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Re: regression confidence intervals


From: Rudolf Widmer-Schnidrig
Subject: Re: regression confidence intervals
Date: Tue, 08 Feb 2011 10:11:59 +0100
User-agent: Mozilla/5.0 (Macintosh; U; Intel Mac OS X 10.6; en-US; rv:1.9.2.13) Gecko/20101207 Lightning/1.0b2 Thunderbird/3.1.7

On 08.02.11 08:50, CdeMills wrote:
Hello,

my math courses are long away ... could someone tell me which of these two
approaches is the right one? An experiment produced n pairs of points (xi,
yi) for which a first order model is postulated
yi = [1 xi] [b; a]

The parameters are found by letting
A = [ ones(size(x)) x]; B = y; th  = A\B; [b, a] = deal(th)

A covariance matrix can be found by
S2y= sumsq(B-A*th)/(length(y)-2); %# a posteriori variance
Cth = inv(A'*A)*S2y;

 From here, I want to get, for a given, errorless x0, what is an 90%
confidence interval for y0.
First way: from error propagation technique, I can compute the variance  of
y0 as
Sy0 = [1 x0]Cth[1; x0]; %# notice that this depends from x0
but then I start from S2y, which has n-2 degrees of freedom, to [db da]
which is bivariate normal, to Sy0 which is univariate normal. Can I assume
that (y-y0)^2/Sy0 is chi-square with one d.f. ?

Second way: [db da] is a bivariate normal, so, applying the concept of
Mahalanobis distance,
[db da] inv(Cth) [db; da] is chi-square with 2 d.f,  confidence intervals
are ellipse. From there, I compute dy = db + x0 da; and I choose the pair
[db, da] belonging to the ellipse which gives rise to an extrema in dy. I
then conclude that the 90% interval on (db, da), the ellipse, is mapped to
[y0-ymin, y0+ymax].

What is the correct approach ? Any pointer to litterature ?

Regards

Pascal

Dear Pascal,

I would use Jackknife technique: remove in turn one out of your n data pairs and with the remaining n-1 data pairs do: th = A\B. This can be done n times and then you look at the distribution of your a's and b's. When Jackknifing you need not assume any fancy statistical properties about the underlying statistics of your data points. My recommendation is to try Jackknifing in parallel with an analytical approach like above. This would give you consistency check.

                             -Ruedi



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