Quote:
Originally Posted by Oblarg
To approximate the standard deviation of the mean (which is what is usually meant by "standard error" of these sorts of measurements), one would then divide this by sqrt(n) (for those interested in a proof of this, simply consider the fact that when summing random variables, variances add), where n is the number of matches used in the team's OPR calculation.
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I am interested in a proof of this, because I don't think the normal assumptions hold. Can you explain this more in the full context of how OPR is computed? [Edit: I spent more time trying to derive this whole thing. See my next posts for an attempt at the derivation].
What you say holds if one is taking a number of independent, noisy measurements of a value and computing the mean of the measurements as the estimate of the underlying value. So that would work if OPR was computed by simply averaging the match scores for a team (and dividing by 3 to accommodate for 1/3 of the match score being due to each team's contribution).
But that's not the way OPR is computed at all. It's computed using linear regressions and all of the OPRs for all of the teams are computed simultaneously in one big matrix operation.
For example, it isn't clear to me what n should be. You say "n is the number of matches used in the team's OPR calculation." But all OPRs are computed at the same time using all of the available match data. Does n count matches that a team didn't play in, but that are still used in the computation? Is n the number of matches a team has played? Or the total matches? OPR can be computed based on whatever matches have already occurred at any time. So if some teams have played 4 matches and some have played 5, it would seem like the OPRs for the teams that have played fewer matches should have more uncertainty than the OPRs for the teams that have played more. And the fact that the computation is all intertwined and that the OPRs for different teams are not independent (e.g., if one alliance has a huge score in one match, that affects 3 OPRs directly and the rest of them indirectly through the computation) seems to make the standard assumptions and arguments suspect.
Thoughts?