Edmund wrote:Thanks to hgm and lucas for the suggestions.

Below you find updated graphs representing the same data.

1) bin-size is 4 elo-points

2) minimum bin-size is 4 samples

3) in the elo-delta graph I shifted all models by 20 elo points to compensate for the white to move advantage

4) I added the cdf of the normal distribution with sd=250

5) I added the function hgm suggested to estimate draws scaling by 40/25

Agreed, the gauss function is a better fit than either the linear function or the logistic function.

Looking at the new graphs I am not so sure about hgms suggestion regarding the progression of the avg-elo score function. You are right that the next step is to take elo-delta into the equation.

I have replicated Edmond's methods with the CCRL 40/4 database. I too used 4 Elo bins. I do have 4 bins with less than 4 samples, but I do have 294 bins with the mean number of samples of 2946 and only 9 samples total with less than 10 samples. With the weighted regression, I judge that these 4 bins have little effect. I shifted the models by 26.973 (which is the computed eloAdvantage). I also have compared the logistic model with a Gaussian cdf, though I have been forced to use an approximation. The site I am using to perform the regressions does not recognize erf(). However, the approximation is quite good (1/(1 + exp(-0.07056*(X**3)-1.5976*X))).

This is White Score versus Average Elo:

The regression equation is White Score = 46.27 + 0.00285 Average Elo

This is Draw Ratio versus Average Elo:

The regression equation is Draw Ratio = -17.23 + 0.0179 Average Elo . I checked to see if the outliers exerted much leverage on the regression line by trimming the points outside the interval (1900, 3100). The confidence interval for the slope of the regression line for the trimmed data includes 0.0179. Therefore, given the weighted data, I believe that the draw ratio regression line is not affected much by the outliers.

Here is Draw Ratio vs Elo Delta (Elo Diff):

Finally, here is White Score vs Elo Delta:

The logistic equation is 100/(1+10^((x+26.973)/~380)).

This is the data with the Gaussian model:

sd=278.18. The approximation used was 100/(1 + exp((-0.07056*(((X+26.973)/278.18)**3)-1.5976*((X+26.973)/278.18))))

The two equations model the the data equally well. The logistic model is compressed ~5%. I believe this is related to the fact that the eloDraw computed from the data is 102.647, which is slightly higher than the default value.

I believe my next step is to recompute the Elo ratings for the 40/40 database and examine the cause for its compression. It is compressed noticeably more than the 40/4 ratings. Any results will be more definitive with this database. As I said in the previous post, I believe the cause is the use of the default eloDraw value (the default eloAdvantage causes the shift). It will take me a day or two to do this. There is one part of the data extraction that I have to do by hand, and it took several hours to do this for the 40/4 database.