3.6 Three versions of b1

We’ve just shown that you can think of b1 as a weighted average of the response values – weighted by where the x values are relative to ˉx:

b1=i(xiˉxSxx)yi The original version of b1 that we mentioned looked different: b1=rsysx

In that formulation, we can think of b1 as the correlation between X and Y, but “scaled to match” X and Y by taking their standard deviations into account.

There’s a third way to write b1 as well: b1=Sxy/Sxx

This one makes it look like b1 is related to the covariance of X and Y, but scaled relative to the amount of variance in X.

I know I said I didn’t really care about b0, but in case you’re wondering (and can’t find this in your old stats notes), one way to write the least-squares estimate of b0 is: b0=ˉyb1ˉx This is mildly interesting as well: the intercept relates to the actual means of X and Y, as well as the relationship between them.

You may also recognize this as related to the old “point-slope form” for defining a line – what is the point we know the line must pass through?

They’re all true! You can get from any of these formulations to the others with a bit of algebra. What they all have in common is the idea of the slope coefficient reflecting both the relationship between X and Y, and the amount of variation/scale of X and Y. (That’s why b1 changes if you change the units of your variables!) Depending on the situation, you may find it helpful to think about the slope in any of these ways :)