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Conclusion

Posted By: Maik Stiebler
Date: Sunday, 14 August 2016, at 3:46 p.m.

In Response To: Conclusion (Casper Van der Tak)

Let X the raw outcome of a trial and Y the measured luck of a trial. Let var the variance, cov the covariance, sig the standard deviation and cor the correlation. Then literally, the variance reduction in absolute terms is

var_X - var_{X+Y}.

Using the definitions and elementary algebra, this simplifies to

2*cov_{X,Y}-var_Y.

In terms of standard deviation and correlation,

absolute var. red. = 2*cor_{X,Y}*sig_X*sig_Y - sig_Y*sig_Y =sig_Y*(2*cor_{X,Y}*sig_X - sig_Y).

So variance reduction 'works' when

2*cor_{X,Y} > sig_Y/sig_X.

It is interesting to look at the relative variance reduction, which is obtained by dividing the absolute one by (sig_X)^2:

rel. var. red. = sig_Y/sig_X*(2*cor_{X,Y} - sig_Y/sig_X)

For a given correlation, this is optimized when

sig_Y/sig_X=cor_{X,Y}.

As simple linear scaling of Y changes sig_Y/sig_X, but not the correlation, I wonder if that can actually be used to fine-tune variance reduction.

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