Follow-up: [R] Inverse prediction with R?

Prof Brian Ripley ripley at
Tue Feb 22 15:12:56 CET 2000

> Date: Tue, 22 Feb 2000 14:18:19 +0000 (GMT)
> From: Bill Simpson <wsi at>
> You fit
> y = a+b*x
> Then you get the inverse regression estimate
> x = (y-a)/b
> I don't see what is hard about that.
> As for the confidence interval for x, try bootstrapping:
> - randomly sample your (x,y) data
> - do the y = a+b*x fit
> - get the estimate x = (y-a)/b
> - use the bootstrapped x distribution to get your CI

But that is not the right bootstrap if the sample x's were fixed,
as we were told (although I find it hard to believe), and if that
is the right bootstrap, it is the wrong inverse prediction procedure.
In most regression problems you need to bootstrap the residuals.

That CI only takes into account the uncertainty in the fitted curve,
not in that the future y, which will surely dominate.  Strictly
speaking, one wants a prediciton interval.

> Or use the formulas you saw in a book.

Yes, reading a good book on this is highly recommended. There are lots
of subtleties that provide puzzles for cognescenti of inference procedures.

Brian D. Ripley,                  ripley at
Professor of Applied Statistics,
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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