It is quite some time since I last made a foray into any football related matters but I was motivated to revisit the application of statistical techniques to predict match outcomes after reading Ian Graham’s fascinating book How to Win The Premier League. Graham was previously head of data analytics at Liverpool FC and he described in detail how the collection of data and its use in tracking player performance and value made a significant contribution to Liverpool’s resurgence as a footballing force under Jürgen Klopp. This culminated in the club winning the Champions League in 2019, and in 2020 it secured its first league title in 30 years.
One area of the book that piqued my interest was the discussion of how a couple of academics[1] in the 1990s set out to assess whether it was possible to predict the outcome of football matches in order to beat the odds offered by bookmakers (answer: yes, but only in certain circumstances). The so-called Dixon-Coles model (no relation) used a Poisson distribution to estimate the number of goals scored and conceded by each team in a head-to-head match. This was particularly interesting to me because I used a similar approach when trying to assess who would win the World Cup in 2022 and I was motivated to see whether I could improve the code to extend the analysis to Premier League football. You may ask why an economist would want to do that. One reason is that it satisfies my long-standing interest in statistics and growing interest in data science methods, and with time on my hands during the summer holiday, why not? It also provided an opportunity to use ChatGPT to fill in any gaps in my coding knowledge in order to assess whether it really is the game changer for the coding industry that people say (my experience in this regard was very positive).
Turning to our problem, the number of goals scored by each team is a discrete number (i.e. we cannot have fractions of goals) so we have to use a discrete probability distribution and it turns out that the Poisson distribution does this job rather well. As Ian Graham put it, “the Poisson distribution, also known as the ‘Law of Small Numbers’, governs the statistics of rare events”. And when you think about it, goals are rare events in the context of all the action that takes place on a football pitch. In aggregate the number of goals scored across all teams does indeed follow a Poisson distribution (see above). However, since the sample size for each team is much smaller (each team plays only 19 home and 19 away games), this increases the likelihood of deviations from the expected average.
That caveat aside, following the literature, if a variable follows the Poisson distribution, it is defined by a probability mass function as set out below:
By simulating each game 1000 times we can derive an estimate of what the average result is likely to be and construct the expected league rankings. To add a little spice, we can adjust λ by adding a random number in the range [-n, n, 0<n<1] in a bid to capture the element of luck inherent in any sporting contest on the basis that over the course of a number of games, this element averages out to zero. One thing to take into account is that the performance of promoted teams will not be so strong in the Premier League compared to the Championship, with the evidence suggesting that promoted teams score around 0.7 goals per game fewer and concede 0.7 goals per game more. We thus amend the λ value for promoted teams accordingly. It is possible to add a number of other tweaks[2] but for the purposes of this exercise, this is a good starting point.We construct λ as follows:
I ran the analysis over the past six seasons to assess the usefulness of the model and measured each team’s expected league position against outturn. On average, the model predicts to within three places the expected league position with 64% accuracy. That’s not bad although I would have hoped for a bit better.
But excluding the Covid-impacted season 2019-20, when the absence of crowds had a significant impact on expected results, we can raise that slightly to 66%. But ironically, it is season 2022-23 which imparts a serious downward bias to results – excluding that season raises the accuracy rate to 69%. A popular cliche associated with football is that the table doesn't lie. What we have discovered here is either that it does, or the model needs a bit of improvement and at some point I will try and incorporate a dynamic estimate of λ to give a higher weight to more recent performance, but that is a topic for another day.
What is the point of it all?
As an economist who spends time looking at macro trends, it may seem difficult to justify such an apparently trivial pursuit as predicting football outcomes. But there are some lessons that we can carry over into the field of macro and market forecasting. In the first instance, the outcomes of football matches involve inherent uncertainty and the application of stochastic simulation techniques such as those applied here can equally be used to account for randomness and uncertainty in economic and financial predictions. Indeed they are often used in the construction of error bands around forecast outcomes. A second application is to demonstrate that probabilistic forecasting is a viable statistical technique to use when faced with a priori uncertainty. We do not know the outcome of a football match in advance any more than we know what will be the closing price of the stock market on any given day. But simulating past performance can give us a guide as to the possible range of outcomes. A third justification is to demonstrate the impact of scenario analysis: by changing model parameters we are able to generate different outcomes and with modern computing techniques rendering the cost of such analysis to be trivially small, it is possible to run a huge number of different alternatives to assess model sensitivities.
Forecasting often throws up surprises and we will never be right 100% of the time. If you can come up with a system that predicts the outcome significantly better than 50% of the time, you are on the right track. Sometimes the outlandish does happen and I am hoping that the model’s current prediction that Newcastle United can finish fourth in the Premiership (probability: 37%) will be one of them.
[1] Dixon, M. J. and S. G. Coles (1997) ‘Modelling Association Football Scores and Inefficiencies in the Football Betting Market’ Journal of the Royal Statistical Society Series C: Applied Statistics, Volume 46, Issue 2, 265–280
[2] For example, if we want to model the overall points outcome, a better approximation to historical outturns is derived if we adjust λ by a multiplicative constant designed to reduce it relative to the mean for teams where it is very high and raise it for those where it is low. This is a way of accounting for the fact that a team which performed well in the previous season may not perform so well in the current season, and teams which performed badly may do rather better. This does not have a material impact on the ranking but does reduce the dispersion of points between the highest and lowest ranking teams in line with realised performance. Another tweak is to adjust the random variable to follow a Gaussian distribution rather than the standard uniform assumption.
