Wednesday, November 17, 2021

Crude Team Ratings, 2021

Crude Team Rating (CTR) is my name for a simple methodology of ranking teams based on their win ratio (or estimated win ratio) and their opponents’ win ratios. A full explanation of the methodology is here, but briefly:

1) Start with a win ratio figure for each team. It could be actual win ratio, or an estimated win ratio.

2) Figure the average win ratio of the team’s opponents.

3) Adjust for strength of schedule, resulting in a new set of ratings.

4) Begin the process again. Repeat until the ratings stabilize.

The resulting rating, CTR, is an adjusted win/loss ratio rescaled so that the majors’ arithmetic average is 100. The ratings can be used to directly estimate W% against a given opponent (without home field advantage for either side); a team with a CTR of 120 should win 60% of games against a team with a CTR of 80 (120/(120 + 80)).

First, CTR based on actual wins and losses. In the table, “aW%” is the winning percentage equivalent implied by the CTR and “SOS” is the measure of strength of schedule--the average CTR of a team’s opponents. The rank columns provide each team’s rank in CTR and SOS: 

This was not a great year for the playoff teams representing those that had the strongest W-L records in context as Toronto, Seattle, and Oakland all were significantly better than St. Louis and the world champs from Atlanta. The reason for this quickly becomes apparent when you look at the average aW%s by division (I use aW% to aggregate the performance of multiple teams rather than CTR because the latter is expressed as a win ratio—for a simple example a 90-72 team and a 72-90 team will end up with an average win ratio of 1.025 but their composite and average winning percentages will both be .500):

The NL East, despite being described by at least one feckless prognosticator as “the toughest division in baseball”, was in fact the worst division in baseball by a large margin. Atlanta had the second-weakest SOS in MLB, turning their lackluster 88-73 record into something even less impressive in context. In defense of the Braves, they did lose some significant pieces to injury and have a multi-year track record of being a strong team, as well as looking better when the CTRs are based on expected record (i.e. Pythagenpat using actual runs/runs allowed):

Here we see the Dodgers overtake the Giants by a large margin as MLB’s top team, and it actually lines up better with the playoff participants as the Braves rank highly and the Mariners drop. 

One weakness of CTR is that I use the same starting win metric to calculate both team strength and strength of schedule in one iterative process. But one could make the case that in order to best put W-L records in context, it would make more sense to use each team’s actual W-L record to determine their ranking but use expected W% or some other measure to estimate strength of schedule. Such an approach would simultaneously recognize that a team should be evaluated on the basis of their actual wins and losses (assuming the objective is to measure “championship worthiness” or some similar hard-to-define but intuitively comprehensible standard), but that just because an opponent had “good luck” or were “efficient” in converting runs to wins, they didn’t necessarily represent a stronger foe. This would give a team credit for its own “efficiency” without letting it accrue credit for its opponents “efficiency”

This is what the ratings look like using predicted W% (using runs created/runs created allowed) as the starting point:

Finally, I will close by reverting to CTRs based on actual W-L, but this time taking the playoffs into account. I am not a big fan of including the playoffs - obviously they represent additional games which provide additional information about team quality, but they are played under very different circumstances than regular season games (particular with respect to pitcher usage), and the fact that series are terminated when a team clinches biases the W-L records that emerge from series. Nonetheless, here they are, along with a column showing each team’s percentage change in CTR relative to the regular season W-L only version. Unsurprisingly, the Braves are the big winner, although they still only rank twelfth in MLB. The biggest loser are the Rays, although they still rank #3 and lead the AL. The Dodgers rating actually declined slightly more than the Giants despite winning their series;  they end up with a 6-5 record weighing down their regular season, and with seven of those games coming against teams that are ranked just #12 and #13, the uptick in SOS was not enough to offset it.

Wednesday, November 10, 2021

Hypothetical Award Ballots, 2021


1. LF Randy Arozarena, TB

2. SP Luis Garcia, HOU

3. SP Casey Mize, DET

4. SP Shane McClanahan, TB

5. CF Adolis Garcia, TEX

Arozarena will likely win the award on name recognition if nothing else, but one could very easily make a case for Garcia, who I actually have slightly ahead in RAR 37 to 35. Arozarena’s baserunning and fielding are largely a wash, but Garcia’s RAR using eRA and dRA are slightly lower (32 and 31). That’s enough for me to slide Arozarena ahead. Adolis Garcia is an interesting case, as his standard offensive stats will probably land him high in the voting, but his OBA was only .289 which contributed to him ranking fifth among position players in RAR. But he has excellent fielding metrics (16 DRS and 12 UZR) which gets him back on my ballot. Among honorable mentions, Wander Franco had 21 RAR in just seventy games which is by far the best rate of performance. Ryan Mountcastle’s homer totals will get him on conventional ballots, but he appears to be slight minus as a fielder and was a below average hitter for a first baseman.


1. 2B Jonathan India, CIN

2. SP Trevor Rogers, MIA

3. RF Dylan Carlson, STL

4. SP Ian Anderson, ATL

5. C Tyler Stephenson, CIN

India is the clear choice among position players and Rogers among pitchers, and I see no reason to make any adjustment to their RAR ordering. In fact, it’s pretty much RAR order all the way down.

AL Cy Young:

1. Robbie Ray, TOR

2. Gerrit Cole, NYA

3. Carlos Rodon, CHA

4. Jose Berrios, MIN/TOR

5. Nathan Eovaldi, BOS

The 2021 AL Cy Young race has to be the worst for a non-shortened season in history; while long-term trends are driving down starter workloads, let’s hope that a full previous season will make the 2022 Cy Young race at least a little less depressing. Robbie Ray is the obvious choice, leading the league in innings and ranking second to Carlos Rodon in RRA for a twelve-run RAR lead over Lance McCullers; Ray’s peripherals are less impressive, but are still solid. In addition to the pitchers on my ballot, McCullers, Lance Lynn, and Chris Bassitt could all easily be included as the seven pitchers behind Ray could be reasonably placed in just about any order.

NL Cy Young:

1. Zack Wheeler, PHI

2. Corbin Burnes, MIL

3. Walker Buehler, LA

4. Max Scherzer, WAS/LA

5. Brandon Woodruff, MIL

The NL race is almost the opposite of the AL, with five solid candidates who could be ranked in almost any order, even for a normal season. The easiest way to explain my reasoning is to show each pitcher’s RAR by each of the three metrics:

Wheeler and Burnes get the nods for my top two spots as they were equally good in the peripheral-based metrics, which I feel is sufficient to elevate them above RAR leader Buehler. It’s worth noting that Burnes was the leader in all three of the RA metrics, but Wheeler led the league with 213 innings while Burnes was nineteenth with 167. I suspect Burnes will win the actual vote, and while it’s tempting to side with the guy with spectacular rate stats, a 46 inning gap is enormous.


1. DH/SP Shohei Ohtani, LAA

2. 1B Vladimir Guerrero, TOR

3. 1B Matt Olson, OAK

4. 2B Marcus Semien, TOR

5. 3B Jose Ramirez, CLE

6. SS Carlos Correa, HOU

7. RF Aaron Judge, NYA

8. SP Robbie Ray, TOR

9. RF Kyle Tucker, HOU

10. 2B Brandon Lowe, TB

A first baseman and a DH are the two AL offensive RAR leaders in a season in which no pitcher comes close to a top of the MVP ballot performance. The first baseman hits .305/.394/.589 to the DH’s .256/.372/.589, over 59 additional plate appearances. Under these circumstances, how can the first baseman possible rank second on the ballot, and a distant second at that? When the DH also pitches 130 innings with a RRA 31% lower than league average.

This should seem like a fairly obvious conclusion, and I suspect that Ohtani will handily win the award, but whether out of the need to generate “controversial” content or some other explanation that would indict their mental faculties, talking heads have spent a great deal of time pretending that this was a reasonable debate. I thought it would have been quite fascinating to see Guerrero win the triple crown as a test case of whether twice in a decade the mystical deference to the traditional categories could deny an Angel having a transcendent season of a MVP award.

For the rest of the ballot, if you take the fielding metrics at face value, you can make the case that Marcus Semien was actually the Most Valuable Blue Jay; I do not, with Carlos Correa serving as a prime example. He was +21 in DRS but only +3 in UZR, which is the difference between leading the league in position player bWAR and slotting seventh on my ballot (as he would fall behind Judge if I went solely on UZR). 

The omission of Salvardor Perez will certainly be a deviation from the actual voting. Perez’ OBA was just .315, and despite 48 homers he created “just” 99 runs. Worse yet, his defensive value was -13 runs per Baseball Prospectus. I would rank him not just behind the ten players listed, but Cedric Mullins, Bo Bichette, Xander Bogaerts, Yasmani Grandal, Rafael Devers, and a slew of starting pitchers. I don’t think he was one of the twenty most valuable players in the AL.


1. RF Juan Soto, WAS

2. RF Bryce Harper, PHI

3. SS Trea Turner, WAS/LA

4. SP Zack Wheeler, PHI

5. SS Fernando Tatis, SD

6. SP Corbin Burnes, MIL

7. SP Walker Buehler, LA

8. 1B Paul Goldschmidt, STL

9. RF Tyler O’Neill, STL

10. SP Brandon Woodruff, MIL

Having not carefully examined the statistics during the season, two things surprised me about this race, which it was quickly apparent would come down to the well-matched right fielders, each of whom were among the best young players ever when they burst on the scene, one of whom inherited the other’s job more or less, and both of whom still toil in the same division. The first was that Soto, despite his dazzling OBA, actually ranked a smidge behind Harper offensively; the second was that Soto had a significant advantage in the fielding metrics that elevated him to the top.

Taking the more straightforward comparison first, Soto and Harper had essentially the same batting average (I’m ignoring park factors as WAS and PHI helpfully had a 101 PF, so it won’t change the comparison between the two), .313 to .309. Soto had the clear edge in W+HB rate despite the pair ranking one-two in the NL (22.7% of PA to Harper’s 17.8%), while Harper had a sizeable edge in isolated power (.305 to .221; Harper had only six more homers than Soto, but 22 more doubles). The walks and power essentially cancel out (Harper had a .520 Secondary Average to Soto’s .514, again ranking one-two in the circuit). Each created 116 runs, but despite his OBA edge Soto made twelve more outs as he had fifty six  more plate appearances. That leaves Harper with a narrow two RAR lead.

Fangraphs estimates that Soto’s non-steal baserunning was one run better than average, Harper’s zero. So it comes down to fielding, where Soto has +3 DRS and +2 UZR to Harper’s -6/+2. As a crude combination with regression to put the result on an equal footing with offensive value, I typically sum the two and divide by four, which leaves Soto +1 and Harper -1, to create a total value difference of two runs in favor of Soto.

Obviously, this difference is so narrow that one should barely even feel the need to address a choice to put Harper on top of their ballot. One could easily reason that the Phillies were in the race, and Harper contributed to keeping them in said race with his September/October performance (1157 September OPS). But I have been pretty consistent in not giving any consideration to a team’s position in the standings, so my only sanity check was to take a closer look at fielding using very crude but accessible metrics. My non-scientific impression would be that Harper might be something like a B- fielder and Soto a C.

I looked at the putout rate for each, dividing putouts by team AB – HR – K – A + SF (this essentially defines the outfielder’s potential plays as any balls including hits put in play, removing plays actually made by infielders of which assists serve as an approximation. Obviously there is much that is not considered even that might be approximated from the standard Baseball Guide data, like actual GB/FB ratio, handedness of pitchers and opposing batters, etc.) and multiplying by each player’s innings in the outfield divided by the team’s total innings. Viewed in this manner, Soto made a putout on 13.7% of potential plays to Harper’s 11.2%. 

A second crude check which may be free of unknown team-level biases but that introduces its own problems in that the other players are very different is to compare each player’s putout rate to that of his team’s other right fielders. For this, we can just look at per 9 innings as we have to assume that the other team level inputs in our putout % (HR, K, A, SF) were uniformly distributed between Soto/Harper’s innings and those played by other Nationals/Phillies right fielders. Soto recorded 2.17 PO/9 innings while other Nationals RF recorded 1.98: Harper 1.64 to other Phillies 1.51, so Soto recorded 10% more putouts than his teammates and Harper 9%.

Is any of this remotely conclusive? Of course not, but it is sufficient to convince me that the proposition that “Juan Soto was two runs more valuable than Bryce Harper in the field” is reasonable, and that in turn is enough to make Soto seem a whisker more valuable than Harper. It’s a very close race, much more interesting than the more discussed AL race (which in truth is interesting only because of Ohtani’s remarkable season and not any comparison to other players). 

I think the rest of the ballot follows RAR very closely with the pitchers mixed in. Max Scherzer ranked ahead of Brandon Woodruff on my Cy Young list, but they flip here as Woodruff was merely bad offensively (-1 run created); Scherzer didn’t reach base in 59 plate appearances (-5).

Wednesday, October 20, 2021

Rate Stat Series, pt. 14: Relativity for the Theoretical Team Framework

Before jumping into win-equivalent rate stats for the theoretical team framework, I think it would be helpful to re-do our theoretical team calculations on a purely rate basis. This is, after all, a rate stat series. In discussing the TT framework in pts. 9-11, I started by using the player’s PA to define the PA of the team, as Bill James chose to do with his TT Runs Created. This allowed our initial estimate of runs created or RAA to remain grounded in the player’s actual season. 

An alternative (and as we will see, equivalent) approach would be to eschew all of the “8*PA” and just express everything in rates to begin with. When originally discussing TT, I didn’t show it that way, but maybe I should have. I found that my own thinking when trying to figure out the win equivalent TT rates was greatly aided by walking through this process first.

Again, everything is equivalent to what we did before – if you just divide a lot of those equations by PA, you will get to the same place a lot quicker than I’m going to. The theoretical team framework we’re working with assumes that the batter gets 1/9 of the PA for the theoretical team. It’s also mathematically true that for Base Runs:

BsR/PA = (A*B/(B+C) + D)/PA = (A/PA)*(B/PA)/(B/PA + C/PA) + D/PA

If for the sake of writing formulas we rename A/PA as ROBA (Runners On Base Average), B/PA as AF (Advancement Factor; I’ve been using this abbreviation long before it came into mainstream usage in other contexts), C/PA as OA (Out Average), and D/PA as HRPA (Home Runs/PA), we can then write:


Since it is also true that R/O = R/PA/(1 – OBA), in this case it is true that:

BsR/O = (BsR/PA)/OA

We can use these equations to calculate the Base Runs per out for a theoretical team (I’m going to skip over “reference team” notation and just assume that the reference team is a league average team):

TT_ROBA = 1/9*ROBA + 8/9*LgROBA

TT_AF = 1/9*AF + 8/9*LgAF

TT_OA = 1/9*OA + 8/9*LgOA

TT_HRPA = 1/9*HRPA + 8/9*LgHRPA



Here’s a sample calculation for 1994 Frank Thomas:

To calculate a win-equivalent rate stat, we can use the TT_BsR/O figure as a starting point (it suggests that a theoretical team of 1/9 Thomas and 8/9 league average would score .2344 runs/out). We don’t need to go through this additional calculation, though; when we calculated R+/O+ (or R+/PA+, RAA+/O+, or RAA+/PA+), we already had everything we needed for this calculation.

You will see if you do the math that:

TT_BsR/O = (RAA+/O+)*(1/9) + LgR/O


TT_BsR/O = (R+/O+)*(1/9) + (LgR/O)*(8/9)


TT_BsR/O = (RAA+/PA+)/(1 – LgOBA)*(1/9) + LgR/O


TT_BsR/O = (R+/PA+)/(1 - LgOBA)*(1/9) + (LgR/O)*(8/9)

You could view this as a validation of the R+/O+ approach, as it does what it set out to do, which is to isolate the batter’s contribution to the theoretical team’s runs/out. Once we’ve established the team’s runs/out, it is pretty simple to convert to wins. I will just give formulas as I think they are pretty self-explanatory:



TT_x = TT_RPG^.29

TT_W% = (TT_BsR/G)^TT_x/((TT_BsR/G)^TT_x + (LgR/G)^TT_x)

Walking through this for the Franks, we have:

One thing to note here is that if we look at the theoretical team’s R/O (or R/G) relative to the league average, subtract one, multiply by nine, and add one back in, we will Thomas and Robinson’s relative R+/O+. This is not a surprising result given what we saw above regarding the relationship between R+/O+ and theoretical team R/O.

We now have a W% for the theoretical team, which we could leave alone as a rate stat, but it’s not very satisfying to me to have an individual rate stat expressed as a team W%. If we subtract .5, we have WAA/Team G; we could interpret this as meaning that Thomas is estimated to add .0609 wins per game and Robinson .0546 to a theoretical team on which they get 1/9 of PA. Another option would be to convert this WAA back to a total, defining “games” as PA/Lg(PA/G), and then we could have WAA+/PA+ or WAA+/O+ as rates. 

In keeping with the general format established in this series, though, my final answer for a win-equivalent rate stat for the TT framework will be to convert the winning percentage (actually, we’ll use win ratio since it  makes the math easier) back to the reference environment, and calculate a relative adjusted R+/O+. Since everything will be on an outs basis (as we’re using O+), we don’t need to worry about league PA/G when calculating our relative adjusted R+/O+.

Instead of calculating TT_W%, we could have left it in the form of team win ratio:

TT_WR = ((TT_BsR/G)/(LgR/G))^(TT_x)

We can convert this back to an equivalent run ratio in the reference environment (which for this series we’ve defined as having Pythagorean exponent r = 1.881) by solving for AdjTT_RR in the equation:

TT_WR = AdjTT_RR^r


AdjTT_RR = TT_WR^(1/r)

We could convert this run ratio back to a team runs/game in the reference environment, and then to a team runs/out, and then use our equation for tying individual R+/O+ to theoretical team R/O to get an equivalent R+/O+ ratio. But why bother with all that, when we will just end up dividing it by the reference environment R/O to get our relative adjusted R+/O+? I noted above that there was a direct relationship between the theoretical team’s run ratio (which is equal to the theoretical team’s R/O divided by league R/O) and the batter’s relative R+/O+:

Rel R+/O+ = (TT_RR – 1)*9 + 1

So our Relative Adjusted R+/O+ can be calculated as:

RelAdj R+/O+ = (AdjTT_RR  - 1)*9 + 1

I brought back our original relative R+/O+ (prior to going through the win-equivalent math) for comparison. Thomas gains slightly and Robinson loses more, because the value of his relative runs is lower in a high scoring environment. This is a similar conclusion to what we saw when comparing relative R+/PA and the relative adjusted R+/PA for Robinson and Thomas. Nominal runs are more valuable when the run scoring environment is lower, because it takes fewer marginal runs to create a marginal win. Relative runs are more valuable when the scoring environment is higher, because the win ratio expected to result from a given run ratio increases due to the higher Pythagenpat exponent.

At this point, we have exhausted my thoughts and ideas concerning the theoretical issues in designing individual batter rate stats. Next time I will discuss mixing up our rate stats and the frameworks within which I assert each should ideally be used.

Tuesday, October 05, 2021

End of Season Statistics, 2021

While this edition of End of Season Statistics will more closely resemble the reports I published through 2019 than the 2020 edition did, there are still a number of issues created by the revised rules, particularly the extra innings rule and seven-inning doubleheaders. Seeing as that both of these changes could be walked back for 2022, I have not attempted to revise my approach to take them into account – should they become permanent, then and only then will I invest time in trying to make the necessary adjustments (some of which I outlined here) to fit the data they produce within traditional sabermetric structures.

In the mean time, there will be three main consequences of the rules:

1) While I will provide relief pitcher statistics this year, I will base the value metrics on eRA rather than RA or ERA. RA is hopelessly polluted by the Manfred runners, while ERA is hopelessly polluted by virtue of being ERA. While I would prefer to base the value metric on a measure that reflects runs actually allowed, they are messy enough to begin with in the case of relievers that I am not too concerned about it.

2) When computing value metrics for starting pitchers, I will be comparing their RRA to the estimated league average RA rather than the actual one, since the latter is polluted by Manfred runners even though starters’ statistics themselves are immune from the impact. 

3) Team run per game metrics will be expressed per 9 innings (27 outs), and a per 9 innings approach will be used to calculate expected winning percentages. As such, these will not exactly be an attempt to estimate what any given team’s W% should have been, but rather a theoretical estimate of what their W% would have been had they been playing under normal rules. For actual runs and runs allowed, these will still be distorted by Manfred runners, but accounting for that is again much more trouble than a (hopefully) two-year interlude justifies.

The data comes from a number of different sources. Most of the data comes from Baseball-Reference; I will try to note exceptions as they come up.

The basic philosophy behind these stats is to use the simplest methods that have acceptable accuracy. Of course, "acceptable" is in the eye of the beholder, namely me. I use Pythagenpat not because other run/win converters, like a constant RPW or a fixed exponent are not accurate enough for this purpose, but because it's mine and it would be kind of odd if I didn't use it.

If I seem to be a stickler for purity in my critiques of others' methods, I'd contend it is usually in a theoretical sense, not an input sense. So when I exclude hit batters, I'm not saying that hit batters are worthless or that they *should* be ignored; it's just easier not to mess with them and not that much less accurate (note: hit batters are actually included in the offensive statistics now).

I also don't really have a problem with people using sub-standard methods (say, Basic RC) as long as they acknowledge that they are sub-standard. If someone pretends that Basic RC doesn't undervalue walks or cause problems when applied to extreme individuals, I'll call them on it; if they explain its shortcomings but use it regardless, I accept that. Take these last three paragraphs as my acknowledgment that some of the statistics displayed here have shortcomings as well, and I've at least attempted to describe some of them in the discussion below.

The League spreadsheet is pretty straightforward--it includes league totals and averages for a number of categories, most or all of which are explained at appropriate junctures throughout this piece. The advent of interleague play has created two different sets of league totals--one for the offense of league teams and one for the defense of league teams. Before interleague play, these two were identical. I do not present both sets of totals (you can figure the defensive ones yourself from the team spreadsheet, if you desire), just those for the offenses. The exception is for the defense-specific statistics, like innings pitched and quality starts. The figures for those categories in the league report are for the defenses of the league's teams. However, I do include each league's breakdown of basic pitching stats between starters and relievers (denoted by "s" or "r" prefixes), and so summing those will yield the totals from the pitching side. 

I added a column this year for “ActO”, which is actual (rather than estimated) outs made by the team offensively. This can be determined from the official statistics as PA – R – LOB. I have then replaced the column I usually show for league R/G (“N”) with R/9, which is actually R*27/ActO, which is equivalent to R*9/IP. This restates the league run average in the more familiar per nine innings. I’ve done the same for “OG”, which is Outs/Game but only for those outs I count in the individual hitter’s stats (AB – H + CS) ,“PA/G”, which is normally just (AB + W)/G, and “KG” and “WG” (normally just K/G and W/G) – these are now “O/9”, “PA/9”, still “KG”/”WG” and are per 27 actual outs.

The Team spreadsheet focuses on overall team performance--wins, losses, runs scored, runs allowed. The columns included are: Park Factor (PF), Winning Percentage (W%), Expected W% (EW%), Predicted W% (PW%), wins, losses, runs, runs allowed, Runs Created (RC), Runs Created Allowed (RCA), Home Winning Percentage (HW%), Road Winning Percentage (RW%) [exactly what they sound like--W% at home and on the road], R/9, RA/9, Runs Created/9 (RC/9), Runs Created Allowed/9 (RCA/9), and Runs Per Game (the average number of runs scored an allowed per game). For the offensive categories, runs/9 are based on runs per 27 actual outs; for pitching categories, they are runs/9 innings.

I based EW% and PW% on R/9 and RA/9 (and RC/9 and RCA/9) rather than the actual runs totals. This means that what they are not estimating what a team’s winning percentage should have been in the actual game constructions that they played, but what they should have been if playing nine inning games but scoring/allowing runs at the same rate per inning. EW%, which is based on actual R and RA, is also polluted by inflated runs in extra inning games; PW%, which is based on RC and RCA, doesn’t suffer from this distortion.

The runs and Runs Created figures are unadjusted, but the per-game averages are park-adjusted, except for RPG which is also raw. Runs Created and Runs Created Allowed are both based on a simple Base Runs formula. The formula is:

A = H + W - HR - CS

B = (2TB - H - 4HR + .05W + 1.5SB)*.76

C = AB - H

D = HR

Naturally, A*B/(B + C) + D.

Park factors are based on five years of data when applicable (so 2017 - 2021), include both runs scored and allowed, and they are regressed towards average (PF = 1), with the amount of regression varying based on the number of games in total in the sample. There are factors for both runs and home runs. The initial PF (not shown) is:

iPF = (H*T/(R*(T - 1) + H) + 1)/2

where H = RPG in home games, R = RPG in road games, T = # teams in league (14 for AL and 16 for NL). Then the iPF is converted to the PF by taking x*iPF + (1-x), where x = .1364*ln(G/162) + .5866. I will expound upon how this formula was derived in a future post. 

It is important to note, since there always seems to be confusion about this, that these park factors already incorporate the fact that the average player plays 50% on the road and 50% at home. That is what the adding one and dividing by 2 in the iPF is all about. So if I list Fenway Park with a 1.02 PF, that means that it actually increases RPG by 4%.

In the calculation of the PFs, I did not take out “home” games that were actually at neutral sites (of which there were a rash in 2020). The Blue Jays multiple homes make things very messy, so I just used their 2021 data only. 

There are also Team Offense and Defense spreadsheets. These include the following categories:

Team offense: Plate Appearances, Batting Average (BA), On Base Average (OBA), Slugging Average (SLG), Secondary Average (SEC), Walks and Hit Batters per At Bat (WAB), Isolated Power (SLG - BA), R/G at home (hR/G), and R/G on the road (rR/G) BA, OBA, SLG, WAB, and ISO are park-adjusted by dividing by the square root of park factor (or the equivalent; WAB = (OBA - BA)/(1 - OBA), ISO = SLG - BA, and SEC = WAB + ISO).

Team defense: Innings Pitched, BA, OBA, SLG, Innings per Start (IP/S), Starter's eRA (seRA), Reliever's eRA (reRA), Quality Start Percentage (QS%), RA/G at home (hRA/G), RA/G on the road (rRA/G), Battery Mishap Rate (BMR), Modified Fielding Average (mFA), and Defensive Efficiency Record (DER). BA, OBA, and SLG are park-adjusted by dividing by the square root of PF; seRA and reRA are divided by PF.

The three fielding metrics I've included are limited it only to metrics that a) I can calculate myself and b) are based on the basic available data, not specialized PBP data. The three metrics are explained in this post, but here are quick descriptions of each:

1) BMR--wild pitches and passed balls per 100 baserunners = (WP + PB)/(H + W - HR)*100

2) mFA--fielding average removing strikeouts and assists = (PO - K)/(PO - K + E)

3) DER--the Bill James classic, using only the PA-based estimate of plays made. Based on a suggestion by Terpsfan101, I've tweaked the error coefficient. Plays Made = PA - K - H - W - HR - HB - .64E and DER = PM/(PM + H - HR + .64E)

Next are the individual player reports. I defined a starting pitcher as one with 15 or more starts. All other pitchers are eligible to be included as a reliever. If a pitcher has 40 appearances, then they are included. Additionally, if a pitcher has 50 innings and less than 50% of his appearances are starts, he is also included as a reliever (this allows some swingmen type pitchers who wouldn’t meet either the minimum start or appearance standards to get in). This would be a good point to note that I didn't do much to adjust for the opener--I made some judgment calls (very haphazard judgment calls) on which bucket to throw some pitchers in. This is something that I should definitely give some more thought to in coming years.

For all of the player reports, ages are based on simply subtracting their year of birth from 2021. I realize that this is not compatible with how ages are usually listed and so “Age 27” doesn’t necessarily correspond to age 27 as I list it, but it makes everything a heckuva lot easier, and I am more interested in comparing the ages of the players to their contemporaries than fitting them into historical studies, and for the former application it makes very little difference. The "R" category records rookie status with a "R" for rookies and a blank for everyone else. Also, all players are counted as being on the team with whom they played/pitched (IP or PA as appropriate) the most.

For relievers, the categories listed are: Games, Innings Pitched, estimated Plate Appearances (PA), Run Average (RA), Relief Run Average (RRA), Earned Run Average (ERA), Estimated Run Average (eRA), DIPS Run Average (dRA), Strikeouts per Game (KG), Walks per Game (WG), Guess-Future (G-F), Inherited Runners per Game (IR/G), Batting Average on Balls in Play (%H), Runs Above Average (RAA), and Runs Above Replacement (RAR).

IR/G is per relief appearance (G - GS); it is an interesting thing to look at, I think, in lieu of actual leverage data. You can see which closers come in with runners on base, and which are used nearly exclusively to start innings. Of course, you can’t infer too much; there are bad relievers who come in with a lot of people on base, not because they are being used in high leverage situations, but because they are long men being used in low-leverage situations already out of hand.

For starting pitchers, the columns are: Wins, Losses, Innings Pitched, Estimated Plate Appearances (PA), RA, RRA, ERA, eRA, dRA, KG, WG, G-F, %H, Pitches/Start (P/S), Quality Start Percentage (QS%), RAA, and RAR. RA and ERA you know--R*9/IP or ER*9/IP, park-adjusted by dividing by PF. The formulas for eRA and dRA are based on the same Base Runs equation and they estimate RA, not ERA.

* eRA is based on the actual results allowed by the pitcher (hits, doubles, home runs, walks, strikeouts, etc.). It is park-adjusted by dividing by PF.

* dRA is the classic DIPS-style RA, assuming that the pitcher allows a league average %H, and that his hits in play have a league-average S/D/T split. It is park-adjusted by dividing by PF.

The formula for eRA is:

A = H + W - HR

B = (2*TB - H - 4*HR + .05*W)*.78

C = AB - H = K + (3*IP - K)*x (where x is figured as described below for PA estimation and is typically around .93) = PA (from below) - H - W

eRA = (A*B/(B + C) + HR)*9/IP

To figure dRA, you first need the estimate of PA described below. Then you calculate W, K, and HR per PA (call these %W, %K, and %HR). Percentage of balls in play (BIP%) = 1 - %W - %K - %HR. This is used to calculate the DIPS-friendly estimate of %H (H per PA) as e%H = Lg%H*BIP%.

Now everything has a common denominator of PA, so we can plug into Base Runs:

A = e%H + %W

B = (2*(z*e%H + 4*%HR) - e%H - 5*%HR + .05*%W)*.78

C = 1 - e%H - %W - %HR

cRA = (A*B/(B + C) + %HR)/C*a

z is the league average of total bases per non-HR hit (TB - 4*HR)/(H - HR), and a is the league average of (AB - H) per game.

Also shown are strikeout and walk rate, both expressed as per game. By game I mean not nine innings but rather the league average of PA/G. I have always been a proponent of using PA and not IP as the denominator for non-run pitching rates, and now the use of per PA rates is widespread. Usually these are expressed as K/PA and W/PA, or equivalently, percentage of PA with a strikeout or walk. I don’t believe that any site publishes these as K and W per equivalent game as I am here. This is not better than K%--it’s simply applying a scalar multiplier. I like it because it generally follows the same scale as the familiar K/9.

To facilitate this, I’ve finally corrected a flaw in the formula I use to estimate plate appearances for pitchers. Previously, I’ve done it the lazy way by not splitting strikeouts out from other outs. I am now using this formula to estimate PA (where PA = AB + W):

PA = K + (3*IP - K)*x + H + W

Where x = league average of (AB - H - K)/(3*IP - K)

Then KG = K*Lg(PA/G) and WG = W*Lg(PA/G).

G-F is a junk stat, included here out of habit because I've been including it for years. It was intended to give a quick read of a pitcher's expected performance in the next season, based on eRA and strikeout rate. Although the numbers vaguely resemble RAs, it's actually unitless. As a rule of thumb, anything under four is pretty good for a starter. G-F = 4.46 + .095(eRA) - .113(K*9/IP). It is a junk stat. JUNK STAT JUNK STAT JUNK STAT. Got it?

%H is BABIP, more or less--%H = (H - HR)/(PA - HR - K - W), where PA was estimated above. Pitches/Start includes all appearances, so I've counted relief appearances as one-half of a start (P/S = Pitches/(.5*G + .5*GS). QS% is just QS/(G - GS); I don't think it's particularly useful, but Doug's Stats include QS so I include it.

I've used a stat called Relief Run Average (RRA) in the past, based on Sky Andrecheck's article in the August 1999 By the Numbers; that one only used inherited runners, but I've revised it to include bequeathed runners as well, making it equally applicable to starters and relievers. One thing that's become more problematic as time goes on for calculating this expanded metric is the sketchy availability of bequeathed runner data for relievers. As a result, only bequeathed runners left by starters (and "relievers" when pitching as starters) are taken into account here. I use RRA as the building block for baselined value estimates for all pitchers. I explained RRA in this article, but the bottom line formulas are:

BRSV = BRS - BR*i*sqrt(PF)

IRSV = IR*i*sqrt(PF) - IRS

RRA = ((R - (BRSV + IRSV))*9/IP)/PF

Given the difficulties of looking at the league average of actual runs due to Manfred rules, I decided to use eRA to calculate the baselined metrics for relievers. So they are no longer based on actual runs allowed by the pitcher, but rather on the component statistics. For starters, I will use the actual runs allowed in the form of RRA, but compared to the league average eRA. Starters’ statistics are not influenced by the Manfred runners, but the league average RA is still artificially inflated by them, so the league eRA should be a better measure of what the league average RRA would be in lieu of Manfred runners. I say “should” as this assumes that the eRA formula is properly calibrated, and it’s hard to calibrate any runs created formula when you don’t know what the league average runs should be. I remain unconvinced that most saberemtricians have fully grasped all of the implications of the Manfred runners on the 2020-2021 statistics, and if these rules are maintained going forward it will require much more effort to maintain basic sabermetric measures. In any event, the RAA/RAR formulas I’m using are:

RAA (relievers) = (.951*Lg(eRA) - eRA)*IP/9

RAA (starters) = (1.025*Lg(eRA) - eRA)*IP/9

RAR (relievers) = (1.11*Lg(eRA) - RRA)*IP/9

RAR (starters) = (1.28*Lg(eRA) - RRA)*IP/9

All players with 250 or more plate appearances (official, total plate appearances) are included in the Hitters spreadsheets (along with some players close to the cutoff point who I was interested in). Each is assigned one position, the one at which they appeared in the most games. The statistics presented are: Games played (G), Plate Appearances (PA), Outs (O), Batting Average (BA), On Base Average (OBA), Slugging Average (SLG), Secondary Average (SEC), Runs Created (RC), Runs Created per Game (RG), Speed Score (SS), Hitting Runs Above Average (HRAA), Runs Above Average (RAA), and Runs Above Replacement (RAR).

Starting in 2015, I'm including hit batters in all related categories for hitters, so PA is now equal to AB + W+ HB. Outs are AB - H + CS. BA and SLG you know, but remember that without SF, OBA is just (H + W + HB)/(AB + W + HB). Secondary Average = (TB - H + W + HB)/AB = SLG - BA + (OBA - BA)/(1 - OBA). I have not included net steals as many people (and Bill James himself) do, but I have included HB which some do not.

BA, OBA, and SLG are park-adjusted by dividing by the square root of PF. This is an approximation, of course, but I'm satisfied that it works well (I plan to post a couple articles on this some time during the offseason). The goal here is to adjust for the win value of offensive events, not to quantify the exact park effect on the given rate. I use the BA/OBA/SLG-based formula to figure SEC, so it is park-adjusted as well.

Runs Created is actually Paul Johnson's ERP, more or less. Ideally, I would use a custom linear weights formula for the given league, but ERP is just so darn simple and close to the mark that it’s hard to pass up. I still use the term “RC” partially as a homage to Bill James (seriously, I really like and respect him even if I’ve said negative things about RC and Win Shares), and also because it is just a good term. I like the thought put in your head when you hear “creating” a run better than “producing”, “manufacturing”, “generating”, etc. to say nothing of names like “equivalent” or “extrapolated” runs. None of that is said to put down the creators of those methods--there just aren’t a lot of good, unique names available.

For 2015, I refined the formula a little bit to:

1. include hit batters at a value equal to that of a walk

2. value intentional walks at just half the value of a regular walk

3. recalibrate the multiplier based on the last ten major league seasons (2005-2014)

This revised RC = (TB + .8H + W + HB - .5IW + .7SB - CS - .3AB)*.310

RC is park adjusted by dividing by PF, making all of the value stats that follow park adjusted as well. RG, the Runs Created per Game rate, is RC/O*26. For a very long time, dating back to the Jamesian era, 25.5 has been a good approximation for the number of (AB – H + CS) per game, but it has been creeping up, and per 9 innings this year it was right around 26, so I am using that value now. 

I do not believe that outs are the proper denominator for an individual rate stat, but I also do not believe that the distortions caused are that bad. (I still intend to finish my rate stat series and discuss all of the options in excruciating detail, but alas you’ll have to take my word for it now).

Several years ago I switched from using my own "Speed Unit" to a version of Bill James' Speed Score; of course, Speed Unit was inspired by Speed Score. I only use four of James' categories in figuring Speed Score. I actually like the construct of Speed Unit better as it was based on z-scores in the various categories (and amazingly a couple other sabermetricians did as well), but trying to keep the estimates of standard deviation for each of the categories appropriate was more trouble than it was worth.

Speed Score is the average of four components, which I'll call a, b, c, and d:

a = ((SB + 3)/(SB + CS + 7) - .4)*20

b = sqrt((SB + CS)/(S + W))*14.3

c = ((R - HR)/(H + W - HR) - .1)*25

d = T/(AB - HR - K)*450

James actually uses a sliding scale for the triples component, but it strikes me as needlessly complex and so I've streamlined it. He looks at two years of data, which makes sense for a gauge that is attempting to capture talent and not performance, but using multiple years of data would be contradictory to the guiding principles behind this set of reports (namely, simplicity. Or laziness. You're pick.) I also changed some of his division to mathematically equivalent multiplications.

The baselined stats are calculated in the same basic manner the pitcher stats are, using the league average RG:

HRAA = (RG – LgRG)*O/26

RAA = (RG – LgRG*PADJ)*O/26

RAR = (RG – LgRG*PADJ*.73)*O/26

PADJ is the position adjustment, based on 2010-2019 offensive data. For catchers it is .92; for 1B/DH, 1.14; for 2B, .99; for 3B, 1.07; for SS, .95; for LF/RF, 1.09; and for CF, 1.05. As positional flexibility takes hold, fielding value is better quantified, and the long-term evolution of the game continues, it's right to question whether offensive positional adjustments are even less reflective of what we are trying to account for than they were in the past. But while I do not claim that the relationship is or should be perfect, at the level of talent filtering that exists to select major leaguers, there should be an inverse relationship between offensive performance by position and the defensive responsibilities of the position. Not a perfect one, but a relationship nonetheless. An offensive positional adjustment than allows for a more objective approach to setting a position adjustment. Again, I have to clarify that I don’t think subjectivity in metric design is a bad thing - any metric, unless it’s simply expressing some fundamental baseball quantity or rate (e.g. “home runs” or “on base average”) is going to involve some subjectivity in design (e.g linear or multiplicative run estimator, any myriad of different ways to design park factors, whether to include a category like sacrifice flies that is more teammate-dependent).

The replacement levels I have used here are very much in line with the values used by other sabermetricians. This is based both on my own "research", my interpretation of other's people research, and a desire to not stray from consensus and make the values unhelpful to the majority of people who may encounter them.

Replacement level is certainly not settled science. There is always going to be room to disagree on what the baseline should be. Even if you agree it should be "replacement level", any estimate of where it should be set is just that--an estimate. Average is clean and fairly straightforward, even if its utility is questionable; replacement level is inherently messy. So I offer the average baseline as well.

For position players, replacement level is set at 73% of the positional average RG (since there's a history of discussing replacement level in terms of winning percentages, this is roughly equivalent to .350). For starting pitchers, it is set at 128% of the league average RA (.380), and for relievers it is set at 111% (.450).

The spreadsheets are published as Google Spreadsheets, which you can download in Excel format by changing the extension in the address from "=html" to "=xlsx", or in open format as "=ods", or in csv as "=csv". That way you can download them and manipulate things however you see fit.

League -- 2021

Park Factors -- 2021

Teams -- 2021

Team Defense -- 2021

Team Offense -- 2021

AL Relievers -- 2021

NL Relievers -- 2021

AL Starters -- 2021

NL Starters -- 2021

AL Hitters -- 2021

NL Hitters -- 2021

Monday, October 04, 2021

Crude Playoff Odds--2021

These are very simple playoff odds, based on my crude rating system for teams using an equal mix of W%, EW% (based on R/RA), PW% (based on RC/RCA), and 69 games of .500. They account for home field advantage by assuming a .500 team wins 54.2% of home games (major league average 2006-2015). They assume that a team's inherent strength is constant from game-to-game. They do not generally account for any number of factors that you would actually want to account for if you were serious about this, including but not limited to injuries, the current construction of the team rather than the aggregate seasonal performance, pitching rotations, estimated true talent of the players, etc.

The CTRs that are fed in are:

Wildcard game odds (the least useful since the pitching matchups aren’t taken into account, and that matters most when there is just one game):



World Series:

Everything combined:

If the Dodgers win the wildcard game, they become the World Series favorites at 19.9%, with the Giants falling to 19.0%; the Rays fall to 16.1%, so the Dodgers don't have a huge impact on the AL odds (the Dodgers are given a 32.6% to win the pennant should they win the wildcard game). If the Cardinals win, the Giants jump to 25.9% and the Rays to 17.6% (of course the Giants benefit greatly because they have an estimated 68% chance to beat the Cardinals in the NLDS but only 50% to beat the Dodgers). Ranging into the realm of the subjective, I personally favor Houston to win the AL pennant and think Milwaukee will benefit from concentrating innings in front-line pitchers (even sans Devin Williams) and from being on the opposite side of the bracket from the NL West.

I don't have the energy for a rant about what a preposterous proposition it is to make the Dodgers play the Cardinals, or about how the much-hyped four-way battle for the AL wildcard yesterday would never happen under Rob Manfred's desired system (or how if it did it would be between teams struggling to reach .500). The playoffs simultaneously manage to be one of the best and worst things about baseball, and every expansion will serve to enhance the latter.

Wednesday, September 29, 2021

Rate Stat Series, pt. 13: Relativity for the Linear Weights Framework

Of the three frameworks for evaluating individual offense, linear weights offers the simplest calculation of runs created or RAA, but will be the hardest to convert to a win-equivalent rate – mentally if not computationally. In order to do this, we need to consider what our metrics actually represent and make our choices accordingly. The path that I am going to suggest is not inevitable – it makes sense to me, but there are certainly valid alternative paths.

In attempting to measure the win value of a batter’s performance in the linear weights framework, we could construct a theoretical team and measure his win impact on it. In so doing, one could argue that the batter’s tertiary impact (which would be ignored under such an approach) is immaterial, perhaps even illusory, and that the process of converting runs to wins is independent from the development of the run estimate. Thus we could use a static approach for estimating runs and a dynamic team approach for converting those runs to wins.

I would argue in turn that the most consistent approach is to continue to operate under the assumption that linear weights represents a batter’s impact on a team that is average once he is added to it, and thus not allow any dynamism in the runs to wins conversion. Since under this school of thought all teams are equal, whether we add Frank Thomas or Matt Walbeck, there is no need to account for how those players change the run environment and the run/win conversion – because they both ultimately operate in the same run environment.

One could argue that I am taking a puritanical viewpoint, and that this would become especially clear in a case in which one compared the final result of the linear weights framework to the final result of the theoretical team framework. As we’ve seen, RAA is very similar between the two approaches, but the run/win conversions will diverge more if in one case we ignore the batter’s impact on the run environment. In any event, the methodology we’ll use for the theoretical team framework will be applicable to linear weights as well, if you desire to use it.

Since we will not be modeling any dynamic impact of the batter upon the team’s run environment, it is an easy choice to start with RAA and convert it to wins above average (WAA) by dividing by a runs per win (RPW) value. An example of this is the rule of thumb that 10 runs = 1 win, so 50 RAA would be worth 5 WAA. 

There are any number of methods by which we could calculate RPW, and a couple philosophical paths to doing so. On the latter, I’m assuming that we want our RPW to be represent the best estimate of the number of marginal runs it would take for a .500 team (or more precisely a team with R = RA) to earn a marginal win. Since I’ve presumed that Pythagenpat is the correct run to win conversion, the most consistent is to use the RPW implied by Pythagenpat, which is:

RPW = 2*RPG^(1 – z) where z is the Pythagenpat exponent

so when z = .29, RPW = 2*RPG^.71

For the 1966 AL, this produces 8.588 RPW and for the 1994 AL it is 10.584. So we can calculate LW_WAA = LW_RAA/RPW, and LW_WAA/PA seems like the natural choice for a rate stat:

This tightens the gap between Robinson and Thomas as compared to a RAA/PA comparison, and since we’ve converted to wins, we can look at WAA/PA without having to worry about the underlying contextual differences (note: this is actually not true, but I’m going to pretend like it is for a little bit for the sake of the flow of this discussion). 

There is another step we could take, which is to recognize that the Franks do influence the context in which their wins are earned, driving up their team’s RPGs and thus RPWs and thus their own WAAs. Again, I would contend that a theoretically pure linear weights framework assumes that the team is average after the player is added. Others would contend that by making that assertion I’m elevating individual tertiary offensive contributions to a completely unwarranted level of importance, ignoring a measurable effect of individual contribution because the methodology ignores an immaterial one. This is a perfectly fair critique, and so I will also show how we can adjust for the hitter’s impact on the team RPW in this step. Pete Palmer makes this adjustment as part of converting from Batting Runs (which is what I’m calling LW_RAA) to Batting Wins (what I’m calling LW_WAA), and far be it from me to argue too vociferously against Pete Palmer when it comes to a linear weights framework.

What Palmer would have you do next (conceptually as he uses a different RPW methodology) is take the batter’s RAA, divide by his games played, and add to RPG to get the RPG for an average team with the player in question added. It’s that simple because RPG already represents average runs scored per game by both teams and RAA already captures a batter’s primary and secondary contributions to his team’s offense. One benefit or drawback of this approach, depending on one’s perspective, is that unlike the theoretical team approach it is tethered to the player’s actual plate appearances/games. Using the theoretical team approach from this series, a batter always gets 1/9 of team PA. Under this approach, a batter’s real world team PA, place in the batting order, frequency of being removed from the game, etc. will have a slight impact on our estimate of his impact on an average team. We could also eschew using real games played, and instead use something like “team PA game equivalents”. For example, in the 1994 AL the average team had 38.354 PA/G; Thomas, with 508 PA, had the equivalent of 119.21 games for an average hitter getting 1/9 of an average team’s PA (508/38.354*9). I’ve used real games played, as Palmer did, in the examples that follow.

Applying the Palmerian approach to our RPW equation:


TmRPW = 2*TmRPG^.71


For the Franks, we get:

The difference between Thomas and Robinson didn’t change much, but both lose WAA and WAA/PA due to their effect on the team’s run environment as each run is less valuable as more are scored. 

I have used WAA/PA as a win-equivalent rate without providing any justification for doing so. In fact, there is very good theoretical reason for not doing so. One of the key underpinnings of all of our rate stats is that plate appearances are not fixed across contexts – they are a function of team OBA. Wins are fixed across contexts – always exactly one per game. Thus when we compare Robinson and Thomas, it’s not enough to simply look at WAA/PA; we also need to adjust for the league PA/G difference or else the denominator of our win-equivalent rate stat will distort the relativity we have so painstakingly tried to measure.

In the 1966 AL, teams averaged 36.606 PA/G; in 1994, it was 38.354. Imagine that we had two hitters from these leagues with identical WAA and PA. We don’t have to imagine it; in 1966 Norm Siebern had .847 WAA in 399 PA, and in 1994 Felix Jose had .845 WAA in 401 PA (I’m using Palmer-style WAA in this example). It seems that Siebern had a minuscule advantage over Jose. But while wins are fixed across contexts (one per game, regardless of the time and place), plate appearances are not. A batter using 401 PA in 1994 was taking a smaller share of the average PA than one taking 399 in 1966 (you might be yelling about the difference in total games played between the two leagues due to the strike, but remember that WAA already has taken into account the performance of an average player – whether over a 113 or 162 game team-season is irrelevant when comparing their WAA figures). In 1994, 401 PA represented 10.46 team games worth of PA; in 1966, 399 represented 10.90 worth. In fact, Siebern’s WAA rate was not higher than Jose’s; despite having two fewer PA, Siebern took a larger share of his team’s PA to contribute his .85 WAA than Jose did.

If we do not make a correction of this type and just use WAA/PA, we will be suggesting that the hitters of 1966 were more productive on a win-equivalent rate basis than the hitters of 1994 (although this is difficult to prove as by definition the average player’s WAA/PA will be 0, regardless of the environment in which they played). I don’t want to get bogged down in this discussion too much, so I will point you here for a discussion focused just on this aspect of comparing across league-seasons.

There are a number of different ways you could adjust for this; the “team games of PA” approach I used would be one. The approach I will use is to pick a reference PA/G, similar to our reference Pythagenpat exponent from the last installment, and force everyone to this scale. For all seasons 1961-2019, the average PA/G is 37.359 which I will define as refPA/G. The average R/G is 4.415, so the average RPG is 8.83 and the refRPW is 9.390.

If we calculate:

adjWAA/PA = WAA/PA * Lg(PA/G)/ref(PA/G)

Then we will have restated a hitter’s WAA rate in the reference environment. This is an option as our final linear weight win stat:

This increases Thomas’ edge over Robinson, while giving Jose a miniscule lead over Siebern. As a final rate stat, I find it a little unsatisfying for a couple of reasons:

1. while the ultimate objective of an offense is to contribute to wins, runs feels like a more appropriate unit

2. related to #1, wins compress the scale between hitters. There’s nothing wrong with this to the extent that it forces us to recognize that small differences between estimates fall squarely within the margin of error inherent to the exercise, but it makes quoting and understanding the figures more of a challenge.

3. WAA/PA, adjusted for PA context or not, is only differentially comparable; ideally we’d like to have a comparable ratio

My solution to this is to first convert adjusted WAA/PA to an adjusted RAA/PA, which takes care of objections #1 and 2, then to convert it to an adjusted R+/PA, which takes care of objection #3. At each stage we have a perfectly valid rate stat; it’s simply a matter of preference. 

To do this seems simple (let’s not get too attached to this approach, which we’ll revisit in a future post):

adjRAA/PA = adjWAA/PA*refRPW (remember, by adjusting WAA/PA using the refPA/G, we’ve restated everything in the terms of the reference league)

adjR+/PA = adjRAA/PA + ref(R/PA)  (reference R/PA is .1182, which can be obtained by dividing the ref R/G by the ref PA/G)

We can also compute a relative adjusted R+/PA:

reladjR+/PA = (adjR+/PA)/(ref(R/PA))

= ((RAA/PA)/TmRPW * Lg(PA/G)/ref(PA/G) * refRPW + Ref(R/PA))/Ref(R/PA)

= (RAA/PA)/TmRPW * Lg(PA/G)/ref(PA/G) * refRPW/ref(R/PA) + 1

I included raw R+/PA and its ratio the league average (relative R+/PA) to compare to this final relative adjusted R+/PA. For three of the players, the differences are small; it is only Frank Robinson whose standing is significantly diminished. This may seem counterintuitive, but remember that the more ordinary hitters have much smaller impact on RPW than the Franks. Relative to Thomas, Robinson gets more of a boost from his lower RPW (his team RPW was 19% lower than Thomas) than Thomas does from the PA adjustment (the 1994 AL had 4.8% more PA/G than the 1966 AL).

We could also return to the puritan approach (which I actually stubbornly favor for the linear weights framework) and make these adjustments as well. The equations are the same as above except where we use TmRPW, we will instead use LgRPW – reverting to assuming that the batter has no impact on the run/win conversion.

Here the impact on the Franks is similar; both are hurt of course when we consider their impact on TmRPW. Next time, we will quit messing around with half-measures – no more mixing linear run contributions with dynamic run/win converters. We’re going full theoretical team.

Wednesday, September 15, 2021

Rate Stat Series, pt. 12: Relativity for the Player as a Team Framework

I have now covered all of the ground I wished to cover regarding the construction of rate stats for various frameworks for evaluating individual offense, so I think this would be a good point to take stock of the conclusions I have drawn. From here on out, I will write about these tenets as if they are inviolable, which is not actually the case (“sound” and “well-reasoned to me” are as far as I’m willing to go), but I’m moving on to other topics:

* For team offense, Runs/Out is the only proper rate stat

* If evaluating individuals as teams (i.e. applying a dynamic run estimator like Runs Created or Base Runs directly to an individual’s statistics), Runs/Out is also the proper rate stat. Any other choice breaks consistency with treating the individual as a team, and consistency is the only thing going for such an approach. On a related note, don’t treat individuals as teams.

* If using a linear weights framework, the proper rate stat is RAA/PA, R+/PA, or restatement of those (wOBA is the one commonly used). These metrics all produce consistent rank orders and can be easily restated from one another. Any of them are valid choices at the user’s discretion, with the only objective distinction being whether you want ratio and differential comparability (R+/PA), only care about differential comparability (R+/PA or RAA/PA), or would prefer a different scale altogether at the sacrifice of direct comparisons without futher modifications (wOBA).

* If using a theoretical team framework, R+/O+ and R+/PA+ are equivalent, with the user needing to decide whether they want to think about individual performance in terms of R/O or R/PA.

Throughout this series, I have not worried about context, and originally envisioned a final installment that would briefly discuss some of the issues with comparing player’s rates across league-seasons. In putting it together, I decided that it will take a few installments to do this properly. It will also take another league-season to pair with the 1994 AL in order to make cross-context comparisons.

I have chosen the 1966 AL to fill this role. In the expansion era (1961 – 2019), the AL has averaged 4.52 R/G. 1994 was third-highest at 5.23, 15.6% higher than average. The closest league to being an inverse relative to the average is the 1966 AL (3.89 R/G, 13.8% lower than average). The obvious choice would have been 1968, since it was the most extreme, but as 1994 was not the most extreme I thought a league that was similarly low scoring would be more appropriate.

The other reason I like the 1966 AL for his purpose is that, just as was the case in 1994, the superior offensive player in the league was a future Hall of Fame slugger named Frank. In making comparisons across the twenty-eight year and (more importantly) 1.34 R/G differences between these two league-seasons, the two Franks will serve as our primary reference points.

To look at the 1966 AL we will first need to define our runs created formulas. I will not repeat the explanations of how these are calculated – I used the exact same approach as for the 1994 AL, which are demonstrated primarily in the parts 1, 9, and 10.

Base Runs:

A = H + W – HR = S + D + T + W

B = (2TB - H – 4HR + .05W)*.79920 = .7992S + 2.3976D + 3.996T + 2.3976HR + .0400W

C = AB – H = Outs

D = HR

BsR = (A*B)/(B + C) + D

Linear Weights:

LW_RC = .4560S + .7751D + 1.0942T + 1.4787HR + .3044W - .0841(outs)

LW_RAA = .4560S + .7751D + 1.0942T + 1.4787HR + .3044W - .2369(outs)

wOBA = (.8888S + 1.2981D + 1.7075T + 2.2006HR + .6944W)/PA

Theoretical Team:

TT_BsR = (A + 2.246PA)*(B + 2.346PA)/(B + C + 7.915PA) + D - .6658PA

TT_BsRP = ((A + 2.246PA)*(B + 2.346PA)/(B + C + 7.915PA) + D + .185PA)*PAR – .8519PA

TT_RAA = ((A + 2.246PA)*(B + 2.346PA)/(B + C + 7.915PA) + D + .185PA)*PAR – .9572PA

Using these equations, let’s look at the leaderboards for the key stats for each framework. First, for the player as a team:

Of course, we’ll get slightly different results from the different frameworks, but two things should be obvious: Robinson was the best offensive player in the league (his lead over Mantle in second place is bigger than the gap from Mantle to ninth-place Curt Blefary, and he was also among the leaders in PA), and the difference in league offensive levels carried through to individuals.

Next for the linear weights framework:

And finally for the theoretical team framework:

Now comes the hard part – how do we compare these performances from the 1966 AL against those from the 1994 AL? It should be implied, but in all comparisons that follow, I am only concerned about the value of a given player’s contributions relative to the context in which he played; this is not about whether/to what extent the overall quality of play differed between 1966 and 1994, or about how the existence of pitcher hitting in 1966 but not in 1994 impacted the league average, etc. I also am not going all the way in accounting for context, as I’ve still done nothing to park-adjust. Park adjustments are important, but whether you adjust or not is irrelevant to the question of which rate stat you should use. I’m also not interested in how “impressive” a given rate is due to the variation between players (e.g. z-scores) – I’m simply trying to quantify the win value of the runs a player has contributed.

The obvious first step to compare players across two league-seasons is to compare the difference or ratio of their rate to the league rate. As we’ve discussed throughout this series, some stats can be compared using both differences and ratios, but others can only be compared (at least meaningfully) with one or the others, and others still can be compared but the ratio or difference no longer has any baseball meaning unless some transformation is carried out (wOBA is the most prominent example we’ve touched on). 

The table below shows the rates for our two Franks, the respective league rate, and the difference and ratio (if applicable) for each of the key rate stats we’ve looked at:

One of the other reasons I was drawn to the pairing of these two league-seasons is how close Thomas and Robinson are in the key metrics when compared to the league using a ratio. Thomas has a slight advantage in each metric, except for BsR/O, where the flaw of treating an individual as a team can be seen. Playing in a high offense context, Thomas’ crazy rates (to put it in simple terms related to but not the run unit stats used in this series, Thomas hit .353/.492/.729 to Robinson’s .316/.406/.637) result in a large estimated tertiary contribution when treating him as his own team. Expressing these metrics as ratios hammers home that even for extreme performers (defining “extreme” within the usual confines of a major league season), there isn’t much difference between using the linear weights and theoretical team frameworks, and both are reasonable choices of framework. Treating an individual as a team, not so much.

We’ve expressed each player’s rate relative to the league average, but we haven’t answered the question of which construct is appropriate – the difference or the ratio, or does the answer differ depending on the metric? And how can we make that determination? In lieu of some guiding principle, it is just a matter of personal preference. Most people are drawn to ratios, and the widespread adoption of metrics like ERA+, ERA-, OPS+, and wRC+ speak to that preference. This case illustrates one of the reasons why ratios make sense intuitively – while the Franks are very close when comparing ratios, Thomas has a healthy lead when comparing differences as the league average R/PA and R/O for 1994 are much higher than the same for 1966. It seems intuitive that a batter will be able to exceed the league average by a larger difference when it is .2419 rather than .1528.

Still, in order to really understand how to compare players across contexts, and ensure that our intuition is grounded in reality, we need to think about how runs relate to wins. All of the metrics in that table which I consider the key metrics discussed in this series have been expressed in runs, and this is a natural starting point as the goal of an offense is to score runs. Of course the ultimate reason why an offense wants to score runs is so that they might contribute to wins.  

I will break one of the rules I’ve tried to adhere to by begging the question and asserting that Pythagenpat and associated constructs are the correct way to convert runs to wins. Of course it is just a model, and while it is a good one, it is not perfect, but I think it is the best choice given its accuracy and its seemingly reasonable results for extreme situations. 

Using Pythagorean also lends itself to adoption of a ratio rate stat, as one way to express the Pythagorean Theorem is that the expected ratio of wins to losses is equal to the ratio of runs to runs allowed raised to some power. If we start by thinking about the team/player as team framework for evaluating offense, it is natural to construct a win-unit rate stat by treating the team’s R/O as the numerator and the league average as the denominator. The ratio of these two, raised to some power, is then the equivalent win ratio that results. This is exactly the approach that Bill James took in his early Baseball Abstracts.

We can easily use this relationship to demonstrate that a simple difference of R/O does not capture the differences in win values between players. If one player exceeds the league average R/O by 10% and another 50%, then assuming a Pythagorean exponent of 2, player A will have a “win ratio” of 1.21 and player B will have a win ratio of 2.25. Even if we convert those to winning percentages (.5475 and .6923), the difference between the two player’s R/O does not capture the win value difference.

The ratio doesn’t either, of course, since it would need to be squared, but if we assume a constant Pythagorean exponent like 2, this is simply a matter of scaling, with no impact on the rank order to players. However, if we use a custom Pythagorean exponent, this assumption breaks down, as we can illustrate by comparing the two Franks. Since we’re starting with the assumption that Pythagenpat is correct, this means that we will always need to make some kind of adjustment in order to convert our run rate into an equivalent win rate.

Since we are treating the players as teams, the consistent approach is to first calculate the RPG for each player as a team, then calculate the Pythagenpat exponent for the situation, then convert their relative BsR/O to an equivalent win value. The RPG for the 1966 AL was 3.893 with 25.482 O/G, while in the 1994 AL it was 5.226 with 25.188 O/G.

T_RPG = LgR/G + BsR/O*LgO/G

x = T_RPG^.29 (for a Pythagenpat z value of .29)

Relative BsR/O = (BsR/O)/(LgBsR/O)

Win Ratio = (Relative BsR/O)^x

Offensive Winning Percentage = Win Ratio/(Win Ratio + 1) = (BsR/O)^x/((BsR/O)^x + (LgBsR/O)^x)

Here, we conclude that just looking at Relative BsR/O understates Thomas’ superiority, as Pythagenpat estimates more wins for a given run ratio when the RPG increases (e.g. a run ratio of 8/4 will produce more wins than a run ratio of 7/3.5). 

If we were actually going to use this framework, we could leave our final relative rate stat in the form of a win ratio or OW%, but I would prefer to convert it back to a run rate. Even within the conceit of the player as a team framework, it’s hard to know what to do with an OW% (or the even less familiar win ratio). We can say that Thomas’ OW% of .903 means that a team that hit like Thomas and had league average defense (runs allowed) would be expected to have a .903 W%, but even if you follow the player as team framework, you would probably like to have a result that can be more easily tied back to the player’s performance as the member of a team, not what record the Yuma Mutant Clone Franks would have. One way to return the win ratio to a more familiar scale is to convert it back to a run ratio.

Let’s define “r” to be the Pythagenpat exponent for some reference context, which I will define to be 8.83 RPG – the major league average for the expansion era (1961 – 2019). We can then easily convert our estimated win ratios for the Franks to run ratios that would produce the same estimated win ratio in the reference (8.83 RPG) environment.

The calculation is simply (Win Ratio)^(1/r). Since r = 8.83^.29 = 1.881, this becomes Win Ratio^.5316, which produces 2.614 for Robinson and 3.282 for Thomas. Following the time-honored custom of dropping the decimal place, we wind up with a win-equivalent relative BsR/O of 261 for Robinson and 328 for Thomas, compared to their initial values of 217 and 260 respectively. These both increased because both players would radically alter their run environments, increasing the win value of their relative BsR/O. I would demonstrate the lesser impact on more typical performers if I cared about this framework beyond being thorough.

While I do think that this example demonstrates that just looking at a ratio of R/O does not appropriately capture the win impact between players, the individual OW% approach goes many steps too far, but it is the logical conclusion of the player as a team methodology. If you were on that train until we got to the end of the line, I’d encourage you to consider jumping off at one of the earlier stations (I’d prefer you get on the green or red linear weights or theoretical team lines rather than ever embarking on the player as team blue line). Next time, I’ll explore those paths to win-unit rate stats using those frameworks.

Wednesday, September 01, 2021

Rate Stat Series, pt. 11: Rate Stats for the Theoretical Team Framework II

Once PAR has been incorporated, it should be clear that a different approach will be needed as our run estimate already includes the batter’s secondary contribution – it is starting out on a similar basis to wRC.  We also can fall back on our necessary test for a rate stat – it must produce the same RAA as the RAA produced by the full implementation of the framework we are looking at.

To argue why this should be the gateway criteria in a slightly different way than I did previously, consider the place of RAA in the theoretical team framework. The theoretical team framework is an attempt to value the batter by estimating the difference between the runs scored for a team on which he accumulates an even share of the plate appearances, and the runs scored for a team on which he does not play. While we have constructed an absolute estimate of runs created using this approach, it inherently is screaming out for a marginal approach – taking the difference between the team with the player and the team without the player. That is exactly what RAA is supposed to represent – and by calculating TT_RAA, we have (to the best of our ability) captured the batter’s primary, secondary, and tertiary contributions. TT_RAA is the marginal impact of the player on a team, and any rate stat that starts by using TT_BsRP should produce the same RAA figure as TT_RAA.

So let’s look at the key figures for our hitters:

Here, “RAA” is what we calculated above, based on TT_BsR/O or TT_BsR+/PA. As you can see, it does not match TT_RAA. If it did, we wouldn’t need to worry about a special set of rate stats for the TT framework – we could just piggyback on the same methodology used for linear weights. It’s clear that we need to apply something different to TT_BsRP when we use the full-blown theoretical team approach including PAR. I would also argue that this is evidence that the theoretical team approach should not be viewed simply as an alternative path to using linear weights, but rather a third unique framework for evaluating a batter’s contribution.

I would now go through a tortured explanation of the logic behind finding a denominator that achieves the desired result, but there is no need – David Smyth developed the answer and explained it clearly and concisely in a June 21, 2001 FanHome post:

On the subject of a rate stat such as R+/PA or R+/O, etc....

The whole idea behind this method is to compute the impact of a player on a theoretical team. On the team level (even a theoretical one), impact is in terms of runs and outs. The R+ generated by the procedure on this thread [NOTE: R+ is theoretically equivalent to what I am calling TT_BsRP] tells us the difference in runs between the theoretical team without the indicated batter (that is, a team of 8 average hitters), and the theoretical team with the indicated batter added. We can also compute the difference in the OUTS between these two teams, and use that total as the rate stat denominator. Call it O+.

And it's easy to calculate. You simply multiply the batter PA by the out percentage (1-OBA) of the reference team...It seems to me that the preferred rate stat for the R+ framework is not R+/PA nor R+/O; it's R+/O+. 

Frank Thomas had 149.9 TT_BsRP in 508 PA. The reference team had a .3433 OBA (same as the league average). Thus, his TT_BsRP/O+ (a specific implementation of Smyth’s R+/O+) would be:

O+ = PA*(1 – LgOBA) = 508*(1 - .3433) = 333.6

TTBsRP/O+ = TT_BsRP/O+ = 149.9/333.6 = .4494

What does this represent? It is not easy to explain it in a sentence, but I will try: Frank Thomas contributed .4494 runs to the theoretical team’s offense for each of its outs distributed to him. When Smyth said that PA*(1 – RefOBA) was equal to the difference in the team’s outs, he was referring to the theoretical team construct, in which the difference in team plate appearances is the batter’s own PA, since in the left hand term (T_BsR) the team has 9*PA, and in the right hand term (R_BsR), the team has 8*PA. The difference in outs is this difference (1*PA) times the team’s out rate.

In this sense, O+ can be thought of in terms of “freezing team outs”. We know that for a team (excluding the list of complicating circumstances like rainouts, foregone batting in the bottom of the ninth, and walkoffs), outs are fixed quality. Regardless of whether we add Frank Thomas or Matt Walbeck to a reference team, the team’s outs are fixed. In the TT/PAR approach, we start by capturing the batter in question’s secondary contribution through the PAR multiplier, but we never directly change the number of PA we start from. Thus, Thomas’ 508 PA and Walbeck’s 355 PA will turn into outs at the same rate for the purpose of the calculation. In reality, Thomas’ team will compile more PA thanks to his greater secondary contribution, but the equation handles this with a multiplier, freezing the original value.

You may not find that explanation convincing – I am struggling to articulate a concept that I may be fooling myself into believing I understand. You might be more convinced by the demonstration of what Thomas’ RAA is under this approach:

TT_RAA = (TT_BSRP/O+ - LgR/O)*O+

which for Thomas = (.4494 - .2075)*333.6 = 80.7 which is the same as his TT_RAA

In fact, I jumped the gun by calling it TT_RAA before I proved it was equal to what we previously called TT_RAA. It is in fact:

This leaves unanswered the question of what the rate stat using TT_RAA should be. Similar to how we had R+/PA and RAA/PA when working with linear weights, we should have an appropriate rate stat for the RAA figure. By manipulating the TT_RAA equation, we can see that TT_RAA/O+ should be consistent with R+/O+:

TT_RAA = (R+/O+ - LgR/O)*O+ 

so divide both sides by O+ to get:

TT_RAA/O+ = R+/O+ - LgR/O

Thus TT_RAA/O+ and R+/O+ are analogous to RAA/PA and R+/PA, except that the difference is LgR/O rather than LgR/PA.

So we’re done, right? Or is there something vaguely unsatisfying about all this, that after an entire series in which I argued that R/O was a team measure and not an individual one, does it bother you that we have left our rate stat for an individual in the form of R/O?

On one hand, it absolutely shouldn’t – R/O is the correct choice of rate stat for a team, and in this case we have modeled the player’s impact on the team and its R/O, and so there’s nothing wrong with expressing a final result in terms of R/O. The problem was not with R/O per se – it was with the inputs that we were putting in for individual batters. Additionally, R+/O+ allows our team and individual rate stats to converge. Team R+/O+ will reduce to team R/O if accept the premise that the proper “reference team” for a team is itself, and that it’s R+ is equal to its actual runs scored since actual runs scored obviously accounts for the team’s primary, secondary, and tertiary offensive actions. Not to mention any quaternary actions we could dream up or the fact that using those terms doesn’t even make sense when talking about teams. 

On the other hand, O+ is not in any way an intuitive metric – just re-read my tortured explanation of what it represents. A batter’s R+/O+ can be contextualized in any number of meaningful ways, just as regular old R/O can, but I’m not sure even those presentations (ala Runs Created/25.2 Outs) are truly relatable to a player’s performance except as a scaling device.

There is a very simple and equivalent alternative, though, and you may have already noticed what it might be from the formulas. We defined O+ as PA*(1 – LgOBA), which is really just plate appearances times a constant. If we just divide by that constant (1 – LgOBA), we can restate everything on the basis of PA, with no loss of ratio comparability. 

Doing this, we now have:


and TT_BsRP/PA = R+/O+ * (1 – LgOBA)


I am going to call TT_BsRP/PA (R+/PA+) even though PA+ is just equal to PA. I’m doing this primarily for ease of discussion, so that R+/PA+ will represent the theoretical team calculation, while R+/PA will represent the linear weights equivalent—and they are equivalent (which also means that R+/O+ could be applied to the linear weights framework – more on this in a later installment). My other justification is that in theoretical team framework, the actual number of plate appearances we plug in is not of importance when dealing with a rate stat, as we are always defining the reference team’s PA to be eight times the batter’s PA. We use the batter’s PA because it allows estimates like TT_BsR to reflect his actual playing time, but if all we care about is the rate at the end, we could use 1 plate appearance or 650 or any number we wanted. Thus I prefer to think about this quantity of plate appearances as the player’s share of the reference team’s PA rather than really being tied to his own, and feel justified in distinguishing it through the abbreviation PA+.

This approach to developing a rate stat for the TT framework can be thought of as “freezing plate appearances” as compared to the “freezing outs” approach of R+/O+. I first became aware of it through a FanHome post in 2007 by David Smyth (surprise). By that time it was over five years since Smyth had published the R+/O+ methodology and I had adopted it myself, so by the time we discussed what I am calling R+/PA+ I was in the interesting position of advocating one Smyth construct against the newer. We quickly verified their equivalence and left it there, which is the position I’m taking now. 

By definition, R+/O+ and R+/PA+ are perfectly correlated since the only difference between the two is multiplying by a constant. One allows us to express a rate stat in terms of R/O, which is consistent with how we would state a team rate stat; the other allow us to express it in terms of R/PA, which is how we would state a rate state from the linear weight framework. Both can be compared using ratios, which will be equivalent; both can be compared using differences, with the question of what the denominator for that difference should be up to the user. Both denominators can be used with RAA as well for RAA+/O+ or RAA+/PA+ (using RAA+ to refer RAA calculated within the full-blown theoretical team framework), and since R+/O+ and R+/PA+ can both be used to calculate RAA, they will also be consistent with RAA+ per O+ or PA+.

I’ll close with a sample calculation. Frank Thomas had 149.9 TT_BsRP in 508 PA, which is .2951 R+/PA+. The league average R/PA was .1363, so Thomas’ TT_RAA was (.2951 - .1363)*508 = 80.7, same as calculated previously using R+/O+ or directly from the original TT_RAA formula. The league OBA was .3433, so Thomas’ R+/O+  is equal to .2951/(1 - .3433) = .4494 as calculated previously.