How are Runs Really Created
by Tangotiger
Run estimator, run estimator, here, there, and everywhere
Every week, someone comes out with a new run estimator. You know those right? Take various components, like hits, HR, walks, outs, apply some constant to each of them, or maybe multiply and divide these components in some fashion, and the result is the runs a team was expected to score.
Let's take the basic version of Bill James Runs Created: take the Total Bases and multiply it by the On Base Average and you get Runs Scored. So, in a 9 inning game with 15 total bases, and you have 13 of 39 batters safe, that gives you an expected 5 runs scored. And if you look at a large enough sample, a team with 15 TB and a .333 OBA will probably score around 5 RPG. So, what's the problem?
The problem is that this does not model reality. If you add a single to this game, you get 16 TB, a .350 OBA for a total of 5.6 RPG. That extra single added .6 runs to this game. Sounds about right, no?
No. Pete Palmer's Linear Weights says that an average single will add .47 runs to a game. Gee, that sounds about right, no?
No. The problem with these run estimators (and all their offshoots and close family members like Extrapolated Runs and Equivalent Runs) is that they don't model reality. The provide an ESTIMATE based on a formula that they hope capture reality. They don't.
Modeling Reality
How can we show this? What does model reality? How can we take any system (not necessarily baseball), and create a model? The best way is to determine all of your known variables, allow some allowances for unknown variables, and create a simulator that uses every variable you know and how each variable interacts with each other. (Think flight simulators.)
Ok, hot shot. So, how much run value does a simulator say a single is worth?
If you were to construct your model perfectly (or as close as possible), you can create a system where 15 TB and a .333 OBA will produce 5 RPG. That is your controlled environment. If you then add a single randomly inside this system, one variable, and look at the result, you will conclude that the effect of adding that variable resulted in a change in the output of the system (over a large enough sample cases). In this case, adding a single would add .48 runs in this game.
Ok, so a single is worth .48 runs. Linear Weights is better, right?
Wrong. What that .48 means is that the MARGINAL effect of the single to a SPECIFIC run environment has a positive run effect of .48 runs. If you had a different run environment, say like the one Pedro or Randy provide their opposition, a single will not add .48 runs. It'll be more like .38 runs. This again can be shown with a perfectly constructed simulator.
Even worse, adding TWO singles does not necessarily add .96 runs, nor does adding 100 singles add 48 runs. Baseball run construction is non-linear interdependent.
But do we have to run a simulator every time? Isn't there some relationship that we can find that'll show us that it's .38 or .58 or whatever?
Yes. And for that we have to understand how runs really are created.
An Interlude
Bill James has made several modifications to his basic Runs Created. These changes were made to better capture reality. Actually, they were made to improve accuracy over a sample of team totals. The changes made are good overall, but he makes one glaring error. While James has the correct basis for capturing the model of interdependence and non-linearity (on a team level anyway), this one error is his downfall.
Pete Palmer strangely enough showed us the dynamic values of each hitting component by era in his book The Hidden Game of Baseball. However, he took the easy way out and decided to use static (non-changing) values for the positive hitting components, and applied a dynamic (changing) value only to the out component. This gives the impression that hits and HR and walks have static values regardless of run environment, while the out is the only one that changes. The easy way out is not only the wrong way out, but it doesn't even capture reality.
Runners on, and runners over
Runs are created by getting runners on base, and moving them over. Most people look at the OBA as the first part, and the SLG as the second part. And the combination of the two should generate runs scored. It's not that simple.
Using common sense on an uncommon example
Have you ever played in a softball league where the typical team scores upwards of 20 runs in a 7-inning game? Such a team will send say 51 batters to the plate, 21 of which will be out, and 20 of the other 30 batters will score. If a runner that gets on base will score two-thirds of the time, how much more valuable is the home run compared to a single? In softball, because the run environment is so high, it is important just to be able to get on base, because you know that you have a good chance at scoring.
Take an even more extreme example. Imagine playing in a run environment with 100+ runs scored in every game. 90% of the baserunners end up scoring. In this environment, there is very little difference between a home run and a single. Just by virtue of getting on base, you are bound to score. There is far more value in getting on base, than of moving runners over.
Take an extreme example the other way. Pedro Martinez provides his opposition with a very low run environment. Getting on base is not enough. Very few of those runners will end up scoring. However, if you can hit a home run, you will be adding alot of run potential to the runners on base. Not only that, but when you hit a home run, you are always guaranteed 1 run.
The run value of various hitting events
Here is what our common sense tells us, in graph form. 
This however is contrary to what Bill James' Runs Created tells us. If you have a formula that says TB x OBA, then this implies that you have an always-rising value for each hitting event. Suppose that you have 100 total bases in a game, and the OBA is .800 (40/50), for a total of 80 runs. If we add 1 HR to this system, that gives 104 TB, and an OBA of .804, for a total of 83.6 runs. This one HR is now worth an astonishing 3.6 runs! But common sense has shown us above that this is wrong.
This is what Runs Created tells us, in graph form. 
What is very important to note at this point is that if we concentrate at the above two graphs between the OBA points of .300 to .400 (where MLB teams perform in reality), we see that Bill James' Runs Created does conform somewhat to what we perceive as common sense. However, the reason that Runs Created "works" is not because of its construction. It's purely an accident that it works. It just so happens that the points at which Runs Created and common sense intersect is exactly at the same points at which MLB teams play at!
Furthermore, note that what applies to teams does not apply to individuals. Individuals need their own run construction formula.
Don't like common sense? Let's try some math
Based on my analysis of the play-by-play data from 1974 to 1990 provided by Retrosheet and software provided by Ray Kerby, here is the likelihood of a runner scoring, based on which base he is on, and the number of outs
Chance of scoring, from each base/out state
0 outs 1 out 2 outs
1B .38 .25 .12
2B .61 .41 .21
3B .86 .68 .29
This simply means that if you have a runner at 3B with 0 outs, then he has an 86% chance of scoring (that's the "getting on base" value). If someone can drive him in, that batter will add .14 runs (to the already established .86 for a total of 1.00; this is the "driving him in" value).
As you can see the higher the chances of scoring, the less value there is in driving in a runner. Let's look at the run-driving value of the walk.
Run Driving value of the walk, from each base/out state
0 outs 1 out 2 outs
1B (to 2B) +.23 +.16 +.09
2B (to 3B) +.25 +.27 +.08
3B (to home)+.14 +.32 +.71
Obviously, the most valuable walk in terms of moving runners over is the walk that scores the runner from 3B and 2 outs. This happens rarely, as the bases would be loaded. So, on top of this table, we need a "frequency" table that shows how often a walk occurs in each of the above scenarios.
Frequency of walk in moving runners over, from each base/out state 0 outs 1 out 2 outs 1B (to 2B) .053 .084 .114 2B (to 3B) .012 .027 .042 3B (to home) .002 .006 .010
As you can see, walks are not given out in random fashion. A large portion of them occur with 2 outs, when they do the least damage. Multiplying these two tables will give you the "moving over" value of the walk. This works out to +.06 runs.
The "getting on" value of the walk can be determined using the "chance of scoring" table presented above, with the appropriate frequency at which walks occurs in those states.
Frequency of walk occuring, by outs 0 outs 1 out 2 outs .316 .326 .359
Doing a similar multiplication, and we see that the "getting on" value of the walk works out to +.24 runs. The run value of the walk is therefore equal to +.30 runs.
We could have performed this analysis in several other ways, each of which would yield the same result of +.30 runs. One is to look at the run expectancy (RE) before the walk, the RE after the walk, take the difference, add the number of runs that score, and you get the run value of the walk. Doing that and we get a run value of +.30 runs. Another way is to construct a simulator, insert a walk, and look at the difference. You will find that given the run environment of 1974-1990 you will get a run value of +.30 runs.
The important point to remember is that the run value of all the hitting events is dependent on the run environment. The walk is worth more today than in 1968. It is worth more in Coors than at the Astrodome.
Using the RE approach, here is the run values of all offensive events.
Run values, 1974-1990, using the RE approach Single Double Triple HR Walk IBB HBP Reached Base On Error Interference OtherSafe .46 .750 1.033 1.402 .303 .176 .33 .478 .357 .631 Sac Strikeout Out -.09 -.269 -.265 SB CS Pickoff Pickoff Error Balk PB WP DefensiveIndiff OtherAdvance .193 -.437 -.228 -.182 .25 .276 .278 .132 -.362
Another Interlude
Remember what the run values represent. They represent the MARGINAL effect of the offensive events GIVEN a specific environment. Remember that. Repeat that.
If you get an out in a run environment that scores 3 runs per INNING, that is very costly to your team. It has a negative effect only because of the expectations of future runs. The out is not very costly when Bob Gibson's 1.12 ERA is on the mound, simply because the expectation is low that a run would be scored at all. -.27 runs doesn't mean that you will score negative runs, but rather that your team's run potential has been decreased by .27, GIVEN the environment in which the out was created.
I will talk more about how to understand the out in the frame of reference of Runs Created and Linear Weights in my next article. And I will apply David Smyth's BaseRuns, a constructor that models reality in almost all run environments.
August 12, 2002 - Devin McCullen
I like this approach, but two things.
1) When you're criticizing James' use of RC, I think you're muddying cause and effect a little. If the results of the RC formula didn't correspond roughly to actual runs, James wouldn't be using it.
2) Okay, my common sense has a problem with a run value system that has events with the same outcome (walk, HPB, interference) having different run values. I realize there are slight differences in how they come to pass, but I can't believe it makes that much of a difference in the expected runs.
August 12, 2002 - tangotiger
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Devin, excellent points.
...If the results of the RC formula didn't correspond roughly to actual runs, James wouldn't be using it.
As I mentioned, as long as you are using typical teams in the .300 to .400 OBA range, and as long as the HR/game hit is around the norm, then RC works fine as something useful.
The problem is when you try to extend that to Barry Bonds types of teams (not that they exist) or Pedro Martinez types of teams (and they exist plenty, as Pedro, when on the mound, is his own team).
My point is to make sure that just because the results of Runs Created works on a particular set of samples doesn't mean that you can extend that methodology to other types of things you may be doing.
There's a reaons RC fails, and it's in its treatment of the HR.
2) Okay, my common sense has a problem with a run value system that has events with the same outcome (walk, HPB, interference) having different run values.
Let's take a real simple example: a regular walk v IBB. Since an IBB walk occurs almost always with first base open, then an IBB has zero "moving over" value. Since the IBB is given out much more with 2 outs than with 0 outs, the "run scoring" value of the IBB is much less than a regular walk.
So, based on the frequency of when the events happen, and the effect of each event, the values can change drastically.
As for a regular walk v HBP, HBP occur in more or less random fashion. A walk occurs with more frequency with 2 outs than 0 outs, and with more frequency with no runners on 1B than expected in random fashion. The effect of these two things reduce the "moving over" value of the walk and the "run scoring potential" of the walk.
If you are thirsty for more, I've published PRELIMINARY results on the run values of various hitting events by the 24 base-out states. (I should be publishing an updated table in a few weeks.) From there you will see there are virtually no differences between the walk, IBB, and HBP, as you'd expect.
http://www.tangotiger.net/lwtsrobo.html
Thanks, Tom
August 12, 2002 - Vinay Kumar
Great article Tangotiger (though I expected to see Base Runs or something in there; is there going to be a part 2?).
Let me see if I can answer Devin's second question, in a slightly different manner than TT's response. In any given situation, a walk and an HBP are worth the same thing. However, because walks and HBPs are distributed differently, the average walk is not exactly the same as the average HBP. And Tango's example of the intentional walk is a great way to demonstrate this.
August 12, 2002 - Patriot
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Vinay: "I will talk more about how to understand the out in the frame of reference of Runs Created and Linear Weights in my next article. And I will apply David Smyth's BaseRuns, a constructor that models reality in almost all run environments. "
Excellent article, Tango.
August 12, 2002 - Rob Wood
So you are saying that the "value" of any event depends on the underlying run environment. But it also surely depends upon the specific base-out situation, the identity of the batter, who is on deck, etc. Simulations over zillions of such situations are used to estimate the "average" value. Accordingly, this approach necessarily smears a whole bunch of disparate situations into one number. For example, a single being worth 0.46 runs.
I wonder how much variability there is in these event values over the different possible situations. Base-out should be fairly easy to do, and you may have already done the analysis.
It may also be interesting to look at where in the lineup the batter hits (for example, if the cleanup hitter is up next or the woeful hitting pitcher). Maybe you could break the lineup up into four groups. 1-2 hitters, 3-4-5 hitters, 6-7-8-9 (for the AL), and 1-2, 3-4-5, 6-7-8, and 9 (for the NL).
The basic question is how good of an estimate of the event's run value is any specific estimate. If the true run value varies widely from situation to situation, then we would place less credibility in any one formula (there would be a fair amount of variance around the estimate). I realize that the runs created formulas do a good job of predicting actual runs scored, so something must be at work to help. But it may be that frequencies are sufficiently rich for events to "even out" over the course of a season and over an entire team. If that were the case, the variance around any one player's runs created may still be rather high.
I further realize that the main goal of this approach is to derive the single best linear-weights formula (using only seasonal situation-independent stats) to estimate each player's offensive contribution. But it appears that you have all the ingredients to investigate whether this approach could be significantly improved by considering other detailed information as well. Recall that Bill James incorporates a hitter's batting average with runners in scoring position and his home runs with runners on base along with his runs created in James' win share system.
Thanks much.
August 12, 2002 - John Warren
Great work, lots of stuff to think about in there. One thing I'm wondering about - looking at the relative marginal values of SB and CS, should this change our thinking as to what the appropriate individual success rate should be to justify an attempted steal? Taking it a bit further, since each individual pitcher creates such a different expected runs environment, do these averages go so far as to say that every runner that gets on should attempt to steal against Pedro, and that under no circumstances should even Ichiro attempt to steal against Chan Ho (laying aside for the moment Pudge's arm)? Thanks in advance for your thoughts on this, folks.
August 12, 2002 - Mike Tamada
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Very nice article. One thing that I wonder about though: sure, the negative value of an out is less in a Pedro Martinez environment than it is in Chan Ho Park environment (talk about Park effects!). As measured in terms of run expectancy.
But what about *win* expectancy?
That is, the formulas all seem to be geared to calculating what happens to a team's expected number of runs. But the VALUE of those runs will presumably be different in the different environments, just as the value of an out or a single is different in the different environments.
Possible example: the value of a home run. In terms of expected runs added, it might be higher in a Chan Ho Park environment than in a Pedro Martinez environment -- there's likely to be more players on base when the HR is hit. But in terms of expected wins added, I'd guess that the HR against Pedro is exceptionally valuable, whereas the HR against Chan Ho is, well, more Ho-hum.
--MKT
August 13, 2002 - tangotiger
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Rob, your question on base-out differences in run value can be found here http://www.geocities.com/tmasc/lwtsrobo.html
I looked at the batting order differences of run values, and there was a long thread posted on fanhome. It is not easily digestable, and someday I'll write an article on the discoveries there. But yes, as you'd expect the leadoff hitter's HR value was 1.30 while the #5 hitter was somewhere around 1.47.
John Warren: the steal is an interesting point. The run value of the SB is very independent of the run environment, as the additive value of the SB is around .17 to .21 for the most part. The CS however changes HIGHLY, as the out is the most dependent on the run environment. The break-even point is therefore much lower with Pedro, and more steals should be attempted against him.
Mike: I've previously published charts on win expectancy which I have to update in the near future. There's no doubt that win expectancy is really the most important aspect of analysis since that's what we are after. Again, for those thirsty for more, you can consult my prelimiary chart on WE here: http://www.geocities.com/tmasc/we.htm . Again, where this comes most into play is the IBB. While the run value of a regular walk is .30 runs and the run value of the IBB is .17 runs, the win values are far different. Because the IBB occurs in game situations where it is "controlled" to minimize the impact of win/loss, then it's win value would also decrease.
Thanks for all your great comments.
August 13, 2002 - Jason
While I admire the academic desire to design the perfect theoritical evaluation tools, I find it a bit dishonest to harp to the point of obsession on things that happen to work well. Newtonian Physics has been shown to be fundamentally unsound for almost 100 years and yet it is still heavily used becuase it works so well. Until their are 3 or 4 Barry Bonds on a team I don't see the particular need to be so detailed. OK, so I lied I do see one use. Evaluating relief pitchers. I think it's great that the Prospectus guys have run with a system that credits guys for the work they do based on when the enter the game, but I'd still ike to see a system that accounts for the actual skill of the batters faced. There's a big difference between coming in with the bases loaded and the 8 hitter up vs. having to pitch to Bonds or Sosa in that situation.
August 13, 2002 - Bobby
What about GIDP? What is its value in terms of RE?
August 13, 2002 - tangotiger
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GIDP: it's worth around -.45 runs. I was thinking of breaking up the "outs" PA into "outs 1, outs 2, outs 3", but decided against it. Maybe I will fix that.
Jason: what I am presenting is how runs are really created. It's the building blocks to whatever it is you want answered. From this, you can generate win expectancy tables, if you like, or the more detailed run values by the 24 base-out states. You can then further extend this to a 24x9 run values that ALSO includes batting order. And from that standpoint, you can evaluate the #9 v Bonds with the bases loaded.
These other run evaluators give no option to do this simply because they are the end to the means. They were built to answer a specific question, and therefore are not very extendable. Play-by-play analysis is very extendable.
August 13, 2002 - Rob Wood
Thanks Tango for the link to the base-out run values. I notice that there is a fair amount of variability in the values across the different base-out situations, as expected.
That brings me back to an issue I raised in my previous post. Have you (or can you) used your simulator to estimate the variability around a player's runs expectancy figure? Say a player creates 100 runs by the linear weights formula. Is the "actual" number of runs he contributed 100 +/- 2 or 100 +/- 20? You could use your simulator to derive these error bars for various levels of RE.
I think that would be an important finding.
August 13, 2002 - Michael Humphreys
How is what's been described any different from a Lindsey-Palmer "change-in-state" model based upon "base-out" run expectation states that are customized per player, per team or per league? See "Curve Ball," the recent book by Jim Albert and Jay Bennett. I'm looking forward to the next installment of your article, but it seems that if you customize a model for every possible situation, it no longer has the conceptual simplicity and practical applicability of a model, and is merely a highly intricate description.
August 14, 2002 - Arvid Engen
Okay, I'm confused.
Tango, if the goal of the work you describe here is in fact to derive optimal situation-neutral values for each batting event, then what really distingishies it from Extrapolated Runs, or another linear-weights based method?
It seems that you're going to wind up with coefficients that, while perhaps more defensible aesthetically, are less valid empirically than those derived via linear regression. You can't beat linear regression, baby.
August 14, 2002 - tangotiger
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Linear Regression
There are certain things that must be understood about linear regression and using it to determine the relationship between hitting events and runs scored.
First, a little background on linear regression. If you have two things, say, the price of a stock and the earnings per share, you can probably find a relationship between these two variables. The higher the earnings, the higher the price of the stock. You will end up with a formula like P = m times E + b, where P is price, E is earnings, b is some constant and m is the slope. The price of a stock, and runs in baseball, is influenced by more than one variable. You end up with an equation that says y = m1a1 + m2a2 + m3a3 + ... + b. Linear regression lets you input the independent variables a1, a2, a3..., the dependent variable y, and solve for m1, m2, m3..., and b.
Here are 4 major problems with using this in baseball: 1 - Linear regression is LINEAR. Linear as in a straight line. While there is a somewhat linear relationship between runs and singles, doubles, triples, and walks, there is NOT a linear relationship between runs and HR, or runs and everything else like SB, WP, BK, etc. Baseball is non-linear.
2 - The independent variables are not independent. There is an interdependence between all these variables. A walk is only worth what it is because of the other things that happen. Linear regression attempts to "freeze" all the other variables when calculating the value of the unfrozen variable. As your run environment increases however, we know that the values of these variables change. Baseball is interdependent.
3 - Even if you assume for ease that run creation is linear and independent (a safe assumption for very controlled environments), what sample data will you use to run your regression against? Most people will use team season totals, which is an aggregate of individual games, which is an aggregate of individual innings. If you want to run a proper regression analysis, at the very least run it on a game or inning level. Your sample size will explode to something much more reliable.
4 - Not accounting for all the variables. Triples have a strong relationship to speed. If you don't have SB in your sets of variables, the regression analysis will award more weight to the triples as a stand-in (because of its relationship to steals). It is possible, based on some samples, that the value of a triple could exceed the value of the HR! What other variables are you not accounting for?
Arvid - Let me get back to your post. The purpose of this article is to explain the building blocks of run creation at the team level. I have not shown how to extrapolate this to individual players. The end-result is not to end up with linear values for each hitting event, since these linear values only apply to a given run environment. We need to determine the linear values for EVERY run environment! As I said, the value of a single in Pedro's run environment is far less than a single in an average pitcher's run environment.
I am interested in the pieces of how runs get created, an actual model. I am not interested in a formula that estimates runs based on whatever variables that ONLY works for a given run environment. Runs Created and Linear Weights work fine for that. BaseRuns is the key, and I will present this hopefully by the end of this month.
Michael - The building blocks of run creation does lie in run expectancy tables for the 24 base-out states. I am not introducing anything new here, but rather showing how we should extend this to other run environments. I have not read Curve Ball. Please clarify your post further so that I can properly answer you.
Rob - Are you asking me what would a player's run value be using a context-neutral approach (i.e., the final weighted average values I presented) compared to a context-specific approach (i.e., the specific values by the 24 base-out states)? If this is the case, the answer is about +/- 10 runs at the extremes. I looked at this last year, with regards to Ichiro. You can find that article here http://www.geocities.com/tmasc/lwbymob.htm though I only looked at the 8 base states. If this is not what you are talking about, please clarify further.
August 14, 2002 - Rob Wood
Here is what I was thinking. The standard linear weights (runs created, etc.) formulas use average run-values over all the disparate contexts a batter faces during a season. Let's just talk about the 24 base-out situations, though conceivably other information could also be taken into account. Tango has shown that there is quite a bit of variability in the run-values of events across the different contexts, as we all intuitively understand.
Thus, when it is reported that a player had Y linear weights runs (or runs created, etc.), we know that this is an estimate of his actual runs contributions which could be more accurately measured by looking at the player's performance in each of the 24 base-out contexts. This is what Tango did for some of the 2001 MVP candidates in the linked article.
I am hoping that you can use your simulator to estimate the inherent variability (due to context) in the linear weights runs estimates. I would suggest fixing a player's seasonal batting line (so many singles, doubles, triples, home runs, etc.). Then put this player into zillions of different team contexts, different places in the lineup, etc., and for each plate appearance randomly select one of his fixed outcomes.
[As a technical aside, I would suggest doing the random selection without replacement so as to have the exact same number of events in his seasonal batting line in each simulation trial.]
Then use your RE machinery to calculate the player's actual runs contribution for each trial. Do zillions of trials. Derive the sampling distribution, if you want to call it that, of the player's runs contribution. Of course the mean of this distribution should be the original linear weights runs we started with. However, there will be a fair amount of variability around this value. How large is that variability is what I think would be extremely interesting to investigate.
Maybe get separate estimates of the variability due to the base-out context (holding the team context fixed) and the variability due to different team contexts (holding the base-out situation fixed).
August 14, 2002 - Michael Humphreys
Thanks for your response to my question, which you answered. You are absolutely right that run-expectancies vary t r e m e n d o u s l y based upon who the pitcher is, who his fielders are, and who is coming up behind the batter. In addition, I believe one of the postings had a good point that w i n-expectancies also vary tremendously depending on the inning and the amount of a lead. With computers, it is now possible to calculate the extent to which a batter increases or decreases run (or win) expectancy each time he comes to the plate, and any such evaluation is clearly the most accurate. The point I was trying to make is that the computational complexity of the model, which is a cinch for a computer, is overwhelming for a human being, and the rating generated therefore has a sort of "black box" quality to the average fan. Simple formulas such as OPS or Palmer linear weights can be apprehended more easily and probably, over the long haul, tend to provide values that approach the virtually perfect values determined under your model. Thanks again.
August 14, 2002 - tangotiger
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Michael: I agree that the easily most digestable measure of run creating is one that is context-neutral, and therefore, I am not adding anything new here, except more perfect values to use (and adding values to the obscure events like RBOE or BK).
My interest lies "under the hood", and the how and the why.
The important point that I'm also trying to get across is that even if you stick to a linear context-neutral measure like linear weights, that you should use a custom version, based on the run environment. It really makes no sense to apply the same formula to Mel Rojas as to Pedro Martinez. We only do this, because it's easy for us. And if we keep doing it, we will forget to question why we do it. Runs Created, as great as it was then, is an example of this. It completely fails us at the extreme player level.
I think I am in basic agreement with your point of view.
Rob: OUCH! First of all, I did look at the batting order about 2 years ago, and there was an effect of something like 15-20 runs for Rickey Henderson in the leadoff spot. That is, putting a player whose skillset is uniquely qualify for a batting spot that has the most variability (which is Rickey to a tee) with his best season I think had a variability of close to 20 runs (against putting Rickey say in the #5 spot). The #2 hitter also showed great variability, and I concluded elsewhere that in certain (many!) situations, your best hitter should bat #2.
With the MVP/Ichiro thread, I showed that batting great with men on base, or being given alot of men on base will add 10 runs. Give both, and you're close to 20 runs as well.
I really don't need to run a simulator to determine all this though. This is a simple problem of determing the frequency of facing the 24 base-out states, and your success in those same states.
I wouldn't be surprised if you have a player who is ideally qualified for a particular spot (say Ichiro for #2, though I don't know that), who faces more than normal high-leverage situations, who is one of the best hitters in the game and who performs far above his "neutral" performance level would add 30 more runs than if placed in a "neutral" spot and performing at his normal high level. This is of course a rarety, and I would guess in practical terms that 1 standard deviation would be +/- 4 runs.
This issue however is very interesting to look at, but it would be something that I would have to prioritize in with the other equally interesting things I'm looking at.
August 14, 2002 - Michael Humphreys
Tangotiger--Thanks again for your response. Another question. I've begun to wonder recently whether OPS, Runs Created and Linear Weights falsely inflate the impact of an exceptional player, such as Bonds, not only for the (correct) reasons you cite, but also because such formulas effectively treat the offensive events of *one* player as if they were "spread around" the whole batting order. In other words, something tells me that if Barry could "give" his exceptional marginal performance to each of his teammates, SF would score a lot more runs. A regression analysis I recently did of 1999-2001 major league team data team resulted in a formula that, although generally consistent with Linear Weights, yielded a projected run total for San Francisco that was *much* higher than the number of runs they actually scored--the error might have been as much as 40 or 50 runs. Do you have the same suspicion about Barry's *real* marginal contribution that I have? I have the feeling that your model could help us answer that question.
August 14, 2002 - tangotiger
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Michael, I would not look at SF actual run output to determine anything since 6000 PA is not a very small sample.
Anyway, I once ran sims where I had a team of 9 .333 OBA guys, and another team with 8 .300 OBA guys and 1 .600 OBA guy. Overall, both groups are the same. I also made the SLG average about 30 or 40% higher for each player.
I then moved this Bonds type player through the batting order.
From what I remember, I did not notice much difference between the 9 equal guys and the Bonds + bad team.
I'll have to redo that study now that I have better data available. It is again another interesting question that I must look at.
August 14, 2002 - tangotiger
I meant IS a small sample.
August 14, 2002 - Michael Humphreys
Thanks, Tangotiger. The way I was thinking about the question, I wonder what would be the difference in team runs scored if one ran the following two types of simulations: first, simulations with Barry and a lineup of 8 guys each with an average OBP and slugging percentage, then, second, simulations with *nine* guys *each* of whom has an OBP and a slugging percentage equal to the *weighted* average in the first simulation. I think my question was less whether Barry has the same impact with a good or bad team, but rather whether "Barry Plus" eight average guys is less productive than a "Barry Blend" of nine guys each with a bit of Barry's marginal magic. Very interested to see what you think. Thanks.
August 15, 2002 - tangotiger
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Michael, I guess I didn't make myself very clear, since what you replied is exactly what I said.
"Anyway, I once ran sims where I had a team of 9 .333 OBA guys, and another team with 8 .300 OBA guys and 1 .600 OBA guy. Overall, both groups are the same. I also made the SLG average about 30 or 40% higher for each player."
So, the 9 equals of .333 OBA had a team weighted team average of .333 OBA. The 8 equals of .300 OBA plus the Bonds-like .600 OBA would have a team weighted average of .333. So, the first team has the Bonds magic spread around. We are talking about two equal teams in terms of overall talent, except that the spread is far different.
As I mentioned, I don't remember seeing any noticeable difference. It might have been maybe 2% difference (say 15 runs over a season) only to the extent that you'd be able to optimize the batting order so that the .600 guy could do the most damage. I will redo the study at some point in the future though to get more accurate results.
Here is a link to the results of the study I did last year. Please take it as preliminary and crude. Spreading the Bonds magic
August 15, 2002 - Jason
tango I think your crude Bonds analysis goofed up in exactly the types of ways you intended to prevent with your article. On a small scale the answer was more or less answered about having a skewed lineup kind of. The problems I see are that as you so elegantly noted walks are valuable only because others drive you in. By using just OBA you missed that completely as most of Bonds exceptional value is in the walks. While the Giants biggest weakness has been driving him in. Similarily, and I know your going to kick yourself for this one, but if the other 8 guys in the order are all the same, how would you get any difference in production aside from lineup effects, the run environment "Bonds" hit in was the same no matter what. On the other hand think ofthe drastic difference in run environment between those players who hit in front of Barry and those who don't? It would be interesting to analyze all of the line-ups that have been tried to see which would be the most effective based on the run environment concept, and of course see if you could find a better one.
August 15, 2002 - tangotiger
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tango I think your crude Bonds analysis goofed up in exactly the types of ways you intended to prevent with your article.
I was doing my best to avoid lone gunmen types like Bonds. I did that analysis ONLY to show the effect of runs at a team level, with having either 9 guys equals, or 8 guys equals, and 1 outlier, even though overall, they have the same stats. I did not want to talk about the "run environement" because...
The problems I see are that as you so elegantly noted walks are valuable only because others drive you in. By using just OBA you missed that completely as most of Bonds exceptional value is in the walks.
...because Bonds doesn't get to partake in his own run environment. Bonds's run environment, the chances that the runners ahead of him will score, and the chances that he himself will score is derived by all the other batters. You can't measure Bonds value of moving runners over, if those chances include partly Bonds' effect.
So, I was hoping that everyone would overlook this, because the Bonds effect to the run environment is outside the scope here. However, since you brought it up, what you have to do, in this case, is establish a run environment for each batting spot for this particular team, such that if Bonds is the #3 hitter, then the run environment of the #2 hitter includes Bonds, but the run environment of the #3 hitter should "assume" an "average" type of ballplayer.
I went into this into great and deathly details in the batting order thread on fanhome. I really want to avoid talking about that here, because we are going to get away from the basics too fast.
Your point is well-taken and accurate.
It would be interesting to analyze all of the line-ups that have been tried to see which would be the most effective based on the run environment concept, and of course see if you could find a better one.
The run environment concept applies to the basic building blocks of run creation, and I did apply this to the above mentioned thread on batting order.
The correct and proper way to do what you are suggesting is to use the proper model (a simulator) to go through all the variations. The run environment concept with its building blocks of run creation however will reduce the different combinations of players to look for greatly.
For those interested in the batting order thread, drop me a line, and I'll point you there. tom@tangotiger.net
August 15, 2002 - Michael Humphreys
Thanks, Tangotiger. Now I get it.
August 15, 2002 - newsense
The impression I am getting is this: 1) Each defensive environment (pitcher, fielders, ballpark)produces a different set of run expectations for an offensive player. 2) The defensive environment can be described in a 24 cell run expectation table. 3) Each offensive event (1B, 2B, BB, GIDP, etc.) produces one or more(depending on what runners can do) possible transitions in the run expectation table. The event can be valued by the changes in run expectations weighted by the probabilities of initial states and outcomes.
If this is correct, the challenge, it seems to me, is to estimate both the run expectation table and the probabilities of the initial states as a function of the defensive environment. A reasonable proxy for the defensive environment might be the pitcher's park-adjusted runs allowed per 9 innings (RA). These could be estimated for typical levels of RA and interpolated as necessary.
August 15, 2002 - tangotiger
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new: you pretty much have got it, except near the end. The run environment is established by the overall offense + pitching + fielding. You CAN create the run expectancy tables and all that with a little programming. You can also extend this into win expectancy tables, which is where the real fun and learning experience lies.
August 16, 2002 - Voros McCracken
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As far as the "there'll never be a whole team like Bonds" argument, as Tango pointed out, this isn't exactly true for pitchers.
One thing I'm still struggling with for DIPS is converting the numbers into a consistent ERA. It seems (just from eyeballing it) that DIPS might underrate pitcher in the John Wasdin family (good at preventing baserunners, gives up homers at a rate of one per every sip of beer you take) and might overrate the guys from the Shawn Estes/Russ Ortiz mold (lots of baserunners, keeps the ball in the park). A non-linear method might ("might" being the operable word) handle this better.
I currently use linear methods because deriving a new run estimator wasn't the point of the project, and trying the various non-linear methods out there (like James's RC) produced on average worse results across the board. Plus with a few recent changes to DIPS, the linear method solves a lot of DIPS-specific problems (like the balancing act between singles per balls in play and extra-base hits per balls in play) that would have to be answered with a "standard" run estimator.
So I'm always wondering if maybe Base Runs or a relative might make it more effective, or whether I could customize a formula like it for specific use with DIPS (since as I mentioned it has a few unique problems that normally don't come up in run estimator discussions).
August 16, 2002 - Walt Davis
1 - Linear regression is LINEAR. Linear as in a straight line. While there is a somewhat linear relationship between runs and singles, doubles, triples, and walks, there is NOT a linear relationship between runs and HR, or runs and everything else like SB, WP, BK, etc. Baseball is non-linear.
Not exactly. Linear regression is linear because it's "linear in the parameters." There are many ways to model non-linearities among the variables using multiple regression. For example:
y= b0 + b1*X + b1*X^2 + e
(where X^2 means X-squared) will model a nice quadratic curve for you. Or something like:
y= b0 + b1*out2 + b2*out3 + b3*X + b4*X*out2 + b5*X*out3 + e
where out2 and out3 are dummy variables representing different out situations (out1 is the "omitted category") would model 3 different lines for each out situation. Or even something like:
y = b0*X^b1*e
can be converted into a linear in the parameters equation by taking the natural log:
ln(y) = ln(b0) + b1*ln(X) + ln(e).
There's no problem mixing linear and non-linear terms in the same equation (assuming you have the rationale to back it up):
y = b0 + b1*ln(X1) + b2*X2 + b3*ln(X1)*X2 + b4*X3 + b5*X3^2 + e
2 - The independent variables are not independent. There is an interdependence between all these variables. A walk is only worth what it is because of the other things that happen. Linear regression attempts to "freeze" all the other variables when calculating the value of the unfrozen variable. As your run environment increases however, we know that the values of these variables change. Baseball is interdependent.
Your terminology here is unclear. First, "independent variables" need not be independent of one another -- statistically, independence means not correlated, whereas multiple regression is useful precisely because the variables in the model need not be independent of one another (as long as they aren't perfectly correlated). In regression, a coefficient gives you the impact of adding that particularly variable to the model, after having removed all the influence of the other variables from both the dependent variable and the independent variable in question (aka "statistical control"). I don't see any inherent problem with doing that here, but perhaps I'm missing something.
Now it is true that the coefficient estimates "control" for the effects of the other variables, and there may be times when such control is impossible. For example take the equation above:
y = b0 + b1*X + b2*X^2 + e
Now obviously we can't hold X^2 constant while changing X. But this doesn't mean that linear regression is an inappropriate model here, it just means it doesn't make sense to use b1 alone to estimate the impact of a change in X. Instead, using derivatives, we can see that the impact of a change in X is given by:
dy/dx = b1 + 2*b2*X
But I'm really not seeing what this has to do with how slopes change by run environment. Modeling that would suggest other possibilities like a series of dummy variables representing different run-scoring eras or a multi-level random effects model.
3 - Even if you assume for ease that run creation is linear and independent (a safe assumption for very controlled environments), what sample data will you use to run your regression against? Most people will use team season totals, which is an aggregate of individual games, which is an aggregate of individual innings. If you want to run a proper regression analysis, at the very least run it on a game or inning level. Your sample size will explode to something much more reliable.
This is a good point, but of course this has nothing to do with the appropriateness or inappropriateness of multiple regression, but rather with what the proper unit of analysis is. Mulitple regression is an extremely robust technique and sample sizes of a few hundred are quite sufficient -- in fact, sample size plays no role in the assumptions that make "ordinary least squares (OLS) regression" the "best linear unbiased estimator" as long as you have more cases than variables. The sample size will impact the magnitude of your standard errors -- i.e. if you have a small sample, your errors may be so large that you can't say anything with precision.
The inappropriateness of what you're talking about here is in applying coefficients derived from team-level regressions to individual-level data, and you're completely correct that this is statistically inappropriate. On the other hand, I'm fairly confident that aggregation bias is fairly small in the particular example of baseball.
The upshot being that it's not inappropriate to use a linear regression model at the team level nor would it be inappropriate to use one at the game level. It is inappropriate to use the coefficients from one to derive estimated run values for the other.
The above point also suggests that a multi-level or other model that controls for autocorrelation is the appropriate method to use -- such a model would produce similar coefficients but would produce more reliable standard errors.
On the other hand, if we model at the game or inning level, then it is no longer kosher to assume a continuous dependent variable which means that OLS is no longer an appropriate model. A poisson or negative binomial distribution is probably a reasonable fit, and it's really not that hard to fit autocorrelational structures to these models either. (poisson and negative binomial aren't easy to explain, but basically they are appropriate where the possible outcomes consist of a small set of integers -- such as runs scored per game or inning. As the number of possibilities increase, the difference between these distributions and a continuous one are usually trivial. And for the record, poisson and negative binomial models are also linear in the parameters.)
4 - Not accounting for all the variables. Triples have a strong relationship to speed. If you don't have SB in your sets of variables, the regression analysis will award more weight to the triples as a stand-in (because of its relationship to steals). It is possible, based on some samples, that the value of a triple could exceed the value of the HR! What other variables are you not accounting for?
This of course is a problem for any model, including yours. After all, the value of stealing 2nd with 2 outs is dependent not just on who's on the mound, but who's at bat and on-deck, how often do they hit singles, how good are the outfielders' arms, how good is the catcher at blocking the plate, the score and inning of the game (will the outfielders throw home?), who's warming up in the bullpen, etc. That's a lot of conditional RE tables. :-)
Essentially by definition, any model is only good at giving you an "average" or "typical" sort of estimate. To get more precise requires such a high degree of specificity that it's no longer a model.
Note that, technically speaking, not accounting for all variables in a regression is only a problem if (1) those variables impact the dependent variable AND (2) the omitted variables are correlated with one or more independent variables. As you note, triples and steals will be correlated and therefore omitting one would be a problem. However, if things like HBP and interference are truly random, omitting them from the model will not bias the coefficients for variables included in the model.
All this aside, chances are none of this will have much impact. Baseball scoring is not all that variable, most of the important variables have been identified, etc. Chances are the best we can hope for is minor improvement in the level of error. The proof's in the pudding there and I hope a future article will compare the accuracy of your method to the existing ones.
August 16, 2002 - tangotiger
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Voros: BaseRuns does not fall into the trap that RC does nor LWTS. You will find it an appropriate measure, though you lose the great additive advantages that LWTS affords you. Readers of fanhome know what I am talking about here. For the others, please bear with me until the end of the month.
Walt: your terrific dissection deserves a generous response. I will in due course. I do want to make three specific points in the meantime though: 1 - the linear models that are presented with regards to baseball are almost always to the power of 1, and therefore that was my basis for my statement
2 - John Jarvis did a regression analysis on I believe the 1976-2000 TEAM SEASONAL totals and came away with a regression value of .62 (or something) for a double, and .87 (or so) for a triple. Those values are nonsensical in reality. It doesn't matter that his r-squared was 90% or that the standard error was very low. It's wrong. I've done regression analysis on team totals by era, and the results also were strange in some cases.
3 - But I'm really not seeing what this has to do with how slopes change by run environment. Modeling that would suggest other possibilities like a series of dummy variables representing different run-scoring eras or a multi-level random effects model.
Yes, we've tried that, but it doesn't work. As I've shown, each element would have to have its own best-fit linear or parabolic or whatever equation, with respect to the run environment. And the run environment itself would have to be known before the fact. Since we are attempting to determine what is the actual run environment without knowing the number of runs scored, we're stuck. This is where BaseRuns comes in. An elegant, simple and accurate equation.
I will reply to your lengthy post soon. Thanks...
August 16, 2002 - tangotiger
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Not exactly. Linear regression is linear because it's "linear in the parameters." There are many ways to model non-linearities among the variables using multiple regression.
Thanks for clarifying some points. I should then say that baseball is virtually linear in the parameters, but is non-linear to its environment.
In regression, a coefficient gives you the impact of adding that particularly variable to the model, after having removed all the influence of the other variables from both the dependent variable and the independent variable in question (aka "statistical control"). I don't see any inherent problem with doing that here, but perhaps I'm missing something.
The problem is that if you freeze say all the hits, HR, etc, but leave the walk to be the independent variable in question, its value is dependent on the values of hits and HR. So, you freeze hits and HR at say 10 and 2, then the value of 1 walk might be .30, the value of 2 walks might average .32, the value of 3 walks might average .34. Furthermore, if you then freeze the hits at 11 and the HR at 1, all these values change. So, exactly what is the value of the walk?
But I'm really not seeing what this has to do with how slopes change by run environment. Modeling that would suggest other possibilities like a series of dummy variables representing different run-scoring eras or a multi-level random effects model.
Yes. But that's really really hard.
This is a good point, but of course this has nothing to do with the appropriateness or inappropriateness of multiple regression, but rather with what the proper unit of analysis is.
Yes, my third statement was exactly this point.
...That's a lot of conditional RE tables. :-)
Yes, the 24 basic states is the least amount of states that you should accept. 24 x 9 to include the batters would be better. The point of about the fielders etc should be factored into the RE tables before the game so that you have a customized set of a 24 x 9 RE tables that is based on the actual 9 hitters, the pitcher, and the fielders.
However, if things like HBP and interference are truly random, omitting them from the model will not bias the coefficients for variables included in the model.
Things like HBP may be an indication of poor or wild pitching and therefore before we omit anything, we have to determine if they are truly random. Interference I'm sure we can ignore.
All this aside, chances are none of this will have much impact. Baseball scoring is not all that variable, most of the important variables have been identified, etc. Chances are the best we can hope for is minor improvement in the level of error. The proof's in the pudding there and I hope a future article will compare the accuracy of your method to the existing ones.
John Jarvis has gone through the exercise of comparing the various estimators, so I don't need to rehash that.
As I said in the article, as long as you adhere to typical MLB teams who play at the typical OBA levels, then really any run estimator will "work". That's because at a very narrow given specific run environment, what you say is correct, and there is not much variability.
For a "team" like Pedro, this does not apply whatsoever. And Voros is correct that while there is no 9 Bonds hitters, there is effectively 9 Bonds hitters when a really bad pitcher is on the mound. This pitcher would provide his opposition with a Bonds environment.
This is why it is important to understand the building block of run creation, and its high dependence to the run environment (which itself is determined by the various offensive events working together in a non-linear interdependent fashion).
Great comments Walt, and I hope that my lack of knowledge on specific statistical concepts did not take away from the comments I have presented. Thanks.
August 16, 2002 - Mike Tamada
Walt Davis wrote an extremely good explanation of linear regression models, including this passage:
* * * In regression, a coefficient gives you the impact of adding that particularly variable to the model, after having removed all the influence of the other variables from both the dependent variable and the independent variable in question (aka "statistical control"). I * * *
To which Tangotiger replied:
* * * The problem is that if you freeze say all the hits, HR, etc, but leave the walk to be the independent variable in question, its value is dependent on the values of hits and HR. So, you freeze hits and HR at say 10 and 2, then the value of 1 walk might be .30, the value of 2 walks might average .32, the value of 3 walks might average .34. Furthermore, if you then freeze the hits at 11 and the HR at 1, all these values change. So, exactly what is the value of the walk? * * *
That is no problem either, for problems which are "linear" (using the broad definition of "linear in the parameters", as explained by Walt Davis). Tangotiger is describing an interactive situation, where the value of a walk VARIES, depending on whether hits/HR are 10/2, or 11/1, or whatever.
The solution (not that it will always work, but it frequently works) is already contained in Walt Davis' original posting, in his replies to Points 1 and 2, especially his Point 1 reply:
* * * There's no problem mixing linear and non-linear terms in the same equation (assuming you have the rationale to back it up):
y = b0 + b1*ln(X1) + b2*X2 + b3*ln(X1)*X2 + b4*X3 + b5*X3^2 + e * * *
That 4th term, ln(X1)*X2, is an example of looking at interactive terms: you simply multiply X1 times X2 (you can take logs if your model calls for it, or not take logs), and include that interactive term in the righthand side along with X1 and X2.
The resulting regression equation will then measure not just the effects of X1 and X2 (where X1 might be walks for example, and X2 home runs), but ALSO the effects of X1 and X2 for DIFFERENT LEVELS of X1 and X2 (the effects of walks, CORRECTED for the different home run environments).
The model is still linear and thus still has some limitations. Even including these interactive terms (and the 2d-degree polynomial terms, as described by Walt Davis) may still not yield the correct model. Walt Davis's original reply however pointed the way to solutions which might cover those situations, while still remaining within a linear model:
* * * Modeling that would suggest other possibilities like a series of dummy variables representing different run-scoring eras or a multi-level random effects model. * * *
It is possible that the different environments could become so different that separate equations or at least separate parameters might be required to adequately model the differences in those environments. And some situations are simply so non-linear that they require nonlinear models.
But Walt Davis's points and suggestions are very good ones. "Linear" models in fact have a lot of flexibility to cover seemingly non-linear situations.
--MKT
August 16, 2002 - Walt Davis
Mike T said pretty much anything I was going to say to Tango. I will add that I don't mean to be disagreeing with Tango here -- there are lots of ways to model things and linear regression is certainly not always the best solution (as Mike points out). I just wanted to clarify what I thought were some misconceptions about what linear regression can and can't do.
I will simply add that the sorts of autocorrelation models that I discussed are not really all that tough as Tango seems to think. The Stata package has several nice ways of dealing with these models very easily, as does LIMDEP (though LIMDEP is kind of a pain overall). S+, SAS, and maybe even SPSS can also handle these models, as well as more specialized packages such as HLM, MLn, and I'm sure somebody has written Gauss and mathematica modules for them as well. Not that there's any reason for sabermetricians to really dive deep into this stuff ... just that I'm a bit disappointed to find folks who are willing to calculate run expectations tables for something like 60,000 different scenarios would be reluctant to include a few dozen dummy variables and interaction terms in their equations. :-)
August 16, 2002 - Walt Davis
Oh, one other thing...
2 - John Jarvis did a regression analysis on I believe the 1976-2000 TEAM SEASONAL totals and came away with a regression value of .62 (or something) for a double, and .87 (or so) for a triple. Those values are nonsensical in reality. It doesn't matter that his r-squared was 90% or that the standard error was very low. It's wrong. I've done regression analysis on team totals by era, and the results also were strange in some cases.
Well, actually, what the standard error was is always important. And it's important to know what's meant by "low." I'm not familiar with this research, but let's for the moment assume that the standard error for the doubles coefficient was .15. That would make the doubles coefficient highly statistically significant (t>4). But it would also give us a 95% confidence interval of roughly .32 to .92. While .92 seems obviously too high, .32 is too low, but we're pretty confident that the real value of a double is in there somewhere.
Now, I've gotten too technical already, but it's also likely that Jarvis did not control for the autocorrelation of the data. I haven't really explained this yet, so let me try. In OLS regression, we assume that the observations are independent of one another -- i.e. not correlated. In the case of team seasonal regressions, it's not a safe assumption that the 1976 Yankees are independent of the 1977 Yankees. They won't be identical, but to suggest they're unrelated is a stretch.
The thing is, if your data have autocorrelation and you don't correct for it, then your standard error estimates are biased (it may also be the result of omitted variables which could introduce bias). Generally speaking, your standard error estimates will be smaller than they should be and therefore your confidence intervals narrower than they should be. So while the doubles coefficient may not have a reported .15 standard error, after correcting for autocorrelation, we might find out that .15 is its 'true' standard error.
And back to an earlier point. The .62 coefficient for doubles should only be applied to team-seasonal data, not individual data or game data or what have you.
Finally and somewhat self-contradictory, I suspect Tango is right in this particular example. Those coefficients do suggest that there's something wrong with the model. Whether that's omitted variables or the wrong functional form (i.e. we need to add some non-linearities) I can't say.
August 17, 2002 - tangotiger
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Here are the results of a basic linear regression, using team totals from 1969-1999 (808 teams). The second line is the standard error.
outs 1b 2b 3b hr bb k sb cs (0.11) 0.51 0.72 1.10 1.47 0.34 (0.10) 0.21 (0.19) 0.00 0.01 0.03 0.08 0.03 0.01 0.01 0.03 0.07
The r-squared is 95.5%. I still wouldn't take those numbers. They are nice guidelines. Very good ones in fact. But when we have access to play-by-play that tells us exactly what each event, on average, is worth, what does looking at the aggregated seasonal line tell us?
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