English Premier League Managers

The Best and Worst EPL Managers Using Emprirical Bayes

By Ifeoma Egbogah in Football Bayes R

April 17, 2023

Introduction

In the last post I mentioned I read an article where the English Premier League managers' were ranked based on their raw winning averages. I felt this was a bit unfair. In this post I am going to attempt to rank each managers' performance in the premier league using the Empirical Bayes method to estimate their winning average as against using their raw winning averages and the manager with the highest average is our best manager.This borrows from Dave’s post on “Understanding Empirical Bayes Estimation Using Baseball Statistics”. (http://varianceexplained.org/). It is a great read.

So we are going to use the empirical Bayes method to: ** estimate the winning average of each manager ** estimate their credible interval

Let’s load our data set (data used in the Data-Exploration blog post (https://ifeoma-egbogah.netlify.app/blog/english-premier-league-ranking-series/02-data-exploration/data-exploration/)) and packages into R.

Raw Winning Average

The primary measure of a manager’s performance is the outcome of the matches under their charge, which can result in a win, draw, or loss. When victory appears unlikely, managers tend to prefer a draw over a loss, as the team would earn a point instead of none. Winning a match rewards the team with three points.

So, who are the best and worst managers? Let’s find out…..

     Raw Winning Average = Games Won/Total Games Managed

These are the ten managers with the highest raw winning average.

Top 10 Managers based on Raw Winning Average

Managers Total Games Managed Games Won Games Drawn Games Lost Average
Jim Barron 1 1 0 0 1.000
Eddie Newton 1 1 0 0 1.000
Josep Guardiola 76 55 13 8 0.724
Antonio Conte 76 51 10 15 0.671
Trevor Brooking 3 2 1 0 0.667
Alex Ferguson 810 528 168 114 0.652
Jose Mourinho 288 183 65 40 0.635
Carlo Ancelotti 76 48 13 15 0.632
Roberto Mancini 133 82 27 24 0.617
Manuel Pellegrini 114 70 21 23 0.614

Code

rwa_top <- epl %>%
  arrange(desc(average)) %>%
  dplyr::select(managers, total_games_managed, games_won, games_drawn, games_lost, average) %>%
  head(10)


knitr::kable(rwa_top, col.names = str_to_title(str_replace_all(names(rwa_top), "_", " ")))
Managers Total Games Managed Games Won Games Drawn Games Lost Average
Jim Barron 1 1 0 0 1.000
Eddie Newton 1 1 0 0 1.000
Josep Guardiola 76 55 13 8 0.724
Antonio Conte 76 51 10 15 0.671
Trevor Brooking 3 2 1 0 0.667
Alex Ferguson 810 528 168 114 0.652
Jose Mourinho 288 183 65 40 0.635
Carlo Ancelotti 76 48 13 15 0.632
Roberto Mancini 133 82 27 24 0.617
Manuel Pellegrini 114 70 21 23 0.614

…..and these are the ten managers with lowest raw winning average.

Bottom 10 Managers Based on Raw Winning AVerages

Managers Total Games Managed Games Won Games Drawn Games Lost Average
Eric Black 9 0 1 8 0
Archie Knox 1 0 0 1 0
Staurt McCall 2 0 0 2 0
Steve Holland 1 0 0 1 0
Frank de Boer 4 0 0 4 0
Billy McEwan 1 0 0 1 0
Jimmy Gabriel 7 0 1 6 0
Ray Lewington 3 0 2 1 0
Paul Goddard 2 0 2 1 0
Tony Book 1 0 1 0 0

Code

rwa_bottom <- epl %>%
  arrange(average) %>%
  dplyr::select(managers, total_games_managed, games_won, games_drawn, games_lost, average) %>%
  head(10)


knitr::kable(rwa_bottom, col.names = str_to_title(str_replace_all(names(rwa_bottom), "_", " ")))
Managers Total Games Managed Games Won Games Drawn Games Lost Average
Eric Black 9 0 1 8 0
Archie Knox 1 0 0 1 0
Staurt McCall 2 0 0 2 0
Steve Holland 1 0 0 1 0
Frank de Boer 4 0 0 4 0
Billy McEwan 1 0 0 1 0
Jimmy Gabriel 7 0 1 6 0
Ray Lewington 3 0 2 1 0
Paul Goddard 2 0 2 1 0
Tony Book 1 0 1 0 0

Wow, isn’t this interesting. Observing the table, we can conclude that these managers are neither the best nor the worst in the league. The raw winning averages are therefore not a reliable indicator for identifying the best and worst managers in the league. This measure only takes into account managers who have led a single game and happened to be either ‘lucky’ or ‘unlucky’ in winning or losing the match.

The average raw winning percentage across all managers is 0.282.

We definitely need a better estimate. Let’s get it.

Distribution of the Raw Winning Average

To start, we will analyze the raw winning averages of the managers by observing their distribution, typically accomplished by creating a histogram. After reviewing the histogram, we can conclude that the beta distribution is a suitable option for both our prior distribution and our binomial data.

Raw Winning Average Distribution of Managers

Code

epl%>%
  dplyr::select(managers, total_games_managed, games_won, games_drawn, games_lost)%>%
  mutate(average = games_won/total_games_managed)%>%
  ggplot(aes(average)) + 
  geom_histogram(binwidth = 0.1, fill="pink") + 
  labs(x = "Winning Averages",
       y = "Counts",
       title = "Distribution of EPL Managers' Winning Average from 1992 to 2018") +
  theme_light() +
  theme(plot.title = element_text(hjust = 0.5, size = 45),
        axis.title = element_text(size = 30),
        axis.text = element_text(size = 30))

What is Bayesian Inference?

Bayes typically refers to Thomas Bayes, an 18th-century British mathematician and theologian who developed Bayes' theorem, a fundamental concept in Bayesian inference. Bayes' theorem is a mathematical formula that describes the probability of an event based on prior knowledge or beliefs about the event and new evidence or data. Bayesian inference is a statistical approach that uses Bayes' theorem to update probabilities and make predictions based on new data or evidence

Empirical Bayes

Empirical Bayes is a statistical approach that combines frequentist and Bayesian methods to estimate the parameters of a statistical model. Empirical Bayes methods use a two-stage process to estimate parameters: first, they estimate the prior distribution of the parameters based on observed data, and then they use Bayesian methods to estimate the posterior distribution.

The key idea behind empirical Bayes is to estimate the prior distribution of the parameters based on the same data that is used to estimate the parameters themselves. This approach is useful when there is not enough data to estimate the prior distribution independently, or when the prior distribution is too broad and uninformative. By using the observed data to estimate the prior distribution, the empirical Bayes approach can often provide more accurate parameter estimates than standard frequentist methods.

Empirical Bayes methods have been widely used in many applications, including sports analytics, genomics, and economics. In sports analytics, empirical Bayes is used to estimate the performance of players based on their past performance and the performance of their opponents. In genomics, it is used to estimate the expression of genes based on DNA microarray data. In economics, it is used to estimate the impact of policies or programs on outcomes such as employment or income.

Step 1: Prior Distribution

The initial stage of Bayesian analysis involves selecting a prior distribution, which, in the case of empirical Bayes, is derived from the data. While this approach may not adhere strictly to Bayesian principles, I opted for it as I had a substantial amount of data available. Thus, our model takes the form of:

$$Y\sim\mbox{Binomial} (Total\ Games\ Managed,\ X)$$

$$X\sim\mbox{Beta}(\alpha,\ \beta)$$

The “hyper parameters” of our model are denoted by $$\alpha$$ and $$\beta$$. To determine these hyper parameters, we will employ the method of moments or maximum likelihood estimation. In this instance, we will be utilizing the method of moments, which involves calculating the mean and variance of the data. The formula for the method of moments is as follows:

$$\mu\ = mean(X)$$ $$\sigma\ = var(X)$$

$$\alpha = ((1 - \mu)/\sigma - 1/\mu) * \mu^2$$ $$\beta = \alpha * (1 /\mu -1 )$$

The fitdist function from the fitdistrplus package can be used to calculate the method of moments. I filtered out the total games managed that were less than 10 to reduce the noise. This gives us $$\alpha$$ = 3.77, $$\beta$$ = 8.631.

Maximum Likelihood Estimation of Shape Parameters

## Fitting of the distribution ' beta ' by matching moments 
## Parameters : 
##        estimate
## shape1     3.77
## shape2     8.63
## Loglikelihood:  -Inf   AIC:  Inf   BIC:  Inf

## shape1 
##   3.77

## shape2 
##   8.63

Code

epl_man <- epl %>%
  filter(total_games_managed >= 10)


e <- fitdist(epl_man$average, "beta", method="mme")

summary(e)

## Fitting of the distribution ' beta ' by matching moments 
## Parameters : 
##        estimate
## shape1     3.77
## shape2     8.63
## Loglikelihood:  -Inf   AIC:  Inf   BIC:  Inf

alpha <- e$estimate[1]
beta <- e$estimate[2]

alpha

## shape1 
##   3.77

beta

## shape2 
##   8.63

We can visualize the fit of the shape parameters for the beta distribution to the distribution of the raw winning averages shown in the histogram by plotting the distribution and examining its alignment

Prior Distribution

Code

epl_man%>%
  ggplot() + 
  geom_histogram(aes(average, y = after_stat(density)), binwidth=0.1, fill = "pink") + 
  stat_function(fun = function(x) dbeta(x, alpha, beta), colour="grey35", linewidth = 0.8) + 
  labs(x = "Winning Averages",
  y = "Density",
  title = "Prior Distribution Using Maximum Likelihood Estimation") +
  theme_light() +
  theme(plot.title = element_text(hjust = 0.5, size = 45),
        axis.title = element_text(size = 30),
        axis.text = element_text(size = 30))

Not perfect but not to bad. Next stop empirical Bayes estimate, then posterior probability and finally credible interval.

Step 2: Applying the Beta Distribution as a Prior for the Winning Average of Each Manager.

Since we have our prior probability distribution we can now calculate our posterior winning average for each manager by updating based on individual evidence. We do this by:

$$Posterior\ Winning\ Estimate = \frac{\alpha + Games\ Won}{\alpha + \beta + Total\ Games\ Managed}$$

Simple right.

Results

Managers Average Eb Winning Average
Eddie Howe 0.298 0.299
Brue Rioch 0.447 0.412
Arsene Wenger 0.575 0.571
Stewart Houston 0.368 0.343
David O’Leary 0.431 0.427
Jim Barron 1.000 0.356
Tim Sherwood 0.422 0.397
Graham Taylor 0.236 0.244
Brian Little 0.368 0.363
Remi Garde 0.100 0.178

Code

epl <-epl %>%
  mutate(eb_winning_average = (games_won + alpha)/(total_games_managed + alpha + beta))


epl%>%
  dplyr::select(managers, average, eb_winning_average)%>%
  head(10)%>%
  knitr::kable(col.names = str_to_title(str_replace_all(names(.), "_", " ")))
Managers Average Eb Winning Average
Eddie Howe 0.298 0.299
Brue Rioch 0.447 0.412
Arsene Wenger 0.575 0.571
Stewart Houston 0.368 0.343
David O’Leary 0.431 0.427
Jim Barron 1.000 0.356
Tim Sherwood 0.422 0.397
Graham Taylor 0.236 0.244
Brian Little 0.368 0.363
Remi Garde 0.100 0.178

Results

According to this estimate, the top ten managers are in! Drum roll, please. And the winner is Pep Josep Guadiola with a remarkable winning average of 0.665 (66.5%), even though he has only spent two seasons in the league based on this data. Coming in at a close second is Alex Ferguson with a winning average of 0.647 (64.7%). Another notable mention goes to the Prof (Arsene Wenger), who secures the number 8 spot with a winning average of 0.571 (57.10%).

Top 10 Managers Using Empirical Bayes

Managers Total Games Managed Games Won Average Eb Winning Average
Josep Guardiola 76 55 0.724 0.665
Alex Ferguson 810 528 0.652 0.647
Jose Mourinho 288 183 0.635 0.622
Antonio Conte 76 51 0.671 0.620
Roberto Mancini 133 82 0.617 0.590
Carlo Ancelotti 76 48 0.632 0.586
Manuel Pellegrini 114 70 0.614 0.584
Arsene Wenger 828 476 0.575 0.571
Rafael Benitez 302 156 0.517 0.508
Jurgen Klopp 106 56 0.528 0.505

Code

epl_man2 <- epl %>%
  arrange(desc(eb_winning_average)) %>%
  dplyr::select(managers, total_games_managed, games_won, average, eb_winning_average) %>%
  head(10)

knitr::kable(epl_man2, col.names = str_to_title(str_replace_all(names(epl_man2), "_", " ")))
Managers Total Games Managed Games Won Average Eb Winning Average
Josep Guardiola 76 55 0.724 0.665
Alex Ferguson 810 528 0.652 0.647
Jose Mourinho 288 183 0.635 0.622
Antonio Conte 76 51 0.671 0.620
Roberto Mancini 133 82 0.617 0.590
Carlo Ancelotti 76 48 0.632 0.586
Manuel Pellegrini 114 70 0.614 0.584
Arsene Wenger 828 476 0.575 0.571
Rafael Benitez 302 156 0.517 0.508
Jurgen Klopp 106 56 0.528 0.505

Lollipop Plot of Top 10 Managers Using Empirical Bayes

Code

epl %>%
  arrange(desc(eb_winning_average)) %>%
  dplyr::select(managers, total_games_managed, games_won, average, eb_winning_average) %>%
  head(10)%>%
  mutate(managers = fct_reorder(managers, eb_winning_average))%>%
  ggplot(aes(eb_winning_average, managers, colour = managers)) +
  geom_point(show.legend = FALSE, size = 10) +
  geom_segment(aes(xend = 0, yend = managers), linewidth = 1.5, show.legend = FALSE) +
  scale_color_scico_d(palette = "bam") +
  labs(x = "Empirical Bayes Average", 
       y = "Managers",
       title = "Top 10 Managers Using Empirical Bayes from 1992 to 2018") +
  theme_light() +
  theme(plot.title = element_text(hjust = 0.5, size = 50),
        axis.title = element_text(size = 30),
        axis.text = element_text(size = 30))

Let’s move on to the bottom ten managers in the Premier League based on our estimates. According to the data, Terry Connor had the lowest winning average of 0.1504 (15.04%). Interestingly, we can observe that the empirical Bayes estimation did not include managers who only managed one or two games, unlike the raw winning average.

Bottom 10 Managers Using Empirical Bayes

Managers Total Games Managed Games Won Average Eb Winning Average
Terry Connor 13 0 0.000 0.149
John Gorman 42 5 0.119 0.161
Roy McFarland 22 2 0.091 0.168
Steve Wigley 16 1 0.062 0.168
Aidy Boothroyd 38 5 0.132 0.174
Eric Black 9 0 0.000 0.176
Remi Garde 20 2 0.100 0.178
Billy Davies 14 1 0.071 0.181
Mark McGhee 24 3 0.125 0.186
Chris Hutchings 24 3 0.125 0.186

Code

Lollipop Plot of Bottom 10 Managers Using Empirical Bayes

Code

epl %>%
  arrange(eb_winning_average) %>%
  dplyr::select(managers, total_games_managed, games_won, average, eb_winning_average) %>%
  head(10)%>%
  mutate(managers = fct_reorder(managers, eb_winning_average))%>%
  ggplot(aes(eb_winning_average, managers, colour = managers)) +
  geom_point(show.legend = FALSE, size = 10) +
  geom_segment(aes(xend = 0, yend = managers), linewidth = 1.5, show.legend = FALSE) +
  scale_color_scico_d(palette = "bam") +
  labs(x = "Empirical Bayes Average", 
       y = "Managers",
       title = "Bottom 10 Managers Using Empirical Bayes from 1992 to 2018") +
  theme_light() +
  theme(plot.title = element_text(hjust = 0.5, size = 50),
        axis.title = element_text(size = 30),
        axis.text = element_text(size = 30))

Posterior Distribution

Now that we have obtained our posterior winning average, it is possible to determine the shape parameters of the posterior probability distribution for each manager. Updating the shape parameters of the prior beta distribution can achieve this with ease.

Posterior Distribution

Managers Alpha2 Beta2
Eddie Howe 37.77 88.63
Brue Rioch 20.77 29.63
Arsene Wenger 479.77 360.63
Stewart Houston 10.77 20.63
David O’Leary 151.77 203.63
Jim Barron 4.77 8.63
Tim Sherwood 22.77 34.63
Graham Taylor 24.77 76.63
Brian Little 56.77 99.63
Remi Garde 5.77 26.63

Code

epl <- epl%>%
  mutate(alpha2 = games_won + alpha,
         beta2 = total_games_managed - games_won + beta)

top_10 <- epl%>%
  head(10)

top_10%>%
  dplyr::select(managers, alpha2, beta2)%>%
knitr::kable(col.names = str_to_title(names(.)))
Managers Alpha2 Beta2
Eddie Howe 37.77 88.63
Brue Rioch 20.77 29.63
Arsene Wenger 479.77 360.63
Stewart Houston 10.77 20.63
David O’Leary 151.77 203.63
Jim Barron 4.77 8.63
Tim Sherwood 22.77 34.63
Graham Taylor 24.77 76.63
Brian Little 56.77 99.63
Remi Garde 5.77 26.63

Now that we have the $$\alpha$$ and $$\beta$$ for each manager we can now visualize their posterior probability distribution. We will begin by looking at the posterior probability distribution of the top 5 managers based on our empirical Bayes estimation.

Posterior Distribution of Top 5 Managers

Code

epl_man4 <- epl %>%
  arrange(desc(eb_winning_average)) %>%
  top_n(5, eb_winning_average) %>%
  crossing(x = seq(0.01, 0.8, 0.001))%>%
  ungroup()%>%
  mutate(density = dbeta(x, games_won + alpha2,
         total_games_managed - games_won + beta2)) 

epl_man4%>%  
  ggplot(aes(x, density, colour = managers)) + 
  geom_line(linewidth = 1.8) +
  stat_function(fun=function(x) dbeta(x, alpha, beta), lty = 2, colour = "grey35", linewidth = 2) + 
  scale_colour_scico_d(palette = "bam") +
  labs(x = "Winning Averages", 
       y = "Density", 
       title="Prior and Posterior Distribution", 
       subtitle="Posterior Distribution of the Top Five Managers Using Emprical Bayes",
       colour = "Managers") + 
  theme(plot.title = element_text(hjust = 0.5, size = 50),
        plot.subtitle = element_text(size = 35),
        axis.title = element_text(size = 30),
        axis.text = element_text(size = 30),
        legend.title = element_text(size = 40),
        legend.key.size  = unit(3, "cm"),
        legend.text = element_text(size = 30))

Sir Alex Ferguson has a narrower posterior probability distribution (since he managed more games, we have enough evidence) compared to the broader posterior probability distribution of Josep Guardiola and Antonio Conte (both managed fewer games than Sir Ferguson). The dashed curve is the prior distribution.

Just for fun lets compare the posterior probability distribution of a few managers outside the top 5.

Posterior Distribution of Some Managers

Code

manager <- c("Arsene Wenger", "Alex Ferguson", "Ryan Giggs", "Josep Guardiola", "Eric Black", "Alan Shearer")

manager_career <- epl %>%
  filter(managers %in% manager)

manager <- manager_career %>%
  crossing(x= seq(0.01, 0.9, 0.001)) %>%
  ungroup() %>%
  mutate(density = dbeta(x, alpha2, beta2))
  

manager%>%
  ggplot(aes(x, density, colour = managers)) + 
  geom_line(linewidth = 1.8) + 
  stat_function(fun=function(x) dbeta(x, alpha, beta), lty =2, colour="grey35", linewidth = 2) +  
  scale_colour_scico_d(palette = "bam") +
  labs(x ="Winning Averages", 
       y = "Density", 
       title = "Prior and Posterior Distribution", 
       subtitle="Posterior Distribution of Some Selected Managers",
       colour = "Managers") + 
  theme(plot.title = element_text(hjust = 0.5, size = 50),
        plot.subtitle = element_text(size = 35),
        axis.title = element_text(size = 30),
        axis.text = element_text(size = 30),
        legend.title = element_text(size = 40),
        legend.key.size  = unit(3, "cm"),
        legend.text = element_text(size = 30))

Empirical Bayes Estimste vs Raw Average

How did our empirical Bayes winning average change compared to the raw winning average? Let’s make another plot. The red dashed horizontal line marks $$y=\frac{\alpha}{\alpha + \beta}$$ this shows what each manager’s estimate would be if we had no evidence. Points above and below the line move towards this line. The diagonal line marks $$x=y$$. Points close to the line are estimates that didn’t get shrunk by empirical Bayes because we have enough evidence so their values are close to the raw estimate and they are managers that over saw more than 300 games.

Empirical Bayes Estimate vs Raw Winning Average

Code

library(ggrepel)

epl%>%
  ggplot(aes(average, eb_winning_average, colour = total_games_managed)) + 
  geom_hline(yintercept = alpha/(alpha + beta), colour=  "red", lty=2, linewidth = 1.5)  + 
  geom_point(size = 10) + 
  geom_text_repel(data = subset(epl, total_games_managed >= 450), aes(label = managers), box.padding=unit(0.5, "lines"), colour="black", size = 12) + 
  geom_abline(colour = "red", linetype = 1, linewidth = 1.5) + 
  #scale_color_gradient(trans= "log", low="midnightblue", high="pink", name="Games Managed", breaks = 10^(1:5)) + 
  scale_colour_scico(palette = "bam", trans = "log", breaks = 10^(1:5), direction = -1) +
  labs(x = "Raw Winning Averages",
       y = "Empirical Bayes Estimate of Winning Average",
       title = "Empirical Bayes Estimate vs Raw Winning Average", 
       subtitle = "Names of Managers Who Managed More Than 450 games",
       colour = "Total Games Managed") + 
 theme(plot.title = element_text(hjust = 0.5, size = 55),
        plot.subtitle = element_text(size = 35),
        axis.title = element_text(size = 35),
        axis.text = element_text(size = 35),
        legend.title = element_text(size = 35),
        legend.key.size  = unit(3, "cm"),
        legend.text = element_text(size = 30))

The estimates are only different for managers who over saw fewer games (less than 2 games). The raw winning average is 0 or 1 for such managers however, the empirical Bayes estimate for such managers are close to the overall mean of the prior beta distribution. This is known as shrinkage.

Managers Who Oversaw Less Than Two Games

Code

epl%>%
  ggplot(aes(average, eb_winning_average, colour = total_games_managed)) +
  geom_hline(yintercept = alpha/(alpha + beta), colour=  "red", lty=2, linewidth = 1.5)  +
  geom_point(size = 10) + 
  geom_text_repel(data=subset(epl, total_games_managed <= 2), aes(label =  managers, colour="grey50"), box.padding = unit(0.5, "lines"), force=1.9, colour="black", size= 12) + 
  geom_abline(colour = "red", linetype=1, linewidth = 1.5) + 
 # scale_color_gradient(trans= "log", low="midnightblue", high="pink", name ="Games Managed", breaks = 10^(1:5)) + 
  scale_colour_scico(palette = "bam", trans = "log", breaks = 10^(1:5), direction = -1) +
  labs(x = "Winning Averages",
       y = "Empirical Bayes Estimate of Winning Averages",
       title = "Empirical Bayes Estimate vs Raw Winning Average", 
       subtitle = "Managers Who Managed Less Than Two games",
       colour = "Games Managed") +
 theme(plot.title = element_text(hjust = 0.5, size = 55),
        plot.subtitle = element_text(size = 35),
        axis.title = element_text(size = 35),
        axis.text = element_text(size = 35),
        legend.title = element_text(size = 35),
        legend.key.size  = unit(3, "cm"),
        legend.text = element_text(size = 30))

Step 4: Credible Interval

Credible interval tells what percentage (for this example 95%) of the the posterior winning average distribution lies within the interval. It shows how much uncertainty is present in our estimate. This computed using qbeta.

Credible Interval

Table: Table 1: Credible Interval

Managers Games Won Total Games Managed Eb Winning Average Low High
Josep Guardiola 55 76 0.665 0.632 0.759
Alex Ferguson 528 810 0.647 0.635 0.679
Jose Mourinho 183 288 0.622 0.603 0.676
Antonio Conte 51 76 0.620 0.585 0.717
Roberto Mancini 82 133 0.590 0.563 0.668
Carlo Ancelotti 48 76 0.586 0.551 0.686
Manuel Pellegrini 70 114 0.584 0.554 0.668
Arsene Wenger 476 828 0.571 0.559 0.604
Rafael Benitez 156 302 0.508 0.489 0.563
Jurgen Klopp 56 106 0.505 0.474 0.594

Code

credible_interval <- epl%>%
  arrange(desc(eb_winning_average))%>%
  dplyr::select(managers, games_won, total_games_managed, eb_winning_average, low, high)%>%
  head(10)


knitr::kable(credible_interval, caption = "Credible Interval", col.names = str_to_title(str_replace_all(names(credible_interval), "_", " ")))

Table: Table 2: Credible Interval

Managers Games Won Total Games Managed Eb Winning Average Low High
Josep Guardiola 55 76 0.665 0.632 0.759
Alex Ferguson 528 810 0.647 0.635 0.679
Jose Mourinho 183 288 0.622 0.603 0.676
Antonio Conte 51 76 0.620 0.585 0.717
Roberto Mancini 82 133 0.590 0.563 0.668
Carlo Ancelotti 48 76 0.586 0.551 0.686
Manuel Pellegrini 70 114 0.584 0.554 0.668
Arsene Wenger 476 828 0.571 0.559 0.604
Rafael Benitez 156 302 0.508 0.489 0.563
Jurgen Klopp 56 106 0.505 0.474 0.594

Managers in the top 20, their credible intervals are narrow. Managers such as Alex Ferguson and Arsene Wenger who have overseen over 800 games our uncertainty is low however managers such as Felipe Scolari and Guus Hiddink have a much broader credible interval as they have managed fewer games (less than 40 games). Let’s visualize this in a plot.

Alex Ferguson vs Felipe Scolari

Code

manager_ci <- c("Alex Ferguson", "Felipe Scolari")

ci <- epl %>%
  filter(managers %in% manager_ci)

ci2 <- ci %>%
  crossing(x = seq(0.005, 0.9, 0.001)) %>%
  ungroup() %>% 
  mutate(density1 = dbeta(x, alpha2, beta2))
 
  
ci3 <- ci2 %>%
  mutate(cum = pbeta(x, alpha2, beta2)) %>%
  filter(cum > .025, cum < .975)
  

ci2 %>%
  ggplot(aes(x, density1, colour = managers)) + 
  geom_line(show.legend = FALSE, linewidth = 1.5) + 
  geom_ribbon(aes(ymin = 0, ymax = density1), data= ci3, alpha = 0.5, fill = "pink", show.legend = FALSE) + 
  stat_function(fun=function(x) dbeta(x, alpha, beta), colour="grey35", lty=2, linewidth = 1.5) + 
  geom_errorbarh(aes(xmin= qbeta(0.025, alpha2, beta2), xmax = qbeta(0.975, alpha2, beta2), y = 0), height = 2, colour = "red", linewidth = 1.3) + 
  facet_wrap(~managers) + 
  scale_colour_scico_d(palette = "acton") +
  labs(x = "Credible Interval",
       y = "Posterior Distribution",
       title = "Posterior Distribution and Credible Interval\nfor Sir Alex Ferguson and Felipe Scolari") +  
  theme(plot.title = element_text(hjust = 0.5, size = 50),
        plot.subtitle = element_text(size = 35),
        axis.title = element_text(size = 35),
        axis.text = element_text(size = 35),
        strip.text = element_text(size = 40, colour = "black"),
        legend.title = element_text(size = 35),
        legend.key.size  = unit(3, "cm"),
        legend.text = element_text(size = 30))

Since it is difficult to visualize the credible interval and posterior probability distribution density plot for all the managers in the top 20 lets make a point and error bar plot instead.

Top Twenty

Code

epl_man5 <- epl %>%
  arrange(desc(eb_winning_average)) %>%
  top_n(20, eb_winning_average) %>%
  mutate(managers = reorder(managers, eb_winning_average)) 

  
epl_man5%>%
  ggplot(aes(eb_winning_average, managers, colour = managers)) +
  geom_errorbarh(aes(xmin= low, xmax= high), show.legend = FALSE, linewidth = 1.5) +
  geom_point(show.legend = FALSE, size = 10) + 
  geom_vline(xintercept = alpha/(alpha + beta), colour="grey50", lty=2, linewidth = 1.5)  +
  scale_colour_scico_d(palette = "bam", direction = -1) +
  labs(x = "Empirical Bayes Winning Averages with 95% Credible Interval",
       y = "Managers",
       title = "Empirical Bayes Winning Averages and Credible Interval\nfor the Top 20 Managers",
       colour = "Managers")+
  theme_light() +
  theme(plot.title = element_text(hjust = 0.5, size = 55),
        plot.subtitle = element_text(size = 35),
        axis.title = element_text(size = 35),
        axis.text = element_text(size = 35),
        legend.title = element_text(size = 35),
        legend.key.size  = unit(3, "cm"),
        legend.text = element_text(size = 30))

Appendix

During our analysis, we made several assumptions. Firstly, we assumed that all the winning averages originated from a single distribution, and secondly, we assumed that each manager managed the same team since their league debut. However, in reality, except for Alex Ferguson and Arsene Wenger who have managed only one club since their debut in the league, other managers have managed several clubs over the years, resulting in changes to their winning averages over time depending on the club they managed.

We could estimate separate beta priors for each team managed as well as for every five years.