chiSq df pValue
home 13.4194 5 0.0197
away 23.4891 5 0.0003Women’s World Cup 2023 Prediction Competition
Milt Mavrakakis & Iain Gourlay / Smartodds
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Are goals Poisson?
Motivation: A Bivariate Weibull Count Model for Forecasting Association Football Scores by Boshnakov et al. (an excellent paper!).
Few natural alternatives to Poisson distribution for count data, other than the Negative Binomial.
Solution: Build count distribution from Weibull interarrival times between goals.
How do we test this?
Fit Poisson distribution to home and away goals (separately).
Compare observe and expected goals.
Perform goodness-of-fit test.
(This is the most common approach in the literature.)
Some results (EPL 2010-2015)
A hypothesis test
- What are we actually testing?
\[ Y^{(h)}_i \stackrel{\text{iid}}\sim \text{Poisson}(\lambda); \; i=1,...,N \\ Y^{(a)}_i \stackrel{\text{iid}}\sim \text{Poisson}(\mu); \; i=1,...,N \]
- Is this a sensible test?
A better assumption
- It makes more sense to assume that
\[ Y^{(h)}_i \stackrel{\text{iid}}\sim \text{Poisson}(\lambda_i); \; i=1,...,N \\ Y^{(a)}_i \stackrel{\text{iid}}\sim \text{Poisson}(\mu_i); \; i=1,...,N \]
We want \(\lambda_i\) and \(\mu_i\) to depend on some covariates \(X_i\) and some parameters \(\theta\).
This can easily look like overdispersion in the aggregated data.
A simple model
Each team has an attack rating (\(\alpha\)) and a defence rating (\(\beta\)).
We also have home advantage (\(\eta\)) and a global mean (\(\gamma\)).
All of these are constant over time.
We set
\[ \log(\lambda_i) = \alpha_{\text{home}(i)} + \beta_{\text{away}(i)} + \gamma + \eta/2 \\ \log(\mu_i) = \alpha_{\text{away}(i)} + \beta_{\text{home}(i)} + \gamma - \eta/2 \]
A simulated example
Sample \(\alpha\), \(\beta\) from a Gaussian. Assume \(\alpha\) and \(\beta\) are positively correlated.
Set reasonable values for \(\eta\) and \(\gamma\).
Generate 5 seasons’ worth of data using the independent Poisson assumption.
Summary statistics:
mean(HG) sd(HG) mean(AG) sd(AG)
EPL 1.564 1.305 1.183 1.150
Sim 1.556 1.330 1.186 1.149Simulation results
chiSq df pValue
home 24.3305 5 0.0002
away 12.6575 5 0.0268Moral of the story
Don’t rely too much on empirical/aggregated data.
Make sure that plots and tests are relevant to your modelling assumptions.
Answer the question by formulating a model.
Always a good idea to simulate from your model.
Modelling women’s football
Conventional wisdom: extreme scorelines are overrepresented (relative to independent Poisson).
Different to men’s football, where low scorelines are overrepresented.
Is this true?
International results
Women:
awayGoals
homeGoals 0 1 2 3 4 5 6+
0 5.68 7.02 4.70 3.29 2.62 1.90 3.24
1 8.58 7.61 4.67 2.37 1.16 0.57 0.82
2 6.85 5.78 2.59 1.29 0.54 0.13 0.15
3 4.89 3.11 1.51 0.52 0.12 0.08 0.03
4 4.00 1.68 0.60 0.25 0.08 0.02 0.02
5 2.62 1.16 0.29 0.08 0.02 0.03 0.00
6+ 5.96 1.07 0.29 0.02 0.00 0.00 0.00Men:
awayGoals
homeGoals 0 1 2 3 4 5 6+
0 9.20 7.87 5.17 2.32 1.15 0.42 0.62
1 11.74 10.64 5.12 1.93 0.85 0.26 0.22
2 8.60 7.64 3.67 1.38 0.41 0.14 0.08
3 4.90 3.44 1.60 0.42 0.17 0.02 0.01
4 2.75 1.63 0.70 0.23 0.06 0.00 0.00
5 1.34 0.65 0.19 0.07 0.00 0.00 0.00
6+ 1.63 0.62 0.14 0.01 0.01 0.00 0.00International results - comparison
Difference (women minus men):
awayGoals
homeGoals 0 1 2 3 4 5 6+
0 -3.52 -0.85 -0.46 0.97 1.47 1.48 2.63
1 -3.16 -3.03 -0.45 0.43 0.31 0.31 0.60
2 -1.74 -1.86 -1.08 -0.09 0.13 0.00 0.07
3 -0.01 -0.33 -0.08 0.10 -0.05 0.06 0.02
4 1.24 0.05 -0.09 0.02 0.02 0.02 0.02
5 1.28 0.51 0.09 0.01 0.02 0.03 0.00
6+ 4.33 0.45 0.15 0.00 -0.01 0.00 0.00- What can we conclude?