[R] error propagation - hope it is correct subject

[Thomas W Blackwell]

Petr  -

Very briefly, I think of three ways to approximate the standard
deviation of  y = f(x1,x2,x3).

  (1) linearise f() and use the covariance matrix of [x1,x2,x3].
  (2) simulate draws from the joint distribution of [x1,x2,x3],
	then compute the sample std dev of resulting f()s.
  (3) go back to the original data set from which [x1,x2,x3] were
	estimated as parameters, re-sample rows with replacement,
	estimate [x1,x2,x3] and compute f, then take sample std dev.

Other names for these three would be (1) the "delta method" or
Taylor series expansion, (2) parametric bootstrap, (3) bootstrap.

Different choices are appropriate in different situations.

If the std devs of x1,x2,x3 are small relative to the curvature
(2nd derivative) in f(), then use (1) and compute by matrix algebra

Var(f(x1,x2,x3))  approx  t(grad f) %*% Cov(x1,x2,x3) %*% grad f.

If the curvature in f() is an issue, use (2) with draws of x1,x2,x3
from some parametric distribution (eg, rnorm()) with each component
properly conditioned on the ones already drawn.

Only if there were no set of intermediate parameters [x1,x2,x3]
would I use (3) to get the precision of f directly.  I'm sure
Brad Efron would say something different.  (3) is the only one
that is canned in R, simply because the other two are practically
one-liners.

-  tom blackwell  -  u michigan medical school  -  ann arbor  -

On Mon, 22 Dec 2003, Petr Pikal wrote:

> Dear all
>
> Please, can you advice me how to compute an error, standard deviation or
> another measure of variability of computed value.
>
> I would like to do something like:
>
> var(y) = some.function(var(x1),var(x2),var(x3))
>
> for level F1 (2,3,...)
>
> Let say I have some variables - x1, x2, x3 (two computed for levels of factor F
> and one which is same for all levels) and I want to compute
>
> y = f(x1,x2,x3)
>
> for some levels of factor F
>
> I can compute variation of variables for levels of F, I know a variation of one
> variable but I am not sure how to transfer it to variation of y within respective
> levels.
>
> I found some methods which I can use but I wonder if  there is some method
> implemented in R (Manly B.F. Biom.J.28,949,(1986), some local statistical books
> available to me).
>
> I have a feeling I could use bootstrap method for this but I am not sure how.
>
> Thank you and merry Christmas to all
>
> Petr Pikal
> petr.pikal at precheza.cz
>
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