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Suppose we have a table of finishing times for an number of 100 metre races where each competition has a different mix of entrants. We suspect that some races are slower than others due to, say, a headwind, but this wasn't measured. How do I find the underlying time for each runner and the correction for each race?

I have devised my own crude technique to solve this problem. I assume

Tobs(e,c) = Tund(e) + Tcor(c) + E(e,c)

where 

Tobs(e,c) = observed time of entrant e in competition c
Tund(e) = underlying time of entrant e
Tcor(c) = time correction for competition c
E(e,c) = error matrix 

sum of all Tcor = 0
sum of all rows in E(e,c) = 0
sum of all columns in E(e,c) = 0

I then iteratively choose values for Tcor until the above relations are satisfied.

Is there a better way? Using R? Please be gentle. As you might have guessed by now, I don't much much about stats or maths. Thanks!

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You are on the right path, I guess. (This is me being gentle.)

If you are looking for a quick and dirty solution (this is me being helpful), you may want to fit a linear model, specifically an ANOVA. Your response would be Tobs(e,c) and your factors of interest would be Ecor (time correction for entrant e) and Ccor (time correction for competition c). This is different from what you had posted with Tund (an underlying time of entrant e), because by default the ANOVA model includes an overall mean (mu). Make sure that both factors of interest (Ecor and Ccor) are treated as "factors" or "categorical variables" in whatever package you use. If you treat them as "continuous" or "numeric" variables, you will get nonsensical results. (Since you seem to be interested in the estimates for each race and each runner, you avoid going down the wormhole of random effects or mixed effects models.)

If you are looking for a more thorough and vetted solution (this is me being realistic), you should consult with a statistician. S/he can help ensure that your model answers your questions, suits the "design" of the data, meets all the necessary assumptions (independence, homogeneity, normality), and avoids commonly made mistakes.

If you are looking for a long term solution (this is me being encouraging), take a statistics or data analysis class. It can be very rewarding, especially when you have specific analysis needs in mind when taking the course.

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