Thank you very much for this response. Here are some remarks to your
questions and some further information:
> Q1. Are the uncertainties {s(a) s(b) s(c)} standard errors of the
> estimates of the coefficients, or 95% confidence intervals, or what?
The uncertainties {s(a) s(b) s(c)} are the 95% confidence intervals.
> Q2. Are the coefficients estimated assumed to be distributed
> normally, or to follow some other distribution?
The coefficients estimated are assumed to be distributed normally.
I used interval arithmetic as you proposed and indeed, uncertainties
become quite large.
What I am really interested in are the values of m(p), s(p) and m(q),
s(q). Hopefully someone has some idea on how to determine that.
Thanks in advance...
J�rgen U. Fritz ( [EMAIL PROTECTED] )
-----Urspr�ngliche Nachricht-----
Von: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
Gesendet: Montag, 19. April 2004 17:09
An: Juergen Fritz
Cc: [EMAIL PROTECTED]
Betreff: Re: Division of confidence interval limits
Here is a start on analyzing your problem. It doesn't get all the way
to your goal, but perhaps others on the list can help... -- DFB.
------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
------------------------------------------------------------
The problem rephrased:
Given a A + b B ---> c X
To find {p, q} such that p A + q B ---> X
When {a, b, c} are fixed numbers, p = a/c and q = b/c
When {a, b, c} contain known uncertainties
{a+-s(a), b+-s(b), c+-s(c)}: one desires {p+-s(p), q+-s(q)}
where {s(p), s(q)} are defined consistently with {s(a) s(b) s(c)}.
Q1. Are the uncertainties {s(a) s(b) s(c)} standard errors of the
estimates of the coefficients, or 95% confidence intervals, or what?
Q2. Are the coefficients estimated assumed to be distributed
normally, or to follow some other distribution?
In the absence of better strategy, one can always use interval
arithmetic to obtain numerical uncertainties about {p, q}:
since {a, b, c} are all positive, AND their uncertainties are all
smaller than their estimates,
p_max = [a + s(a)] / [c - s(c)], p_min = [a - s(a)] / [c + s(c)];
q_max = [b + s(b)] / [c - s(c)], q_min = [b - s(b)] / [c + s(c)].
and the interval [p_min, p_max] can be expressed as p*+-d(p);
but this is not a statistical interval, and d(p) cannot be expected
to
be the same sort of thing as {s(a) s(b) s(c)}: it is probably larger,
perhaps considerably larger, than s(p) would be. However, it may be
better than nothing. (Similarly for q .)
For this approach, it is a serious problem that the uncertainty in b
is almost as large as the estimate of b : q will have very wide
boundaries, q_min being near 0 and q_max being more than twice the
value
(0.18) calculated as though {b c} were known with no uncertainty.
(But that will be a serious problem in any case.)
For a proper confidence interval on p one would need to know the
presumed distributions of a and c (e.g., normal (= Gaussian) with
mean m(a) and standard deviation s(a), etc.), AND what that implies
about the distribution of p = a/c (e.g., normal with mean m(p) and
standard deviation s(p); or chi with parameters {m(p), s(p), df};
or whatever; where m(p) and s(p) are functions of the parameters of
a and c. (This last bit I've never had occasion to work out, but
probably someone else on the list would know.)
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
