Calculating CI from a nonlinear least squares regression

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Guest

Posted: Tue Jan 23, 2007 7:57 pm

Hello,

I am going through some data that we have acquired in multiple
experiments we've done over the past several years, and I'm not sure
that I'm estimating the error correctly.

In each experiment, we had an input variable which we varied and looked
for a binary outcome. We then used probit regression to calculate the
LD50 value for this outcome with 95% CI.

We repeated this experiment several times, each time changing another
variable (g).

Thus, we data that looks like this:

g LD50 CI
--------------------
g1 T1 (CI1a - CI1b)
g2 T2 (CI2a - CI2b)
g3 T3 (CI3a - CI3b)
g4 T4 (CI4a - CI4b)
g5 T5 (CI5a - CI5b)

Now, I think that the LD50 should follow this equation:

LD50(g) = constant*sqrt(g)

and I am using non-linear least squares regression to fit my data to
this equation.

My question is: how do I properly calculate the CI for this regression?
I would like to plot the results of the fit with CI, and I would also
like to report the constant with CI.

Earlier, I asked this question in a matlab newsgroup, and a suggestion
was made to use a weighted nonlinear regression using (1/CI)^2 as the
weights (normalized so the sum of the weights=1), and they pointed me
to a link where it's shown how to do a weighted nonlinear least
squares fit in matlab:

http://www.mathworks.com/products/statistics/demos.html?file=/products/demos/shipping/stats/wnlsdemo.html

However, they weren't 100% sure this is correct for my situation.
I've consulted with a statistician at our institution, and she also
wasn't able to answer this.

Does anyone here have any thoughts on this? Can I use the CI as
described above as weights to a nonlinear least squares fit?

-N

Guest

Posted: Tue Jan 23, 2007 8:01 pm

Sorry for replying to my own post, but I fear that I may have been
unclear. The suggestion is to do the nonlinear least squares regression
of the above equation using the (1/CI)^2 values as weights, where these
are CI's for the LD50 estimates.

-N

Ray Koopman

Posted: Tue Jan 23, 2007 8:33 pm

Guest

On Jan 23, 4:01 pm, ndgr...@yahoo.com wrote:

Quote:

That wouldn't be totally wrong, but it does assume that the CIs (which
I assume are based on the usual asymptotic estimates of the covariance
matrices of the regression coefficients) have negligible error.

You could also try analyzing the merged data from all the experiments,
with LD50 constrained to being proportional to sqrt(g), and comparing
the deviance to the sum of the deviances from the separate analyses.

Anon.

Posted: Wed Jan 24, 2007 8:53 am

Guest

Ray Koopman wrote:

Quote:

On Jan 23, 4:01 pm, ndgr...@yahoo.com wrote:
Sorry for replying to my own post, but I fear that I may have been
unclear. The suggestion is to do the nonlinear least squares regression
of the above equation using the (1/CI)^2 values as weights, where these
are CI's for the LD50 estimates.

-N

That wouldn't be totally wrong, but it does assume that the CIs (which
I assume are based on the usual asymptotic estimates of the covariance
matrices of the regression coefficients) have negligible error.

You could also try analyzing the merged data from all the experiments,
with LD50 constrained to being proportional to sqrt(g), and comparing
the deviance to the sum of the deviances from the separate analyses.

To follow up on this, why not just fit a model with concentration,

sqrt(g) and the interaction?

Incidentally, non-linear least squares wouldn't be needed: the model is
linear in sqrt(g), so you could use that in a weighted regression forced
through the origin.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax: +358-9-191 51400
WWW: http://www.RNI.Helsinki.FI/~boh/
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