## Sports Geek Mecca: Recap and Thoughts, Part 2

This is part 2 of my “recap” of the Sloan Sports Analytics Conference that I attended in March (part 1 is here), mostly covering Day 2 of the event, but also featuring my petty way-too-long rant about Bill James (which I’ve moved to the end).

### Day Two

First I attended the Football Analytics despite finding it disappointing last year, and, alas, it wasn’t any better. Eric Mangini must be the only former NFL coach willing to attend, b/c they keep bringing him back:

Overall, I spent more time in day 2 going to niche panels, research paper presentations and talking to people.

The last, in particular, was great. For example, I had a fun conversation with Henry Abbott about Kobe Bryant’s lack of “clutch.” This is one of Abbott’s pet issues, and I admit he makes a good case, particularly that the Lakers are net losers in “clutch” situations (yes, relative to other teams), even over the periods where they have been dominant otherwise.

Kobe is kind of a pivotal case in analytics, I think. First, I’m a big believer in “Count the Rings, Son” analysis: That is, leading a team to multiple championships is really hard, and only really great players do it. I also think he stands at a kind of nexus, in that stats like PER give spray shooters like him an unfair advantage, but more finely tuned advanced metrics probably over-punish the same. Part of the burden of Kobe’s role is that he has to take a lot of bad shots—the relevant question is how good he is at his job.

Abbott also mentioned that he liked one of my tweets, but didn’t know if he could retweet the non-family-friendly “WTF”:

I also had a fun conversation with Neil Paine of Basketball Reference. He seemed like a very smart guy, but this may be attributable to the fact that we seemed to be on the same page about so many things. Additionally, we discussed a very fun hypo: How far back in time would you have to go for the Charlotte Bobcats to be the odds-on favorites to win the NBA Championship?

As for the “sideshow” panels, they’re generally more fruitful and interesting than the ESPN-moderated super-panels, but they offer fewer easy targets for easy blog-griping. If you’re really interested in what went down, there is a ton of info at the SSAC website. The agenda can be found here. Information on the speakers is here. And, most importantly, videos of the various panels can be found here.

### Box Score Rebooted

Featuring Dean Oliver, Bill James, and others.

This was a somewhat interesting, though I think slightly off-target, panel. They spent a lot of time talking about new data and metrics and pooh-poohing things like RBI (and even OPS), and the brave new world of play-by-play and video tracking, etc. But too much of this was discussing a different granularity of data than what can be improved in the current granularity levels. Or, in other words:

James acquitted himself a bit on this subject, arguing that boatloads of new data isn’t useful if it isn’t boiled down into useful metrics. But a more general way of looking at this is: If we were starting over from scratch, with a box-score-sized space to report a statistical game summary, and a similar degree of game-scoring resources, what kinds of things would we want to include (or not) that are different from what we have now?  I can think of a few:

1. In basketball, it’s archaic that free-throws aren’t broken down into bonus free throws and shot-replacing free throws.
2. In football, I’d like to see passing stats by down and distance, or at least in a few key categories like 3rd and long.
3. In baseball, I’d like to see “runs relative to par” for pitchers (though this can be computed easily enough from existing box scores).

In this panel, Dean Oliver took the opportunity to plug ESPN’s bizarre proprietary Total Quarterback Rating. They actually had another panel devoted just to this topic, but I didn’t go, so I’ll put a couple of thoughts here.

First, I don’t understand why ESPN is pushing this as a proprietary stat. Sure, no-one knows how to calculate regular old-fashioned quarterback ratings, but there’s a certain comfort in at least knowing it’s a real thing. It’s a bit like Terms of Service agreements, which people regularly sign without reading: at least you know the terms are out there, so someone actually cares enough to read them, and presumably they would raise a stink if you had to sign away your soul.

As for what we do know, I may write more on this come football season, but I have a couple of problems:

One, I hate the “clutch effect.” TQBR makes a special adjustment to value clutch performance even more than its generic contribution to winning. If anything, clutch situations in football are so bizarre that they should count less. In fact, when I’ve done NFL analysis, I’ve often just cut the 4th quarter entirely, and I’ve found I get better results. That may sound crazy, but it’s a bit like how some very advanced Soccer analysts have cut goal-scoring from their models, instead just focusing on how well a player advances the ball toward his goal: even if the former matters more, its unreliability may make it less useful.

Two, I’m disappointed in the way they “assign credit” for play outcomes:

Division of credit is the next step. Dividing credit among teammates is one of the most difficult but important aspects of sports. Teammates rely upon each other and, as the cliché goes, a team might not be the sum of its parts. By dividing credit, we are forcing the parts to sum up to the team, understanding the limitations but knowing that it is the best way statistically for the rating.

I’m personally very interested in this topic (and have discussed it with various ESPN analytics guys since long before TQBR was released). This is basically an attempt to address the entanglement problem that permeates football statistics.  ESPN’s published explanation is pretty cryptic, and it didn’t seem clear to me whether they were profiling individual players and situations or had created credit-distribution algorithms league-wide.

At the conference, I had a chance to talk with their analytics guy who designed this part of the metric (his name escapes me), and I confirmed that they modeled credit distribution for the entire league and are applying it in a blanket way.  Technically, I guess this is a step in the right direction, but it’s purely a reduction of noise and doesn’t address the real issue.  What I’d really like to see is like a recursive model that imputes how much credit various players deserve broadly, then uses those numbers to re-assign credit for particular outcomes (rinse and repeat).

### Deconstructing the Rebound With Optical Tracking Data

Rajiv Maheswaran, and other nerds.

This presentation was so awesome that I offered them a hedge bet for the “Best Research Paper” award. That is, I would bet on them at even money, so that if they lost, at least they would receive a consolation prize. They declined. And won. Their findings are too numerous and interesting to list, so you should really check it out for yourself.

Obviously my work on the Dennis Rodman mystery makes me particularly interested in their theories of why certain players get more rebounds than others, as I tweeted in this insta-hypothesis:

Following the presentation, I got the chance to talk with Rajiv for quite a while, which was amazing. Obviously they don’t have any data on Dennis Rodman directly, but Rajiv was also interested in him and had watched a lot of Rodman video. Though anecdotal, he did say that his observations somewhat confirmed the theory that a big part of Rodman’s rebounding advantage seemed to come from handling space very well:

1. Even when away from the basket, Rodman typically moved to the open space immediately following a shot. This is a bit different from how people often think about rebounding as aggressively attacking the ball (or as being able to near-psychically predict where the ball is going to come down.
2. Also rather than simply attacking the board directly, Rodman’s first inclination was to insert himself between the nearest opponent and the basket. In theory, this might slightly decrease the chances of getting the ball when it heads in toward his previous position, but would make up for it by dramatically increasing his chances of getting the ball when it went toward the other guy.
3. Though a little less purely strategical, Rajiv also thought that Rodman was just incredibly good at #2. That is, he was just exceptionally good at jockeying for position.

To some extent, I guess this is just rebounding fundamentals, but I still think it’s very interesting to think about the indirect probabilistic side of the rebounding game.

### Live B.S. Report with Bill James

Quick tangent: At one point, I thought Neil Paine summed me up pretty well as a “contrarian to the contrarians.”  Of course, I’m don’t think I’m contrary for the sake of contrariness, or that I’m a negative person (I don’t know how many times I’ve explained to my wife that just because I hated a movie doesn’t mean I didn’t enjoy it!), it’s just that my mind is naturally inclined toward considering the limitations of whatever is put in front of it. Sometimes that means criticizing the status quo, and sometimes that means criticizing its critics.

So, with that in mind, I thought Bill James’s showing at the conference was pretty disappointing, particularly his interview with Bill Simmons.

I have a lot of respect for James.  I read his Historical Baseball Abstract and enjoyed it considerably more than Moneyball.  He has a very intuitive and logical mind. He doesn’t say a bunch of shit that’s not true, and he sees beyond the obvious. In Saturday’s “Rebooting the Box-score” panel, he made an observation that having 3 of 5 people on the panel named John implied that the panel was [likely] older than the rest of the room.  This got a nice laugh from the attendees, but I don’t think he was kidding.  And whether he was or not, he still gets 10 kudos from me for making the closest thing to a Bayesian argument I heard all weekend.  And I dutifully snuck in for a pic with him:

James was somewhat ahead of his time, and perhaps he’s still one of the better sports analytic minds out there, but in this interview we didn’t really get to hear him analyze anything, you know, sportsy. This interview was all about Bill James and his bio and how awesome he was and how great he is and how hard it was for him to get recognized and how much he has changed the game and how, without him, the world would be a cold, dark place where ignorance reigned and nobody had ever heard of “win maximization.”

Bill Simmons going this route in a podcast interview doesn’t surprise me: his audience is obviously much broader than the geeks in the room, and Simmons knows his audience’s expectations better than anyone. What got to me was James’s willingness to play along, and everyone else’s willingness to eat it up. Here’s an example of both, from the conference’s official Twitter account:

Perhaps it’s because I never really liked baseball, and I didn’t really know anyone did any of this stuff until recently, but I’m pretty certain that Bill James had virtually zero impact on my own development as a sports data-cruncher.  When I made my first PRABS-style basketball formula in the early 1990’s (which was absolutely terrible, but is still more predictive than PER), I had no idea that any sports stats other than the box score even existed. By the time I first heard the word “sabermetrics,” I was deep into my own research, and didn’t bother really looking into it deeply until maybe a few months ago.

Which is not to say I had no guidance or inspiration.  For me, a big epiphanous turning point in my approach to the analysis of games did take place—after I read David Sklansky’s Theory of Poker. While ToP itself was published in 1994, Sklansky’s similar offerings date back to the 70s, so I don’t think any broader causal pictures are possible.

More broadly, I think the claim that sports analytics wouldn’t have developed without Bill James is preposterous. Especially if, as i assume we do, we firmly believe we’re right.  This isn’t like L. Ron Hubbard and Incident II: being for sports analytics isn’t like having faith in a person or his religion. It simply means trying to think more rigorously about sports, and using all of the available analytical techniques we can to gain an advantage. Eventually, those who embrace the right will win out, as we’ve seen begin to happen in sports, and as has already happened in nearly every other discipline.

Indeed, by his own admission, James liked to stir controversy, piss people off, and talk down to the old guard whenever possible. As far as we know, he may have set the cause of sports analytics back, either by alienating the people who could have helped it gain acceptance, or by setting an arrogant and confrontational tone for his disciples (e.g., the uplifting “don’t feel the need to explain yourself” message in Moneyball). I’m not saying that this is the case or even a likely possibility, I’m just trying to illustrate that giving someone credit for all that follows—even a pioneer like James—is a dicey game that I’d rather not participate in, and that he definitely shouldn’t.

On a more technical note, one of his oft-quoted and re-tweeted pearls of wisdom goes as follows:

Sounds great, right? I mean, not really, I don’t get the metaphor: if the sea is full of ignorance, why are you collecting water from it with a bucket rather than some kind of filtration system? But more importantly, his argument in defense of this claim is amazingly weak. When Simmons asked what kinds of things he’s talking about, he repeatedly emphasized that we have no idea whether a college sophomore will turn out to be a great Major League pitcher.  True, but, um, we never will. There are too many variables, the input and outputs are too far apart in time, and the contexts are too different.  This isn’t the sea of ignorance, it’s a sea of unknowns.

Which gets at one of my big complaints about stats-types generally.  A lot of people seem to think that stats are all about making exciting discoveries and answering questions that were previously unanswerable. Yes, sometimes you get lucky and uncover some relationship that leads to a killer new strategy or to some game-altering new dynamic. But most of the time, you’ll find static. A good statistical thinker doesn’t try to reject the static, but tries to understand it: Figuring out what you can’t know is just as important as figuring out what you can know.

On Twitter I used this analogy:

Success comes with knowing more true things and fewer false things than the other guy.

## Google Autocomplete Error in My Favor

So I was scanning for funny search terms that have led wary surfers to the blog, but stumbled into the following instead (click to enlarge):

In case you’re wondering, yes, I signed out of Google and turned off search personalization first.  The URL of the search just leads to “the case for dennis rodman” results, so if you want to duplicate it, you have to enter “+/- for Dennis Rodman” yourself (without pressing enter or the search button, obv).  Incidentally, this site is only the #6 result for the original search.

I understand that my humble offering may be the only study of Dennis Rodman’s +/- stats in existence (I have no idea), but, regardless, this seems like a clear flaw in the autocomplete algorithm to me. Personally, I would like to see Google get better at making semantic distinctions, while this seems to flub one of the most basic: between search term and search result.

Incidentally, I was just going to title this post “Dennis Rodman Still Looks Like the Scariest Clown Ever,” but I didn’t want to set expectations too high.

## The Case for Dennis Rodman, Part 4/4(b): The Finale (Or, “Rodman v. Jordan 2”)

[ADDED: Unsurpisingly, this post has been getting a lot of traffic, which I assume includes a number of new readers who are unfamiliar with my “Case For Dennis Rodman.” So, for the uninitiated, I’d like to (at least temporarily) repeat a few of my late-comer intro points from Part 4(a): “The main things you need to know about this series are that it’s 1) extremely long (sprawling over 13 sections in 4 parts), 2) ridiculously (almost comically) detailed, and 3) only partly about Dennis Rodman.  There is a lot going on, so to help new and old readers alike, I have a newly-updated “Rodman Series Guide,” which includes a broken down list of articles, a sampling of some of the most important graphs and visuals, and a giant table summarizing the entire series by post, including the main points on both sides of the analysis.”]

So it comes down to this: With Rodman securely in the Hall of Fame, and his positive impact conclusively demonstrated by the most skeptical standards of proof I can muster, what more is there to say? Repeatedly, my research on Rodman has led to unexpectedly extreme discoveries: Rodman was not just a great rebounder, but the greatest of all time—bar none. And despite playing mostly for championship contenders, his differential impact on winning was still the greatest measured of any player with data even remotely as reliable as his. The least generous interpretation of the evidence still places Rodman’s value well within the realm of the league’s elite, and in Part 4(a) I explored some compelling reasons why the more generous interpretation may be the most plausible.

Yet even that more generous position has its limitations. Though the pool of players I compared with Rodman was broadly representative of the NBA talent pool on the whole, it lacked a few of the all-time greats—in particular, the consensus greatest: Michael Jordan. Due to that conspicuous absence, as well as to the considerable uncertainty of a process that is better suited to proving broad value than providing precise individual ratings, I have repeatedly reminded my readers that, even though Rodman kept topping these lists and metrics, I did NOT mean to suggest that Rodman was actually greater than the greatest of them all. In this final post of this series, I will consider the opposite position: that there is a plausible argument (with evidence to back it up) that Rodman’s astounding win differentials—even taken completely at face value—may still understate his true value by a potentially game-changing margin.

# A Dialogue:

First off, this argument was supposed to be an afterthought. Just a week ago—when I thought I could have it out the next morning—it was a few paragraphs of amusing speculation. But, as often seems to be the case with Dennis Rodman-related research, my digging uncovered a bit more than I expected.

The main idea has its roots in a conversation I had (over bruschetta) with a friend last summer. This friend is not a huge sports fan, nor even a huge stats geek, but he has an extremely sharp analytical mind, and loves, loves to tear apart arguments—and I mean that literally: He has a Ph.D. in Rhetoric. In law school, he was the guy who annoyed everyone by challenging almost everything the profs ever said—and though I wouldn’t say he was usually right, I would say he was usually onto something.

That night, I was explaining my then-brand new “Case for Dennis Rodman” project, which he was naturally delighted to dissect and criticize. After painstakingly laying out most of The Case—of course having to defend and explain many propositions that I had been taking for granted and needing to come up with new examples and explanations on the fly, just to avoid sounding like an idiot (seriously, talking to this guy can be intense)—I decided to try out this rhetorical flourish that made a lot of sense to me intuitively, but which had never really worked for anyone previously:

“Let me put it this way: Rodman was by far the best third-best player in NBA History.”

As I explained, “third best” in this case is sort of a term of art, not referring to quality, but to a player’s role on his team. I.e., not the player a team is built around (1st best), or even the supporting player in a “dynamic duo” (like HOF 2nd-besters Scotty Pippen or John Stockton), but the guy who does the dirty work, who mostly gets mentioned in contexts like, “Oh yeah, who else was on that [championship] team? Oh that’s right, Dennis Rodman”).

“Ah, so how valuable is the best third-best player?”

At the time, I hadn’t completely worked out all of the win percentage differentials and other fancy stats that I would later on, but I had done enough to have a decent sense of it:

“Well, it’s tough to say when it’s hard to even define ‘third-best’ player, but [blah blah, ramble ramble, inarticulate nonsense] I guess I’d say he easily had 1st-best player value, which [blah blah, something about diminishing returns, blah blah] . . . which makes him the best 3rd-best player by a wide margin”.

“How wide?”

“Well, it’s not like he’s as valuable as Michael Jordan, but he’s the best 3rd-best player by a wider margin than Jordan was the best 1st-best player.”

“So you’re saying he was better than Michael Jordan.”

“No, I’m not saying that. Michael Jordan was clearly better.”

“OK, take a team with Michael Jordan and Dennis Rodman on it. Which would hurt them more, replacing Michael Jordan with the next-best primary scoring option in NBA history, or replacing Rodman with the next-best defender/rebounder in NBA history?”

“I’m not sure, but probably Rodman.”

“So you’re saying a team should dump Michael Jordan before it should dump Dennis Rodman?”

“Well, I don’t know for sure, I’m not sure exactly how valuable other defender-rebounders are, but regardless, it would be weird to base the whole argument on who happens to be the 2nd-best player. I mean, what if there were two Michael Jordan’s, would that make him the least valuable starter on an All-Time team?”

“Well OK, how common are primary scoring options that are in Jordan’s league value-wise?”

“There are none, I’m pretty sure he has the most value.”

“BALLPARK.”

“I dunno, there are probably between 0 and 2 in the league at any given time.”

“And how common are defender/rebounder/dirty workers that are in Rodman’s league value-wise?”

“There are none.”

“BALLPARK.”

“There are none. Ballpark.”

“So, basically, if a team had Michael Jordan and Dennis Rodman on it, and they could replace either with some random player ‘in the ballpark’ of the next-best player for their role, they should dump Jordan before they dump Rodman?”

“Maybe. Um. Yeah, probably.”

“And I assume that this holds for anyone other than Jordan?”

“I guess.”

“So say you’re head-to-head with me and we’re drafting NBA All-Time teams, you win the toss, you have first pick, who do you take?”

“I don’t know, good question.”

“No, it’s an easy question. The answer is: YOU TAKE RODMAN. You just said so.”

“Wait, I didn’t say that.”

“O.K., fine, I get the first pick. I’ll take Rodman. . . Because YOU JUST TOLD ME TO.”

“I don’t know, I’d have to think about it. It’s possible.”

Up to this point, I confess, I’ve had to reconstruct the conversation to some extent, but these last two lines are about as close to verbatim as my memory ever gets:

“So there you go, Dennis Rodman is the single most valuable player in NBA History. There’s your argument.”

“Dude, I’m not going to make that argument. I’d be crucified. Maybe, like, in the last post. When anyone still reading has already made up their mind about me.”

And that’s it. Simple enough, at first, but I’ve thought about this question a lot between last summer and last night, and it still confounds me: Could being the best “3rd-best” player in NBA history actually make Rodman the best player in NBA history? For starters, what does “3rd-best” even mean? The argument is a semantic nightmare in its own right, and an even worse nightmare to formalize well enough to investigate. So before going there, let’s take a step back:

# The Case Against Dennis Rodman:

At the time of that conversation, I hadn’t yet done my league-wide study of differential statistics, so I didn’t know that Rodman would end up having the highest I could find. In fact, I pretty much assumed (as common sense would dictate) that most star-caliber #1 players with a sufficient sample size would rank higher: after all, they have a greater number of responsibilities, they handle the ball more often, and should thus have many more opportunities for their reciprocal advantage over other players to accumulate. Similarly, if a featured player can’t play—potentially the centerpiece of his team, with an entire offense designed around him and a roster built to supplement him—you would think it would leave a gaping hole (at least in the short-run) that would be reflected heavily in his differentials. Thus, I assumed that Rodman probably wouldn’t even “stat out” as the best Power Forward in the field, making this argument even harder to sell. But as the results revealed, it turns out feature players are replaceable after all, and Rodman does just fine on his own. However, there are a couple of caveats to this outcome:

First, without much larger sample sizes, I wouldn’t say that game-by-game win differentials are precise enough to settle disputes between players of similar value. For example, the standard deviation for Rodman’s 22% adjusted win differential is still 5% (putting him less than a full standard deviation above some of the competition). This is fine for concluding that he was extremely valuable, but it certainly isn’t extreme enough to outright prove the seemingly farfetched proposition that he was actually the most valuable player overall. The more unlikely you believe that proposition to be, the less you should find this evidence compelling—this is a completely rational application of Bayes’ Theorem—and I’m sure most of you, ex ante, find the proposition very very unlikely. Thus, to make any kind of argument for Rodman’s superiority that anyone but the biggest Rodman devotees would find compelling, we clearly need more than win differentials.

Second, it really is a shame that a number of the very best players didn’t qualify for the study—particularly the ultimate Big Three: Michael Jordan, Magic Johnson, and Larry Bird (who, in maybe my favorite stat ever, never had a losing month in his entire career). As these three are generally considered to be in a league of their own, I got the idea: if we treated them as one player, would their combined sample be big enough to make an adequate comparison? Well, I had to make a slight exception to my standard filters to allow Magic Johnson’s 1987 season into the mix, but here are the results:

Adjusted Win percentage differential is Rodman’s most dominant value stat, and here, finally, Herr Bjordson edges him. Plus this may not fully represent these players’ true strength: the two qualifying Jordan seasons are from his abrupt return in 1994 and his first year with the Wizards in 2001, and both of Bird’s qualifying seasons are from the last two of his career, when his play may have been hampered by a chronic back injury. Of course, just about any more-conventional player valuation system would rank these players above (or way above) Rodman, and even my own proprietary direct “all-in-one” metric puts these three in their own tier with a reasonable amount of daylight between them and the next pack (which includes Rodman) below. So despite having a stronger starting position in this race than I would have originally imagined, I think it’s fair to say that Rodman is still starting with a considerable disadvantage.

So let’s assume that at least a few players offer more direct value than Dennis Rodman. But building a Champion involves more than putting together a bunch of valuable players: to maximize your chances of success, you must efficiently allocate a variety of scare resources, to obtain as much realized value as possible, through a massively complicated set of internal and external constraints.

For example, league rules may affect how much money you can spend and how many players you can carry on your roster. Game rules dictate that you only have so many players on the floor at any given time, and thus only have so many minutes to distribute. Strategic realities require that certain roles and responsibilities be filled: normally, this means you must have a balance of talented players who play different positions—but more broadly, if you hope to be successful, your team must have the ability to score, to defend, to rebound, to run set plays, to make smart tactical maneuvers, and to do whatever else that goes into winning. All of these little things that your team has to do can also be thought of as a limited resource: in the course of a game, you have a certain number of things to be done, such as taking shots, going after loose balls, setting up a screens, contesting rebounds, etc. Maybe there are 500 of these things, maybe 1000, who knows, but there are only so many to go around—and just as with any other scarce resource, the better teams will be the ones that squeeze the most value out of each opportunity.

Obviously, some players are better at some things than others, and may contribute more in some areas than others—but there will always be trade-offs. No matter how good you are, you will always occupy a slot on the roster and a spot on the floor, every shot you take or every rebound you get means that someone else can’t take that shot or get that rebound, and every dollar your team spends on you is a dollar they can’t spend on someone else. Thus, there are two sides to a player’s contribution: how much surplus value he provides, and how much of his team’s scarce resources he consumes.

The key is this: While most of the direct value a player provides is observable, either directly (through box scores, efficiency ratings, etc.) or indirectly (Adjusted +/-, Win Differentials), many of his costs are concealed.

## Visible v. Invisible Effects

Two players may provide seemingly identical value, but at different costs. In very limited contexts this can be extremely clear: thought it took a while to catch on, by now all basketball analysts realize that scoring 25 points per game on 20 shots is better than scoring 30 points a game on 40 shots. But in broader contexts, it can be much trickier. For example, with a large enough sample size, Win Differentials should catch almost anything: everything good that a player does will increase his team’s chances of winning when he’s on the floor, and everything bad that he does will decrease his team’s chances of losing when he’s not. Shooting efficiency, defense, average minutes played, psychological impact, hustle, toughness, intimidation—no matter how abstract the skill, it should still be reflected in the aggregate.

No matter how hard the particular skill (or weakness) is to identify or understand, if its consequences would eventually impact a player’s win differentials, (for these purposes) its effects are visible.

But there are other sources of value (or lack thereof) which won’t impact a player’s win differentials—these I will call “invisible.” Some are obvious, and some are more subtle:

### Example 1: Money

“Return on Investment” is the prototypical example of invisible value, particularly in a salary-cap environment, where every dollar you spend on one player is a dollar you can’t spend on another. No matter how good a player is, if you give up more to get him than you get from him in return, your team suffers. Similarly, if you can sign a player for much less than he is worth, he may help your team more than other (or even better) players who would cost more money.

This value is generally “invisible” because the benefit that the player provides will only be realized when he plays, but the cost (in terms of limiting salary resources) will affect his team whether he is in the lineup or not. And Dennis Rodman was basically always underpaid (likely because the value of his unique skillset wasn’t fully appreciated at the time):

Note: For a fair comparison, this graph (and the similar one below) includes only the 8 qualifying Shaq seasons from before he began to decline.

Aside from the obvious, there are actually a couple of interesting things going on in this graph that I’ll return to later. But I don’t really consider this a primary candidate for the “invisible value” that Rodman would need to jump ahead of Jordan, primarily for two reasons:

First, return on investment isn’t quite as important in the NBA as it is in some other sports: For example, in the NFL, with 1) so many players on each team, 2) a relatively hard salary cap (when it’s in place, anyway), and 3) no maximum player salaries, ROI is perhaps the single most important consideration for the vast majority of personnel decisions.  For this reason, great NFL teams can be built on the backs of many underpaid good-but-not-great players (see my extended discussion of fiscal strategy in major sports here).

Second, as a subjective matter, when we judge a player’s quality, we don’t typically consider factors that are external to their actual athletic attributes. For example, a great NFL quarterback could objectively hurt his team if he is paid too much, but we still consider him great. When we ask “who’s the best point guard in the NBA,” we don’t say, “IDK, how much more does Chris Paul get paid than Jason Kidd?” Note this is basically a social preference: It’s conceivable that in some economically-obsessed culture, this sort of thing really would be the primary metric for player evaluation. But personally, and for the purposes of my argument, I prefer our more traditional values on this one.

### Example 2: Position

In the “perfect timing” department, a commenter “Siddy Hall” recently raised a hypothetical very similar to my friend’s:

You get 8 people in a room, all posing as GM’s. We’re allowed to select 5 players each from the entire history of the NBA. Then we’ll have a tournament. At PF, I would grab Rodman. And I’m confident that I’d win because he’s on my team. He’d dominate the glass and harass and shutdown a superstar. I think he’s the finest PF to ever play the game.

Of course, you need to surround him with some scorers, but when is that ever a problem?

The commenter only explicitly goes so far as to say that Rodman would be the most valuable power forward. Yet he says he is “confident” that he would win, with the only caveat being that his team gets other scorers (which is a certainty). So, he thinks Rodman is the best PF by a wide enough margin that his team would be a favorite against the team that got Michael Jordan. Let me play the role of my friend above: whether he means to or not, he’s basically saying that Rodman is more valuable than Jordan.

In this example, “position” is the scarce resource. Just as a player can be valuable for the amount of money the team must spend on him, he can also be valuable for his position. But this value can be visible, invisible, or both.

This is probably easiest to illustrate in the NFL, where positions and responsibilities are extremely rigid. An example I used in response to the commenter is that an NFL kicker who could get you 2 extra wins per season could be incredibly valuable. These two extra wins obviously have visible value: By definition, this is a player for whom we would expect to observe a 2 game per season win differential. But there’s another, very important way in which this player’s value would be much greater. As I said in response to the commenter, a +2 kicker could even be more valuable than a +4 quarterback.

In order to play the 2 win kicker, the only cost is your kicker slot, which could probably only get you a fraction of a win even if you had one of the best in the league on your team (relevant background note: kickers normally don’t contribute much, particularly since bad kickers likely influence their teams to make better tactical decisions, and vice-versa). But to play a 4-win quarterback, the cost is your quarterback slot. While the average QB and the average kicker are both worth approximately 0 games, good quarterbacks are often worth much more, and good kickers are worth very little.

Put most simply, because there are no other +2 kickers, that kicker could get 2 wins for virtually ANY team. The +4 QB would only provide 2 wins for teams who would be unable to acquire a +2 quarterback by other means. Or you can think about it conversely: Team A signs the kicker, and Team B signs the QB. For the moment, Team B might appear better, but the most value they will ever be able to get out of their QB/Kicker tandem is +4 games plus epsilon. Team A, on the other hand, can get more value out of their QB/kicker combo than Team B simply by signing any QB worth +2 or greater, who are relatively common.

Why does this matter? Well, in professional sports, we care about one thing more than any other: championships. Teams that win championships do so by having the best roster with the most value. Players like our special kicker provide unique avenues to surplus value that even great other players can’t.

To generalize a bit, you could say that value vs. a replacement player is generally visible, as it will be represented in win differentials no matter who you play for. But a player’s value relative to the entire distribution of players at his position can lead to substantial invisible benefits, as it can substantially improve his team’s ability to build a championship contender.

## Formalizing “I-Factor”

Unfortunately, in basketball, such distinctions are much more nebulous. Sure, there are “positions,” but the spot where you line up on the floor is very different from the role you play. E.g., your primary scoring responsibilities can come from any position. And even then “roles” are dynamic and loosely defined (if at all)—some roles that are crucial to certain teams don’t even exist on others. Plus, teams win in different ways: you can do it by having 5 options on offense with 5 guys that can do everything (OK, this doesn’t happen very often, but the Pistons did it in 03-04), or you can be highly specialized and try to exploit the comparative advantages between your players (this seems to be the more popular model of late).

Rodman was a specialist. He played on teams that, for the most part, didn’t ask him to do more than what he was best at—and that probably helped him fully leverage his talents. But the truly amazing part is how much of a consistent impact he could have, on such a variety of different teams, and with seemingly so few responsibilities.

So let’s posit a particular type of invisible value and call it “I-Factor,” with the following elements:

1. It improves your team’s chances of building a championship contender.
2. It wouldn’t be reflected in your game-to-game win differential.
3. It stems from some athletic or competitive skill or attribute.

In the dialogue above, I suggested that Rodman had an inordinate positive impact for a “3rd-best” player, and my friend suggested (insisted really) that this alone should vault him above great but more ordinary “1st-best” players, even if they had significantly more observable impact. Putting these two statements together, we have an examinable hypothesis: That Dennis Rodman’s value relative to his role constituted a very large “I-Factor.”

# Evaluating the Hypothesis:

Because the value we’re looking for is (by definition) invisible, its existence is ridiculously hard—if not impossible—to prove empirically (which is why this argument is the dessert instead of the main course of this series).

However, there could be certain signs and indicators we can look for that would make the proposition more likely: specifically, things that would seem unusual or unlikely if the hypothesis were false, but which could be explainable either as causes or effects of the hypothesis being true.

Since the hypothesis posits both an effect (very large I-Factor), and a cause (unusually high value for his role), we should primarily be on the lookout for two things: 1) any interesting or unusual patterns that could be explainable as a consequence of Rodman having a large I-Factor, and 2) any interesting or unusual anomalies that could help indicate that Rodman had an excessive amount of value for his role.

## Evidence of Effect:

To lighten the mood a bit, let’s start this section off with a riddle:

Q. What do you get for the team that has everything?

A. Dennis Rodman.

Our hypothetical Rodman I-Factor is much like that of our hypothetical super-kicker in the NFL example above. The reason that kicker was even more valuable than the 2 wins per season he could get you is that he could get those 2 wins for anyone. Normally, if you have a bunch of good players and you add more good players, the whole is less than the sum of its parts. In the sports analytics community, this is generally referred to as “diminishing returns.” An extremely simple example goes like this: Having a great quarterback on your team is great. Having a second great quarterback is maybe mildly convenient. Having a third great quarterback is a complete waste of space. But if you’re the only kicker in the league who is worth anywhere near 2 wins, your returns will basically never be diminished. In basketball, roles and responsibilities aren’t nearly as wed to positions as they are in football, but the principle is the same. There is only one ball, and there are only so many responsibilities: If the source of one player’s value overlaps the source of another’s, they will both have less impact. Thus, if Rodman’s hypothetical I-Factor were real, one thing we might expect to find is a similar lack of diminishing returns—in other words, an unusual degree of consistency.

And indeed, Rodman’s impact was remarkably consistent. His adjusted win differential held at between 17% and 23% for 4 different teams, all of whom were championship contenders to one extent or another. Obviously the Bulls and Pistons each won multiple championships. The two years that Rodman spent with the pre-Tim-Duncan-era Spurs, they won 55 and 62 games respectively (the latter led the league that season, though the Spurs were eliminated by eventual-champion Houston in the Western Conference Finals). In 1999, Rodman spent roughly half of the strike-shortened season on the Lakers; in that time the Lakers went 17-6, matching San Antonio’s league-leading winning percentage. But, in a move that was somewhat controversial with the Lakers players at the time, Rodman was released before the playoffs began, and the Lakers fell in the 2nd round—to the eventual-champion Spurs.

But consistency should only be evidence of invisible value if it is unusual—that is, if it exists where we wouldn’t expect it to. So let’s look at Rodman’s consistency from a couple of different angles:

### Angle 1: Money (again)

The following graph is similar to my ROI graph above, except instead of mapping the player’s salary to his win differential, I’m mapping the rest of the team’s salary to his win differential:

Note: Though obviously it’s only one data point and doesn’t mean anything, I find it amusing that the one time Shaq played for a team that had a full salary-cap’s worth of players without him, his win differential dropped to the floor.

So, basically, whether Rodman’s teams were broke or flush, his impact remained fairly constant. This is consistent with unusually low diminishing returns.

### Angle 2: Position (again)

A potential objection I’ve actually heard a couple of times is that perhaps Rodman was able to have the impact he did because the circumstances he played in were particularly well-suited to never duplicating his skill-set: E.g., both Detroit and Chicago lacked dominant big men. Indeed, it’s plausible that part of his value came from providing the defense/rebounding of a dominant center, maximally leveraging his skill-set, and freeing up his teams to go with smaller, more versatile, and more offense-minded players at other positions (which could help explain why he had a greater impact on offensive efficiency than on defensive efficiency). However, all of this value would be visible. Moreover, the assumption that Rodman only played in these situations is false. Not only did Rodman play on very different teams with very different playing styles, he actually played on teams with every possible combination of featured players (or “1st and 2nd-best” players, if you prefer):

As we saw above, Rodman’s impact on all 4 teams was roughly the same. This too is consistent with an unusual lack of diminishing returns.

## Evidence of Cause:

As I’ve said earlier, “role” can be very hard to define in the NBA relative to other sports. But to find meaningful evidence that Rodman provided an inordinate amount of value for his role, we don’t necessarily need to solve this intractable problem: we can instead look for “partial” or “imperfect” proxies. If some plausibly related proxy were to provide an unusual enough result, its actual relationship to the posited scenario could be self-reinforced—that is, the most likely explanation for the extremely unlikely result could be that it IS related to our hypothesis AND that our hypothesis is true.

So one scarce resource that is plausibly related to role is “usage.” Usage Rate is the percentage of team possessions that a player “uses” by taking a shot or committing a turnover. Shooters obviously have higher usage rates than defender/rebounders, and usage generally has little correlation with impact. But let’s take a look at a scatter-plot of qualifying players from my initial differential study (limited to just those who have positive raw win differentials):

The red dot is obviously Dennis Rodman. Bonus points to anyone who said “Holy Crap” in their heads when they saw this graph: Rodman has both the highest win differential and the lowest Usage Rate, once again taking up residence in Outlier Land.

Let’s look at it another way: Treating possessions as the scarce resource, we might be interested in how much win differential we get for every possession that a player uses:

Let me say this in case any of you forgot to think it this time:

“Holy Crap!”

Yes, the red dot is Dennis Rodman. Oh, if you didn’t see it, don’t follow the blue line, it won’t help.

This chart isn’t doctored, manipulated, or tailored in any way to produce that result, and it includes all qualifying players with positive win differentials. If you’re interested, the Standard Deviation on the non-Rodman players in the pool is .19. Yes, that’s right, Dennis Rodman is nearly 4.5 standard deviations above the NEXT HIGHEST player. Hopefully, you see the picture of what could be going on here emerging:  If value per possession is any kind of proxy (even an imperfect one) for value relative to role, it goes a long way toward explaining how Rodman was able to have such incredible impacts on so many teams with so many different characteristics.

The irony here is that the very aspect of Rodman’s game that frequently causes people to discount his value (“oh, he only does one thing”) may be exactly the quality that makes him a strong contender for first pick on the all-time NBA playground.

# Conclusions:

Though the evidence is entirely circumstantial, I find the hypothesis very plausible, which in itself should be shocking. While I may not be ready to conclude that, yes, in fact, Rodman would actually be a more valuable asset to a potential championship contender than Michael freaking Jordan, I don’t think the opposite view is any stronger: That is, when you call that position crazy, conjectural, speculative, or naïve—as some of you inevitably will—I am fairly confident that, in light of the evidence, the default position is really no less so.

In fact, even if this hypothesis isn’t exactly true, I don’t think the next-most-likely explanation is that it’s completely false, and these outlandish outcomes were just some freakishly bizarre coincidence—it would be more likely that there is some alternate explanation that may be even more meaningful. Indeed, on some level, some of the freakish statistical results associated with Rodman are so extreme that it actually makes me doubt that the best explanation could actually stem from his athletic abilities. That is, he’s just a guy, how could he be so unusually good in such an unusual way? Maybe it actually IS more likely that the groupthink mentality of NBA coaches and execs accidentally DID leave a giant exploitable loophole in conventional NBA strategy; a loophole that Rodman fortuitously stumbled upon by having such a strong aversion to doing any of the things that he wasn’t the best at. If that is the case, however, the implications of this series could be even more severe than I intended.

# Series Afterword:

Despite having spent time in law school, I’m not a lawyer. Indeed, one of the reasons I chose not to be one is because I get icky at the thought of picking sides first, and building arguments later.

In this case, I had strong intuitions about Rodman based on a variety of beliefs I had been developing about basketball value, combined with a number of seemingly-related statistical anomalies in Rodman’s record. Though I am naturally happy that my research has backed up those intuitions—even beyond my wildest expectations—I felt prepared for it to go the other way. But, of course, no matter how hard we try, we are all susceptible to bias.

Moreover, inevitably, certain non-material choices (style, structure, editorial, etc.) have to be made which emphasize the side of the argument that you are trying to defend. This too makes me slightly queasy, though I recognize it as a necessary evil in the discipline of rhetoric. My point is this: though I am definitely presenting a “case,” and it often appears one-sided, I have tried to conduct my research as neutrally as possible. If there is any area where you think I’ve failed in this regard, please don’t hesitate to let me know. I am willing to correct myself, beef up my research, or present compelling opposing arguments alongside my own; and though I’ve published this series in blog form, I consider this Case to be an ongoing project.

If you have any other questions, suggestions, or concerns, please bring them up in the comments (preferably) or email me and I will do my best to address them.

Finally, I would like to thank Nate Meyvis, Leo Wolpert, Brandon Wall, James Stuart, Dana Powers, and Aaron Nathan for the invaluable help they provided me by analyzing, criticizing, and/or ridiculing my ideas throughout this process. I’d also like to thank Jeff Bennett for putting me on this path, Scott Carder for helping me stay sane, and of course my wife Emilia for her constant encouragement.

## The Case for Dennis Rodman, Part 4/4(a): All-Hall?

First of all, congrats to Dennis for his well-deserved selection as a 2011 Hall of Fame inductee—of course, I take full credit.  But seriously, when the finalists were announced, I immediately suspected that he would make the cut, mostly for two reasons:

1. Making the finalists this year after failing to make the semi-finalists last year made it more likely that last year’s snub really was more about eligibility concerns than general antipathy or lack of respect toward him as a player.
2. The list of co-finalists was very favorable.  First, Reggie Miller not making the list was a boon, as he could have taken the “best player” spot, and Rodman would have lacked the goodwill to make it as one of the “overdue”—without Reggie, Rodman was clearly the most accomplished name in the field.  Second, Chris Mullen being available to take the “overdue” spot was the proverbial “spoonful of sugar” that allowed the bad medicine of Rodman’s selection go down.

Congrats also to Artis Gilmore and Arvydas Sabonis.  In my historical research, Gilmore’s name has repeatedly popped up as an excellent player, both by conventional measures (11-time All-Star, 1xABA Champion, 1xABA MVP, led league in FG% 7 times), and advanced statistical ones (NBA career leader in True Shooting %, ABA career leader in Win Shares and Win Shares/48, and a great all-around rebounder).  It was actually only a few months ago that I first discovered—to my shock—that he was NOT in the Hall [Note to self: cancel plans for “The Case for Artis Gilmore”].  Sabonis was an excellent international player with a 20+ year career that included leading the U.S.S.R. to an Olympic gold medal and winning 8 European POY awards.  I remember following him closely when he finally came to the NBA, and during his too-brief stint, he was one of the great per-minute contributors in the league (though obviously I’m not a fan of the stat, his PER over his first 5 season—which were from age 31-35—was 21.7, which would place him around 30th in NBA history).  Though his sample size was too small to qualify for my study, his adjusted win percentage differential over his NBA career was a very respectable 9.95%, despite only averaging 24 minutes per game.

I was hesitant to publish Part 4 of this series before knowing whether Rodman made the Hall or not, as obviously the results shape the appropriate scope for my final arguments. So by necessity, this section has changed dramatically from what I initially intended.  But I am glad I waited, as this gives me the opportunity to push the envelope of the analysis a little bit:  Rather than simply wrapping up the argument for Rodman’s Hall-of-Fame candidacy, I’m going to consider some more ambitious ideas.  Specifically, I will articulate two plausible arguments that Rodman may have been even more valuable than my analysis so far has suggested.  The first of these is below, and the second—which is the most ambitious, and possibly the most shocking—will be published Monday morning in the final post of this series.

# Introduction

I am aware that I’ve picked up a few readers since joining “the world’s finest quantitative analysts of basketball” in ESPN’s TrueHoop Stat Geek Smackdown.  If you’re new, the main things you need to know about this series are that it’s 1) extremely long (sprawling over 13 sections in 4 parts, plus a Graph of the Day), 2) ridiculously (almost comically) detailed, and 3) only partly about Dennis Rodman.  It’s also a convenient vehicle for me to present some of my original research and criticism about basketball analysis.

Obviously, the series includes a lot of superficially complicated statistics, though if you’re willing to plow through it all, I try to highlight the upshots as much as possible.  But there is a lot going on, so to help new and old readers alike, I have a newly-updated “Rodman Series Guide,” which includes a broken down list of articles, a sampling of some of the most important graphs and visuals, and as of now, a giant new table summarizing the entire series by post, including the main points on both sides of the analysis.  It’s too long to embed here, but it looks kind of like this:

As I’ve said repeatedly, this blog isn’t just called “Skeptical” Sports because the name was available: When it comes to sports analysis—from the mundane to the cutting edge—I’m a skeptic.  People make interesting observations, perform detailed research, and make largely compelling arguments—which is all valuable.  The problems begin when then they start believing too strongly in their results: they defend and “develop” their ideas and positions with an air of certainty far beyond what is objectively, empirically, or logically justified.

With that said, and being completely honest, I think The Case For Dennis Rodman is practically overkill.  As a skeptic, I try to keep my ideas in their proper context: There are plausible hypotheses, speculative ideas, interim explanations requiring additional investigation, claims supported by varying degrees of analytical research, propositions that have been confirmed by multiple independent approaches, and the things I believe so thoroughly that I’m willing to write 13-part series’ to prove them.  That Rodman was a great rebounder, that he was an extremely valuable player, even that he was easily Hall-of-Fame caliber—these propositions all fall into that latter category: they require a certain amount of thoughtful digging, but beyond that they practically prove themselves.

Yet, surely, there must be a whole realm of informed analysis to be done that is probative and compelling but which might fall short of the rigorous standards of “true knowledge.”  As a skeptic, there are very few things I would bet my life on, but as a gambler—even a skeptical one—there are a much greater number of things I would bet my money on.  So as my final act in this production, I’d like to present a couple of interesting arguments for Rodman’s greatness that are both a bit more extreme and a bit more speculative than those that have come before.  Fortunately, I don’t think it makes them any less important, or any less captivating:

Read the rest of this entry »

## The Case for Dennis Rodman, Part 3/4(d)—Endgame: Statistical Significance

The many histograms in sections (a)-(c) of Part 3 reflect fantastic p-values (probability that the outcome occurred by chance) for Dennis Rodman’s win percentage differentials relative to other players, but, technically, this doesn’t say anything about the p-values of each metric in itself.  What this means is, while we have confidently established that Rodman’s didn’t just get lucky in putting up better numbers than his peers, we haven’t yet established the extent to which his being one of the best players by this measure actually proves his value.  This is probably a minor distinction to all but my most nitpicky readers, but it is exactly one of those nagging “little insignificant details” that ends up being a key to the entire mystery.

# The Technical Part (Feel Free to Skip)

The challenge here is this: My preferred method for rating the usefulness and reliability of various statistics is to see how accurate they are at predicting win differentials.  But, now, the statistic I would like to test actually is win differential.  The problem, of course, is that a player’s win differential is always going to be exactly identical to his win differential. If you’re familiar with the halting problem or Gödel’s incompleteness theorem, you probably know that this probably isn’t directly solvable: that is, I probably can’t design a metric for evaluating metrics that is capable of evaluating itself.

To work around this, our first step must be to independently assess the reliability of win predictions that are based on our inputs.  As in sections (b) and (c), we should be able to do this on a team-by-team basis and adapt the results for player-by-player use.  Specifically, what we need to know is the error distribution for the outcome-predicting equation—but this raises its own problems.

Normally, to get an error distribution of a predictive model, you just run the model a bunch of times and then measure the predicted results versus the actual results (calculating your average error, standard deviation, correlation, whatever).  But, because my regression was to individual games, the error distribution gets “black-boxed” into the single-game win probability.

[A brief tangent: “Black box” is a term I use to refer to situations where the variance of your input elements gets sucked into the win percentage of a single outcome.  E.g., in the NFL, when a coach must decide whether to punt or go for it on 4th down late in a game, his decision one way or the other may be described as “cautious” or “risky” or “gambling” or “conservative.”  But these descriptions are utterly vapid: with respect to winning, there is no such thing as a play that is more or less “risky” than any other—there are only plays that improve your chances of winning and plays that hurt them.  One play may seem like a bigger “gamble,” because there is a larger immediate disparity between its possible outcomes, but a 60% chance of winning is a 60% chance of winning.  Whether your chances comes from superficially “risky” plays or superficially “cautious” ones, outside the “black box” of the game, they are equally volatile.]

For our purposes, what this means is that we need to choose something else to predict: specifically, something that will have an accurate and measurable error distribution.  Thus, instead of using data from 81 games to predict the probability of winning one game, I decided to use data from 41 season-games to predict a team’s winning percentage in its other 41 games.

To do this, I split every team season since 1986 in half randomly, 10 times each, leading to a dataset of 6000ish randomly-generated half-season pairs.  I then ran a logistic regression from each half to the other, using team winning percentage and team margin of victory as the input variables and games won as the output variable.  I then measured the distribution of those outcomes, which gives us a baseline standard deviation for our predicted wins metric for a 41 game sample.

Next, as I discussed briefly in section (b), we can adapt the distribution to other sample sizes, so long as everything is distributed normally (which, at every point in the way so far, it has been).  This is a feature of the normal distribution: it is easy to predict the error distribution of larger and smaller datasets—your standard deviation will be directly proportional to the square-root of the ratio of the new sample size to the original sample size.

Since I measured the original standard deviations in games, I converted each player’s “Qualifying Minutes” into “Qualifying Games” by dividing by 36.  So the sample-size-adjusted standard deviation is calculated like this:

=[41GmStDev]*SQRT([PlQualGames]/41)

Since the metrics we’re testing are all in percentages, we then divide the new standard deviation by the size of the sample, like so:

=([41GmStDev]*SQRT([PlQualGames]/41))/[PlQualGames]

This gives us a standard deviation for actual vs. predicted winning percentages for any sample size.  Whew!

# The Good, Better, and Best Part

The good news is: now that we can generate standard deviations for each player’s win differentials, this allows us to calculate p-values for each metric, which allows us to finally address the big questions head on: How likely is it that this player’s performance was due to chance?  Or, put another way: How much evidence is there that this player had a significant impact on winning?

The better news is: since our standard deviations are adjusted for sample size, we can greatly increase the size of the comparison pool, because players with smaller samples are “punished” accordingly.  Thus, I dropped the 3-season requirement and the total minutes requirement entirely.  The only remaining filters are that the player missed at least 15 games for each season in which a differential is computed, and that the player averaged at least 15 minutes per game played in those seasons.  The new dataset now includes 1539 players.

Normally I don’t weight individual qualifying seasons when computing career differentials for qualifying players, because the weights are an evidentiary matter rather than an impact matter: when it comes to estimating a player’s impact, conceptually I think a player’s effect on team performance should be averaged across circumstances equally.  But this comparison isn’t about whose stats indicate the most skill, but whose stats make for the best evidence of positive contribution.  Thus, I’ve weighted each season (by the smaller of games missed or played) before making the relevant calculations.

So without further ado, here are Dennis Rodman’s statistical significance scores for the 4 versions of Win % differential, as well as where he ranks against the other players in our comparison pool:

Note: I’ve posted a complete table of z scores and p values for all 1539 players on the site.  Note also that due to the weighting, some of the individual differential stats will be slightly different from their previous values.

You should be careful to understand the difference between this table of p-values and ranks vs. similar ones from earlier sections.  In those tables, the p-value was determined by Rodman’s relative position in the pool, so the p-value and rank basically represented the same thing.  In this case, the p-value is based on the expected error in the results.  Specifically, they are the answer to the question “If Dennis Rodman actually had zero impact, how likely would it be for him to have posted these differentials over a sample of this size?”  The “rank” is then where his answer ranks among the answers to the same question for the other 1538 players.  Depending on your favorite flavor of win differential, Rodman ranks anywhere from 1st to 8th.  His average rank among those is 3.5, which is 2nd only to Shaquille O’Neal (whose differentials are smaller but whose sample is much larger).

Of course, my preference is for the combined/adjusted stat.  So here is my final histogram:

Note: N=1539.

Now, to be completely clear, as I addressed in Part 3(a) and 2(b), so that I don’t get flamed (or stabbed, poisoned, shot, beaten, shot again, mutilated, drowned, and burned—metaphorically): Yes, actually I AM saying that, when it comes to empirical evidence based on win differentials, Rodman IS superior to Michael Jordan.  This doesn’t mean he was the better player: for that, we can speculate, watch the tape, or analyze other sources of statistical evidence all day long.  But for this source of information, in the final reckoning, win differentials provide more evidence of Dennis Rodman’s value than they do of Michael Jordan’s.

The best news is: That’s it.  This is game, set, and match.  If the 5 championships, the ridiculous rebounding stats, the deconstructed margin of victory, etc., aren’t enough to convince you, this should be:  Looking at Win% and MOV differentials over the past 25 years, when we examine which players have the strongest, most reliable evidence that they were substantial contributors to their teams’ ability to win more basketball games, Dennis Rodman is among the tiny handful of players at the very very top.

## The Case for Dennis Rodman, Part 3/4(c)—Beyond Margin of Victory

In the conventional wisdom, winning is probably overrated.  The problem ultimately boils down to information quality: You only get one win or loss per game, so in the short-run, great teams, mediocre team, or teams that just get lucky can all achieve the same results.  Margin of victory, on the other hand, has a whole range of possible outcomes that, while imperfectly descriptive of the bottom line, correlate strongly with team strength.  You can think about it like sample size: a team’s margin of victory over a handful of games gives you a lot more data to work with than their won-loss record.  Thus, particularly when the number of games you draw your data from is small, MOV tends to be more probative.

Long ago, the analytic community recognized this fact, and has moved en masse to MOV (and its ilk) as the main element in their predictive statistics.  John Hollinger, for example, uses margin exclusively in his team power ratings—completely ignoring winning percentage—and these ratings are subsequently used for his playoff prediction odds, etc.  Note, Hollinger’s model has a lot of baffling components, like heavily weighting a team’s performance in their last 10 games (or later in their last 25% of games), when there is no statistical evidence that L10 is any more predictive than first 10 (or any other 10).  But this particular choice is of particular interest, especially as it is indicative of an almost uniform tendency among analysts to substitute MOV-style stats for winning percentage entirely.

This is both logically and empirically a mistake.  As your sample size grows, winning percentage becomes more and more valuable.  The reason for this is simple:  Winning percentage is perfectly accurate—that is, it perfectly reflects what it is that we want to know—but has extremely high variance, while MOV is an imperfect proxy, whose usefulness stems primarily from its much lower variance.  As sample sizes increase, the variance for MOV decreases towards 0 (which happens relatively quickly), but the gap between what it measures and what we want to know will persist in perpetuity.  Thus, after a certain point, the “error” in MOV remains effectively constant, while the “error” in winning percentage continuously decreases.  To get a simple intuitive sense of this, imagine the extremes:  after 5 games, clearly you will have more faith in a team that has won 2 but has a MOV of +10 over a team that has won 3 but has a MOV of +1.  But now imagine 1000 games with the same MOV’s and winning percentages: one team has won 400 and the other has won 600.  If you had to place money on one of the two teams to win their next game, you would be a fool to favor the first.  But beyond the intuitive point, this is essentially an empirical matter: with sufficient data, we should be able to establish the relative importance of each for any given sample-size.

So for this post, I’ve employed the same method that I used in section (b) to create our MOV-> Win% formula (logistic regression for all 55,000+ team games since 1986), except this time I included both Win % and MOV (over the team’s other 81 games) as the predictive variables.  Here, first, are the coefficients and corresponding p-values (probability that the variable is not significant):

It is thus empirically incontrovertible that, even with an 81-game predictive sample, both MOV and Win% are statistically significant predictive factors.  Also, for those who don’t eat logistic regression outputs for breakfast, I should be perfectly clear what this means: It doesn’t just mean that both W% and MOV are good at predicting W%—this is trivially true—it means that, even when you have one, using the other as well will make your predictions substantially better.  To be specific, here is the formula that you would use to predict a team’s winning percentage based on these two variables:

$\large{PredictedWin\% = \dfrac{1}{1+e^{-(1.43wp+.081mv-.721)}}}$

Note: Again, e is euler’s number, or ~2.72.  wp is the variable for winning % over the other 81 games, and mv is the variable for Margin of Victory over the other 81 games.

And again, for your home-viewing enjoyment, here is the corresponding Excel formula:

=1/(1+EXP(-(1.43*[W%]+.081[MOV]-.721)))

Finally, in order to visualize the relative importance of each variable, we can look at their standardized coefficients (shown here with 95% confidence bars):

Note: Standardized coefficients, again, are basically a unit of measurement for comparing the importance of things that come in different shapes and sizes.

For an 81-game sample (which is about as large of a consistent sample as you can get in the NBA), Win% is about 60% as important as MOV when it comes to predicting outcomes.  At the risk of sounding redundant, I need to make this extremely clear again: this does NOT mean that Win% is 60% as good at predicting outcomes as margin of victory (actually, it’s more like 98% as good at that)—it means that, when making your ideal prediction, which incorporates both variables, Win % gets 60% as much weight as MOV (as an aside, I should also note that the importance of MOV drops virtually to zero when it comes to predicting playoff outcomes, largely—though not entirely—because of home court advantage).

This may not sound like much, but I think it’s a pretty significant result:  At the end of the day, this proves that there IS a skill to winning games independent of the rates at which you score and allow points.  This is a non-obvious outcome that is almost entirely dismissed by the analytical community.  If NBA games were a random walk based on possession-to-possession reciprocal advantages, this would not be the case at all.

Now, note that this is formally the same as the scenario discussed in section (b): We want to predict winning percentages, but using MOV alone leaves a certain amount of error.  What this regression proves is that this error can be reduced by incorporating win percentage into our predictions as well.  So consider this proof-positive that X-factors are predictively valuable.  Since the predictive power of Win% and MOV should be equivalent no matter their source, we can now use this regression to make more accurate predictions about each player’s true impact.

Adapting this equation for individual player use is simple enough, though slightly different from before:  Before entering the player’s Win% differential, we have to convert it into a raw win percentage, by adding .5.  So, for example, if a player’s W% differential were 21.6%, we would enter 71.6%.  Then, when a number comes out the other side, we can convert it back into a predicted differential by subtracting .5, etc.

Using this method, Rodman’s predicted win differential comes out to 14.8%.  Here is the new histogram:

Note: N is still 470.

This histogram is also weighted by the sample size for each player (meaning that a player with 100 games worth of qualifying minutes counts as 100 identical examples in a much larger dataset, etc.).  I did this to get the most accurate distribution numbers to compute P values (which, in this case, work much like a percentile) for individual players.  Here is a summary of the major factors for Dennis Rodman:

For comparison, I’ve also listed the percentage of eligible players that match the qualifying thresholds of my dataset (minus the games missed) who are in the Hall of Fame.  Specifically, that is, those players who retired in 2004 or earlier and who have at least 3 seasons since 1986 with at least 15 games played in which they averaged at least 15 minutes per game.  This gives us a list of 462 players, of which 23 are presently IN the Hall. The difference in average skill between that set of players and the differential set is minimal, and the reddish box on the histogram above surrounds the top 5% of predicted Win% differentials in our main data.

While we’re at it, let’s check in on the list of “select” players we first saw in section (a) and how they rank in this metric, as well as in some of the others I’ve discussed:

For fun, I’ve put average rank and rank of ranks (for raw W% diff, adjusted W% diff, MOV-based regression, raw W%/MOV-based regression, raw X-Factor, adjusted X-Factor, and adjusted W%/MOV-based regression) on the far right.  I’ve also uploaded the complete win differential table for all 470 players to the site, including all of the actual values for these metrics and more.  No matter which flavor of metric you prefer (and I believe the highlighted one to be the best), Rodman is solidly in Hall of Fame territory.

Finally, I’m not saying that the Hall of Fame does or must pick players based on their ability to contribute to their team’s winning percentages.  But if they did, and if these numbers were accurate, Rodman would deserve a position with room to spare.  Thus, naturally, one burning question remains: how much can we trust these numbers (and Dennis Rodman’s in particular)?  This is what I will address in section (d) tomorrow.

## The Case for Dennis Rodman, Part 3/4(b)—Rodman’s X-Factor

The sports analytical community has long used Margin of Victory or similar metrics as their core component for predicting future outcomes.  In situations with relatively small samples, it generally slightly outperforms win percentages, even when predicting win percentages.

There are several different methods for converting MOV into expected win-rates.  For this series, I took the 55,000+ regular-season team games played since 1986 and compared their outcomes to the team’s Margin of Victory over the other 81 games of the season.  I then ran this data through a logistic regression (a method for predicting things that come in percentages) with MOV as the predictor variable.  Here is the resulting formula:

$\large{PredictedWin\% = \dfrac{1}{1+e^{-(.127mv-.004}}}$

Note: e is euler’s number, or ~2.72.  mv is the variable for margin of victory.

This will return the probability between 0 and 1, corresponding to the odds of winning the predicted game.  If you want to try it out for yourself, the excel formula is:

1 / (1 + EXP(-(-0.0039+0.1272*[MOV])))

So, for example, if a team’s point differential (MOV) over 81 games is 3.78 points per game, their odds of winning their 82nd game would be 61.7%.

Of course, we can use this same formula to predict a player’s win% differential based on his MOV differential.  If, based on his MOV contribution alone, a player’s team would be expected to win 61.7% of the time, then his predicted win% differential is what his contribution would be above average, in this case 11.7% (this is one reason why, for comparison purposes, I prefer to use adjusted win differentials, as discussed in Part 3(a)).

As discussed in the part 2(b) of this series (“With or Without Worm”), Dennis Rodman’s MOV differential was 3.78 points, which was tops among players with at least a season’s worth of qualifying data, corresponding to the aforementioned win differential of 11.7%.  Yet this under-predicts his actual win percentage differential by 9.9%.  This could be the result of a miscalibrated prediction formula, but as you can see in the following histogram, the mean for win differential minus predicted win differential for our 470 qualifying player dataset is actually slightly below zero at –0.7%:

Rodman has the 2nd highest overall, which is even more crazy considering that he had one of the highest MOV’s (and the highest of anyone with anywhere close to his sample size) to begin with.  Note how much of an outlier he is in this scatterplot (red dot is Rodman):

I call this difference the “X-Factor.”  For my purposes, “X” stands for “unknown”:  That is, it is the amount of a player’s win differential that isn’t explained by the most common method for predicting win percentages.  For any particular player, it may represent an actual skill for winning above and beyond a player’s ability to contribute to his team’s margin of victory (in section (c), I will go about proving that such a skill exists), or it may simply be a result of normal variance.  But considering that Rodman’s sample size is significantly larger than the average in our dataset, the chances of it being “error” should be much smaller.  Consider the following:

Again, Rodman is a significant outlier:  no one with more than 2500 qualifying minutes breaks 7.5%.  Rodman’s combination of large sample with large Margin of Victory differential with large X-Factor is remarkable.  To visualize this, I’ve put together a 3-D scatter plot of all 3 variables:

It can be hard to see where a point stands in space in a 2-D image, but I’ve added a surface grid to try to help guide you: the red point on top of the red mountain is Dennis Rodman.

To get a useful measure of how extreme this is, we can approximate a sample-size adjustment by comparing the number of qualifying minutes for each player to the average for the dataset, and then adjusting the standard deviation for that player accordingly (proportional to the square root of the ratio, a method which I’ll discuss in more detail in section (d)).  After doing this, I can re-make the same histogram as above with the sample-adjusted numbers:

No man is an island.  Except, apparently, for Dennis Rodman.  Note that he is about 4 standard deviations above the mean (and observe how the normal distribution line has actually blended with the axis below his data point).

Naturally, of course, this raises the question:

Where does Rodman’s X-Factor come from?

Strictly speaking, what I’m calling “X-Factor” is just the prediction error of this model with respect to players.  Some of that error is random and some of it is systematic.  In section (c), I will prove that it’s not entirely random, though where it comes from for any individual player, I can only speculate.

Margin of Victory treats all contributions to a game’s point spread equally, whether they came at the tail end of a blowout, or in the final seconds of squeaker.  One thing that could contribute to a high X-factor is “clutch”ness.  A “clutch” shooter (like a Robert Horry), for example, might be an average or even slightly below-average player for most of the time he is on the floor, but an extremely valuable one near the end of games that could go either way.  The net effect from the non-close games would be small for both metrics, but the effect of winning close games would be much higher on Win% than MOV.  Of course, “clutch”ness doesn’t have to be limited to shooters:  e.g., if one of a particular player’s skill advantages over the competition is that he makes better tactical decisions near the end of close games (like knowing when to intentionally foul, etc.), that would reflect much more strongly in his W% than in his MOV.

Also, a player who contributes significantly whenever they are on the floor but is frequently taken out of non-close games as a precaution again fatigue or injury may have a Win % that accurately reflects his impact, but a significantly understated MOV.  E.g., in the Boston Celtics “Big 3” championship season, Kevin Garnett was rested constantly—a fact that probably killed his chances of being that season’s MVP—yet the Celtics won by far the most games in the league.  In this case, the player is “clutch” just by virtue of being on the floor more in clutch spots.

The converse possibility also exists:  A player could be “reverse clutch,” meaning that he plays worse when the game is NOT on the line.  This would ultimately have the same statistical effect as if he played better in crunch time.  And indeed, based on completely non-rigorous and anecdotal speculation, I think this is a possible factor in Rodman’s case.  During his time in Chicago, I definitely recall him doing a number of silly things in the 4th quarter of blowout games (like launching up ridiculous 3-pointers) when it didn’t matter—and in a game of small margins, these things add up.

Finally, though it cuts a small amount against the absurdity of Rodman’s rebounding statistics, I would be derelict as an analyst not to mention the possibility that Rodman may have played sub-optimally in non-close games in order to pad his rebounding numbers.  The net effect, of course, would be that his rebounding statistics could be slightly overstated, while his value (which is already quite prodigious) could be substantially understated.  To be completely honest, with his rebounding percentages and his X-Factor both being such extreme outliers, I have to think that at least some relationship existing between the two is likely.

If you’re emotionally attached to the freak-alien-rebounder hypothesis, this might seem to be a bad result for you.  But if you’re interested in Rodman’s true value to the teams he played for, you should understand that, if this theory is accurate, it could put Rodman’s true impact on winning into the stratosphere.  That is, this possibility gives no fuel to Rodman’s potential critics: the worst cases on either side of the spectrum are that Rodman was the sickest rebounder with a great impact on his teams, or that he was a great rebounder with the sickest impact.

In the next section, I will be examining the relative reliability and importance of Margin of Victory vs. Win % generally, across the entire league.  In my “endgame” analysis, this is the balance of factors that I will use.  But the league patterns do not necessarily apply in all situations:  In some cases, a player’s X-factor may be all luck, in some cases it may be all skill, and in most it is probably a mixture of both.  So, for example, if my speculation about Rodman’s X-Factor were true, my final analysis of Rodman’s value could be greatly understated.

## The Case for Dennis Rodman, Part 3/4(a)—Just Win, Baby (in Histograms)

First off, congratulations to Dennis for making the Hall of Fame finalist list for 2011. The circumstances seem favorable to his making it, and if I had to guess I’d say he probably will. While his under-appreciated status has been a useful vehicle for my analytical agenda, I certainly hope he will be voted in—though I might prefer it be with a copy of my series in the voters’ hands.

Second, I apologize for the delay in getting this section out. I’m reminded of the words of the always brilliant Detective Columbo:

I worry. I mean, little things bother me. I’m a worrier. I mean, little insignificant details – I lose my appetite. I can’t eat. My wife, she says to me, “You know, you can really be a pain.”

Of course, as Columbo understood, the “insignificant” details that nag at you are usually anything but. Since Part 3 of this series should be the last to include heavily-quantitative analysis—and because it is so important to understanding Rodman’s true value—I really tried to tie up all the loose ends (even those that might at first seem to be redundant or obvious).

As a result, what began as a simple observation grew into something painfully detailed and extremely long (even by my standards)—but well worth it. So, once again, I’ve decided to break it down into 4 sections—however, each of these will be relatively short, and I’ll be posting them back-to-back each morning from now through Saturday. Here is the Cliff’s Notes version:

1. Rodman had an observably great impact on his teams’ winning percentages.
2. This impact was much greater than his already great impact on Margin of Victory would have predicted.
3. Contrary to certain wisdom in the analytical community, Margin of Victory and Win% are both valuable indicators predictively, and combining Rodman’s differentials in both put him deep in Hall of Fame territory.
4. Rodman’s differentials are statistically significant at one of the highest levels in NBA history.

Now, on with the show:

# Introduction

One of the most common doubts I hear about Dennis Rodman’s value stems from the belief that his personal successes—5 NBA championships, freakish rebounding statistics, etc.—were probably largely a result of his having played for superior teams. For example, his prodigious rebounding may have been something he was “allowed” to do because he played for good offensive teams and (as the argument goes) had few other offensive responsibilities.

In it’s weaker form, I think this argument is plausible but irrelevant: Perhaps Rodman would not have been able to put up the numbers that he did if he were “required” to do more on offense. But the implication that this diminishes his value is absurd—it would be like saying that Cy Young wasn’t a particularly valuable baseball player because he couldn’t have put up such a great ERA if he were “required” to hit every night.

The stronger form, however, suggests that Rodman’s anomalous rebounding statistics probably weren’t due to any particularly anomalous talent or contribution, but were merely (or at least mostly) a byproduct of his fortunate circumstances.

If this were true, however, one of the following things would necessarily have to follow:

1. His rebounding must not have contributed much value to his teams, or
2. The situations he played in must have been uniquely favorable to leveraging value from a designated rebounder, or
3. The choice to use a designated rebounder on an offensively strong team must have been an extremely successful exploitative strategy.

The third, I technically cannot disprove: It is theoretically possible that Rodman’s refusal to take a lot of shots on offense unintentionally caused his teams to stumble upon an amazing exploitative strategy that no one had discovered before and that no-one has duplicated since (though, if that were the case, he still might deserve some credit for forcing their hands).

But 1 and 2 simply aren’t supported by the data: As I will show, Rodman had wildly positive impacts on 4 different teams that had little in common, except of course for being solid winners with Rodman in the lineup.

# Rodman’s Win % Differential

As I’ve discussed previously, a player’s differential statistics are simply the difference in their team’s performance in the games they played versus the games they missed. One very important differential stat we might be interested in is winning percentage.
To look at Rodman’s numbers in this area, I used exactly the same process that I described in Part 2(b) to look at his other differentials. However, for comparison purposes, I’ve greatly expanded the pool of players by dropping the qualifying minutes requirement from 3000 to 1000. This grows the pool from 164 players to 470.

Why expand? Honestly, because Rodman’s extreme win % differential allows it. I think the more stringent filters produce a list that is more reliable from top to bottom—but in this case, I am mostly interested in (literally) the top. There are some players on the list with barely 1/3 of a season’s worth of qualifying playing time to back up their numbers—which should produce extreme volatility—yet still no one is able to overtake Rodman.

Here is Rodman’s raw win differential, along with those of a number of select players (including a few whose styles are often compared to Rodman’s, some Hall of Famers, some future first-ballot Hall of Fame selections, and Rodman’s 2011 Hall of Fame co-finalists Chris Mullin and Maurice Cheeks):

I will put up a table of the entire list of 470 players—including win differentials and a number of other metrics that I will discuss throughout the rest of Part 3—along with section (c) on Friday.
Amazingly, this number may not even reflect Rodman’s true impact, because he generally played for extremely good teams, where it is not only harder to contribute, but where a given impact will have less of an effect on win percentage (for example, if your team normally wins 90% of its games, it is clearly impossible to have a win% differential above 10%). To account for this, I’ve also created “adjusted” win% differentials, which attempt to normalize a player’s percentage increase/decrease to what it would be on a .500 team.

This adjustment is done somewhat crudely, by measuring how far the player gets you toward 100% (for positive impacts) or toward 0% (for negative). E.g., if someone plays for a team that normally wins 70%, and they win 85% with him in the lineup, that is 50% of the way to 100%. Thus, as 50% of the way from 50% to 100% is 75%, that player’s adjusted differential is 25% (as opposed to their raw value of 15%).
A few notes about this method: While I prefer the adjusted numbers for this situation, they have their drawbacks. They are most accurate when dealing with consistently good or bad teams, over multiple seasons, and with bigger sample sizes. They are less accurate with smaller sample sizes, in individual seasons, and with uncertain team quality. This is because regression to the mean can become an interfering factor. When looking at individual seasons in a void, it is relatively easy to account for both effects, which I do for my league-wide win differential analysis. But when aggregating independent seasons that have a common related element—such as the same team or player—you basically have to pick your poison (of course, there may be some way to deal with this issue that I just don’t know or haven’t thought of yet). I will tend to use the adjusted numbers for this analysis, but though they are slightly more favorable to Rodman, either metric leads to the same bottom line. In any case, the tables I will be posting include both metrics (as well as other options).

Dennis Rodman’s adjusted numbers boost his win differential to 21.6%, widening the margin between him and 2nd place. I know I will be flamed if I don’t add that (just as I noted in part 2(b)) I am not claiming that Rodman was actually the best player in the last 25 years. This is a volatile statistic, and Rodman merely happening to have the best win differential among the group of 470 qualifying players does not mean he was actually the best player overall, or even that he was the best player in the group. That said, we should not dismiss the extremeness of the result either:

I will be using a number of (eerily similar) histograms through the rest of Part 3 as well. If you’re not familiar, histograms are one of the simplest and most useful graphical representations of single-variable data (yet, inexplicably, they aren’t built into Excel): each bar represents the number of data points of the designated value. If the variable is continuous (as it is in this case), each bar is basically a “container” that tells you how many data points fit in between the left and right values of the bar (technically it tells you the “density” of points near the center of the container, but those are effectively the same in most circumstances). Their main purpose is to eyeball how the variable is distributed—in this case, as you can see it is distributed normally.

The red line is an overlay of the normal distribution of the sample, which has a mean of –0.5% and standard deviation of 6.3%. This puts Rodman just over 3.5 standard deviations above the mean, a value that should occur about once in every 4000 instances—and he does this based on a standard deviation that is derived from a pool that includes the statistics of many players that have as little as 1/4th as much relevant data as he has.

Moreover, as I will discuss in section (b) tomorrow, his win % differential is not only extreme relative to the rest of the NBA, it is even extreme relative to himself—and this has important implications in its own right.

## The Case for Dennis Rodman, Part 2/4 (b)—With or Without Worm

I recently realized that if I don’t speed up my posting of this series, Rodman might actually be in the Hall of Fame before I’m done.  Therefore, I’m going to post this section now, and Part 3 (which will probably only be one post) in the next few days.

This blog is called “Skeptical” Sports Analysis for a reason: I’m generally wary of our ability to understand anything definitively, and I believe that most people who confidently claim to know a lot of things other than facts—whether in sports, academics, or life—are either lying, exaggerating, or wrong.  I don’t accept this as an a priori philosophical tenet (in college I was actually very resistant to the skeptics), but as an empirical conclusion based on many years of engaging and analyzing various people’s claims of knowledge.  As any of you who happen to know me will attest, if I have any talent on this earth, it is finding fault with such claims (even when they are my own).

Keeping that in mind—and keeping in mind that, unlike most sports commentators, I don’t offer broadly conclusive superlatives very often—I offer this broadly conclusive superlative:  Dennis Rodman was the greatest rebounder of all time. If there has been any loose end in the arguments I’ve made already, it is this: based on the evidence I’ve presented so far, Rodman’s otherworldly rebounding statistics could, theoretically, be a result of shenanigans.  That is, he could simply have been playing at the role of rebounder on his teams, ignoring all else and unnaturally inflating his rebounding stats, while only marginally (or even negatively) contributing to his team’s performance.  Thus, the final piece of this puzzle is showing that his rebounding actually helped his teams.  If that could be demonstrated, then even my perversely skeptical mind would be satisfied on the point—else there be no hope for knowledge.

This is where “The Case for Dennis Rodman Was a Great Rebounder” and “The Case for Dennis Rodman” join paths: Showing that Rodman got a lot of rebounds without also showing that this significantly improved his teams proves neither that he was a great player nor that he was a great rebounder.  Unfortunately, as I discussed in the last two sections, player value can be hard to measure, and the most common conventional and unconventional valuation methods are deeply flawed (not to mention unkind toward Rodman).  Thus, in this post and the next, I will take a different approach.

# Differential (Indirect) Statistics

For this analysis, I will not be looking at Dennis Rodman’s (or any other player’s) statistics directly at all.  Instead, I will be looking at his team’s statistics, comparing the games in which he played to the games that he missed. I used a similar (though simpler) method in my mildly popular Quantum Randy Moss post last fall, which Brian Burke dubbed WOWRM, or “With or Without Randy Moss.”  So, now I present that post’s homophonic cousin: WOWWorm, or “With or Without Worm.”

The main advantages to indirect statistics are that they are all-inclusive (everything good or bad that a player does is accounted for, whether it is reflected in the box score or not), empirical (what we do or don’t know about the importance of various factors doesn’t matter), and they can get you about as close as possible in this business to isolating actual cause and effect.  These features make the approach especially trenchant for general hypothesis-testing and broader studies of predictivity that include league-wide data.

The main disadvantage for individual player analysis, however, is that the samples are almost always too small to be conclusive (in my dream universe, every player would be forced to sit out half of their team’s regular-season games at random).  They are also subject to bias based on quality of the player’s team (it is harder to have a big impact on a good team), or based on the quality of their backup—though I think the latter effect is much smaller in the basketball than in football or baseball.  In the NBA, teams rotate in many different players and normally have a lot of different looks, so when a starter goes out, they’re rarely just replaced by one person—the whole roster (even the whole gameplan) may shift around to exploit the remaining talent.  This is one reason you almost never hear of an NBA bench player finally “getting his shot” because the player in front of them was injured—if someone has exploitable skills, they are probably going to get playing time regardless.  Fortunately, Dennis Rodman missed his fair share of games—aided by his proclivity for suspensions—and the five seasons in which he missed at least 15 games came on four different teams.

Note, for the past few years, more complete data has allowed people to look at minute-by-minute or play-by-play +/- in basketball (as has been done for some time in hockey).  This basically eliminates the sample size problem, though it introduces a number of potential rotational, strategic and role-based biases.  Nevertheless, it definitely makes for a myriad of exciting analytical possibilities.

# Margin of Victory

For structural reasons, I’m going to hold off on Rodman’s Win % differentials until my next post in this series.  In this post, however, I will look at everything else, starting with team point differential differential—a.k.a. “Margin of Victory”:

Note: Table is only the top 25 players in the dataset.

First, the nitty-gritty:  This data goes back to 1986, starting with all players who missed and played at least 15 games in a single season while averaging at least 20 minutes per game played.  The “qualifying games” in a season is the smaller of games played or games missed.  E.g., if someone played 62 games and missed 20, that counts as 20 qualifying games, the same as if someone played 20 games and missed 62.  Their “qualifying minutes” are then their average of minutes per game played multiplied by their total number of qualifying games.  For the sample, I set the bar at 3000 qualifying minutes, or roughly the equivalent of a full season for a typical starter (82 games * 36 minutes/game is 2952 minutes), which leaves 164 qualifying players.  I then calculated differentials for each team-season:  I.e., per-game averages were calculated separately for the set of games played and the set of games missed by each player from within a particular season, and each season’s “differentials” were created for each stat simply by subtracting the second from the first.  Finally, I averaged the per-season differentials for each qualifying season for each player.  This is necessarily different from how multiple-season per-game stats are usually calculated (which is just to sum up the stats from the various seasons and divide by total games).  As qualifying games may come from different teams and different circumstances, to isolate a player’s impact it is crucially important that (as much as possible) their presence or absence is the only variable that changes, which is not even remotely possible across multiple seasons.  In case anyone is interested, here is the complete table with all differential stats for all 164 qualified players.

I first ran the differentials for Dennis Rodman quite some time ago, so I knew his numbers were very good.  But when I set out to do the same thing for the entire league, I had no idea that Rodman would end up literally on top.  Here is a histogram of the MOV-differential distribution for all qualified players (rounded to the nearest .5):

Note: Red is Dennis Rodman (and Ron Artest).

3.8 points per game may not sound like much compared to league-leading scorers who score 30+, but that’s both the beauty of this method and the curse of conventional statistics:  When a player’s true impact is actually only a few points difference per night (max), you know that the vast majority of the “production” reflected in their score line doesn’t actually contribute to their team’s margin.

This deserves a little teasing out, as the implications can be non-obvious: If a player who scores 30 points per game is only actually contributing 1 or 2 points to his team’s average margin, that essentially means that at least 28 of those points are either 1) redundant or 2) offset by other deficiencies.  With such a low signal-to-noise ratio, you should be able to see how how it is that pervasive metrics like PER can be so unreliable: If a player only scores 10 points a night, but 4 of them are points his team couldn’t have scored otherwise, he could be contributing as much as Shaquille O’Neal.  Conversely, someone on the league leaderboard who scores 25 points per game could be gaining his team 2 or 3 points a night with his shooting, but then be giving it all back if he’s also good for a couple of unnecessary turnovers.

Professional basketball is a relatively low-variance sport, but winners are still determined by very small margins.  Last year’s championship Lakers team had an average margin of victory of just 4.7 points.  For the past 5 years, roughly three quarters of teams have had lower MOV’s than Dennis Rodman’s differential in his 5 qualifying seasons:

Now, I don’t want to suggest too much with this, but I would be derelict if I didn’t mention the many Hall of Fame-caliber players who qualified for this list below Rodman (my apologies if I missed anyone):

• Hakeem Olajuwon
• Scottie Pippen<
• Clyde Drexler
• Dominique Wilkins

HoF locks:

• Shaquille O’Nea
• Jason Kidd
• Allen Iverson
• Ray Allen

HoF possible:

• Yao Ming
• Pau Gasol
• Marcus Camby
• Carlos Boozer
• Alonzo Mourning

Not in HoF but probably should be:

• Toni Kukoc
• Chris Mullin
• Tim Hardaway
• Dikembe Mutumbo

The master list also likely includes many players that are NOT stars but who quietly contributed a lot more to their teams than people realize.  Add the fact that Rodman managed to post these differentials while playing mostly for extremely good, contending, teams (where it is harder to have a measurable impact), and was never ostensibly the lynchpin of his team’s strategy—as many players on this list certainly were—and it is really quite an amazing outcome.

Now, I do not mean to suggest that Rodman is actually the most valuable player to lace up sneakers in the past 25 years, or even that he was the most valuable player on this list: 1) It doesn’t prove that, and 2) I don’t think that.  Other more direct analysis that I’ve done typically places him “only” in the top 5% or so of starting players.  There is a lot of variance in differential statistics, and there are a lot of different stories and circumstances involved for each player. But, at the very least, this should be a wake-up call for those who ignore Rodman for his lack of scoring, and for those who dismiss him as “merely” a role-player.

# Where Does His Margin Come From?

As I have discussed previously, one of the main defenses of conventional statistics—particularly vis a vis their failures w/r/t Dennis Rodman—is that they don’t account for defense or “intangibles.”  As stated in the Wikipedia entry for PER:

Neither PER nor per-game statistics take into account such intangible elements as competitive drive, leadership, durability, conditioning, hustle, or WIM (wanting it more), largely because there is no real way to quantitatively measure these things.

This is true, for the most part—but not so much for Rodman.  He does very well with indirect statistics, which actually DO account for all of these things as part of the gestalt that goes into MOV or Win% differentials.  But these stats also give us a very detailed picture of where those differences likely come from.  Here is a table summarizing a number of Rodman’s differential statistics, both for his teams and their opponents.  The “reciprocal advantage” is the difference between his team’s differential and their opponent’s differential for the same statistic:

Note: Some of the reciprocals were calculated in this table, and others are taken from the dataset (like margin of victory).  In the latter case, they may not necessarily match up perfectly, but this is for a number of technical and mathematical reasons that have no significant bearing on the final outcomes.

Rodman’s Margin of Victory differential comes in part from his teams scoring more points on offense and in part from their allowing fewer points on defense.  Superficially, this may look like the majority of Rodman’s impact is coming on the defensive side (-2.4 vs. + 1.3), but that’s deceptive.  As you can find in the master table, Rodman also has a significant negative effect on “Pace”—or number of possessions per game—which basically applies equally to both teams.  This is almost certainly due to his large number of possession-extending offensive rebounds, especially as he was known (and sometimes criticized) for “kicking it out” and resetting the offense rather than trying to shoot it himself or draw a foul.  “Scoring opportunities” are total possessions plus offensive rebounds.  As you might expect intuitively, his teams generally had about the same number of these with or without him, because the possessions weren’t actually lost, they were only restarted.

As we can see from the reciprocal table, Rodman had a slightly positive effect on his teams scoring efficiency (points per opportunity), but also had a small positive (though nearly negligible) effect on his opponents’.  Thus, combining the effect his rebounding had on number of scoring opportunities with any other effects he had on each side’s scoring efficiency, we can get a fairly accurate anatomy of his overall margin.  In case that confused you, here it is broken down step-by-step:

So, roughly speaking, his 3.7ish margin of victory breaks down to roughly 2.8 points from effect on offensive and defensive scoring opportunities and .9 points from the actual value of those opportunities—or, visually:

Furthermore, at least part of that extra offensive efficiency likely stems from the fact that a larger proportion of those scoring opportunities began as offensive rebounds, and post-offensive-rebound “possessions” are typically worth slightly more than normal (though this may actually be less true with Rodman due to the “kicking”).  Otherwise, the exact source of the efficiency differences is much more uncertain, especially as the smaller margins in the other statistics are that much more unreliable because of the sample-size issues inherent in this method.

The next-strongest reciprocal effects on the list above appear to be personal fouls and their corresponding free throws: with him in the lineup, his teams had fewer fouls and more free throws, and his opponents the opposite.  This is particularly peculiar because Rodman himself got a lot of fouls and was a terrible free throw shooter (note: this is yet another reason why including personal fouls in your player valuation method—yes, I’m looking at you, PER—is ridiculous).

Whether Rodman was a “role player” or not is irrelevant: whatever his role, he did it well enough to contribute more to his teams than the vast majority of NBA players (role players or not) contributed to theirs. For some reason, this simple concept seems to be better understood in other sports: No-one would say that Mariano Rivera hasn’t contributed much to the Yankees winning because he is “merely” a closer (though I do think he could contribute more if he pitched more innings), just as no-one would say that Darrelle Revis hasn’t contributed much to the Jets because he is “merely” a cornerback.

So does this mean I am conceding that Rodman was just a very good, but one-dimensional, player?  Not that there would be anything wrong with that, but definitely not.  That is how I would describe it if he had hurt his team in other areas, but then made up for it—and then some—through excellent rebounding. This is actually probably how most people would predict that Rodman’s differentials would break down (including, initially, myself), but they don’t.  E.g., the fact that his presence on the court didn’t hurt his team’s offensive efficiency, despite hardly ever scoring himself, is solid evidence that he was actually an excellent offensive player.  Even if you take the direct effects of his rebounds out of the equation entirely, he still seems to have made three different championship contenders—including one of the greatest teams of all time—better.  While the majority of his value added—that which enabled him to significantly improve already great teams—came from his ability to grab rebounds that no one else would have gotten, the full realization of that value was made possible by his not hurting those teams significantly in any other way.

As it wasn’t mystical intangibles or conveniently immeasurable defensive ability that made Rodman so valuable, I think it is time we rescind the free pass given to the various player valuation metrics that have relied on that excuse for getting this one so wrong for so long.  However, this does not prove that even a perfectly-designed metric would necessarily be able to identify this added value directly.  Though I think valuation metrics can be greatly improved (and I’m trying to do so myself), I can’t say for certain that my methods or any others will definitely be able to identify which rebounds actually helped a team get more rebounds and which points actually helped a team score more points.  Indeed, a bench player who scores 8 points per game could be incredibly valuable if they were the right 8 points, even if there were no other direct indications (incidentally, this possibility has been supported by separate research I’ve been doing on play-by-play statistics from the last few seasons, in which I’ve found that a number of bench players have contributed much more to their teams than most people would have guessed possible).  But rather than throwing our hands in the air and defending inadequate pre-existing approaches, we should be trying to figure out how and whether these sorts of problems can be addressed.

# Defensive Stalwart or Offensive Juggernaut?

As an amusing but relevant aside, you may have already noticed that the data—at least superficially—doesn’t even seem to support the conventional wisdom that, aside from his rebounding, Rodman was primarily a defensive player.  Most obviously, his own team’s points per scoring opportunity improved, but his opponents’ improved slightly as well.  If his impact were primarily felt on the defensive side, we would probably expect the opposite.  Breaking down the main components above into their offensive and defensive parts, our value-source pie-chart would look like this.

The red is actually slightly smaller than his contribution from defensive rebounds alone, as technically defensive efficiency was slightly lower with Rodman in the games.  For fun, I’ve broken this down a bit further into an Offense vs. Defense “Tale of the Tape,” including a few more statistics not seen above:

Note: Differentials that help their respective side are highlighted in blue, and those that hurt their respective side are highlighted in Red.  The values for steals and blocks are each transposed from their team and opponent versions above, as these are defensive statistics to begin with.

Based on this completely ridiculous and ad-hoc analysis, it would seem that Rodman was more of an offensive player than a defensive one.

Including rebounding, I suspect it is true that Rodman’s overall contribution was greater on offense than defense.  However, I wouldn’t read too much into the breakdowns for each side.  Rodman’s opponents scoring slightly more per opportunity with him in the game does NOT prove that he was a below-average defender.  Basketball is an extremely dynamic game, and the effects of success in one area may easily be realized in others.  For example, a strong defensive presence may free up other players to focus on their team’s offense, in which case the statistical consequences could be seen on the opposite side of the floor from where the benefit actually originated.

There are potential hints of this kind of possibility in this data, such as:  Why on earth would Rodman’s teams shoot better from behind the arc, considering that he was only a .231 career 3-point shooter himself?  This could obviously just be noise, but it’s also possible that some underlying story exists in which more quality long-range shots opened up as a result of Rodman’s successes in other assignments.  Ultimately, I don’t think we can draw any conclusions on the issue, but the fact that this is even a debatable question has interesting implications, both for Dennis Rodman and for basketball analytics broadly.

# Conclusions

While I am the first to admit that the dataset this analysis is based on might not be sufficiently robust to settle the entire “Case” on its own, I still believe these results are powerful evidence of the truth of my previous inferences—and for very specific reasons:

Assessing the probability of propositions that have a pre-conceived likelihood of being true in light of new evidence can be tricky business.  In this case, the story goes like this: I developed a number of highly plausible conclusions about Rodman’s value based on a number of reasonable observations and empirical inquiries, such as: 1) the fact that his rebounding prowess was not just great, but truly extreme, 2) the fact that his teams always seemed to do extremely well on both ends of the floor, and 3) my analysis (conducted for reasons greater than just this series) suggesting that A) scoring is broadly overrated, B) rebounding is broadly underrated, and C) that rebounding has increasing marginal returns (or is exponentially predictive).  Then, to further examine these propositions, I employed a completely independent method—having virtually no overlap with the various factors involved in those previous determinations—and it not only appears to confirm my prior beliefs, but does so even more than I imagined it would.

Now, technically, it is possible that Rodman just got extremely lucky in the differential data—in fact, for this sample size, getting that lucky isn’t even a particularly unlikely event, and many of his oddball compatriots near the top of the master list probably did just that.  But this situation lends itself perfectly to Bayes’ Theorem-style analysis.  That is, which is the better, more likely explanation for this convergence of results: 1) that my carefully reasoned analysis has been completely off-base, AND that Rodman got extremely lucky in this completely independent metric, or 2) that Dennis Rodman actually was an extremely valuable player?

## Graph of the Day: NBA Player Stats v. Team Differentials (Follow-Up)

In this post from my Rodman series, I speculated that “individual TRB% probably has a more causative effect on team TRB% than individual PPG does on team PPG.”  Now, using player/team differential statistics (first deployed in my last Rodman post), I think I can finally test this hypothesis:

Note: As before, this dataset includes all regular season NBA games from 1986-2010.  For each player who both played and missed at least 20 games in the same season (and averaged at least 20 minutes per game played), differentials are calculated for each team stat with the player in and out of the lineup, weighted by the smaller of games played or games missed that season.  The filtered data includes 1341 seasons and a total of 39,162 weighted games.

This graph compares individual player statistics to his in/out differential for each corresponding team statistic.  For example, a player’s points per game is correlated to his team’s points per game with him in the lineup minus their points per game with him out of the lineup.  Unlike direct correlations to team statistics, this technique tells us how much a player’s performance for a given metric actually causes his team to be better at the thing that metric measures.

Lower values on this scale can potentially indicate a number of things, particularly two of my favorites: duplicability (stat reflects player “contributions” that could have happened anyway—likely what’s going on with Defensive Rebounding %), and/or entanglement (stat is caused by team performance more than it contributes to team performance—likely what’s going on with Assist %).

In any case, the data definitely appears to support my hypothesis: Player TRB% does seem to have a stronger causative effect on team TRB% than player PPG does on team PPG.