Is there a way to analytically determine the new coefficients without re-estimating the regression? To give you a concrete example that might make answering easier, consider the following two models. Let's say I have b1 and b2, and would like to know what b3 and b4 would be?

(Model 1) y = b1 + b2*x

(Model 2) log(y) = b3 + b4*x


1 Answer 1


With simple linear regression, you can save a little calculation -- essentially those quantities from the first regression that don't depend on $y$ can be reused (so $n$ and the mean and variance of $x$, or equivalent information). Except in some very paricular situations, the mean of the transformed $y$, the variance of the transformed $y$ and the covariance of $x$ with the transformed $y$ (or equivalent information) will have to be obtained from the data.

However, with larger models (multiple regression) and large sample sizes it can save a fair bit of time to save the corresponding information from the first regression (in particular, you could save the QR decomposition of the $X$-matrix, avoiding the need to recalculate it). If numerical stability is not going to be an issue you can sometimes go a bit further. In extremely stable cases with few predictors and large sample size and many different response vectors to calculate the regression on, you might even consider progressing to $(X^\top X)^{-1}$, but this is usually not a suitable strategy.

  • $\begingroup$ +1 for constructive ideas. Wouldn't it be better, though, to signal a plain answer by starting this post with a great big "NO"? $\endgroup$
    – whuber
    Commented Jun 8, 2018 at 12:44
  • 1
    $\begingroup$ The gist of my answer is nearer to "sort of" than "no" -- you don't need to re-do everything from scratch, because you can reuse a substantive fraction of the calculations involving only the predictors (but of course no matter what you do you re-estimate a regression, since that's entirely the point; any shortcut to the coefficients is still in some sense re-estimation, so it's always both 'ýes' and 'no). $\endgroup$
    – Glen_b
    Commented Jun 8, 2018 at 14:00

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