ability to avoid turnovers is as well.
Finally, free throw rate was in the middle of the pack in terms of correlations. I expected it to be high, alongside OR% and AsR, but it makes sense that it would be lower when you consider that it at least partially represents a player's ability to get shots close to the basket... Just like the ratio of high-percentage shots to low-percentage ones decreases when you take on a bigger role (as evidenced by TS%'s low correlation), it stands to reason that the ratio of close shots to longer ones also decreases with an increase in usage, albeit at a slower rate.