# 4. Colley’s Method

Massey’s Method of rating is defined as

$$Mr = p$$,

where,

• $$M$$ is a $$n$$ x $$n$$ matrix,

• each $$M_{ii}$$ is the number of games played by the i-th team

• each $$M_{ij}$$ is the negation of games played by the i-th team agains the j-th team

• $$r$$ are the ratings we are trying to estimate, and

• $$p$$ is the point differentials across games played.

Colley’s Method is another method of rating defined as

$$Cr = b$$,

where,

• $$C = 2I + M$$, and

• $$b = 1 + \frac{1}{2}(w - l)$$.

• $$w$$ is the number of wins

• $$l$$ is the number of losses

The benefits of Colley’s Method are stated as follows.

• Colley’s Method is bias-free since it does not use point differential and just the wins and losses. Point differential may distort the ratings as teams the dominant team may overwhelm the opponent in a game (generate a lot of points) or hold their score steady so as not to appear overwhelming the opponent (sportsmanship).

• Colley’s Method also considers the ratings together; when one team’s rating improves, another must suffer.

• Colley’s Method is applicable when point differential is not available or undesireable.

## 4.1. NCAAF ACC 2005

Let’s look at the NCAAF ACC 2005 data.

[1]:

import pandas as pd
import numpy as np
from sklearn.linear_model import LinearRegression

def get_ncaaf():

def get_teams(df):
return sorted(list(set(df.t1) | set(df.t2)))

def get_fap(df):
def get_f(t):
return df[df.t1 == t].s1.sum() + df[df.t2 == t].s2.sum()

def get_a(t):
return -df[df.t1 == t].s2.sum() + -df[df.t2 == t].s1.sum()

teams = get_teams(df)
x = pd.DataFrame([{'for': get_f(t), 'against': get_a(t)} for t in teams], index=teams)
x['differential'] = x['for'] + x['against']
return x

def get_wlb(df):
def get_wins(t):
w1 = df[(df.t1 == t) & (df.s1 > df.s2)].shape[0]
w2 = df[(df.t2 == t) & (df.s2 > df.s1)].shape[0]
return w1 + w2

def get_losses(t):
l1 = df[(df.t1 == t) & (df.s1 < df.s2)].shape[0]
l2 = df[(df.t2 == t) & (df.s2 < df.s1)].shape[0]
return l1 + l2

teams = get_teams(df)

x = pd.DataFrame({
'w': [get_wins(t) for t in teams],
'l': [get_losses(t) for t in teams]
}, index=teams)
x['b'] = 1 + 0.5 * (x.w - x.l)
return x

def get_M(df):
def get_games_played(t1, t2):
if t1 == t2:
return df[(df.t1 == t1) | (df.t2 == t2)].shape[0]
else:
q1 = (df.t1 == t1) & (df.t2 == t2)
q2 = (df.t1 == t2) & (df.t2 == t1)
q = q1 | q2
return -df[q].shape[0]

teams = get_teams(df)
mat = [[get_games_played(t1, t2) for t2 in teams] for t1 in teams]
mat = pd.DataFrame(mat, index=teams, columns=teams)
return mat

def get_C(df):
M = get_M(df)
C = 2 * np.eye(M.shape[0], M.shape[1]) + M
return C

def get_MTP(df):
M = get_M(df)

teams = get_teams(df)
T = pd.DataFrame(np.diag(pd.Series(np.diag(M))), index=teams, columns=teams)
P = T - M
return M, T, P

def get_massey_r(df):
M = get_M(df)
M.iloc[-1,:] = 1

p = get_fap(df).differential

model = LinearRegression()
model.fit(M, p)

return pd.Series(model.coef_, index=M.index)

def get_colley_r(df):
C = get_C(df)
b = b = get_wlb(df).b

model = LinearRegression()
model.fit(C, b)

return pd.Series(model.coef_, index=C.index)

def get_rankings(df):
return pd.DataFrame({c: df[c].sort_values(ascending=False).index for c in df.columns})


Compare $$M$$ to $$C$$.

[2]:

M = get_M(get_ncaaf())
M

[2]:

Duke Miami UNC UVA VT
Duke 4 -1 -1 -1 -1
Miami -1 4 -1 -1 -1
UNC -1 -1 4 -1 -1
UVA -1 -1 -1 4 -1
VT -1 -1 -1 -1 4
[3]:

C = get_C(get_ncaaf())
C

[3]:

Duke Miami UNC UVA VT
Duke 6.0 -1.0 -1.0 -1.0 -1.0
Miami -1.0 6.0 -1.0 -1.0 -1.0
UNC -1.0 -1.0 6.0 -1.0 -1.0
UVA -1.0 -1.0 -1.0 6.0 -1.0
VT -1.0 -1.0 -1.0 -1.0 6.0

Compare $$p$$ to $$b$$.

[4]:

p = get_fap(get_ncaaf())['differential']
p

[4]:

Duke    -124
Miami     91
UNC      -40
UVA      -17
VT        90
Name: differential, dtype: int64

[5]:

b = get_wlb(get_ncaaf())['b']
b

[5]:

Duke    -1.0
Miami    3.0
UNC      1.0
UVA      0.0
VT       2.0
Name: b, dtype: float64


Compare the ratings between Massey and Colley’s Methods.

[6]:

get_massey_r(get_ncaaf()).sort_values(ascending=False)

[6]:

Miami    36.2
VT       36.0
UVA      14.6
UNC      10.0
Duke     -6.8
dtype: float64

[7]:

get_colley_r(get_ncaaf()).sort_values(ascending=False)

[7]:

Miami    2.857143e-01
VT       1.428571e-01
UNC     -2.775558e-17
UVA     -1.428571e-01
Duke    -2.857143e-01
dtype: float64


Massey and Colley’s Methods differ only by the positions of UVA and UNC.

[8]:

pd.DataFrame({
'massey': get_massey_r(get_ncaaf()).sort_values(ascending=False).index,
'colley': get_colley_r(get_ncaaf()).sort_values(ascending=False).index
})

[8]:

massey colley
0 Miami Miami
1 VT VT
2 UVA UNC
3 UNC UVA
4 Duke Duke

## 4.2. NBA, 2021

Let’s apply these methods to the NBA 2021 season up to Thanksgiving.

[9]:

def get_nba():
.rename(columns={
'a_team': 't1',
'h_team': 't2',
'a_score': 's1',
'h_score': 's2'})
x = x[x.preseason == False]\
.drop(columns=['preseason'])\
.reset_index(drop=True)
return x


Here are the ratings.

[10]:

pd.DataFrame({
'massey': get_massey_r(get_nba()),
'colley': get_colley_r(get_nba())
})

[10]:

massey colley
76ers 2.100543 -0.004808
Bucks 1.999356 0.037718
Bulls 4.235154 0.086708
Cavaliers 1.429301 0.051312
Celtics 0.579776 -0.015193
Clippers 3.792290 0.073897
Grizzlies -4.525625 0.011843
Hawks 3.338752 0.038129
Heat 6.488263 0.134159
Hornets 1.227789 0.099810
Jazz 8.549796 0.085264
Kings 0.089514 -0.082525
Knicks -0.306992 0.009597
Lakers -3.357641 -0.049940
Magic -9.818479 -0.263918
Mavericks -0.688728 0.042523
Nets 4.254646 0.170953
Nuggets 0.755934 0.004656
Pacers 3.138730 -0.064910
Pelicans -4.916126 -0.201009
Pistons -7.552551 -0.241295
Raptors 1.055674 -0.033273
Rockets -8.788710 -0.281985
Spurs -3.433295 -0.217864
Suns 6.744632 0.285625
Thunder -5.150785 -0.155606
Timberwolves 1.276979 -0.016636
Trail Blazers 2.456361 0.035358
Warriors 12.899850 0.330721
Wizards 1.180180 0.130689

These are the rankings. If we take point differential out of the picture, then the Jazz fall from second to eighth place!

[11]:

pd.DataFrame({
'massey': get_massey_r(get_nba()).sort_values(ascending=False).index,
'colley': get_colley_r(get_nba()).sort_values(ascending=False).index
})

[11]:

massey colley
0 Warriors Warriors
1 Jazz Suns
2 Suns Nets
3 Heat Heat
4 Nets Wizards
5 Bulls Hornets
6 Clippers Bulls
7 Hawks Jazz
8 Pacers Clippers
9 Trail Blazers Cavaliers
10 76ers Mavericks
11 Bucks Hawks
12 Cavaliers Bucks
13 Timberwolves Trail Blazers
14 Hornets Grizzlies
15 Wizards Knicks
16 Raptors Nuggets
17 Nuggets 76ers
18 Celtics Celtics
19 Kings Timberwolves
20 Knicks Raptors
21 Mavericks Lakers
22 Lakers Pacers
23 Spurs Kings
24 Grizzlies Thunder
25 Pelicans Pelicans
26 Thunder Spurs
27 Pistons Pistons
28 Rockets Magic
29 Magic Rockets

## 4.3. NFL, 2021

Let’s apply these methods to the NFL 2021 season up to Thanksgiving.

[12]:

def get_nfl():
.rename(columns={
'team1': 't1',
'team2': 't2',
'score1': 's1',
'score2': 's2'})\
.drop(columns=['week'])
x['t1'] = x['t1'].apply(lambda s: s.strip())
x['t2'] = x['t2'].apply(lambda s: s.strip())

return x


Here are the ratings.

[13]:

pd.DataFrame({
'massey': get_massey_r(get_nfl()),
'colley': get_colley_r(get_nfl())
})

[13]:

massey colley
49ers 1.565063 0.001040
Bears -9.322540 -0.108254
Bengals -0.607977 0.046633
Bills 8.425259 0.065165
Broncos -2.431469 -0.024019
Browns -2.723871 0.042118
Buccaneers 6.321022 0.144495
Cardinals 8.235687 0.264187
Chargers -0.551756 0.134031
Chiefs 2.706593 0.170924
Colts 3.115332 0.019958
Cowboys 5.819441 0.133962
Dolphins -8.305769 -0.134007
Eagles 1.696797 -0.034021
Falcons -12.409125 -0.112008
Giants -6.524430 -0.136574
Jaguars -10.338981 -0.243478
Jets -14.139647 -0.240181
Lions -12.281915 -0.354927
Packers 1.443518 0.183655
Panthers -1.742287 -0.063501
Patriots 6.349719 0.070305
Raiders -5.414601 0.056728
Rams 2.472228 0.139103
Ravens -1.608072 0.153518
Saints 0.808172 -0.028999
Seahawks -0.880815 -0.117927
Steelers -3.707054 0.035267
Texans -11.365255 -0.221763
Titans 2.584503 0.178996
Vikings 0.737758 0.035146
Washington -3.649900 -0.055569

Here are the rankings. It seems if we take point differential out of the picture, then the Bills fall way out of first place (11-th place)!

[14]:

pd.DataFrame({
'massey': get_massey_r(get_nfl()).sort_values(ascending=False).index,
'colley': get_colley_r(get_nfl()).sort_values(ascending=False).index
})

[14]:

massey colley
0 Bills Cardinals
1 Cardinals Packers
2 Patriots Titans
3 Buccaneers Chiefs
4 Cowboys Ravens
5 Colts Buccaneers
6 Chiefs Rams
7 Titans Chargers
8 Rams Cowboys
9 Eagles Patriots
10 49ers Bills
11 Packers Raiders
12 Saints Bengals
13 Vikings Browns
14 Chargers Steelers
15 Bengals Vikings
16 Seahawks Colts
17 Ravens 49ers
18 Panthers Broncos
19 Broncos Saints
20 Browns Eagles
21 Washington Washington
22 Steelers Panthers
23 Raiders Bears
24 Giants Falcons
25 Dolphins Seahawks
26 Bears Dolphins
27 Jaguars Giants
28 Texans Texans
29 Lions Jets
30 Falcons Jaguars
31 Jets Lions

## 4.4. Movie ratings

Let’s look at sample movie rating data where there are 4 users rating 4 movies. The ratings are from $$[1, 5]$$ and a 0 indicates no rating.

[15]:

def get_movie():
return pd.DataFrame([
[5, 4, 3, 0],
[5, 5, 3, 1],
[0, 0, 0, 5],
[0, 0, 2, 0],
[4, 0, 0, 3],
[1, 0, 0, 4]
], columns=[f'Movie {i}' for i in range(1, 5)], index=[f'User {i}' for i in range(1, 7)])

df = get_movie()
df

[15]:

Movie 1 Movie 2 Movie 3 Movie 4
User 1 5 4 3 0
User 2 5 5 3 1
User 3 0 0 0 5
User 4 0 0 2 0
User 5 4 0 0 3
User 6 1 0 0 4

We can flatten this matrix by considering each pair of rating per user.

[16]:

from itertools import combinations, chain

def get_pairwise_ratings(r):
ratings = [{'t1': m1, 't2': m2, 's1': r[m1], 's2': r[m2]} for m1, m2 in pairs]
ratings = [d for d in ratings if d['s1'] > 0 and d['s2'] > 0]
return ratings

pairs = list(combinations(df.columns, 2))
movie_df = pd.DataFrame(chain(*[get_pairwise_ratings(r) for _, r in df.iterrows()]))
movie_df

[16]:

t1 t2 s1 s2
0 Movie 1 Movie 2 5 4
1 Movie 1 Movie 3 5 3
2 Movie 2 Movie 3 4 3
3 Movie 1 Movie 2 5 5
4 Movie 1 Movie 3 5 3
5 Movie 1 Movie 4 5 1
6 Movie 2 Movie 3 5 3
7 Movie 2 Movie 4 5 1
8 Movie 3 Movie 4 3 1
9 Movie 1 Movie 4 4 3
10 Movie 1 Movie 4 1 4

Here are the ratings.

[17]:

pd.DataFrame({
'massey': get_massey_r(movie_df),
'colley': get_colley_r(movie_df)
})

[17]:

massey colley
Movie 1 -0.634021 0.168605
Movie 2 0.084683 0.127261
Movie 3 -1.486745 -0.150517
Movie 4 -3.649485 -0.145349

Here are the rankings.

[18]:

pd.DataFrame({
'massey': get_massey_r(movie_df).sort_values(ascending=False).index,
'colley': get_colley_r(movie_df).sort_values(ascending=False).index
})

[18]:

massey colley
0 Movie 2 Movie 1
1 Movie 1 Movie 2
2 Movie 3 Movie 4
3 Movie 4 Movie 3

What if we removed ties?

[19]:

pd.DataFrame({
'massey': get_massey_r(movie_df[movie_df.s1 != movie_df.s2]),
'colley': get_colley_r(movie_df[movie_df.s1 != movie_df.s2])
})

[19]:

massey colley
Movie 1 -0.792651 0.173453
Movie 2 0.149606 0.121336
Movie 3 -1.556430 -0.150651
Movie 4 -3.648294 -0.144137
[20]:

pd.DataFrame({
'massey': get_massey_r(movie_df[movie_df.s1 != movie_df.s2]).sort_values(ascending=False).index,
'colley': get_colley_r(movie_df[movie_df.s1 != movie_df.s2]).sort_values(ascending=False).index
})

[20]:

massey colley
0 Movie 2 Movie 1
1 Movie 1 Movie 2
2 Movie 3 Movie 4
3 Movie 4 Movie 3