reconciling PER and Win Shares per minute

[Mike G]

Some of us still use basketball-reference a lot, and the two 'composite stats' listed for players in the "Advanced Statistics" section are often not in very close agreement. A player may look great in one number and mediocre in the other.

My general take is that PER accounts for pace but not for how good the team is, and thus only accounts for how good a player is per possession, regardless of how much better the opponent is. And Win Shares exaggerate the difference between players for good teams and those with lesser teams.

I grabbed 645 player lines from 2010-11, players with multiple teams being separated into their part-seasons.
http://www.basketball-reference.com/lea ... stats.html
I found the following formula to guess a player's WS/48 based on his PER:
WS/48 = (PER - 4.92)/99.0

This minimizes the sum of differences between player WS and the total 'guessed' WS.
It's very tempting to use the formula WS/48 = (PER-5)/100 . And doing so yields this conversion:

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PER   WS/48
35.0   .300
30.0   .250
25.0   .200
20.0   .150
15.0   .100
10.0   .050
5.0    .000
0.0   -.050

Anyway, of players with >1000 minutes, those whose PER most underestimates their WS/48.

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Player          Tm    Min    PER   WS/48   WS   ws48?   WS?  dif  dif48
Joel Anthony   MIA   1463    7.3   .107   3.3   .024    .7   2.5   .083
James Jones    MIA   1549   11.1   .145   4.7   .062   2.0   2.7   .083
Tyson Chandler DAL   2059   18.4   .218   9.4   .136   5.8   3.5   .082
Kurt Thomas    CHI   1178   10.0   .131   3.2   .051   1.3   2.0   .080
Keith Bogans   CHI   1461    9.0   .118   3.6   .041   1.3   2.3   .077

Ryan Anderson  ORL   1424   19.0   .217   6.4   .142   4.2   2.2   .075
Kyle Korver    CHI   1649   13.0   .149   5.1   .082   2.8   2.3   .067
Joakim Noah    CHI   1576   18.8   .205   6.7   .140   4.6   2.1   .065
J.J. Redick    ORL   1513   12.8   .143   4.5   .080   2.5   2.0   .063
Matt Bonner    SAS   1432   13.5   .147   4.4   .087   2.6   1.8   .060

Ronnie Brewer  CHI   1781   13.8   .147   5.5   .090   3.3   2.1   .057
Hedo Turkoglu  ORL   1910   13.5   .143   5.7   .087   3.4   2.2   .056
Nick Collison  OKC   1524   10.8   .113   3.6   .059   1.9   1.7   .054
Paul Pierce    BOS   2774   19.7   .201  11.6   .149   8.6   3.0   .052
Ray Allen      BOS   2890   16.4   .166  10.0   .116   7.0   3.0   .050

All players from very good teams, mostly or entirely with high TS% and low TO rates.

Those whose PER would generally get a much higher WS/48:

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Player           Tm    Min   PER   WS/48   WS    ws48?  WS?   dif   dif48
Jordan Crawford TOT   1027   11.8  -.026   -.6   .069   1.5   2.0   -.095
Mo Williams     CLE   1065   13.8  -.005   -.1   .090   2.0   2.1   -.095
J.J. Hickson    CLE   2256   15.6   .032   1.5   .108   5.1   3.6   -.076
DMarcus Cousins SAC   2309   14.6   .022   1.1   .098   4.7   3.6   -.076
Michael Beasley MIN   2361   15.5   .035   1.7   .107   5.3   3.5   -.072

John Wall       WAS   2606   15.8   .041   2.2   .110   6.0   3.7   -.069
Andray Blatche  WAS   2172   16.9   .053   2.4   .121   5.5   3.1   -.068
Darko Milicic   MIN   1686   12.2   .007    .2   .074   2.6   2.3   -.067
Mo Williams     TOT   1788   13.9   .026   1.0   .091   3.4   2.4   -.065
Sonny Weems     TOR   1413   10.2  -.011   -.3   .053   1.6   1.9   -.064

Andrea Bargnani TOR   2353   16.4   .053   2.6   .116   5.7   3.1   -.063
Linas Kleiza    TOR   1032   10.1  -.010   -.2   .052   1.1   1.3   -.062
Tyreke Evans    SAC   2107   14.4   .036   1.6   .096   4.2   2.6   -.060
Stephen Jackson CHA   2405   14.6   .039   2.0   .098   4.9   2.9   -.059
Antawn Jamison  CLE   1842   16.8   .067   2.6   .120   4.6   2.0   -.053

All players from bad teams, etc.

[Crow]

That is a handy general conversion formula.

Only about half the big WS/48 underestimates using PER are for guys with low usages (<15%) and only about the big overestimates are for guys with high usages (near or over 25%). I had expected those rates to be higher.
Other factors are involved. One big factor in WS/48 but not in PER at all is team shot defense.The underestimate list is all players from good teams and almost all good defensive teams (one average defensive team but probably the best defender on that team- Collison). The overestimate list is all players from bad teams and bad defensive teams.

[Mike G]

Indeed, the 61-win Bulls have the PER's of a 44-win team with average defense; the 21-win Raptors have PER's of a 39-win team. That seems pretty drastic.
One might "adjust" PER by (Tm DRtg/Lg DRtg)^N -- finding an exponent that best fits team PER-wins (from formula above) to Pythagorean wins (or WS) .

[Mike G]

OK, this is definitely going to be a work in progress for a while.
I'm calling this PERWins.
PERW = (PER-5)/100 * (Min/48.4)

Now the suggested adjustment to PER.
PER2 = PER * (LgDRtg/TmDRtg)^4
For whatever reason, an exponent of 4.00 makes a better fit than does 3.99 or 4.01 .

Now an adjusted version of PERWins:
PER2W = (PER2 - 5)/100 * (Min/48.4)

Ranked from best to worst teams, the differences from WS, for PERW and PER2W

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West   WS    perW   per2W  diff  dif2     East   WS    perW  per2W   diff  dif2
LAL   58.0   46.3   54.3  -11.7  -3.7     CHI   63.1   43.4   63.2  -19.7   0.1
SAS   57.0   46.5   51.0  -10.5  -6.0     MIA   63.1   46.3   56.6  -16.8  -6.5
OKC   54.3   46.5   46.8   -7.7  -7.5     ORL   57.2   40.8   55.2  -16.4  -2.1
DAL   52.9   43.6   49.4   -9.2  -3.4     BOS   56.5   41.8   61.1  -14.7   4.5
DEN   52.2   45.2   45.2   -7.0  -7.0     PHI   46.2   42.2   47.9   -4.0   1.7

HOU   47.6   44.7   40.7   -2.9  -6.9     NYK   43.9   44.4   38.1    0.5  -5.8
MEM   47.3   44.0   49.6   -3.3   2.3     MIL   40.8   35.6   46.9   -5.1   6.1
POR   46.3   42.8   43.3   -3.5  -3.0     ATL   39.4   39.1   39.8   -0.3   0.4
NOH   45.8   41.0   46.0   -4.8   0.3     IND   38.9   37.6   40.5   -1.2   1.7
PHO   39.7   42.6   35.8    2.9  -3.9     DET   32.2   41.3   32.1    9.1  -0.1

UTA   36.5   41.9   35.8    5.4  -0.7     CHA   31.7   36.8   35.7    5.1   4.0
GSW   36.3   42.3   34.9    6.0  -1.4     NJN   26.4   34.9   29.8    8.5   3.4
LAC   32.9   37.5   34.6    4.6   1.7     TOR   25.7   38.6   28.1   12.9   2.4
SAC   28.9   36.1   32.7    7.2   3.7     WAS   23.3   36.2   30.5   13.0   7.2
MIN   24.9   35.7   28.4   10.8   3.5     CLE   18.2   33.7   25.5   15.5   7.3

Denver's DRtg was exactly the league average, so it doesn't change anything.
The avg unadjusted absolute difference between WS and PERW is 9.1
Avg diff between WS and PER2W is 3.6
I have noticed that teams averaged 42.2 WS, rather than 41. This seems to have something to do with overtime minutes; perhaps converting from WS/48, even though teams averaged up to 48.8 mpg (Phx, NJ, Okl) ?

[Mike G]

PER2W are constructed such that they more closely match Win Shares -- which are supposed to match pythagorean Wins.
PER is adjusted by DRtg, in some cases dramatically, to achieve this. The top 10 players in minutes:

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Player             Tm    Min    PER   per2    WS    perW  per2W
Monta Ellis       GSW   3227   18.6   16.4    6.0    9.1    7.6
LaMarcus Aldridge POR   3211   21.5   21.7   11.1   10.9   11.1
Luol Deng         CHI   3208   15.5   20.3   10.0    7.0   10.1
Dorell Wright     GSW   3147   15.0   13.2    5.8    6.5    5.4
Blake Griffin     LAC   3112   21.9   20.8    9.9   10.9   10.2

LeBron James      MIA   3063   27.3   31.5   15.6   14.1   16.8
Kevin Durant      OKC   3038   23.6   23.7   12.0   11.7   11.7
Pau Gasol         LAL   3037   23.3   26.1   14.7   11.5   13.2
Derrick Rose      CHI   3026   23.5   30.8   13.1   11.6   16.1
Al Jefferson      UTA   2940   20.1   18.1    7.8    9.2    8.0

All players with poor defensive teams have had their PER adjusted downward.
LeBron and Rose both now have per2W which exceed their WS.

Here are some players who were with 2 teams for at least 300 minutes each.
On the left are players whose PER changed rather drastically, and it may be explained almost entirely by their changed environment: per2 has changed much less.
Players on the right would seem to offer counter-evidence.

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Player           Tm    PER   per2  ws48      Player          Tm    PER   per2  ws48
J Richardson    PHO   19.1   17.0  .118     Vince Carter    ORL   16.1   19.9  .153
J Richardson    ORL   13.2   16.3  .127     Vince Carter    PHO   14.2   12.7  .061

Marcin Gortat   ORL   13.7   16.9  .159     Hedo Turkoglu   PHO   13.1   11.7  .093
Marcin Gortat   PHO   18.8   16.8  .152     Hedo Turkoglu   ORL   13.5   16.7  .144
 
Gilbert Arenas  WAS   14.0   12.6  .025     Nenad Krstic    OKC   12.5   12.5  .097
Gilbert Arenas  ORL    8.6   10.6  .008     Nenad Krstic    BOS   14.3   18.7  .171

Marcus Thornton NOH   14.0   15.2  .057     Jeff Green      OKC   12.9   12.9  .088
Marcus Thornton SAC   18.2   17.1  .124     Jeff Green      BOS   12.9   16.9  .119

Baron Davis     LAC   16.3   15.5  .080     Jarrett Jack    TOR   12.5   10.3  .003
Baron Davis     CLE   19.3   16.4  .087     Jarrett Jack    NOH   14.6   15.8  .083

Mickael Pietrus ORL    8.3   10.2  .084    Kendrick Perkins BOS   10.2   13.4  .089
Mickael Pietrus PHO   10.8    9.6  .039    Kendrick Perkins OKC    9.1    9.1  .046

Shane Battier   HOU   12.9   12.1  .106     Carl Landry     SAC   14.7   13.8  .087
Shane Battier   MEM   10.6   11.5  .098     Carl Landry     NOH   15.2   16.5  .126

WS/48 is given for corroboration: In almost every case, per2 is more closely related.
Of 36 such traded players, the average PER change (absolute) was 2.75, and per2 change was 2.70

[Mike G]

Here are the top 20 PER and PER2 from last season; minimum 1000 minutes.

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Player        Tm    PER  ws48      Player      Tm   per2   ws48  per2w48  WS   per2W
LeBron James MIA   27.3  .246    Dwi Howard   ORL   32.1   .238   .271   14.4   16.4
Dwi Howard   ORL   26.0  .238    LeBron James MIA   31.5   .246   .265   15.6   16.8
Dwyane Wade  MIA   25.6  .220    Derrick Rose CHI   30.8   .210   .258   13.1   16.1
Kevin Love   MIN   24.3  .212    Dwyane Wade  MIA   29.6   .220   .246   12.8   14.3
Kobe Bryant  LAL   23.9  .180    Ke Garnett   BOS   27.0   .196   .220    9.0   10.1

Chris Paul   NOH   23.7  .234    Kobe Bryant  LAL   26.8   .180   .218   10.4   12.5
Kevin Durant OKC   23.6  .191    Pau Gasol    LAL   26.1   .234   .211   14.7   13.2
R Westbrook  OKC   23.6  .160    Paul Pierce  BOS   25.8   .203   .208   11.6   11.9
Derrick Rose CHI   23.5  .210    Chris Paul   NOH   25.6   .234   .206   13.9   12.3
D Nowitzki   DAL   23.4  .215    D Nowitzki   DAL   25.5   .215   .205   11.1   10.6

Pau Gasol    LAL   23.3  .234    Ca Boozer    CHI   24.6   .150   .196    5.8    7.6
A Stoudemire NYK   22.7  .135    Joakim Noah  CHI   24.6   .207   .196    6.7    6.4
Z Randolph   MEM   22.6  .187    Za Randolph  MEM   24.6   .187   .196   10.5   11.0
De Williams  UTA   22.1  .160    Kevin Durant OKC   23.7   .191   .187   12.0   11.7
Bl Griffin   LAC   21.9  .153    Ru Westbrook OKC   23.7   .160   .187    9.4   11.0

Tim Duncan   SAS   21.9  .172    Andrew Bynum LAL   23.6   .212   .186    6.6    5.8
Ma Ginobili  SAS   21.7  .197    Ry Anderson  ORL   23.5   .219   .185    6.4    5.4
L Aldridge   POR   21.5  .167    Tim Duncan   SAS   23.3   .172   .183    7.7    8.2
Kevin Martin HOU   21.4  .176    Ma Ginobili  SAS   23.1   .197   .181    9.9    9.1
Ca Anthony   DEN   21.2  .128    Chris Bosh   MIA   22.4   .178   .174   10.3   10.1

Kevin Love (per2 = 21.1) would be the highest from a bad team.

[Crow]

I compared Mike's per2W values to “Win Shares" using the values on this chart http://dberri.files.wordpress.com/2011/07/untitled9.png for the top 10 minute guys and the average absolute difference was about 3 wins. That is substantial and probably related to how these metrics vary on treatment of usage and scoring efficiency.

I then compared “Wins Produced Combined” (which gives more individualized defensive ratings to players than traditional Wins Produced by looking at presumptive based on boxscore opponent performance against a particular player) to per2W and the average absolute difference was just a bit more than 1 win. Still apples and oranges and this is just a small sample. The closer similarity is probably just coincidence, but I mention it anyways, mainly to get to the next suggestion.

Perhaps it might be worth comparing net counterpart per2w to Wins Produced Combined so that both metrics use the net counterpart basis. Though this is getting pretty twisted and might not interest many, especially not the purists. Adjusting for team defensive performance and counterpart performance might be too much or in a roundabout way might balance those weights and fit with a perspective that shot defense occurs at counterpart and team levels and perhaps should be evaluated at both rather than just one.

Comparison to some form of APM could be added, if inclined. I generally think viewing and considering the pattern of ratings across metrics is more helpful than focusing on just one.

[Crow]

Of the top 20 on PER, only 60% were in a lineup that was in the league's top 20 for minutes used and was positive on raw and / or Adjusted +/-.

5 were on negative performing lineups- D. Williams, Westbrook & Durant (both on the court and the lineup is rated negative? I guess Thabo, Green and Krstic wasn't a good idea as the by far most used lineup for 2+ cumulative seasons), Love and Aldridge. 3 (Anthony, Griffin and Nowitski) were not in a top 20 for minutes used lineup.

Is it that hard to find a good big minute lineup for 40% of the league's top 20 on PER? Some of these players may be tough cases to work well at lineup level, some are missed coaching opportunities (or a different philosophy, intention or not).

WS per 48 for bigger minute players did somewhat better. 75% of them appear in a lineup that was in the league's top 20 for minutes used and was positive on raw and / or Adjusted +/-.

[Crow]

The top 20 on 1 year RAPM appear on positive lineups within the top 20 on minutes played, slightly more than the top 20 on PER but a bit less than the top 20 on WS/48.

But would need to check multi-season to see if that was a stable pattern. WS/48 benefits from having team-based shot defense included instead of just individual stats and that could affect which lineups are positive and which lineups are allowed to get big minutes.

Maybe that is worth something. Never know til you check and think about it.

[Mike G]

This is a bit tangential. I got to checking trends in league dominance by various positions, and why some rather mediocre players got into lots of All-Star games while some really good ones did not.
This isn't at all an in-depth study, but just breaking it down to decades -- the '90s being 1990-1999, for example -- the median PER and WS/48 of All-Star seasons, by position, via:
http://bkref.com/tiny/BPAZy\

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position:  Guards         Forwards         Centers     all All-Stars
decade   PER    ws48    PER     ws48     PER    ws48     PER    ws48
'50s    15.9    .119    20.6    .177    22.5    .158    19.1    .150
'60s    16.9    .134    18.7    .138    19.3    .178    18.1    .144
'70s    18.4    .125    18.5    .138    21.7    .178    19.1    .141
'80s    19.2    .148    21.6    .168    20.4    .168    20.4    .160
'90s    19.9    .170    20.4    .172    22.9    .178    20.7    .172
'00s    21.4    .162    22.2    .179    22.6    .194    22.0    .175
'10s    21.2    .171    22.6    .182    20.3    .181    21.6    .177

Recall that PER and ws/48 tend to be related by ws/48 = (PER-5)/100
That is, a PER of 15 predicts a WS/48 = .100; 20 = .150

Knowing, too, that WS/48 are generally higher (relative to PER) for players with winning teams, the table corroborates the perception that players from better teams are also more likely to be selected to All-Star lineups, over players with similar stats for weaker teams.

The only examples breaking this pattern are centers in the '50s and guards in the '70s.
In all other instances, WS/48 are higher than expected, given the PER's.

The link above is for '70s guards. Pete Maravich was a '79 AllStar with PER 13.4 and .004 WS/48.

[huevonkiller]

The problem with PER is usage rate.

The problem with WS/48 is team defense of course. Both are very helpful figures once you understand their flaws.