Having trouble finding parameters with nlinfit function

Hi! I'm currently studying an article from Andrew JK Phillips (f1metrics.wordpress.com) which proposes a stadistical model to estimate the performance of an Formula One driver and their team across a certain season (Phillips, A. J. (2014). Uncovering Formula One driver performances from 1950 to 2013 by adjusting for team and competition effects. Journal of Quantitative Analysis in Sports, 10(2), 261-278). I'm not sure I can share the article here, so I won't. I'm willing to clarify any matter or confusion about everything I say about it, though.

Introduction

I think I'll not be able to summarize the article as good as I'd like to, but here it goes: Phillips proposes the scoring rate which is bounded between 0 and 10 (the worst and the best possible performance, respectively, for a driver i in a certain team j in a certain season k). Quickly summarized, it is computed as the sum of the points a driver scored in a season divided by the number of races where the driver either finished or crashed on its own, not accounting for mechanical failures. The points are assigned according to a certain system suggested by Phillips itself:

• Positions 1st to 6th: 1991 to 2002 F1 points scoring system is used.
• Positions 7th onwards: , where p is the position.
• Non-mechanical DNF: 0 points.

On the other hand, Phillips also proposes that can be computed as:

Now, is a linear predictor whose proposed stadistical model is:

where , and are, respectively, fixed effects representing driver performance, team performance and difficulty scoring in season k due to competition with other drivers. is a random effect representing variability in performance. As is a expected scoring rate of a population, it can be computed using:

where

(combined effect of a driver and team combination in a certain season)

The questions

Now, to the Matlab part: since can also be computed using the points scored from a driver in a certain season (and thus ), Phillips estimates the two unknown parameters and using the nlinfit function in Matlab. I understand that is the modelfun argument, is equivalent to random effects that could be added (using normrnd, for example), and the result of the nlinfint function would be a vector with the values of and .

• However, my first question is... what represents the x argument? The y argument would be the scoring rate computed usign a driver's results across a certain season, but there's no x value there. It could be the results themselves, but the function does not depend on them.
• Besides, another thing that bugs me is how can I compute beforehand, as it depends on many different values, one for every other driver in the grid. If , where 22 drivers and 11 teams competed (2 drivers on each team), do I have to make nlinfit compute 22 different and 11 different values?

Thank you so much for your time.

Edit: forgot to point out that is the number of drivers in season k, is the number of races a driver m drove for a team n in a season k (a driver can switch teams mid-season), and is the number of races in a season k.

[AndJ]

Have you found a solution?