So I'm sure some remember this tidbit from earlier in the thread:
I was curious, so I went and looked at the top 6000 player seasons in PP TOI in the cap era, to look at the correlation coefficient of PP Points vs. PP Time On Ice.
The correlation coefficient was: +0.92
So by your own definition, there is a very strong positive relationship, and as PP TOI rises, PP production rises very similarly.
If you're looking at a large set of data (ie 6,000 player-seasons), I would definitely expect to see a positive correlation - but not for the reason you think.
Look at the players who got the most PP TOI in a single season since the lockout - Ovechkin, Kovalchuk, Crosby, Malkin, Gonchar, a few seasons of Jagr and Sakic, etc. These are/were some of greatest offensive talents in the NHL. It should be obvious why they received a lot of PP ice time, and why they produce a lot of points on the man advtange.
The player with the 6,000th most PP TOI post-lockout is Peter Harrold, during his 2013 campaign. He played less than 38 minutes on the powerplay that year. The players immediately above him in PP TOI are luminaries like Darren Haydar, Juuso Riikola, Alex Chiasson and penalty killer PJ Axelsson. Right after him are more snipers like Chris Bourque, Derek Meech, Drake Batherson, and another PK'er in Wes Walz.
Effectively, you're comparing Jagr/Ovechkin/Crosby (at the high end of PP ice time and production) and a bunch of players you've probably never even heard of (at the low end of PP ice time and production). Obviously, there will be a correlation. If I understand you correctly, you're presenting this as evidence that Jagr/Ovechkin/Crosby scored more on the PP than Peter Harrold, Darren Haydar, Juuso Riikola as a result of getting more ice time. I would argue that they scored more on the PP because they're much better hockey players.
Why am I dismissing this data, and not the correlational data I presented before? The difference between the two is obvious. The data I presented previously was an apples to apples comparison - it was comparing Auston Matthews to Auston Matthews. The problem with the data presented here is you're comparing Jagr/Ovechkin/Crosby/etc to Harrold/Haydar/Riikola/etc, and drawing a conclusion from that. Instead of isolating the two variables that we're talking about (ice time vs production), you've instead introduced thousands of other variables (the ability of each of these players). (This is something of a simplification as the data I presented combined all four of Matthews' seasons into one, at he's clearly a better player at age 22 than he was at 19 - but the same general trend exists even if you look at each year in isolation).
A much better way to approach this (assuming you want to look at a large set of data) would be to get paired data (ie Jagr 2006 season and Jagr 2007). Look at the player's PP ice time and PP production in Year N. Then look at the percentage change in his production (the dependent variable) in Year N+1, and the percentage change in his ice time that year (independent variable). Calculate the correlation on that. This approach makes sense because, again, it's an apples-to-apples comparison. You're comparing each player against himself, one year down the road. You don't introduce noise by comparing Jaromir Jagr to Peter Harrold. (Of course, there are other factors that could influence the results rather than only ice time - aging, injuries, quality of linemates, etc - but over a sample size of hundreds or thousands of paired player-seasons, the likelihood of this having a material impact would be remote). It would probably take me 3-4 hours to do that analysis - I may have time in the second half of December, but likely not before.
For the record, I agree that, as a general rule, players will score more when given more ice time. That's obvious. But I disagree that this is a simple linear relationship (ie doubling ice time will double production) for the reasons that numerous people in this thread have already explained.
