[R] nonlinear fitting when both x and y having measurement error?

Prof Brian Ripley ripley at stats.ox.ac.uk
Sat Sep 15 16:44:49 CEST 2001

On Sat, 15 Sep 2001 pauljohn at ukans.edu wrote:

> I've seen the answers that point in the measurement model
> direction, but I wonder if there is not a more direct approach.

> In my copy of Pindyck and Rubinfeld's Econometric Models and
> Economic Forecasts, it outlines an instrumental variable
> approach in which the x with error is replaced by an instrument,
> a predicted value from an auxiliary model in which x is
> regressed on other exogenous/predetermined variables.  They
> prove the parameter estimates are consistent, which (I believe)
> is about the best we can hope for.

It's an asymptotic notion.  It may be better to be a little biased and a
lot more precise in small samples.

> One advantage of that strategy is that one need not assume a
> specific distribution for the error terms involved, only
> something general like E(e)=0 and constant variance.  The ML
> approach will require the choice of a precise distribution. Not
> so?

Not so!  There are sum-of-squares formulations and those were thought of
first (in the 19th century).  If the error variances are known to be
constant and known to be equal, the MLE under normality minimizes the sum
of squared perpendicular distances to the curve.  ML theory helps when
those assumptions are not true.

But these arguments are slippery. Least-squares regression does not assume
a normal model.  But it is optimal for a normal model (and for nothing
else), and seriously sub-optimal for very small departures from the normal
model (enter Robust Statistics).  So you can only justify using it if
normality has been tested to be a plausible working assumption and the
damaging departures have been guarded against. Or of course, if no better
tools are available.  That's why I believe projects like are R so
important: to make better tools available.

One major problem with the IV approach as I understand it is that you need
a good instrument. Both in this problem (allometric relationship of
biological data, in my understanding) and the ones I am familiar with
(calibration in chemistry, line-fitting in radio astronomy) I do not think
there are any plausible instruments: everything known is `known' with
considerable error, and all errors on one unit will be correlated.

> They don't show that approach works when the relationship
> between x and y is nonlinear.  Come to think of it, I don't
> recall a treatment of IV applications for nonlinear equations.
> This is a great question and I'm interested to hear more about
> how the project works out in the end.
> pj
> Etsushi Kato wrote:
> >
> > Dear r-help,
> >
> > I want to conduct nonlinear fitting to a data frame having x and y
> > variables.  Because both x and y have measurement error, I want to
> > include error term of x variable in the model.  I'm not sure but I
> > think ordinary nls model only consider error term of y variable.
> --
> Paul E. Johnson                       email: pauljohn at ukans.edu
> Dept. of Political Science
> http://lark.cc.ukans.edu/~pauljohn
> University of Kansas                  Office: (785) 864-9086
> Lawrence, Kansas 66045                FAX: (785) 864-5700
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Brian D. Ripley,                  ripley at 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 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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