## Show the code

```
library(rstan)
library(tidyverse)
library(lubridate)
library(tidybayes)
options(mc.cores = parallel::detectCores())
rstan_options(auto_write = TRUE)
source("R/utils.R")
```

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Author

Maurits Evers

Published

November 16, 2022

A Stan-based Bayesian model is used to predict outcomes of the 2022 FIFA football world cup in Qatar. The model itself is fairly simple, and was adopted from the model introduced by Andrew Gelman on his blog “Statistical Modeling, Causal Inference, and Social Science” here and here. Andrew’s model was also used by Mitzi Morris in the “Bayesian workflows with CmdStanPy” workshop to predict results of the 2019 FIFA women’s world cup in France.

Load necessary R libraries, set Stan options and load R convenience functions.

In a nutshell, the model estimates score differentials and uses those to determine

- the ranking of teams during the group stage, and
- winners and losers during the quarter-finals, semi-finals and final.

The model has some similarities with the Bradley-Terry model, and infers team abilities from historical match data by considering prior team abilities (more precisely: the model estimates team abilities by partially pooling ability estimates toward a prior estimate). Prior team abilities are given by (standardised) ELO ratings. Team abilities are estimated based on the observed goal difference.

The model uses draws from the posterior distribution of team abilities to simulate the outcome of all matches, from the group phase all the way to the final. Predicted outcomes of the group phase matches are used to determine the winners and runner ups in every group that advance to the round of 16, and from there to the quarter finals, semi finals and finals. Therefore, the number of draws mimics the total number of simulated world cup tournaments played. With every draw, a set up teams can be identified at any stage of the world cup tournament all the way to the final winner.

A CSV file with all 32 2022 WC football teams, their group associations and positions within their group is stored in ./data-raw/teams_by_group.csv.

ELO ratings were extracted from https://www.eloratings.net on 15 November 2022 and stored in a CSV file ./data-raw/elo_ratings_Nov22.csv in the code repository.

Team data with group information and ELO ratings are combined to give a `countries`

dataset

Historical football match data are conveniently downloaded from Kaggle here (note that this requires a Kaggle account which is free). The dataset includes international football results from 1872 to 2022, and is stored in a CSV file ./data-raw/results.csv in the code repository. Historical match data are filtered to include only games

- from 2020 onwards,
- with teams now playing in the 2022 football WC, and
- that were “non-friendly” matches (i.e. they were part of a football association’s tournament).

The Stan model file can be found in ./models/wc_qatar_full.stan in the code repository.

As explained previously, the model estimates team abilities and then simulates playing 1 million world cup tournaments from the group phase to the final, based on the inferred distribution of every team’s abilities. Estimates of teams and winners at every stage are stored in the fit object.

Based on the model fit, the end of the group phase has teams with the following final score differential distribution (Bayesian inference gives the full posterior *distribution*). The solid dots show the median values, the thick and thin horizontal lines denote the 66% and 95% quantile intervals, respectively.

```
fit %>%
spread_draws(final_score_diff[group, team_pos]) %>%
mutate(group = recode(
group,
`1` = "A", `2` = "B", `3` = "C", `4` = "D",
`5` = "E", `6` = "F", `7` = "G", `8` = "H")) %>%
left_join(countries, by = c("group", "team_pos")) %>%
ggplot(aes(final_score_diff, y = fct_reorder(team, final_score_diff))) +
stat_pointinterval() +
facet_wrap(~ group, scales = "free_y") +
theme_minimal() +
labs(
title = "Final score differential of teams per group at the end of the group phase",
x = "Final score differential",
y = "") +
theme(plot.title.position = "plot")
```

The two teams with the highest positive score differentials are named the winners and runner-ups of every group and advance to the quarter finals.

The top 20 most probable semi-finals are shown in the following plot.

```
fit %>%
spread_draws(teams_sf[match, team_idx]) %>%
left_join(
countries %>% select(ranking, team),
by = c("teams_sf" = "ranking")) %>%
select(-teams_sf) %>%
pivot_wider(names_from = team_idx, values_from = team) %>%
unite(match, `1`, `2`, sep = " : ") %>%
group_by(match) %>%
summarise(prob = n()) %>%
ungroup() %>%
mutate(prob = prob / sum(prob)) %>%
arrange(desc(prob)) %>%
head(n = 20) %>%
ggplot(aes(prob, fct_reorder(match, prob))) +
geom_point() +
geom_segment(aes(x = 0, xend = prob, yend = fct_reorder(match, prob))) +
theme_minimal() +
scale_x_continuous(labels = scales::percent) +
labs(
title = "Top 10 most probable semi-finals",
x = "Probability",
y = "Teams") +
theme(plot.title.position = "plot")
```

Team probabilities for winning the 2022 football world cup are given below.

```
fit %>%
spread_draws(winner_ff) %>%
left_join(countries, by = c("winner_ff" = "ranking")) %>%
group_by(team) %>%
summarise(prob = n()) %>%
ungroup() %>%
mutate(prob = prob / sum(prob)) %>%
ggplot(aes(prob, fct_reorder(team, prob))) +
geom_point() +
geom_segment(aes(x = 0, xend = prob, yend = fct_reorder(team, prob))) +
theme_minimal() +
scale_x_continuous(labels = scales::percent) +
labs(
title = "Probability of winning the 2022 football WC",
x = "Probability",
y = "") +
theme(plot.title.position = "plot")
```