[Ether]

Guys,

Can we all agree on the following?

Computing OPR, as done here on CD, is a problem in multiple linear regression (one dependent variable and 2 or more independent variables).

The dependent variable for each measurement is alliance final score in a qual match.

Each qual match consists of 2 measurements (red alliance final score and blue alliance final score).

If the game has any defense or coopertition, those two measurements are not independent of each other.

For Archimedes, there were 127 qual matches, producing 254 measurements (alliance final scores).

Let [b] be the column vector of those 254 measurements.


For Archimedes, there were 76 teams, so there are 76 independent dichotomous variables (each having value 0 or 1).

For each measurement, all the independent variables are 0 except for 3 of them which are 1.

Let [A] be the 254 by 76 matrix whose i^th row is a vector of the values of the independent variables for measurement i.


Let [x] be the 76x1 column vector of model parameters. [x] is what we are trying to find.


[A][x]=[b] is a set of 254 simultaneous equations in 76 variables. The variables in those 254 equations are the 76 (unknown) model parameters in [x]. We want to solve that system for [x].

Since there are more equations (254) than unknowns (76), the system is overdetermined, and there is no exact solution for [x].

Since there's no exact solution for [x], we use least squares to find the "best" solution¹. The solution will be a 76x1 column vector of Team OPR. Let that solution be known as [OPR].

Citrus Dad wants to know "the standard error" of each element in [OPR].

Are we in agreement so far? If so, I will continue.


¹Yes, I know there are other ways to define "best", but every OPR computation I've ever on CD uses least squares, so I infer that's what Citrus Dad had in mind.