(Apologies for text-heavy post) When managing the game, a coach has two responsibilities: set forth a strategy on how the team will play/respond to the opposing team, and decide how to allot minutes to his players. But what motivates a coach's allocation of minutes to his players? I was thinking about this question today. I suppose there are two possible motivating factors: (1) maximize probability of winning the games (2) player development via in-game experience For a team trying to make the playoffs and win a championship, (1) becomes the dominant factor. In fact, I'd argue that it really only makes sense to emphasize (2) for teams that are very likely going to miss the playoffs. There are other means to "develop" players than giving them minutes at the cost of winning. But that's not the topic for this thread. So, we assume the overriding motivation is maximizing wins. If we further assume that any variability of player performance is random and not dependent on things like matchups, fatigue, "hot/cold streaks", then the rational choice would be to play your 5 best guys the entire 48 minutes. Of course, this is not reality! And if variability depended only on fatigue and nothing else, then the rational choice would be to come up with a fixed rotation/substitution pattern and not to deviate from it. Of course, this is not reality either! A coach has to make choices in-game based on things like matchups or if he thinks a player (or combination) is "hot" or "cold". And here is where fan discontent often creeps in. If favorite player A gets pulled earlier than expected and/or stays on the bench longer than expected, the natural response from player A's fan is to question what on earth the coach is thinking. Was the matchup really not favorable, or is the coach just underestimating the player's abilities? Is this player really hurting the team (or, just as often, is this scrub who subbed in for my guy really helping the team), or is it just randomness in short minutes that the coach is mistaking for something real? My intention here is not to justify McHale's minute allocations, but rather to understand it. I was curious if the stats could shed any light on why he decides a certain player should get X number of minutes in a given game. Of course, I would expect that there'd generally be a multitude of reasons behind it (how well is he playing?, how well is the other guy at his position is playing?, is it a blowout?, how is the team doing overall when he's on the floor?, is he sick or recovering from an injury?, etc.). I figured that, generally speaking, the two most important factors in theory would be (1) how well is he playing?, (2) how well is the team doing overall when he's on the floor?. So, I took 6 players who's place within the rotation has been basically stable over the course of the season -- Lin, Harden, Parsons, Asik, Douglas, and Delfino. Patterson and Morris switched spots, and Greg Smith has been in and out, so I thought not to include them. For each player, I looked at their game log and picked out 3 stats: Min: how many minutes did the coach give them GmScr/min: proxy for how well the player was performing when he was on the floor +/-/min: proxy for how well the team was playing when he was on the floor Then, I simply looked at how these metrics correlated to eachother over the games they played (not including last night's game against the Celtics). I expected all these stats to be positively correlated to eachother, and they were. But I found some interesting differences across the players that might suggest something about how the coach evaluates each player's ability to help the team within the game. The following chart shows the results for each of the players: The bars are just the r<sup>2</sup> for the different pairs. You can think of the blue bar as representing the percent of the variation in the player's minutes each game that can be explained by his individual stats (GmScr/min). The red bar represents how much the variation in his minutes each game can be explained by his +/- per minute (+/-/min). And the green bar represents how much the variation in the player's +/- per minute can be explained by his individual stats. Again, minutes allocation depends on several different variables, and so the r<sup>2</sup>'s will be low, but nevertheless I'll try to interpret what these results mean. The thing that immediately jumps out is the PG's minute allocation appears to be more sensitive to how they are performing individually. For Lin and Douglas, their per-minute statistical production explains about 20% of the variation in their minutes. This percentage is much less for the other players. For Lin, it appears that how the team performs with him on the court can explain roughly 13.5% of his variation in minutes, which is much higher than for the other players. I'm curious if this is normal for young PGs that have yet to fully earn his coach's trust. For Harden, I suppose it is expected that the variation in his minutes depends very little on how he's performing individually or indeed how the team is performing when he's on the floor. He's the designated "team's best player", and the coach is going to trust that with him on the floor the team will have a better shot at winning irrespective on how he or the team happens to be playing. The variation in Parsons's minutes also seems quite independent of these two metrics. Even though Chandler is inexperienced, I suspect that his versatility (he can often defend 3 positions and play either forward spot on offense) and savvy beyond his years has earned his coach's trust. Interestingly, for Parsons the green bar is quite low, indicating that the variation in his +/- per minute each game depends very little on his individual statistics. The positive/charitable way to interpret this is that Parsons really is the team's "glue guy", and the ways in which he really contributes to the team aren't reflected very well in his box score numbers. Asik's mininutes distribution also seems to not depend very much on his individual production or how well the team happens to be playing with him. I interpret this to be that his minutes are largely dictated by matchups or the coach desiring a defensive presence. Finally we come to Delfino. The low correlation of GmScr/min and +/-/min with respect to his Min distribution is probably to be expected. The coach trusts him to be solid on both ends and values the fact that he's the only player on the team with several years of experience. Anyway, I certainly can't say that any questions have been definitively answered with these results. Just some food for thought.
I can see why the center position wouldn't be reliant on performance and more on matchup, as well as Harden, even Parsons can be explained, but I do wish you had found some way to include Patterson/Morris in your analysis, because I'd like to see if McHale simply sees the PG position as easily interchangeable or he doesn't trust Lin/Douglas to play through a few difficult possessions. It's also possible McHale sees the PG position as the captain of the ship, and so is more likely to swap PGs based on performance/team performance than the other positions. Is it possible to use last year's team (with Dragic, Lowry, Scola, etc.) as a comparison?
I think you're trying to look into things more deeply than they really are. It's simple case of value above your replacement. If you think of it this way, it makes sense that Delfino, who's replacement is Daquan Cook or James Anderson, may be more indispensible than someone like Asik or Lin, who have much more capable replacements. Simple economics. The rarer the product(player), the more its value(playing time).
Interesting analysis, durvasa. thanks. It helps quantify that Lin's minutes might have been more contingent on his performance than has been true for the minutes of the other players, as some have sensed. I am guessing some might seize this as evidence that the coaches don't trust Lin as much, are treating him unfairly, or even are not as interested in his development. I don't think that's true. What I think it might show though is that the coaches are still struggling with how best to optimize this Harden-Lin back court. Remember those stats saying that Harden performs better when Lin is not on the court. Such stats would I think quite naturally lead coaches to scrutinize Lin's play more, and more quickly choose to pull him upon indications he is having a less productive game. While I sympathize with the coaches' dilemma, I am not sure pulling Lin upon signs of early (individual or group) struggle when Lin is on the floor is the best thing for the Rockets. Primarily because I think quite soon Harden will become as productive, if not more so, with Lin on the court if this back court is allowed to develop. When Lin is having a good shooting night, having to defend against a Lin and Harden back court is just harder than having to deal with TD and Harden back court. Lin is just a more potent and unpredictable offensive threat. (See Chicago Christmas game, Memphis game, NYKs game, Cavaliers game, on and on). I think the assumption should be that it is always better for the Rockets to close with them both on the floor. But I don't blame the coaches either, Lin was really not shooting reliably earlier this season. For me, I think his performance since the Dec 17 Knicks game at Madson Square Gardens has dispelled most doubts that he can be a productive PG for this team. But it is a new reality; and sometimes it takes time to adjust to new realities. That game was only 3 1/2 weeks ago. It's going to take some time to work out the Rockets' back court. We always knew that.
Or maybe an even simpler answer is that McHale is still learning as a NBA head coach & still has room to get better. Fans often assume that coaches are perfect once they enter. They get better just like players do. Look at Doc Rivers, or VDN. They got/are getting better. Also, you don't learn everything about your players in less than half a season particularly when your entire roster is brand new & so young.
nice post! How many games did you use? Can you share the data you using for regression? does the result varies a lot if you look at differnt set of games (eg.home/away )? I also want to test the hypothesis whether Sampson and Mchale is using the players the same way.
Regarding that claim about harden performance w or w/o Lin on the court, I think it was not conclusive. They did not show that the diffenfece is statistically significant. I mean the variance could be huge and shadow the diffenfece in mean completely. While I sympathize with the coaches' dilemma, I am not sure pulling Lin upon signs of early (individual or group) struggle when Lin is on the floor is the best thing for the Rockets. Primarily because I think quite soon Harden will become as productive, if not more so, with Lin on the court if this back court is allowed to develop. When Lin is having a good shooting night, having to defend against a Lin and Harden back court is just harder than having to deal with TD and Harden back court. Lin is just a more potent and unpredictable offensive threat. (See Chicago Christmas game, Memphis game, NYKs game, Cavaliers game, on and on). I think the assumption should be that it is always better for the Rockets to close with them both on the floor. But I don't blame the coaches either, Lin was really not shooting reliably earlier this season. For me, I think his performance since the Dec 17 Knicks game at Madson Square Gardens has dispelled most doubts that he can be a productive PG for this team. But it is a new reality; and sometimes it takes time to adjust to new realities. That game was only 3 1/2 weeks ago. It's going to take some time to work out the Rockets' back court. We always knew that.[/QUOTE]
Interesting food for thought, if nothing else. wonder if coach realize this and how he might explain why he is doing so.
But Delfino is a second-stringer. His "replacement" would be the starters in this case (Harden, Parsons), just as Douglas's replacement in our rotation isn't Machado but rather Lin. I would certainly expect the variation in a player's minutes to be explained largely by the variation in his replacement's minutes, which would be a third factor that I haven't yet looked at.
This is one perfect example of use statistics without understand the meaning of statistics. It is really misleading. R square is an indication of how two data sets are correlated, it basically could never be represented with percentage. Perfect correlation set of data you will get 1, and pure random you will get 0. Usually when it is R square goes to 0.35, (as the highest point in your graph), it is pretty random already, no correlation, thus there is no point of comparing a 0.2 to a 0.05, ( in your graph of 20% vs. 5%) doing that you are trying to tell which one is more random than the other random prediction. It just does not make sense. Yet your interpretation is totally wrong, when use R square, higher the number indicate better correlation, not bigger divination from the prediction, thus if you really trying to make sense from this calculation, Lin's time is then better predicted by his score than other people. This calculation just totally don't make sense. I'm totally for using statistic data to analysis Bball, but one really need to understand what they are doing and what the data come out means. Not just throw in some random calculation, and somehow the number come out go with the trend you would like to see, and just use it like that.
A value that ranges from 0 to 1, like R squared, can't be represented as a percentage? I think you misread what I was saying. Yes, this is examining correlation, not asserting causation. Also, I did point out that the R squared values are low. Hence, at the end I said that this is not providing definitive answers. The point is more to raise questions and open up a discussion. I'd also suggest that the threshold for what makes a R squared value too low to be meaningful depends on the nature of the problem one is looking at, and how strong a conclusion one is trying to draw. If I may, I'll quote from the Wikipedia entry on R squared ("coefficient of determination"): [rquoter] R2 is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, R2 = 1 indicates that the fitted model explains all variability in y, while R2 = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope=0, intercept=\bar{y}) between the response variable and regressors). An interior value such as R2 = 0.7 may be interpreted as follows: "Approximately seventy percent of the variation in the response variable can be explained by the explanatory variable. The remaining thirty percent can be explained by unknown, lurking variables or inherent variability." A caution that applies to R2, as to other statistical descriptions of correlation and association is that "correlation does not imply causation." In other words, while correlations may provide valuable clues regarding causal relationships among variables, a high correlation between two variables does not represent adequate evidence that changing one variable has resulted, or may result, from changes of other variables. [/rquoter] If variables A and B are highly correlated, and one theorizes that B is dependent on A, then another way to state this high correlation is that it reflects the extent to which variable A explains the variation in variable B. And I'm not sure what you mean by "not bigger divination from the prediction," if not "better predicted".
without putting in data, just think about it. A consistent player might have higher score with longer play time, thus score/min should be relatively stable. A inconsistent player might score random, not really relate to play time, maybe score a bunch in small stretch of time, but play longer not necessary give you same production, (I think TD would belong to this group). Thus, this kind of player would have pretty random score/min. Then could you imagine any player would or could have a positive correlation of their score/min with their play time? That means when they play longer time, not only their total score will increase, (which is normal), but their unit time score will increase, that is efficiency will increase? I could only see there is potential of negative correlation, the longer time they play, their score/min will drop, and if that correlation is strong, that would be the perfect explanation that coach should take TD out when they catch up by 2, and there is no term as "ride the hot hand". This kind of correlation play time vs. score/min just don't make any sense, as they just should not correlate.
although 0 to 1 fit perfectly to 0% to 100%, but the meaning is different. when you presented into percentage, the following mistake is so easy to make. You quickly convert the R square number to % variation, which again is totally different thing. R square is a indication that how much the real data fit into your prediction, in this case a linear correlation of two data set, and when you got a 0.35 or less, that just says that your model does not fit the data, and you could not use one data set to predict the other one, period. It is a good thing to think out of the box, come up with different theory, and test it, I appreciate that. However, when the calculation comes back indicating that your model does not fit, you should stop there, or try to find new models or new fit, not go a step further trying to make sense of nonsense data. Whatever conclusion come out of a shaky theory, it does not make sense, even you have numbers in it.
You're saying that there is only potential for negative correlation, and yet the correlation between minutes each game and GmScr/min is positive for all 6 players -- especially so for the PGs. Perhaps this wasn't clear, because I posted the R squared rather than the R before. This graph shows the correlation (note, they are all positively correlated for every player):
