Tuesday, June 19, 2007

Homers in (Win) Context

As Barry Bonds closes in on Hank Aaron’s record, there will surely be a lot of discussion about which player’s accomplishments are more impressive, who is more worthy of the record, etc. From a sabermetric perspective, there are a number of different approaches available to shed light on the comparison. These generally attempt to evaluate not only the raw totals but the context in which they were achieved.

One of the most basic examples is the use of relative or adjusted statistics, comparing the frequency of home runs hit by a player to that of his league, and using this to give a plus/minus figure against the league average, or to restate that figure as a new raw number of homers expected in a different context. These results might be based on the difference or ratio of the player’s performance to that of his league.

Another approach along the same lines is to look at the number of standard deviations away from the mean the player’s performance is. This has the presumed effect of accounting for league quality, with the theory that a lower standard deviation means a higher-quality league.

One can also look at ranks or percentiles of performance within the league. Under this approach, leading the league in home runs is considered equally impressive to leading the league in a different year. The best of a given time are assumed to be equal to the best of another time.

There are also more complex approaches that attempt to project performance in a different context while maintaining the same run or win value of the player’s overall performance. Examples of these approaches include Bill James’ “Willie Davis Method” and Clay Davenport’s Davenport Translations.

All of these approaches have their pros and cons. Relative methods are easy to understand, but often give results that are outlandish (ex. Babe Ruth would hit 100+ home runs in a modern context if he bettered the league home run rate by the same ratio as he did in his great seasons). Rank and percentile methods assume that leading a deadball league with 11 homers is inherently equivalent to leading a modern league with 55 homers.

The standard deviation and translation methods have less obvious flaws. The biggest hurdle to the translation methods is complexity; the principles behind Davenport’s have been explained, but not the details. The Willie Davis method is based on a flawed run estimator (basic Runs Created), which makes all of the results somewhat circumspect.

My approach is similar in spirit to those, but cuts out a good deal of complexity with maybe a small loss in theoretical purity. It is hardly a unique or revolutionary approach, and for all I know may have been published by another analyst in nearly identical form, but if it has I do not recall seeing it. As I have explained in previous posts on other subjects, I begin my analysis of baseball performance with the premise that a major league win today is worth the same as a major league win in 1961 is worth the same as a major league win in 1927…you are of course free to disagree with this premise, but it is where I am coming from.

Saying that a win is a win is not immediately helpful when moving away from comparing estimated win contributions of players to comparing player performance in specific component categories, like home runs. However, in a linear weight construct, we can put a run or win value on each home run that the batter hits. By comparing the win value of player’s home runs, we can avoid making any assumptions about how difficult it was to hit home runs in the given contexts, and focus on what the home runs the player actually hit were actually worth in his own context.

After all, home runs are only valuable because they contribute to wins. Hitting 70 home runs, even in a modern high-scoring context, contributes more wins to a player’s team then hitting 10 home runs to lead the league in the dead ball era. It may have been “harder” to hit the ten homers, they may have stuck out more from the other players in that time in terms of standard deviations or differences in frequency, but that does not matter from a pure value perspective. All that matters is the number of wins each player contributed via the home run in his own time and place. And of course we also started with the premise that a major league win is of equal value regardless of the year in which it was achieved.

So how does this approach work? Let’s assume that the run value of a homer is always 1.4 runs. This is obviously not correct; the coefficient of a home run is variable, although it is not nearly as variable as that of some other events. You could of course use Base Runs or some other approach to find a unique coefficient for the HR in each context, but I am going to just use 1.4, while acknowledging that it is imprecise.

Let’s look at Roger Maris’ 1961 season. His 61 home runs are estimated to be worth 1.4*61 = 85.4 runs. We will assume that the runs per win factor is equal to the total number of runs per game in the league that season. This again is imprecise, but not to too great a degree, and it makes things very simple. In 1961 the AL average was 9.06 RPG. We can also account for how Maris’ home park changed the value of runs in terms of wins. I have the ’61 Yankee Stadium PF at .94. That means that the prevailing RPW in Maris’ context was PF*RPG or .94*9.06 = 8.52. Putting it all together, the win value of Maris’ home runs was 1.4*HR/(PF*RPG) or 1.4*61/(.94*9.06) or 85.4/8.52 = 10.02 wins.

We can keep Maris’ figure at 10.02, but homer-wins are not in a scale that we are familiar with, and we can easily convert this into a number of homers in an average major league context of 9 RPG. All we have to do is figure out how many homers one would need in a 9 RPG context to get 10.02 homer-wins. The answer is 10.02*9/1.4 = 64.4, or we can easily express that New HR = HR/(PF*RPG)*9, since we are assuming that the home run is worth 1.4 runs, always.

We can now figure equivalent win-value home runs for all players. Let’s look at a Gavvy Cravath, leading the NL in longballs with 19 in 1913. The NL RPG was 8.3, and Cravath’s park had a PF of 1.04. Cravath therefore had 19.8 adjusted home runs. Since this is kind of a frivolous stat, I’ll give these adjusted home runs a less serious name, Gavvys. I’m using Cravath here because he is an example that leading the lead in home runs did not have the same value in 1913 as it did in 1961. Hitting home runs in 1913 was, as always, beneficial to the team, but 19 homers just do not produce that many wins. Again, it is feasible that Cravath’s performance was more difficult to achieve or more impressive then Maris’. Whether it was or not is irrelevant when considering the win value they had.

I calculated Gavvys for all of the members of the 500 home run club (through 2006, considering only performance up to 2006. Each season was evaluated individually and the totals were summed for a career). Here is the list sorted by Gavvys:

No player’s home runs have had a higher win impact then Hank Aaron’s, and he has a wide enough margin that his “record” may sustain Bonds’ challenge. Ironically, Bonds has 755 Gavvys, matching Aaron’s real total. Only Foxx and Williams played in high enough scoring environments to see their modified figures drop below 500. Here is the list again, this time sorted by Gavvys minus home runs.

The overall trend is for Gavvys>homers, which is most likely because 4.5 is not the true long term mean; we could use 1 as the mean, or 1000, or leave them in terms of homer-wins, and the order of the career leader list would hold, so 4.5 works just fine.

In order to find the best single season Gavvy performance, I also looked at the 50 home run club. Obviously, these seasons will not represent all of the best single seasons in terms of Gavvys, but doubtlessly the record-holder will be found. And the 50 in a season club is a natural one to consider, just like the 500 in a career. The list is sorted by Gavvys, and “DIFF” again is Gavvys less actual home runs:

Bonds’ 2001 and McGwire’s 1998 still top the list, with Maris’ 1961 holding against all of the other times McGwire and Sosa beat the raw total. Ruth’s single season luster fades a bit as does his career total, while multiple seasons tumble out of the 50 HR club.

Take Gavvys for what they are worth--they only focus on one single event and attempt to restate it as a win-equivalent figure in a certain context. They are based on the assumptions I discussed earlier and they are by no means the be all and end all of accounting for context in determining the best home run hitters. I do believe that they represent an interesting way to evaluate home runs while keeping an eye on the ultimate goal of the game (winning), and evaluating them in that light.

Let me stress again that Gavvys are not a projection of how many home runs a player would have hit in a nine RPG context, and do not attempt to adjust for how difficult it was to hit home runs in the player’s environment (RPG coupled with PF is to some extent a proxy for this, but that is not how it is being used here). What they are is a number of home runs in a 9 RPG context that would be equal in win value to the win value of the player’s home runs in his actual context.


  1. Bonds's 755 Gavvys isn't ironic. It's just a coincidence.

    Great piece.

  2. I love vague, anonymous comments (I'm not referring to the first one--correcting my useage of a word is constructive).

  3. It's like somebody putting a bag over their head, sticking it inside the door, telling you the party sucks, and leaves. Who does that?

  4. I like it, but I think most people who dabble in 'context conversions' want to know something along the lines of:

    "What would Ty Cobb's batting line for 1908 look like if he played in 1998?" given assumptions like a win in 1908 is the same as a win in 1998, etc.

    If you can come up with a good method to do those types of context conversions, then you'll REALLY have something.

  5. I certainly don't disagree with Kevin's comment that most people are looking for "what would he do today" type lines. But that is the kind of the point of my post...offering a quick and dirty stab at translating the values and the values only. I personally don't get into the what if scenarios, because there are too many variables that you can't possibly know. Maybe Cobb, surly old cuss that he would be, would refuse to change his ways to the modern game and would look more like an Ichiro and bat .400. Maybe not.

    Those questions are certainly interesting to speculate about, and I have no beef if others want to, it's just not something I feel comfortable doing. But there's no doubt that it is a more ambitious task for a sabermetrician to work on.


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