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index.qmd
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---
title: Faster identification of faster Formula 1 drivers via time-rank duality
subtitle: Companion R code and data
author:
- name: John Fry
affiliations:
- name: University of Hull
department: Department of Mathematics
state: United Kingdom
email: [email protected]
url: https://www.hull.ac.uk/staff-directory/john-fry
- name: Tom Brighton
affiliations:
- name: University of Hull
department: Department of Mathematics
state: United Kingdom
email: [email protected]
- name: Silvio Fanzon
affiliations:
- name: University of Hull
department: Department of Mathematics
state: United Kingdom
email: [email protected]
url: https://www.silviofanzon.com
date: 2024-03-18
# To enter today's date replace by today
# Enter date formatted as dd MM yyyy
date-format: "D MMM YYYY"
---
## Introduction
This page contains the R code and data to reproduce the statistical analysis in
the paper [@f1-paper] named **Faster identification of faster Formula 1 drivers via time-rank duality** by John Fry, Tom Brighton and Silvio Fanzon.
The code should be simple to understand and comments are provided throughout.
For a deeper understanding of the ranking model proposed, and the underlying statistical analysis, please refer to the paper [@f1-paper].
You are free to use and modify the code in accordance with the license
[CC BY-NC 4.0](https://creativecommons.org/licenses/by-nc/4.0/).
We kindly ask our work is credited by citing the paper [@f1-paper].
You can download the BibTeX citation [here](citation.bib).
## The data
Data used for the statistical analysis in the paper [@f1-paper] can be downloaded [here](data/f1seconddata.txt). The latter contains placements of 20 drivers for the 22 races in the 2022 F1 Season plus 3 sprint races ([Source](https://en.wikipedia.org/wiki/2022_Formula_One_World_Championship)).
## The R code
The annotated R code given below reproduces the statistical analysis in
the paper [@f1-paper]. The code is mix of R scripts and interactive R console work.
The code runs in [R version 4.3.3](https://www.r-project.org) and above with
no additional packages.
### Calibration with bookmakers’ odds
The first R function listed below is used to minimise the residual sum of squares between the implied probabilities obtained from bookmakers odds and the win probabilities written as a function of the lambda values. The input is a parameter of lambda values. The dimension of the input vector is the number of unique bookmakers odds. This is an important constraint that needs to be obeyed. Imposing this constraint also improves the speed and smoothness of the computation. The function is then run in conjunction with the ``optim`` command in R to perform the minimisation. Please see below.
```r
#input is the vector of win probabilities
#output is the estimated lambda values
lambdaest4 <- function(x){
input <- c(0.031655049, 0.031655049, 0.673389233, 0.063310099,
0.031655049, 0.028380389, 0.063310099, 0.048413605,
0.001642777, 0.001642777, 0.01016088, 0.001642777,
0.001642777, 0.001642777, 0.001642777, 0.001642777,
0.001642777, 0.001642777, 0.001642777, 0.001642777)
#given 7 input values
target<-sort(input)
lambda<-rep(c(x[1], x[2], x[3], x[4], x[5], x[6], x[7]),
rle(target)$lengths)
pred <- lambda / sum(lambda)
distance <- sum( (target - pred )^2 )
return(distance)
}
```
To optimize ``lambdaest4`` we run ``optim`` on a set of randomly generated values. After
careful randomized restarts, a local minimizer is found to be
```r
x1
```
```verbatim
[1] 0.0004205564 0.0026012171 0.0072654675 0.0081037902 0.0123940343
[6] 0.0162075831 0.1723897481
```
As proof of concept that ``x1`` is a local minimizer we run ``optim`` starting at ``x1``
```r
optim(x1, lambdaest4, control=list(maxit=10000))$par
```
```verbatim
[1] 0.0004205564 0.0026012171 0.0072654675 0.0081037902 0.0123940343
[6] 0.0162075831 0.1723897481
```
### Regression estimation
The regression analysis in the paper proceeds via stepwise regression. Useful background can be found in Fry and Burke [@fry-burke]. However, in sharp contrast to the standard regression examples in Fry and Burke [@fry-burke], a constraint is made so that all considered models have to include the ``driverorder2`` dummy variable distinguishing between teams’ first and second drivers.
The following R code reads in the data on drivers placements found [here](data/f1seconddata.txt). Then it assigns variables and then runs a set of stepwise, forwards and backwards regressions.
```r
f1seconddata <- read.table("F:f1seconddata.txt")
position <- f1seconddata[ , -1]
positionlabel <- c(position[,1], position[,2], position[,3],
position[,4], position[,5], position[,6],
position[,7], position[,8], position[,9],
position[,10], position[,11], position[,12],
position[,13], position[,14], position[,15],
position[,16], position[,17], position[,18],
position[,19], position[,20], position[,21],
position[,22], position[,23], position[,24],
position[,25])
#Parameterise in terms of first driver, second driver
driverorder <- rep(c(1, 2), 10)
driverorder< - rep(driverorder, 25)
#Re-coded the driver dummy variable to lie between 0 and 1
driverorder2 <- driverorder - 1
constructors <- rep(c("Mercedes", "RedBull", "Ferrari", "Mclaren",
"Alpine", "AstonMartin", "Haas", "AlfaTauri",
"AlfaRomeo", "Williams"),
c(2, 2, 2, 2, 2, 2, 2, 2, 2, 2))
constructors <- rep(constructors, 25)
mercedesdummy <- 1 * (constructors == "Mercedes")
redbulldummy <-1 * (constructors == "RedBull")
ferraridummy <- 1 * (constructors == "Ferrari")
mclarendummy <- 1 * (constructors == "Mclaren")
alpinedummy <- 1 * (constructors == "Alpine")
astonmartindummy <- 1 * (constructors == "AstonMartin")
haasdummy <- 1 * (constructors == "Haas")
alfatauridummy <- 1 * (constructors == "AlfaTauri")
alfaromeodummy <- 1 * (constructors == "AlfaRomeo")
full2.lm <- lm(formula = positionlabel ~ driverorder2 + mercedesdummy
+ redbulldummy + ferraridummy + mclarendummy
+ alpinedummy + astonmartindummy + haasdummy
+ alfatauridummy + alfaromeodummy)
b.lm <- lm(positionlabel ~ driverorder2)
#Stepwise regression
step(b.lm,
scope = list(
lower = formula(b.lm),
upper = formula(full2.lm)),
direction = "both")
stepwise.lm <- lm(formula = positionlabel ~ driverorder2 + redbulldummy
+ mercedesdummy + ferraridummy + mclarendummy
+ alpinedummy + astonmartindummy)
#Forward selection
step(b.lm,
scope = list(
lower = formula(b.lm),
upper = formula(full2.lm)),
direction = "forward")
stepforward.lm <- lm(formula = positionlabel ~ driverorder2
+ redbulldummy + mercedesdummy + ferraridummy
+ mclarendummy + alpinedummy + astonmartindummy)
#Backard selection
step(full2.lm,
scope = list(
lower = formula(b.lm),
upper = formula(full2.lm)),
direction = "backward")
stepback.lm <- lm(formula = positionlabel ~ driverorder2
+ mercedesdummy + redbulldummy + ferraridummy
+ mclarendummy + alpinedummy + astonmartindummy
+ haasdummy + alfatauridummy + alfaromeodummy)
```
At this juncture it becomes clear that forwards and stepwise regression choose the same model.
Backwards regression leads to a model with additional variables in it.
The following R code suggests that the larger model does not lead to a significant improvement
over the smaller model chosen by stepwise regression.
```r
anova(stepwise.lm, stepback.lm, test = "F")
```
```verbatim
Analysis of Variance Table
Model 1: positionlabel ~ driverorder2 + redbulldummy + mercedesdummy +
ferraridummy + mclarendummy + alpinedummy + astonmartindummy
Model 2: positionlabel ~ driverorder2 + mercedesdummy + redbulldummy +
ferraridummy + mclarendummy + alpinedummy + astonmartindummy +
haasdummy + alfatauridummy + alfaromeodummy
Res.Df RSS Df Sum of Sq F Pr(>F)
1 492 10117.4
2 489 9978.1 3 139.3 2.2756 0.07903 .
```
The following R code now presents the regression results presented in Table 3 of the paper.
```r
summary(stepwise.lm)
```
```verbatim
Call:
lm(formula = positionlabel ~ driverorder2 + redbulldummy + mercedesdummy +
ferraridummy + mclarendummy + alpinedummy + astonmartindummy)
Residuals:
Min 1Q Median 3Q Max
-9.058 -3.192 -1.058 2.517 15.592
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 13.8420 0.3794 36.484 < 2e-16 ***
driverorder2 0.2160 0.4056 0.533 0.5946
redbulldummy -9.6500 0.7170 -13.459 < 2e-16 ***
mercedesdummy -8.2700 0.7170 -11.534 < 2e-16 ***
ferraridummy -7.6900 0.7170 -10.725 < 2e-16 ***
mclarendummy -3.5500 0.7170 -4.951 1.02e-06 ***
alpinedummy -3.5500 0.7170 -4.951 1.02e-06 ***
astonmartindummy -1.7900 0.7170 -2.496 0.0129 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 4.535 on 492 degrees of freedom
Multiple R-squared: 0.3914, Adjusted R-squared: 0.3828
F-statistic: 45.21 on 7 and 492 DF, p-value: < 2.2e-16
```
## License & Attribution
::: {.column width="65%"}
This work is licensed under [Creative Commons Attribution-NonCommercial 4.0 International License](https://creativecommons.org/licenses/by-nc/4.0/)
:::
::: {.column width="30%"}
{width=2.1in}
:::
This license enables reusers to distribute, remix, adapt, and build upon
the material in any medium or format for noncommercial purposes only,
and only so long as attribution is given to the creator.
We kindly ask our work is credited by citing the paper [@f1-paper] as shown below
> Fry, John and Brighton, Tom and Fanzon, Silvio. *Faster identification of faster Formula 1 drivers via time-rank duality*, Economics Letters, 237:111671, 2024
[https://doi.org/10.1016/j.econlet.2024.111671](https://doi.org/10.1016/j.econlet.2024.111671)
**BibTex citation:** Download [here](citation.bib) or copy from box below
```bibtex
@article{2024-Fry-Bri-Fan,
author = {Fry, John and Brighton, Tom and Fanzon, Silvio},
title = {Faster identification of faster Formula 1 drivers via
time-rank duality},
journal = {Economics Letters},
volume = {237},
pages = {111671},
year = {2024},
doi = {10.1016/j.econlet.2024.111671}
}
```