May 4, 2007

Fatigue

Many people who look at hockey statistics focus on these scoring rates as a measure of performance. Alan Ryder used the difference in scoring rates of players from some marginal scoring rate to determine how much value a player added. I also focus on scoring rates to determine value. Problem is most people in hockey know that players get tired and tired players aren’t as good as fresh players.

There are a number of things to consider when looking at scoring rates, the most commonly mentioned two: who are the other 4 players on the ice with the given player and who are they playing against. However, what might be more significant than both of those issues is how much they play.

Let’s say there are two players who are identical in every way. Let’s say one gets 10 minutes of ice time and the other gets 20 minutes of ice time. Which player’s scoring rate will be higher? At this point the answer should be obvious the 10 minute player shouldn’t do worse under identical conditions as the player with 20 minutes as he’ll likely be a few seconds faster and have a little more jump. So, how does increased ice time affect scoring rates?

Short Mathematical Discussion

Determining the cost of additional ice time is not trivial. A regression on ice time and scoring rates is generally positively correlated, that is to say players with more ice time have higher scoring rates because coaches play the better players more often, but if you give the same player more ice time his scoring rate will fall. So I had to look at how individuals did with extra ice time.

Setup

I ran a regression with 4 variables: power play time [pp], penalty killing time [sh], even strength time [ev] and (power play time/1.5 + penalty killing time + even strength time) [adjusted ice time]. You can see the adjusted ice time variable lowers the value of power play time as I feel that the power play is generally easier than even strength play. This is simply my opinion. The response was how many goals the player was on the ice for in that game (for all situations), I’ll call these pluses. All ice time variables are measure in terms of hours per game.

This setup means that I will get an equation that looks like:


Regression Summary

A player will play in the neighborhood of 70-80 games a season and my model has 4 variables with very small domains, that is to say that ice time will vary by only a couple of minute per game. Due to this, some coefficients can become quite “strange” and aren’t very meaningful on the individual player level as many players do better with more ice time, which could be because they are paired with better players when they get more ice time. But, I’m not trying to discover an individual player’s fatigue pattern; instead I am looking for a trend in the NHL. So I take an average of all the coefficients to get an average coefficient.

Results

I separated forwards and defenseman:

Forwards:
Pluses/game = 4.0 * ev + 7.0 * pp + 3.0 * sh– 4.5 * adjusted ice time2
Defense:
Pluses/game = 3.6 * ev + 6.6 * pp + 3.4 * sh– 2.8 * adjusted ice time2
Time measured in hours.

There is significantly less cost for a defenseman in terms of extra ice time and this makes sense.

How does this effect even strength scoring?
The math part.

Pluses/hour = Pluses/game*games/hour =

Taking the derivative of the above expression:

And for defenseman:

The reason the equations are as complicated as they are is because there are two types of effects: fatigue factor and a different scoring rate. On the power play you get the benefit of an increased scoring rate and a cost of additional ice time. For penalty killing you are hit with a lower scoring rate and the additional ice time.

One simplification I co do is to assume sh = 0 and pp = 0:

In other words the cost of an additional hour of ice time per game is 4.5 goals for per hour. So, an additional minute (one 60th of an hour) is equal to: -4.5/60 = -0.075 goals for per hour. So the difference between a 10 even strength minute player and a 15 minute even strength player is about 0.375 GF/hour. So a player who has a scoring rate of 2.1 with 15 minutes is equivalent to a player who has a scoring rate of 2.5 with 10 minutes. That’s quite a big difference when you start thinking about it.

Application

In order to make this analysis more real I calculated how the average player would do with a given player’s ice time and compared that to how they actually did. The comparison calculated a percentage better (or worse) than average (labeled ‘P’ on the table below). A score of +100 is equivalent to doing twice as well as the average player would do with the given amount of ice time. A score of -83.5 represents playing at 16.5% (= 100+(-83.5)) of expected. Doing poorly on this metric makes you a bad offensive player, but that does not make the player bad, they can easily make up the differences by providing defense. There are other things to consider including the standard line mate problem and opposition issues. And I should note injuries and luck play their part as well.

Table


NLastnameIevppsh | P
1VANEKT12.93.50.3 | 43.6
2ST. LOUISM17.64.81.7 | 41.9
3ALFREDSSOND14.54.82.2 | 36.6
4PREISSINGT11.73.50.1 | 36.1
5SPEZZAJ14.24.90.2 | 35.9
6LECAVALIERV16.34.71.5 | 33.3
7HEATLEYD15.35.00.8 | 32.9
8SELANNET12.94.60.1 | 32.7
9CROSBYS14.75.80.2 | 32.4
10POMINVILLEJ12.52.52.4 | 32.1
11THORNTONJ14.85.00.4 | 31.6
12SAKICJ14.55.10.6 | 31.3
13ROYD12.93.02.5 | 31.1
14DATSYUKP14.04.01.9 | 30.7
15AFINOGENOVM13.23.80.1 | 30.1
16IGINLAJ15.75.11.3 | 29.9
17BRUNETTEA13.34.10.1 | 29.4
18ZETTERBERGH13.94.12.9 | 28.8
19KUNITZC12.74.00.3 | 27.6
20SULLIVANS12.04.42.9 | 26.8
21MALKINE12.95.80.5 | 25.7
22MCDONALDA13.24.30.1 | 25.5
23BRIERED14.14.70.3 | 25.0
24HEJDUKM12.63.71.6 | 24.9
25STASTNYP12.33.62.3 | 24.7
26RADULOVA9.81.80.0 | 24.5
27JOKINENO15.23.91.4 | 24.4
28PHANEUFD18.15.62.0 | 23.9
29JAGRJ15.95.70.2 | 23.8
30CHEECHOOJ13.04.40.2 | 23.0
31HOSSAM14.35.32.1 | 22.4
32STAFFORDD11.81.30.0 | 22.3
33MARLEAUP14.14.30.2 | 22.2
34ARNOTTJ13.44.50.2 | 22.1
35SMYTHR13.54.71.9 | 21.9
36VISNOVSKYL17.56.01.0 | 21.9
37HECHTJ13.32.82.7 | 21.3
38NYLANDERM14.85.50.1 | 21.3
39GABORIKM14.73.71.1 | 21.0
40CAMPBELLB16.93.51.4 | 20.5
41LANGKOWD13.84.51.8 | 19.8
42PRONGERC17.55.83.8 | 19.7
43BOYLED19.96.40.7 | 19.2
44KOIVUS12.34.01.8 | 18.5
45MURRAYG13.44.80.6 | 17.5
46YASHINA13.23.50.4 | 17.2
47HOLMSTROMT11.14.10.0 | 17.0
48RECCHIM14.05.20.5 | 16.9
49LILESJ12.55.00.3 | 16.2
50GONCHARS16.17.43.1 | 16.2
51LIDSTROMN17.35.74.5 | 15.9
52HIGGINSC12.63.22.1 | 15.6
53RYDERM11.63.90.8 | 15.4
54KOZLOVV13.95.11.5 | 15.4
55DUMONTJ12.53.40.2 | 15.3
56OVECHKINA15.75.50.2 | 15.3
57SCHUBERTC9.40.71.1 | 15.3
58CAMMALLERIM13.24.60.2 | 14.5
59WOLSKIW12.72.60.2 | 13.4
60NUMMINENT14.82.53.5 | 13.1
61REDDENW15.94.62.5 | 13.0
62HUSELIUSK12.74.00.8 | 12.9
63BOUWMEESTERJ18.23.44.6 | 12.9
64KNUBLEM13.84.11.7 | 12.3
65NIEDERMAYERS17.75.84.0 | 12.1
66WHITNEYR15.76.02.3 | 12.1
67TIMONENK13.84.83.2 | 12.1
68HAMRLIKR17.24.82.8 | 12.0
69BRIND'AMOURR14.75.03.6 | 11.9
70MICHALEKM12.93.40.4 | 11.7
71SEMINA12.65.60.1 | 11.5
72TALLINDERH16.90.24.0 | 11.2
73SCHNEIDERM16.65.31.7 | 10.4
74ZUBRUSD14.55.10.2 | 10.1
75TANGUAYA13.54.10.0 | 9.9
76CORVOJ13.63.60.9 | 9.8
77HORTONN13.82.81.5 | 9.6
78REINPRECHTS12.13.20.4 | 9.3
79SEDIND13.54.40.1 | 9.2
80STRAKAM14.15.20.7 | 8.9
81LEGWANDD13.82.02.6 | 8.8
82WELLWOODK12.34.30.0 | 8.7
83GETZLAFR10.13.41.5 | 8.4
84SAVARDM14.55.40.3 | 8.4
85SEDINH13.84.40.3 | 8.4
86KOZLOVV12.92.11.4 | 8.3
87GAGNES14.34.52.2 | 7.2
88PAETSCHN13.21.80.2 | 6.9
89GOMEZS14.44.50.0 | 6.0
90CARLEM13.24.60.3 | 5.9
91HEMSKYA12.64.30.1 | 5.8
92EHRHOFFC13.54.11.0 | 5.7
93HEWARDJ12.62.51.3 | 5.7
94FROLOVA13.94.51.5 | 5.6
95KUBAF14.93.02.3 | 5.2
96MALIKM15.70.43.1 | 5.1
97STUMPELJ12.93.23.1 | 5.0
98COMRIEM11.52.90.0 | 4.9
99WHITNEYR13.55.00.2 | 4.6
100SUNDINM13.85.01.7 | 4.5
101WEISSS12.12.52.5 | 4.0
102STAALE14.55.40.3 | 4.0
103KARIYAP15.05.00.4 | 4.0
104CLARKC12.23.52.7 | 4.0
105SHANAHANB12.25.12.5 | 3.8
106KABERLET16.55.83.6 | 3.7
107BLAKEJ14.03.80.3 | 3.6
108VLASICM16.62.33.3 | 3.4
109RAFALSKIB19.54.71.2 | 3.4
110DEMITRAP14.54.41.8 | 3.2
111WEBERS15.92.60.9 | 3.1
112KALININD15.21.03.3 | 3.0
113KLEEK16.41.33.0 | 3.0
114ALLENB16.61.13.9 | 2.7
115MOSSD9.31.90.0 | 2.6
116ANTROPOVN13.72.40.5 | 2.5
117SOURAYS15.34.93.0 | 2.3
118MARTINEKR16.11.72.1 | 2.3
119GELINASM11.42.00.1 | 2.1
120KELLYC12.00.23.1 | 1.9
121PONIKAROVSKYA13.62.61.0 | 1.7
122RICHARDSB16.75.22.2 | 1.6
123KOPITARA13.84.62.1 | 1.5
124STOLLJ10.74.82.7 | 1.2
125MCCABEB17.15.93.8 | 1.0
126ERATM13.74.31.0 | 1.0
127MARKOVA16.54.43.6 | 0.7
128HARTNELLS10.93.41.4 | 0.6
129SPACEKJ14.23.02.0 | 0.6
130BERGERONP12.75.12.9 | 0.6
131BOUCHERP15.84.13.0 | 0.4
132ROCHET13.03.30.7 | 0.3
133KOVALCHUKI14.86.60.1 | 0.2
134VOLCHENKOVA16.60.24.5 | 0.1
135ARMSTRONGD12.22.70.2 | -0.1
136PARISEZ14.33.00.2 | -0.1
137ROBERTSG13.33.70.1 | -0.6
138CLARKB16.63.33.7 | -0.6
139TUCKERD12.25.00.6 | -1.0
140SOPELB15.13.42.5 | -1.3
141PAVELSKIJ12.12.80.1 | -1.4
142COLEE12.94.30.8 | -1.6
143RIBEIROM10.94.00.0 | -1.6
144PERRYC9.72.70.1 | -2.1
145RANGERP16.81.81.8 | -2.3
146STREITM10.42.80.8 | -2.6
147SIMONC9.11.80.1 | -2.6
148GAUSTADP9.71.71.9 | -2.7
149PENNERD11.02.90.0 | -2.8
150ZIDLICKYM15.14.40.3 | -2.9
151WHITEI15.21.81.6 | -3.1
152ELIASP14.04.40.2 | -3.1
153O'BRIENS12.21.40.4 | -3.2
154ROZSIVALM15.94.63.3 | -3.2
155WILLIAMSJ13.64.52.7 | -3.4
156FISHERM13.12.82.5 | -3.9
157HAVLATM15.24.71.5 | -3.9
158PROSPALV16.32.80.0 | -4.3
159DOANS14.34.41.7 | -4.5
160LANGR13.13.60.1 | -4.7
161JACKMANB17.01.23.3 | -4.8
162METROPOLITG9.52.20.0 | -4.9
163KOTALIKA11.82.70.1 | -5.0
164PELTONENV12.20.93.3 | -5.2
165BURNSB13.81.30.7 | -5.4
166LYDMANT16.10.93.6 | -5.5
167ZUBOVS16.95.73.3 | -5.5
168MCCARTHYS10.44.80.0 | -5.6
169DRURYC11.44.42.9 | -5.6
170SAPRYKINO11.61.70.2 | -5.8
171ARNASONT11.92.30.1 | -5.9
172O'NEILLJ10.73.00.0 | -5.9
173SALOS14.63.63.2 | -6.0
174LANGENBRUNNERJ13.74.20.6 | -6.4
175GIONTAB13.94.40.5 | -6.6
176BERGERONM12.94.00.5 | -6.7
177WEIGHTD13.54.50.3 | -6.7
178PLEKANECT12.02.02.0 | -6.9
179KUBINAP16.12.13.1 | -7.1
180CLOWER11.51.60.0 | -7.1
181STEMPNIAKL11.33.20.2 | -7.1
182SUTTONA15.60.43.5 | -7.3
183OUELLETM10.23.20.0 | -7.4
184ARMSTRONGC11.61.43.9 | -7.5
185KOVALEVA13.14.11.1 | -7.6
186BACKMANC15.93.62.7 | -7.6
187PITKANENJ17.64.72.2 | -7.9
188EAVESP10.81.00.4 | -7.9
189LOMBARDIM11.42.52.5 | -8.0
190BIEKSAK16.93.73.6 | -8.0
191WESLEYG11.90.33.4 | -8.0
192MARKOVD15.80.22.9 | -8.1
193JOVANOVSKIE15.75.12.3 | -8.2
194VYBORNYD13.04.62.8 | -8.3
195NAGYL13.24.00.7 | -8.3
196SMOLINSKIB12.83.22.6 | -8.3
197POTIT16.54.64.6 | -8.4
198FRITSCHED11.61.31.2 | -8.7
199PHILLIPSC17.50.24.7 | -8.9
200SKRASTINSK16.70.24.3 | -9.1
201CHRISTENSENE9.12.50.0 | -9.1
202STAALJ10.71.23.0 | -9.1
203GILLH14.90.14.0 | -9.3
204WHITET12.22.52.6 | -9.6
205CHARAZ17.85.24.9 | -9.7
206SVATOSM9.92.60.0 | -9.9
207SUTERR14.32.53.3 | -10.0
208PARRISHM10.53.80.0 | -10.1
209SCHAEFERP12.42.32.1 | -10.1
210LARAQUEG9.60.70.0 | -10.1
211BERNIERS11.22.40.0 | -10.2
212ROBITAILLER12.50.90.4 | -10.3
213GIORDANOM10.32.70.4 | -10.4
214BEAUCHEMINF18.53.04.0 | -10.4
215ZAJACT13.32.70.0 | -10.5
216HILLS16.12.83.5 | -10.8
217BOUCHARDP12.53.50.0 | -11.0
218CULLENM12.12.32.7 | -11.0
219COMMODOREM15.80.43.7 | -11.0
220NASLUNDM13.74.00.0 | -11.0
221SILLINGERM12.53.33.4 | -11.1
222VERMETTEA11.90.92.9 | -11.2
223ROLSTONB13.45.22.6 | -11.3
224SATANM13.63.71.2 | -11.3
225BROWND12.73.72.4 | -11.5
226CAJANEKP11.93.11.2 | -11.8
227HORCOFFS13.93.53.4 | -11.9
228MCAMMONDD8.90.41.7 | -11.9
229MALONER12.01.62.7 | -12.1
230BATTAGLIAB11.10.31.0 | -12.3
231HUDLERJ8.51.50.0 | -12.4
232HAVELIDN17.23.54.6 | -12.5
233MODANOM13.23.71.5 | -12.6
234RICHARDSM13.01.53.4 | -12.8
235SAMUELSSONM12.03.00.1 | -12.8
236KOIVUM12.63.31.6 | -12.9
237MESZAROSA15.43.72.5 | -13.1
238HAMILTONJ9.53.40.0 | -13.1
239COLAIACOVOC15.31.61.1 | -13.2
240LEBDAB13.71.10.1 | -13.4
241WILLIAMSJ11.52.80.1 | -13.4
242LEHTINENJ12.93.53.0 | -13.6
243REGEHRR16.61.14.2 | -13.6
244SYKORAP12.04.60.1 | -13.7
245HAMHUISD16.70.34.3 | -13.9
246HUNTERT11.71.32.9 | -14.0
247MELICHARJ15.00.13.7 | -14.0
248FEDOROVS13.24.42.4 | -14.2
249GUERINB13.24.00.2 | -14.2
250STAJANM12.22.11.9 | -14.3
251POTHIERB16.54.23.1 | -14.4
252WITTB16.70.14.7 | -14.4
253RACHUNEKK14.02.82.5 | -14.5
254SALEIR16.93.33.1 | -14.8
255BALLARDK16.02.03.7 | -14.9
256HNIDYS13.70.51.5 | -15.1
257BLAKER15.55.13.8 | -15.2
258AVERYS12.62.51.8 | -15.4
259CARNEYK13.30.02.2 | -15.4
260KRONVALLN16.52.51.6 | -15.5
261NASHR13.04.51.7 | -15.6
262VAN RYNM15.42.43.4 | -15.6
263KOLNIKJ10.70.90.4 | -15.7
264STUARTB16.72.24.0 | -15.7
265VAANANENO13.80.10.4 | -15.7
266KEITHD18.41.73.5 | -15.8
267CARTERJ13.63.02.3 | -16.0
268CLEARYD10.82.12.5 | -16.1
269PRUCHAP10.12.90.0 | -16.1
270NOLANO10.92.61.9 | -16.1
271MORRISONB12.33.42.3 | -16.2
272RUCINSKYM12.23.41.1 | -16.4
273MORRISD14.72.63.1 | -16.5
274MCCLEMENTJ10.80.62.5 | -16.7
275WIDEMAND14.54.31.4 | -17.1
276MCLARENK17.41.22.8 | -17.1
277HUTCHINSONA8.23.80.2 | -17.2
278STEENA12.21.52.0 | -17.5
279HANNANS17.91.53.4 | -17.5
280STURMM11.94.32.4 | -17.7
281PYATTT11.03.00.0 | -17.7
282VISHNEVSKIV16.10.13.0 | -17.7
283EMINGERS15.50.23.3 | -17.8
284ERIKSSONA14.52.82.9 | -18.1
285NASREDDINEA12.80.12.8 | -18.3
286NUMMELINP15.03.91.3 | -18.4
287DE VRIESG17.12.13.1 | -18.6
288ORPIKB15.21.10.3 | -18.7
289MICHALEKZ16.83.43.5 | -18.8
290WALKERS12.63.30.2 | -18.8
291JOHNSSONK17.22.83.6 | -18.9
292LILJAA12.30.13.0 | -19.0
293MARCHANTT11.70.13.3 | -19.0
294KOMISAREKM15.70.13.5 | -19.1
295TKACHUKK13.43.90.2 | -19.1
296BACKESD10.92.40.0 | -19.3
297ALEXEEVN12.11.11.1 | -19.3
298MCLEANB11.71.10.8 | -19.5
299PICARDA15.22.40.8 | -19.5
300FRANZENJ11.91.22.5 | -19.7
301JOHNSONA11.60.70.4 | -19.8
302PERREAULTY13.33.80.5 | -19.9
303HILBERTA10.20.11.1 | -20.3
304PERRINE13.51.61.9 | -20.4
305SANDERSONG10.53.00.5 | -20.6
306RUUTUT13.83.40.1 | -20.6
307CHIMERAJ11.90.82.6 | -20.7
308TANABED13.33.01.5 | -20.7
309ZANONG13.00.24.2 | -20.8
310O'DONNELLS16.10.13.7 | -21.1
311POHLJ9.51.60.2 | -21.5
312KWIATKOWSKIJ8.60.90.3 | -21.7
313JOKINENJ10.83.10.0 | -21.8
314OLESZR11.61.02.9 | -22.1
315PISANIF12.42.02.6 | -22.1
316CARTERA11.13.20.5 | -22.1
317CRAIGR10.23.71.5 | -22.2
318GRATTONC10.71.10.3 | -22.5
319SKOULAM17.00.72.6 | -22.8
320FOSTERK13.33.70.9 | -22.9
321KLEPISJ8.31.60.0 | -23.1
322LADDA9.41.60.0 | -23.1
323JANIKD13.30.50.7 | -23.3
324KILGERC10.70.33.5 | -23.5
325BOOTHD8.80.10.7 | -23.7
326RUCCHINS13.42.42.5 | -23.7
327KESLERR11.61.13.7 | -23.7
328AXELSSONP13.03.03.2 | -23.7
329BOUILLONF15.20.62.5 | -23.7
330LAPERRIEREI11.80.41.6 | -23.8
331MELLANBYS9.74.00.1 | -23.8
332SCUDERIR14.50.63.7 | -24.0
333FEDOTENKOR14.31.90.1 | -24.0
334HATCHERD17.30.85.6 | -24.1
335OHLUNDM17.14.23.5 | -24.1
336WALLINN15.20.23.2 | -24.1
337PETTINGERM11.52.52.9 | -24.2
338GORDONB11.90.13.9 | -24.2
339YELLES11.00.43.3 | -24.2
340VRBATAR12.73.40.8 | -24.3
341COOKEM12.30.82.5 | -24.3
342SLATERJ9.50.20.5 | -24.6
343SHARPP12.31.92.9 | -24.6
344GAUTHIERD12.00.04.6 | -24.6
345FIDDLERV10.60.22.8 | -24.7
346JONESR12.01.22.9 | -24.7
347CHISTOVS7.70.90.0 | -24.7
348SAMSONOVS11.82.00.1 | -24.9
349RIVETC15.52.23.4 | -25.0
350BABCHUKA13.62.11.8 | -25.0
351CULLIMOREJ12.40.13.8 | -25.3
352DALEYT16.00.72.6 | -25.3
353BOYESB12.42.80.8 | -25.3
354WISNIEWSKIJ15.42.21.5 | -25.8
355ASHAMA9.10.20.0 | -25.9
356FERENCEA13.22.13.2 | -26.0
357BOURQUER12.02.31.6 | -26.0
358GRIERM12.40.73.3 | -26.1
359WALKERM13.40.11.7 | -26.3
360BEECHK9.23.10.1 | -26.3
361MORRISONNS16.40.34.2 | -26.3
362CAMPOLIC12.31.90.6 | -26.3
363MIETTINENA11.12.90.3 | -26.7
364BOOGAARDD4.60.00.0 | -26.8
365HALED9.60.00.0 | -26.9
366HEDICANB15.62.32.1 | -26.9
367AFANASENKOVD11.30.40.0 | -27.1
368RADIVOJEVICB10.92.00.1 | -27.5
369MODRYJ14.41.51.6 | -27.5
370BRASHEARD7.50.40.0 | -27.5
371SAUERK15.70.12.6 | -27.5
372HAINSEYR15.53.73.6 | -27.9
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2 comments:

David Johnson said...

Fatigue is an interesting theory, and I don't doubt it exists, but the way the NHL is played I am not sure you can analyze fatigue without simultaneously analyzing who the players are playing with and against and even the game situation.

First off many teams line match so players on the ice at any given time might accumulate minutes at more or less the same rate.

Second, a team that is winning by a couple of goals in the third period might ease off trying to score goals this making players on good teams look like they fatigue in the third period when in fact they just play a different style. Conversely teams that trail in the third period will play a more offensive style to score goals. If you simultaneously considering goals against might help eliminate change in style of play.

Finally, every player is going to fatigue at different rates based on their style of play, their age, and most importantly their conditioning level.

Again, I don't doubt that fatigue exists but I am just not convinced that the analysis that you are doing considers all the factors necessary. It might be interesting to do a simple test by taking a ratio of third period PF/Min vs First and Second period PF/Min and then see how that ratio correlates with total ice time. You could even take it a step further by only considering 'close' games so that style of play changes are minimal.

Plus, a more important fatigue might is probably playing in back to back games. If every team plays the same number of back to back games and plays the same number of opponents on the second night of back to back games then maybe the results if your analysis won't change much, but for any single game it does matter immensely as can be seen by teams winning percentages in back to back games. Playing on the road probably also contributes to fatigue.

JavaGeek said...

1. Matching in the regular season is not that dominant that it would have a huge effect on this analysis. Its effects wouldn't affect the results, as the players get more ice time against the same competition their scoring rates still fall with increased ice time. (I don't see a problem here).

2. The question to ask here is how is the coach adjusting his lines when he goes into his defensive shell. If he increases the top line minutes in the last period that will mean he has to decrease the minutes of his lower guys. This would imply his lower guys should start scoring more often. In other words the one will cancel out the other.

3. Due to this, some coefficients can become quite “strange” and aren’t very meaningful on the individual player level as many players do better with more ice time, which could be because they are paired with better players when they get more ice time. But, I’m not trying to discover an individual player’s fatigue pattern; instead I am looking for a trend in the NHL. So I take an average of all the coefficients to get an average coefficient.

The question I'm not asking: "Does player performance degrade in the third period, due to fatigue". That would be quite different. Fatigue occurs as soon as you skate down the ice to chase a puck and if you're double shifted you have a lot less time to recover: you wont chase the puck as much and you probably wont score as often. That's all this is trying to do. Players are built to play for 60 minutes so their performance should be reasonably constant under ideal conditions, but if they play outside their conditioning or experience their performance should fall...

If this wasn't the case why then does Radulov and others with similar stats get such little ice time?

I did the defensive regression and it does appear that as players get tired they substitute offense with defense.