My wife and I love to watch the Great British Bake Off on Netflix. The competition is for the most part collegial in general and all around feel good television, especially at night. After watching several seasons of the show, a lingering question came to mind: how good are the judges at estimating talent?

The format of the show is composed of three rounds; the first and third rounds have themes/genre of baked good that the contestants knew about in advance and could decide (and practice) what they wanted to make. The second/middle round is composed of a “technical” challenge where the bakers have all given the same ingrediants and instructions and asked to make something of which they had no prior knowledge. Unlike the other rounds, the judges judge each bake blindly in the technical (and of course the contestants make the same dish).

Enter the Pyschometrics

This set-up is perfect for understanding how well the judges can estimate “ability” to use a psychometric term. Because we have contestants facing the same exact challenge and being judged blindly, we can use pyschometric tools to judge the “ability” of the baker and the “difficulty” of the challenge. There is a lot of noise on these measures due to the fact that contestants are eliminated after each show meaning that they do not get a chance at each challenge, but it will give a little bit of insight the ability of the bakers. We can then compare the outcomes of each round with the judged “technical” ability.

IRT and CRM

High stakes tests like the GRE and GMAT use something called Item Response Theory (IRT) to measure “ability.” The tests work by matching item difficulty (or how hard a question is) to the test-taker’s latent ability (tendency to get the right answer). Test takers should get the correct answer for those items where their ability is greater than the item difficulty, should get those items wrong where the difficulty is greater than their ability, and some distribution due to the measurement error in both ability and difficulty.

Not mentioned here, but a random component for guessing can be added. The literature for IRT is immense and there are many different models (2PL,3PL, Rasch, among others).

The key contribution of IRT over classical test theory (in my opinion) is that there is some latent noise in the test question/item.

IRT typically required a single “correct” answer. When we are looking at the rating of Bakers from 1 to N bakers, we need to observe the continuous data of the data. Enter the Continuous Response Models which allow us to use the principles of IRT for continuous data. In particular we will use Samejima’s continuous response model for the ranking of contestants.

Analysis Plan

So now we can lay out our analysis plan:

Get the baking results

Run the CRM to understand and rank baker ability

Compare the modelled ability to the actual results

Getting the Data

The first part in this analysis is getting the data. Luckily, Wikipedia, the grandest resource on the interweb, provides these data in a regular pattern.

First, we will load the usual suite of packages for webscraping and analysis.

── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
x dplyr::filter() masks stats::filter()
x dplyr::lag() masks stats::lag()

library(rvest)

Attaching package: 'rvest'

The following object is masked from 'package:readr':
guess_encoding

library(data.table)

Attaching package: 'data.table'

The following objects are masked from 'package:dplyr':
between, first, last

The following object is masked from 'package:purrr':
transpose

library(cmdstanr)

This is cmdstanr version 0.4.0

- Online documentation and vignettes at mc-stan.org/cmdstanr

- CmdStan path set to: /Users/michael/.cmdstanr/cmdstan-2.28.2

- Use set_cmdstan_path() to change the path

A newer version of CmdStan is available. See ?install_cmdstan() to install it.
To disable this check set option or environment variable CMDSTANR_NO_VER_CHECK=TRUE.

To run an initial test, I am just going to pull Season 3.

Now the more challenging part is to parse all of the results. I am going to use some loops and index variables because I can’t think of a more expediant way to do it.

Importantly, each baker will appear for as many challenges in which they participated. This means someone who was eliminated after the first show will only have one record (enter measurement error) and those who participated in later rounds will appear multiple times.

technicals <-list()z <-1for(i inseq_along(tables)){ x <- tables[[i]] interesting <-grepl(pattern ="Baker|Technical", names(x))if(sum(interesting)<2){next() } y <- x[,interesting]names(y) <-c("baker", "technical") y$technical_no <- z technicals[[i]] <- y z <-1+z}out_long<-do.call(rbind, technicals)setDT(out_long)knitr::kable(head(out_long,10))

baker

technical

technical_no

Brendan

10th

1

Cathryn

5th

1

Danny

7th

1

James

2nd

1

John

11th

1

Manisha

6th

1

Natasha

12th

1

Peter

3rd

1

Ryan

8th

1

Sarah-Jane

1st

1

Now with a little nore parsing we can extract the result and associated rank of the bakers.

In completely transparency, I utilized code from https://cengiz.me/posts/crm-stan/ which provided an excellent starting point for the analysis.

The code is lightly modified (just to tighten some priors) because of the

writeLines(readLines("irt.stan"))

// From https://cengiz.me/posts/crm-stan/
data{
int J; // number of items
int I; // number of individuals
int N; // number of observed responses
int item[N]; // item id
int id[N]; // person id
real Y[N]; // vector of transformed outcome
}
parameters {
vector[J] b; // vector of b parameters forJ items
real mu_b; // mean of the b parameters
real<lower=0> sigma_b; // standard dev. of the b parameters
vector<lower=0>[J] a; // vector of a parameters for J items
real mu_a; // mean of the a parameters
real<lower=0> sigma_a; // standard deviation of the a parameters
vector<lower=0>[J] alpha; // vector of alpha parameters for J items
real mu_alpha; // mean of alpha parameters
real<lower=0> sigma_alpha; // standard deviation of alpha parameters
vector[I] theta; // vector of theta parameters for I individuals
}
model{
mu_b ~ normal(0,5);
sigma_b ~ normal(0,1);
b ~ normal(mu_b,sigma_b);
mu_a ~ normal(0,5);
sigma_a ~ normal(0,2.5);
a ~ normal(mu_a,sigma_a);
mu_alpha ~ normal(0,5);
sigma_alpha ~ cauchy(0,2.5);
alpha ~ normal(mu_alpha,sigma_alpha);
theta ~ normal(0,1); // The mean and variance of theta is fixed to 0 and 1
// for model identification
for(i in 1:N) {
Y[i] ~ normal(alpha[item[i]]*(theta[id[i]]-b[item[i]]),alpha[item[i]]/a[item[i]]);
}
}

Now we just compile the model and format our data:

fit <- mod$sample(dat_stan,parallel_chains =4,max_treedepth =15, adapt_delta = .99, refresh =0)

Running MCMC with 4 parallel chains...

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 1 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 1 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 1 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 1 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 1

Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 2 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 2

Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 2 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 2

Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 2 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 2

Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 2 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 2

Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 2 Exception: normal_lpdf: Scale parameter is 0, but must be positive! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 32, column 9 to column 37)

Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 2

Chain 2 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 2 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 2 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 2 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 2

Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 3 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 3

Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 3 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 3

Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 3 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 3

Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 3 Exception: normal_lpdf: Scale parameter is 0, but must be positive! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 32, column 9 to column 37)

Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 3

Chain 3 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 3 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 3 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 3 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 3

Chain 4 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 4 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 4 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 4 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 4

Chain 4 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 4 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 4 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 4 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 4

Chain 4 Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:

Chain 4 Exception: normal_lpdf: Location parameter is inf, but must be finite! (in '/var/folders/0x/6bnjy4n15kz8nbfbbwbyk3pr0000gn/T/RtmprGJfRZ/model-a9ba72708057.stan', line 46, column 8 to column 90)

Chain 4 If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,

Chain 4 but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.

Chain 4

Chain 4 finished in 29.5 seconds.
Chain 1 finished in 69.6 seconds.
Chain 3 finished in 105.3 seconds.
Chain 2 finished in 164.6 seconds.
All 4 chains finished successfully.
Mean chain execution time: 92.3 seconds.
Total execution time: 164.8 seconds.

Warning in seq.default(from = 1, len = along - 1): partial argument match of
'len' to 'length.out'

Warning in seq.default(to = N - 1, len = N - along): partial argument match of
'len' to 'length.out'

Warning in seq.default(len = N): partial argument match of 'len' to 'length.out'

Warning in seq.default(along = arg.names): partial argument match of 'along' to
'along.with'

Warning in seq.default(len = length(arg.list)): partial argument match of 'len'
to 'length.out'

Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'

Warning in seq.default(len = ncol(arg.dim)): partial argument match of 'len' to
'length.out'

Warning in seq.default(len = N): partial argument match of 'len' to 'length.out'

Warning in seq.default(along = arg.names): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = arg.names): partial argument match of 'along' to
'along.with'

Warning in seq.default(len = length(arg.names)): partial argument match of 'len'
to 'length.out'

Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'

Warning: 40 of 4000 (1.0%) transitions ended with a divergence.
This may indicate insufficient exploration of the posterior distribution.
Possible remedies include:
* Increasing adapt_delta closer to 1 (default is 0.8)
* Reparameterizing the model (e.g. using a non-centered parameterization)
* Using informative or weakly informative prior distributions

5 of 4000 (0.0%) transitions hit the maximum treedepth limit of 15 or 2^15-1 leapfrog steps.
Trajectories that are prematurely terminated due to this limit will result in slow exploration.
Increasing the max_treedepth limit can avoid this at the expense of more computation.
If increasing max_treedepth does not remove warnings, try to reparameterize the model.

We’re interested here in theta which represents the ability of the bakers.

Warning in seq.default(from = 1, len = along - 1): partial argument match of
'len' to 'length.out'

Warning in seq.default(to = N - 1, len = N - along): partial argument match of
'len' to 'length.out'

Warning in seq.default(len = N): partial argument match of 'len' to 'length.out'

Warning in seq.default(along = arg.names): partial argument match of 'along' to
'along.with'

Warning in seq.default(len = length(arg.list)): partial argument match of 'len'
to 'length.out'

Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'
Warning in seq.default(along = perm): partial argument match of 'along' to
'along.with'

Warning in seq.default(len = ncol(arg.dim)): partial argument match of 'len' to
'length.out'

Warning in seq.default(len = N): partial argument match of 'len' to 'length.out'

Warning in seq.default(along = arg.names): partial argument match of 'along' to
'along.with'
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'along.with'

Warning in seq.default(len = length(arg.names)): partial argument match of 'len'
to 'length.out'

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'along.with'

Joining, by = "baker_id"

out_come_with_rank <- combined_out %>%arrange(desc(median)) %>%mutate(outcome_modelled =row_number()) %>%select(outcome_modelled, baker) %>%left_join(performance) %>%mutate(outcome_realized =case_when( value =="Runner-up"~2, value =="WINNER"~1,TRUE~13-round_num ))

Joining, by = "baker"

knitr::kable(out_come_with_rank)

outcome_modelled

baker

variable

value

round_num

perf

outcome_realized

1

James

10

Runner-up

10

Runner-up 10

2

2

Brendan

10

Runner-up

10

Runner-up 10

2

3

Danny

9

OUT

9

OUT 9

4

4

John

10

WINNER

10

WINNER 10

1

5

Cathryn

8

OUT

8

OUT 8

5

6

Victoria

3

OUT

3

OUT 3

10

7

Sarah-Jane

7

OUT

7

OUT 7

6

8

Ryan

7

OUT

7

OUT 7

6

9

Peter

2

OUT

2

OUT 2

11

10

Manisha

5

OUT

5

OUT 5

8

11

Natasha

1

OUT

1

OUT 1

12

12

Stuart

4

OUT

4

OUT 4

9

Now we can see what the correlation of ability performance is:

Not too bad! It would seem that there is evidence that the performance in the technical is correlated with the final result (thank goodness).

combined_out %>%ggplot(aes(reorder(baker,median), median))+geom_pointrange(aes(ymin = q5, ymax =q95))+geom_point()+coord_flip()+theme_classic() +geom_text(data = performance,aes(x = baker, y =0,label = perf), inherit.aes =FALSE,hjust =0,nudge_x = .2 , size =2)+labs(title ="Who Was the Most Skilled Baker in Season 3?",subtitle = glue::glue("Using Samejima’s Continuous Response Model (CRM)\nBased on Technical Round Performance\nTechnical Performance Correlation to Final Results {correlation_analysis}"),caption = glue::glue("Data: Wikipedia\n See {url}"),x =NULL,y ="Skill" ) ->pp

out_come_with_rank %>%select(baker,outcome_modelled,outcome_realized) %>%mutate(color_use=ifelse(outcome_modelled >outcome_realized,"Better than Skill", "Worse than Skill")) %>%ggplot(aes(y =reorder(baker, outcome_modelled)))+geom_point(aes(x = outcome_modelled), color ="orange")+geom_point(aes(x = outcome_realized), color ="blue")+geom_segment(aes(x = outcome_modelled,xend = outcome_realized, yend = baker,color = color_use))+theme_classic()+labs(title ="Comparison Between Outcome and Modelled Skill",color ="Outcome",y =NULL,x ="Rank" )+scale_color_manual(values =c("green", "red"))+scale_x_continuous(breaks =seq(1,12,1))->p2p2

Next Steps

This analysis only covers one season. It would be neat to come back and do all of the seasons to get a feel for the level of difficulty of the different rounds (i.e., was the technical in round 3 of similar difficulty in each season). Additionally it would be neat to see if this relationship between the technical score and final outcome held up in each season.