Bradley Terry Preference Model on NFL Games

Posted Sep 27, 2024

Team connectivity of the NFL 2023 Season Graph

By Jake Taylor

18 min read

A post on the Bradley Terry model with NFL game data. The Bradley Terry model:

\[\log{\frac{p_{ij}}{1 - p_{ij}}} = \beta_i - \beta_j\]

Is a classical model for ranking or preference data and is sometimes referred to as a preference model.

  
import pandas as pd
import plotly.graph_objects as go
import plotly.express as px
from IPython.display import display, Markdown
import statsmodels.api as sm
from statsmodels.stats.outliers_influence import variance_inflation_factor
import patsy
import numpy as np
from sklearn.metrics import confusion_matrix
from sklearn.metrics import roc_curve, roc_auc_score

  
def render_plotly_html(fig: go.Figure) -> None:
    fig.show()
    display(
        Markdown(
            fig.to_html(
                include_plotlyjs="cdn",
            )
        )
    )


def render_table(df: pd.DataFrame | pd.Series) -> Markdown:
    return Markdown(df.to_markdown())

NFL Game Data.

This data was collected through ESPN’s public API. Our goal will be to model the probability of a home team beating an away team,

\[p_{ij} = \text{Probability that home team $i$ beats away team $j$}\]

We will use the Bradley Terry model using the NFL game data.

  
df = pd.concat([pd.read_csv("games_2023.csv"), pd.read_csv("games_2024.csv")])
print(df.shape)
ALL_TEAMS = list(sorted(set(df.awayTeamAbbr)))
render_table(df.describe())

(332, 60)

	season_type	week	event	firstDowns_away	firstDownsPassing_away	firstDownsRushing_away	firstDownsPenalty_away	totalOffensivePlays_away	totalYards_away	yardsPerPlay_away	totalDrives_away	netPassingYards_away	yardsPerPass_away	interceptions_away	rushingYards_away	rushingAttempts_away	yardsPerRushAttempt_away	turnovers_away	fumblesLost_away	interceptions_away.1	defensiveTouchdowns_away	firstDowns_home	firstDownsPassing_home	firstDownsRushing_home	firstDownsPenalty_home	totalOffensivePlays_home	totalYards_home	yardsPerPlay_home	totalDrives_home	netPassingYards_home	yardsPerPass_home	interceptions_home	rushingYards_home	rushingAttempts_home	yardsPerRushAttempt_home	turnovers_home	fumblesLost_home	interceptions_home.1	defensiveTouchdowns_home	awayScore	homeScore	awayTeamId	homeTeamId	year
count	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332	332
mean	2.03614	8.13855	4.01565e+08	18.7018	10.7831	6.26205	1.65663	62.3886	320.193	5.11566	10.9759	210.798	5.95542	0.792169	109.395	26.4127	4.08253	1.33434	0.542169	0.792169	0.162651	19.8072	11.2229	6.78614	1.79819	63.1416	342.117	5.43283	11.0452	224.039	6.35361	0.746988	118.078	27.2289	4.27861	1.26506	0.518072	0.746988	0.105422	20.4398	23.1355	16.7982	16.4669	2023.14
std	0.186932	5.62623	43744.9	4.85857	3.60864	2.96617	1.36071	8.44189	83.831	1.14303	1.69218	72.2784	1.97181	0.900922	46.2201	7.38179	1.15109	1.19653	0.762429	0.900922	0.393362	4.78153	3.7463	3.26096	1.40723	8.34606	84.0121	1.22029	1.72971	74.2508	2.07376	0.880722	50.733	7.78809	1.21486	1.1327	0.735158	0.880722	0.353277	9.40878	10.3198	9.42348	9.57442	0.352206
min	2	1	4.01547e+08	6	3	0	0	38	58	1.2	6	17	0.6	0	23	9	1.2	0	0	0	0	6	0	0	0	41	119	2.1	6	-9	-0.5	0	17	10	1.5	0	0	0	0	0	0	1	1	2023
25%	2	3	4.01547e+08	15	8	4	1	56	265	4.4	10	160	4.7	0	73	21	3.3	0	0	0	0	17	9	5	1	57.75	283	4.6	10	169.75	4.9	0	82	22	3.5	0	0	0	0	14	17	9	8	2023
50%	2	7.5	4.01548e+08	19	11	6	1	62	323	5.1	11	211	5.8	1	106	25	4	1	0	1	0	20	11	6.5	2	63	351	5.5	11	219	6.3	1	112	27	4.2	1	0	1	0	20	23	17	16	2023
75%	2	13	4.01548e+08	22	13	8	2	68	378	5.9	12	259.25	7.1	1	139	31.25	4.8	2	1	1	0	23	13	9	3	68	396.25	6.1	12	277	7.5	1	147.25	32	5	2	1	1	0	27	29.25	25	24.25	2023
max	3	18	4.01672e+08	32	23	16	6	92	544	8.3	17	466	14.2	4	274	46	7.8	5	4	4	2	37	25	20	8	89	726	10.2	17	472	14.4	5	350	53	9.7	6	3	5	2	48	70	34	34	2024

Treatment Contrasts

For a good review of different contrasts, see this documentation. We will use the Treatment (dummy) coding.

  
# Dummy encode the home team as an indicator variable.
# We `drop_first=True` to avoid collinearity amongst the variables.
# In this case, `ARI` is dropped from the dataset and represents
# the reference level (equivalent with setting \beta_ARI = 0).
X_home = pd.get_dummies(df.homeTeamAbbr, drop_first=True).astype(int)
# Ensure we have linear independence.
assert not (X_home.sum(axis=1) == 1).all()
render_table(X_home.head())

	BAL	CLE	KC	MIN	NYJ
0	0	0	0	0	1
1	0	0	1	0	0
2	1	0	0	0	0
3	0	1	0	0	0
4	0	0	0	1	0

  
# Same for the away team.
X_away = pd.get_dummies(df.awayTeamAbbr, drop_first=True).astype(int)
# Ensure we have linear independence.
assert not (X_away.sum(axis=1) == 1).all()
render_table(X_away.head())

	BUF	CIN	DET	HOU	TB
0	1	0	0	0	0
1	0	0	1	0	0
2	0	0	0	1	0
3	0	1	0	0	0
4	0	0	0	0	1

Bradley Terry Model Contrasts

To specify the Bradley Terry model, we need to model the (logit) of the probability of the home team winning as the difference between the coefficient’s for each team:

\[\log{\frac{p_{ij}}{1 - p_{ij}}} = \beta_i - \beta_j\] \[\iff p_{ij} = \frac{\exp{\beta_i - \beta_j}}{1 + \exp{\beta_i - \beta_j}} = \frac{\exp{\beta_i}}{\exp{\beta_i} + \exp{\beta_j}}\]

We can represent this numerically by taking the difference of the contrasts we created before, X_diff = X_home - X_away. Each team pairing will be uniquely encoded and the parameter estimate will focus only on the difference of those two teams.

  
# We should have the same teams represented in the home and away teams.
assert (X_home.columns == X_away.columns).all()
# X_diff will represent which two teams played that week.
# Specifically, +1 will represent home and -1 will be away (except reference level).
X_diff = X_home - X_away
# Ensure we have linear independence.
assert not (X_away.sum(axis=1) == 0).all()
render_table(X_diff.head())

	BAL	BUF	CIN	CLE	DET	HOU	KC	MIN	NYJ	TB
0	0	-1	0	0	0	0	0	0	1	0
1	0	0	0	0	-1	0	1	0	0	0
2	1	0	0	0	0	-1	0	0	0	0
3	0	0	-1	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0	1	0	-1

  
# We can visualize the "encoding" and see how often one team played another.
def encoding(x: pd.Series) -> str:
    # Just concatenate all of the contrasts into one long string encoding.
    return "".join(str(i) for i in x)


# We can see how the encoding maps back to the original teams
# and the frequency of this matchup.
df["encoding"] = X_diff.apply(encoding, axis=1)
render_table(
    df.groupby("encoding")[["homeTeamAbbr", "awayTeamAbbr"]]
    .agg(["unique", "count"])
    .sort_values(
        [("homeTeamAbbr", "count"), ("awayTeamAbbr", "count")],  # type: ignore
        ascending=False,
    )
    .head()
)

encoding	(‘homeTeamAbbr’, ‘unique’)	(‘homeTeamAbbr’, ‘count’)	(‘awayTeamAbbr’, ‘unique’)	(‘awayTeamAbbr’, ‘count’)
00-10000000000000001000000000000	[‘MIA’]	2	[‘BUF’]	2
000-1000000000000000001000000000	[‘NO’]	2	[‘CAR’]	2
00000-10000000010000000000000000	[‘KC’]	2	[‘CIN’]	2
000000-1000010000000000000000000	[‘HOU’]	2	[‘CLE’]	2
00000000000-11000000000000000000	[‘IND’]	2	[‘HOU’]	2

  
# Tie goes to the home team, since we want to keep the outcomes binary.
y = pd.Series((df.homeScore >= df.awayScore), name="homeTeamWin").astype(int)
render_table(y.head())

	homeTeamWin
0	1
1	0
2	1
3	1
4	0

We also include an intercept $\alpha$, which is a coefficient for the home team advantage. Our final model is then:

\[\log{\frac{p_{ij}}{1 - p_{ij}}} = \alpha + \beta_i - \beta_j = \alpha + \Delta_{ij}\]

Where $\Delta_{ij}$ is the difference of the contrasts we computed before.

Design Matrix

  
formula = "homeTeamWin ~ 1 + " + " + ".join(X_diff.columns)
y, X = patsy.dmatrices(
    formula,
    pd.concat([y, X_diff], axis=1),
    return_type="dataframe",
)
Markdown(formula)

homeTeamWin ~ 1 + ATL + BAL + BUF + CAR + CHI + CIN + CLE + DAL + DEN + DET + GB + HOU + IND + JAX + KC + LAC + LAR + LV + MIA + MIN + NE + NO + NYG + NYJ + PHI + PIT + SEA + SF + TB + TEN + WSH

VIF

Variance inflation factor measures the amount of collinearity in the dataset. It is recommended to keep this under < 5 when performing statistical inference.

  
# Print the VIF to see how much collinearity there are in the features.
# We want to avoid VIF > 5.
render_plotly_html(
    px.histogram(
        pd.Series(
            [variance_inflation_factor(X, i) for i, _ in enumerate(X.columns)],
            name="VIF",
        ),
        nbins=15,
        title="Distribution of VIF",
    )
)

Bradley Terry GLM

Now we fit the BT model. Since we are modeling logits of a probability, this reduces to logistic regression. This is widely available, we will use statsmodel’s GLM.

  
# Fit a GLM with Binomial family and logit link function.
results = sm.GLM(
    endog=y,
    exog=X,
    family=sm.families.Binomial(
        link=sm.families.links.Logit(),
    ),
).fit()

  
results.summary()

Generalized Linear Model Regression Results
Dep. Variable:	homeTeamWin	No. Observations:	332
Model:	GLM	Df Residuals:	300
Model Family:	Binomial	Df Model:	31
Link Function:	Logit	Scale:	1.0000
Method:	IRLS	Log-Likelihood:	-198.10
Date:	Sat, 28 Sep 2024	Deviance:	396.19
Time:	23:13:58	Pearson chi2:	332.
No. Iterations:	4	Pseudo R-squ. (CS):	0.1657
Covariance Type:	nonrobust

	coef	std err	z	P>\|z\|	[0.025	0.975]
Intercept	0.2235	0.122	1.829	0.067	-0.016	0.463
ATL	0.2448	0.724	0.338	0.735	-1.174	1.664
BAL	1.9120	0.717	2.666	0.008	0.506	3.318
BUF	1.5916	0.734	2.167	0.030	0.152	3.031
CAR	-0.9843	0.855	-1.152	0.249	-2.659	0.691
CHI	0.3354	0.727	0.461	0.645	-1.089	1.760
CIN	0.8876	0.713	1.245	0.213	-0.510	2.285
CLE	1.2966	0.708	1.832	0.067	-0.091	2.684
DAL	1.3452	0.711	1.892	0.058	-0.048	2.739
DEN	0.7358	0.740	0.994	0.320	-0.714	2.186
DET	1.7979	0.719	2.499	0.012	0.388	3.208
GB	1.1053	0.718	1.540	0.124	-0.302	2.513
HOU	1.1260	0.714	1.577	0.115	-0.273	2.525
IND	0.7037	0.733	0.960	0.337	-0.733	2.141
JAX	0.7857	0.741	1.060	0.289	-0.667	2.239
KC	2.0205	0.749	2.698	0.007	0.553	3.488
LAC	0.2262	0.760	0.298	0.766	-1.262	1.715
LAR	1.2363	0.670	1.845	0.065	-0.077	2.549
LV	0.6112	0.753	0.812	0.417	-0.864	2.087
MIA	1.1409	0.746	1.530	0.126	-0.321	2.602
MIN	0.9293	0.730	1.272	0.203	-0.502	2.361
NE	-0.2335	0.778	-0.300	0.764	-1.757	1.290
NO	0.7731	0.743	1.041	0.298	-0.682	2.229
NYG	0.2182	0.718	0.304	0.761	-1.190	1.626
NYJ	0.5562	0.744	0.748	0.455	-0.902	2.014
PHI	1.3884	0.714	1.944	0.052	-0.012	2.788
PIT	1.4859	0.716	2.076	0.038	0.083	2.889
SEA	1.3372	0.701	1.907	0.057	-0.037	2.712
SF	1.8241	0.699	2.611	0.009	0.455	3.193
TB	1.0179	0.729	1.396	0.163	-0.411	2.447
TEN	-0.1036	0.763	-0.136	0.892	-1.599	1.392
WSH	-0.0545	0.726	-0.075	0.940	-1.478	1.369

  
# Verify that the Binomial family with a Logit link is equivalent with the Logit binary model.
logit_params = sm.Logit(endog=y, exog=X).fit().params
pd.testing.assert_series_equal(logit_params, results.params)

Optimization terminated successfully.
         Current function value: 0.596675
         Iterations 6

  
# Compute the confusion matrix on the training predictions.
render_table(
    pd.DataFrame(
        confusion_matrix(y_pred=results.predict(X) > 0.5, y_true=y),
        columns=["Home Loss", "Home Win"],
        index=["Home Loss", "Home Win"],
    )
)

	Home Loss	Home Win
Home Loss	89	59
Home Win	43	141

Analysis of GLM Results

  
results_df = pd.DataFrame(
    {
        "p-value": results.pvalues,
        "parameter": results.params,
        "team": results.params.index,
        "Parameter Rank": results.params.rank(ascending=False),
    }
)

P-values

  
# Visualize the parameter estimate vs. p-value.
# As expected, this should roughly trace out a normal distribution.
render_plotly_html(
    px.scatter(
        results_df,
        x="parameter",
        y="p-value",
        color="team",
        title="P-Value vs. Parameter Estimate",
    )
)

Parameter Ranks

  
# Tally all of the team's wins (if y == 1, then home won, otherwise away won).
team_wins = pd.Series(
    np.where(y.squeeze(), df.homeTeamAbbr, df.awayTeamAbbr)
).value_counts()
# Tally games played in total by each team.
team_n = pd.concat([df.homeTeamAbbr, df.awayTeamAbbr]).value_counts()
win_df = pd.DataFrame({"wins": team_wins, "n": team_n})
win_df["win"] = win_df.wins / win_df.n
win_df.sort_values(by="win", ascending=False, inplace=True)
win_df["Win Rank"] = win_df.win.rank(ascending=False)
render_table(win_df.head())

	wins	n	win	Win Rank
KC	17	23	0.73913	1
DET	16	23	0.695652	2
BAL	15	22	0.681818	4
BUF	15	22	0.681818	4
SF	15	22	0.681818	4

  
# Join the parameter results with the raw win percentages.
results_win_df = results_df.merge(win_df, left_index=True, right_index=True)
# Plot the parameter rank vs. the win percentage rank
fig_rank = px.scatter(
    results_win_df,
    x="Parameter Rank",
    y="Win Rank",
    color="team",
    title="Win Percentage Rank vs. Parameter Estimate Rank",
)
# Add a reference line. If points fall on the line, then parameter_rank == win_rank.
# Otherwise, the Bradley Terry model and raw win percentage disagree.
fig_rank.add_scatter(
    x=results_win_df["Parameter Rank"],
    y=results_win_df["Parameter Rank"],
    mode="lines",
    name="Reference",
    line=dict(color="red", width=2, dash="dash"),
    hoverinfo="skip",
)
render_plotly_html(fig_rank)

It seems the BT model is especially useful when there are ties in the standings and in some cases (i.e. LAR) where the parameter rank < win rank (indicating the team may be undervalued).

  
# How often do the rankings agree?
(results_win_df["Parameter Rank"] == results_win_df["Win Rank"]).mean()

0.12903225806451613

Probability of team i beating team j

We can compute the probabilities using the equation:

\[p_{ij} = \frac{\exp{\alpha + \beta_i - \beta_j}}{1 + \exp{\alpha + \beta_i - \beta_j}}\]

  
# Compute the probability of home team i defeating away team j
log_odds = (
    results.params.filter(items=["Intercept"]).sum()
    + results.params.filter(items=ALL_TEAMS).values
    - results.params.filter(items=ALL_TEAMS).values.reshape(-1, 1)
)
probability_matrix = pd.DataFrame(np.exp(log_odds) / (1 + np.exp(log_odds)))
probability_matrix.columns = results.params.filter(items=ALL_TEAMS).index
probability_matrix.index = results.params.filter(items=ALL_TEAMS).index

  
render_plotly_html(
    px.imshow(
        probability_matrix,
        labels={"x": "Home Team", "y": "Away Team"},
        title="Probability of a Home Team Winning",
        aspect="auto",
        color_continuous_scale="RdBu_r",
    )
)

To interpret, we can look at one column (i.e. KC) and see their chances of beating away teams (red scores are better). Conversely, following a row tells us how likely a team is to win on the road.

Adding in Numeric Features

Since the BT model amounts to fitting a logisitic regression model, we can extend our previous attempt with numeric features.

  
# Focus on x_per_y style metrics and turnovers.
numeric_features = [
    i + j
    for i in [
        "thirdDownEff",
        "fourthDownEff",
        "completionAttempts",
        "yardsPerRushAttempt",
        "yardsPerPass",
        "redZoneAttempts",
        "interceptions",
        "fumblesLost",
    ]
    for j in ["_home", "_away"]
]

  
def compute_percentage(x: str, delim: str = "-") -> float:
    first, second = x.split(delim)
    return float(first) / float(second) if float(second) > 0 else 0

  
# Construct a numeric features dataframe.
X_numeric = df[numeric_features].copy()
X_numeric["redZoneAttempts_home"] = X_numeric.redZoneAttempts_home.map(
    compute_percentage
)
X_numeric["thirdDownEff_home"] = X_numeric.thirdDownEff_home.map(compute_percentage)
X_numeric["redZoneAttempts_away"] = X_numeric.redZoneAttempts_away.map(
    compute_percentage
)
X_numeric["fourthDownEff_home"] = X_numeric.fourthDownEff_home.map(compute_percentage)
X_numeric["completionAttempts_away"] = X_numeric.completionAttempts_away.map(
    lambda x: compute_percentage(x, "/")
)
X_numeric["fourthDownEff_away"] = X_numeric.fourthDownEff_away.map(compute_percentage)
X_numeric["thirdDownEff_away"] = X_numeric.thirdDownEff_away.map(compute_percentage)
X_numeric["completionAttempts_home"] = X_numeric.completionAttempts_home.map(
    lambda x: compute_percentage(x, "/")
)
# After these conversions, we should only have numeric features
assert (X_numeric.dtypes != np.object_).all()
render_table(X_numeric.head())

	thirdDownEff_home	thirdDownEff_away	fourthDownEff_home	fourthDownEff_away	completionAttempts_home	completionAttempts_away	yardsPerRushAttempt_home	yardsPerRushAttempt_away	yardsPerPass_home	yardsPerPass_away	redZoneAttempts_home	redZoneAttempts_away	interceptions_home	interceptions_away	fumblesLost_home	fumblesLost_away
0	0.384615	0.384615	1	0	0.636364	0.707317	6.1	4.4	4.7	4.7	0.333333	0.5	1	3	0	1
1	0.357143	0.333333	0	0.333333	0.538462	0.628571	3.9	3.5	5.8	6.9	0.666667	0.666667	1	0	0	1
2	0.533333	0.388889	0	0.25	0.772727	0.636364	3.4	3.1	6	4	0.6	0	1	0	1	1
3	0.285714	0.133333	0	0	0.551724	0.4375	5.2	3.8	4.5	2	0.666667	0	1	0	1	0
4	0.428571	0.352941	0	1	0.75	0.617647	2.4	2.2	7.1	4.8	0.333333	0.5	1	0	2	0

Design Matrix

We expand the design matrix from before with the numeric features.

  
formula_full = formula
formula_full += " + " + " + ".join(X_numeric.columns)
y, X_full = patsy.dmatrices(
    formula_full,
    pd.concat([y, X_diff, X_numeric], axis=1),
    return_type="dataframe",
)
Markdown(formula_full)

homeTeamWin ~ 1 + ATL + BAL + BUF + CAR + CHI + CIN + CLE + DAL + DEN + DET + GB + HOU + IND + JAX + KC + LAC + LAR + LV + MIA + MIN + NE + NO + NYG + NYJ + PHI + PIT + SEA + SF + TB + TEN + WSH + thirdDownEff_home + thirdDownEff_away + fourthDownEff_home + fourthDownEff_away + completionAttempts_home + completionAttempts_away + yardsPerRushAttempt_home + yardsPerRushAttempt_away + yardsPerPass_home + yardsPerPass_away + redZoneAttempts_home + redZoneAttempts_away + interceptions_home + interceptions_away + fumblesLost_home + fumblesLost_away

Full Bradley Terry GLM

  
# Fit a GLM with Binomial family and logit link function.
model = sm.GLM(
    endog=y,
    exog=X_full,
    family=sm.families.Binomial(
        link=sm.families.links.Logit(),
    ),
)
results_full = model.fit()
y_hat = results_full.predict(X_full)

  
fpr, tpr, _ = roc_curve(y, y_hat)
fig_roc = px.line(
    pd.DataFrame({"True Positive Rate": tpr, "False Positive Rate": fpr}),
    x="False Positive Rate",
    y="True Positive Rate",
    title=f"Probability of Home Team Winning ROC Curve, AUC: {round(roc_auc_score(y, y_hat), 4)}",
)
fig_roc.add_scatter(
    x=fpr,
    y=fpr,
    mode="lines",
    name="Reference",
    line=dict(color="red", width=2, dash="dash"),
    hoverinfo="skip",
)
render_plotly_html(fig_roc)

  
render_table(
    pd.DataFrame(
        confusion_matrix(y_pred=y_hat > 0.5, y_true=y),
        columns=["Home Loss", "Home Win"],
        index=["Home Loss", "Home Win"],
    )
)

	Home Loss	Home Win
Home Loss	137	11
Home Win	10	174

We see a large improvement in the confusion matrix and ROC AUC from adding the numerical features.

Summary

The Bradley Terry model is a classic way to rank preferences applicable in spots statistics. More recently, it has been used in LLM training (check out the Direct Preference Optimization paper for one example) to learn human preferences.

Math

statistics math

This post is licensed under CC BY 4.0 by the author.

	BAL	CLE	KC	MIN	NYJ
0	0	0	0	0	1
1	0	0	1	0	0
2	1	0	0	0	0
3	0	1	0	0	0
4	0	0	0	1	0

	BUF	CIN	DET	HOU	TB
0	1	0	0	0	0
1	0	0	1	0	0
2	0	0	0	1	0
3	0	1	0	0	0
4	0	0	0	0	1

	BAL	BUF	CIN	CLE	DET	HOU	KC	MIN	NYJ	TB
0	0	-1	0	0	0	0	0	0	1	0
1	0	0	0	0	-1	0	1	0	0	0
2	1	0	0	0	0	-1	0	0	0	0
3	0	0	-1	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0	1	0	-1

	BAL	CLE	KC	MIN	NYJ
0	0	0	0	0	1
1	0	0	1	0	0
2	1	0	0	0	0
3	0	1	0	0	0
4	0	0	0	1	0

	BUF	CIN	DET	HOU	TB
0	1	0	0	0	0
1	0	0	1	0	0
2	0	0	0	1	0
3	0	1	0	0	0
4	0	0	0	0	1

	BAL	BUF	CIN	CLE	DET	HOU	KC	MIN	NYJ	TB
0	0	-1	0	0	0	0	0	0	1	0
1	0	0	0	0	-1	0	1	0	0	0
2	1	0	0	0	0	-1	0	0	0	0
3	0	0	-1	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0	1	0	-1

NFL Game Data.

Treatment Contrasts

Bradley Terry Model Contrasts

Design Matrix

VIF

Bradley Terry GLM

Analysis of GLM Results

P-values

Parameter Ranks

Probability of team i beating team j

Adding in Numeric Features

Design Matrix

Full Bradley Terry GLM

Summary

Trending Tags

	BAL	CLE	KC	MIN	NYJ
0	0	0	0	0	1
1	0	0	1	0	0
2	1	0	0	0	0
3	0	1	0	0	0
4	0	0	0	1	0

	BUF	CIN	DET	HOU	TB
0	1	0	0	0	0
1	0	0	1	0	0
2	0	0	0	1	0
3	0	1	0	0	0
4	0	0	0	0	1

	BAL	BUF	CIN	CLE	DET	HOU	KC	MIN	NYJ	TB
0	0	-1	0	0	0	0	0	0	1	0
1	0	0	0	0	-1	0	1	0	0	0
2	1	0	0	0	0	-1	0	0	0	0
3	0	0	-1	1	0	0	0	0	0	0
4	0	0	0	0	0	0	0	1	0	-1