Math Forum / Math Software / MATLAB / July 2008

Joe, in regular regression suppose you want to fit a model like this:

  y = a + x1 + b*x2 + error

In other words, you know the coefficient of x1 (we can assume it is 1 here).
It's easy enough to fit this by least squares by re-writing

  y - x1 = a + b*x + error

For generalized linear models, the response isn't a simple sum of a linear
function of the predictors with additive errors, so it's not possible to
re-write in the same way.

Here's a semi-realistic example where this would be useful.  Suppose the
number of defects on a surface should be proportional to the surface area,
or the number of events in an interval of time should be proportional to the
length of time.  The count of defects or events might reasonably be modeled
by a Poisson distribution.  If we subtracted or divided off the area or
time, we'd get something that might not even be integer valued.  Instead, if
we model the expected value as

   E[y] = area * exp(a + b*x)

we can take logs to get

  log(E[y]) = log(area) + a + b*x

The offset parameter allows us to handle the term that doesn't have a
coefficient to be estimated.

-- Tom

McCullagh&Nelder's book has an example of this in the Poisson chapter,
in the ship damage example:  length of service.

Thanks, both replies have been very helpful.

My problem at hand deals with the dose-response of a
population of animals to a toxic agent using the
binomial-probit model. Now I realize that I could use the
average weight of each dose-group of animals to normalize
the dose or use it as an offset. What would be the
difference in these approaches?

Thanks in advance.