On Mon, 19 Dec 2005, David L Lorenz wrote:
We have a situation where we know the variance of observations for a
linear regression problem and need confidence limits on the parameter
estimates.
The documentation in the NOTES section for lm states "In addition,
S-PLUS does not currently support weighted regression when the absolute
precision of the observations is known. This situation arises often in
physics and engineering, when the uncertainty associated with a
particular measurement is known in advance due to properties of the
measuring procedure or device. If you know the absolute precision of
your observations, it is possible to supply them to the weights
argument. This computes the correct coefficients for your model, but the
standard errors and other inference tools will be incorrect."
I have searched the web, done bibliographic searches, and even searched
the documentation for a competing product and I found nothing regarding
the solution of this problem. I would be surprised if this problem has
not been solved because it should be a high priority in physics. At
least I remember it being an issue in physics classes, if not in the
engineering classes I took.
Physicists optimistically think they know their error variances, but
rarely do. Chemists used to think the same, but have learnt about the
many possible sources of error and that the precision quoted by their
instrumentation is nothing more than the between-readings instrumentation
variability, ignoring for example calibration errors, operator
differences let alone sample preparation variability.
Your quote is a bit out of context. You can fit (correctly) a weighted
regression via lm, and lm does not quote standard errors. Rather,
summary.lm does and that does not allow you to specify the error variance.
However, you can fit the same model by glm(), and summary.glm() does allow
you to specify the error variance. As a quick check:
x <- seq(2, 10, len=100)
v <- x
y <- x + sqrt(v)*rnorm(100)
fit <- glm(y ~ x, weights = 1/v)
summary(fit, dispersion = 1)
appears to have done the correct things. (Setting dispersion=1 means that
the inverse weights are the variances and not merely proportional to
them.) If you do that, do check that the residual SS is about the size
predicted by the theory (a chisq distribution, with df=n-p, 98 in my
example). Being large indicates model inadequacy, often in the variance
specification in my experience.
Is anyone aware of an approach to estimating the confidence limits of
parameter estimates when the variance of the observations is known? I
think I can see a solution for the case of equal known variance, but I
don't know that I have the ability to apply that to unequal
variances.Thanks.
Dave
--
Brian D. Ripley, ripley@stats.ox.ac.uk
Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
University of Oxford, Tel: +44 1865 272861 (self)
1 South Parks Road, +44 1865 272866 (PA)
Oxford OX1 3TG, UK Fax: +44 1865 272595
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