R help - Nov 2001 - piecewise linear regression

We sent questionnaires to lots of folks.  The questionnaires
differed in word count (wc).  For each questionnaire, I know wc
and whether the questionnaire was returned or not (status).  It
looks like there is a fixed rate of return when wc is below a
break point, and the rate of return falls off linearly after
that.

I want to do a piecewise linear regression with two pieces
meeting at a common breakpoint.  I think it is important that I
don't specify the breakpoint in advance (although, if I get
desperate, I'll do that).

I've tried to fit a model in nls, several different ways
(including using the "hockey stick" method described in
http://www5.biostat.wustl.edu/s-news/s-news-archive/200004/msg00215.html)
and I keep getting a "singular gradient matrix" error (sometimes
the "initial" matrix, sometimes not).  I also got the same thing
with very simple artificial data (using a continuous dependent
measure).  For example:

nls(status~A+B*wc+C*pmin(A,D-wc)),start=list(A=1,B=0,C=0,D=0))

(I know I should use a logistic version, but if this version
doesn't work, I can't imagine that would work.)

Ideas?

Jon Baron
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On Sat, 10 Nov 2001, Jonathan Baron wrote:
> We sent questionnaires to lots of folks.  The questionnaires
> differed in word count (wc).  For each questionnaire, I know wc
> and whether the questionnaire was returned or not (status).  It
> looks like there is a fixed rate of return when wc is below a
> break point, and the rate of return falls off linearly after
> that.
>
> I want to do a piecewise linear regression with two pieces
> meeting at a common breakpoint.  I think it is important that I
> don't specify the breakpoint in advance (although, if I get
> desperate, I'll do that).
>
One simple approach is a grid search for the best breakpoint. Since the
model with specified breakpoint is linear it's very easy and quick to
try a fairly fine grid of possible breakpoints.

The problem with more sophisticated approaches is that if the two slopes
are equal or the breakpoint is at either end of the data the problem is
not identifiable, so bad luck or bad data can make a Newton-type search
break down.

	-thomas

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